(Cambridge Studies in Mathematical Biology) Jane Cronin-Mathematical Aspects of Hodgkin-Huxley...

CAMBRIDGE STUDIESIN MATHEMATICAL BIOLOGY: 7

EditorsC. CANNINGSDepartment of Probability and Statistics, University of Sheffield,Sheffield, U.K.F. C. HOPPENSTEADT

College of Natural Sciences, Michigan State University,East Lansing, Michigan, U.S.A.

L. A. SEGEL

Weizmann Institute of Science, Rehovot, Israel

MATHEMATICAL ASPECTSOF HODGKIN-HUXLEYNEURAL THEORY

JANE C RON INProfessor of Mathematics, Rutgers University

Mathematical aspectsof Hodgkin-Huxleyneural theory

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Library of Congress Cataloguing in Publication dataCronin, Jane, 1922-

Mathematical aspects of Hodgkin-Huxley neural theory.(Cambridge studies in mathematical biology; 7)Bibliography: p.1. Neural conduction—Mathematical models.2. Purkinje cells—Mathematical models.3. Electrophysiology—Mathematical Models. I. Title. II. SeriesQP363.C76 1987 599'.0188 87-9241

ISBN 978-0-521-33482-2 hardbackISBN 978-0-521-06388-3 paperback

CONTENTS

1 Introduction page 1

2 Nerve conduction: The work of Hodgkinand Huxley 6

2.1 The physiological problem 62.2 A brief summary of Hodgkin and Huxley's

conclusions 122.3 The work of Hodgkin and Huxley 16

2.3.1 The experimental results 162.3.2 Derivation of the differential equations 39

2.4 What the Hodgkin-Huxley equations describe 532.5 Mathematical status of the Hodgkin-Huxley

equations 532.5.1 History 532.5.2 Some successful numerical analysis of the

Hodgkin-Huxley equations 572.5.3 Drawbacks of the Hodgkin-Huxley

equations 65

3 Nerve conduction: Other mathematical models 673.1 Earlier models 673.2 The FitzHugh-Nagumo model 683.3 The Zeeman model 693.4 Modifications of the Hodgkin-Huxley

equations 703.4.1 Modifications in the description of potassium

and sodium conductances 703.4.2 The FitzHugh-Adelman model 713.4.3 The Hunter-McNaughton-Noble models 71

3.5 The Lecar-Nossal stochastic model 73

Contents vi

44.14.24.34.4

55.15.2

5.3

5.4

5.5

66.16.2

Models of other electrically excitable cellsIntroductionThe myelinated nerve fiberStriated muscle fiberCardiac fibers4.4.1 The cardiac Purkinje fiber4.4.2 The Beeler-Reuter model of ventricular

myocardial fiber

Mathematical theoryIntroductionBasic theory5.2.1 Existence theorems and extension theorems5.2.2 Autonomous systems5.2.3 Equilibrium poin ts5.2.4 Stability and asymptotic stability of solutionsPeriodic solutions5.3.1 Autonomous systems5.3.2 Periodic solutions of equations with a periodic

forcing termSingularly perturbed equations5.4.1 Introduction5.4.2 Some examples5.4.3 Some theory of singularly perturbed systemsPartial differential equations

Mathematical analysis of physiological modelsIntroductionModels derived from voltage-clamp experiments6.2.1 Nerve conduction models6.2.2 Analysis of the Noble model of the cardiac

Purkinje fiber

Appendix

References

747475818585

98

101101101101105106114130130

134138138139143178

180180184184

216

249

253

PREFA CE

The quantitative study of electrically active cells received its prin-cipal impetus from the remarkable work of Hodgkin and Huxley,in 1951, on nerve conduction in the squid giant axon. Hodgkin andHuxley used voltage-clamp methods to obtain extensive quantita-tive experimental results and proposed a system of ordinary dif-ferential equations that summarized and organized these data.Since then, their experimental methods have been extended andadapted to the study of other electrically active cells. Also, numer-ous mathematical studies of the Hodgkin-Huxley equations havebeen made. The results, experimental and mathematical, arescattered through the literature in research papers, and the firstpurpose of this book is to provide an organized account of some ofthese results. This account is intended to be accessible to mathema-ticians with little or no background in physiology.

In Chapter 2, a fairly detailed account is given of the experimen-tal results of Hodgkin and Huxley, and similar detail is providedfor the derivation of the Hodgkin-Huxley equations. It is notnecessary to study the experimental results or the derivation of theequations in order to understand the equations themselves, whichare a four-dimensional system of autonomous differential equa-tions containing messy nonlinear functions. (The functions are,however, quite well-behaved: They are, indeed, real analytic func-tions, and the usual existence theorems can be applied to thedifferential equations.)

It is tempting to the mathematician to disregard the derivation ofthe equations and plunge ahead, instead, to the mathematical analy-sis of the equations, a familiar activity made additionally attractive,in this case, by the fact that the equations model an importantsystem in the "real" world. There are, however, several reasonswhy this is an ill-advised procedure. First and perhaps foremost is

Preface viii

that, without a knowledge of how the equations are derived, onehas no understanding of the status of the various terms in theequations. There is a tendency to regard the equations as "writtenin stone" or as axioms, and to proceed from there. This viewpointis often valuable in pure mathematics, but it is worse than uselessin this part of applied mathematics. For this study, one needs tounderstand in what ways the equations are accurate descriptionsand in what ways the equations represent rough descriptions of thephysiological system. This kind of understanding can only begained from some knowledge of how the equations are derived.Second, a knowledge of where the equations come from is essentialin judging what mathematical problems to undertake in studyingthe equations. Without the guidance of that knowledge, themathematician can easily drift into attractive mathematical prob-lems that have no physiological significance. Finally, the greatdepth and ingenuity of the methods used by Hodgkin and Huxleyin deriving their equations makes the study of the derivation agenuine intellectual adventure. It would be regrettable to miss it.

In the latter part of Chapter 2, we discuss some of the physio-logical phenomena that have been studied mathematically, and inChapter 3 we describe some other models of nerve conduction,most of which were inspired by or are modifications of theHodgkin-Huxley equations.

In Chapter 4, we describe some of the other models of electri-cally active cells that have been derived by modifications of andextensions of the methods of Hodgkin and Huxley. These descrip-tions are brief and incomplete. They are intended mainly to givethe reader some idea of the models that exist and their status. In alater chapter, we will be particularly concerned with models of thecardiac Purkinje fiber.

The second objective of this book is to summarize the theoryof ordinary differential equations that is used or may later beused in the study of these mathematical models. This summary(in Chapter 5) brings the mathematician to the research level incertain aspects of the theory and also indicates to the interestedbiologist the kinds of mathematics that may be useful in suchstudies. Our description of the mathematics is only a summary andomits all proofs and most examples.

Preface ix

We have a second purpose in Chapter 5: to emphasize a particu-lar direction for study, that is, to emphasize the application ofsingular perturbation theory. The use of singular perturbationtheory in nerve conduction models is certainly not new. Indeed,one of the first mathematical analyses of the Hodgkin-Huxleyequations (actually a study of a simplified version of theHodgkin-Huxley equations), made by FitzHugh in 1960, is asingular perturbation study in all but name. In the years since1960, singular perturbation theory has been used in research papers,but it has not been used as systematically or as extensively as itcould have been. Later in Chapter 6, we discuss explicitly thenumerous reasons, both physiological and mathematical, for usingthe singular perturbation viewpoint. Our summary of the mathe-matics includes a detailed description of those results from singularperturbation theory that are clearly useful for the study of mathe-matical models of electrically active cells.

Our account of the mathematics to be used may seem disap-pointing for several reasons. To the physiologist, there may seeman undue emphasis on existence theorems and qualitative theoryand not enough discussion of how to find explicit solutions. Theanswer to this may seem equally disappointing. Chapter 5 isintended as an account of the existing mathematics that seems tobe useful in studying these mathematical models. The theory is byno means complete; much work of a purely mathematical natureremains to be done.

The next disappointment concerns the almost exclusive emphasison ordinary differential equations. Partial differential equationsplay an important role in the study of nerve conduction. The mostspectacular success of the mathematical work of Hodgkin andHuxley was the prediction of the speed of the nerve impulse, andthat was obtained by analysis of a system of partial differentialequations. However, it is by no means certain that the techniqueused by Hodgkin and Huxley to derive a system of partial differen-tial equations is valid for other mathematical models of electricallyactive cells. The structure of other cells is more complicated thanthe cylindrical structure of the nerve axon, and this fact alone maypreclude the derivation of similar partial differential equations forother cells. Second, as stated before, there are cogent reasons for

Preface x

making more extensive use of singular perturbation methods. If itturns out that the singular perturbation viewpoint is valuable in thestudy of the models that are ordinary differential equations, thenthis would suggest that singular perturbation techniques shouldalso be used in the study of the partial differential equations. Thus,it seems reasonable to postpone study of the partial differentialequations until more definite knowledge of how to approach theordinary differential equations has been obtained.

In Chapter 6, we discuss the application of the methods, espe-cially the singular perturbation theory described in Chapter 5, tothe models of the electrically active cells derived in the earlierchapters. We give a detailed analysis of the FitzHugh-Nagumoequations, indicate the analysis that could be undertaken for theHodgkin-Huxley equations, and discuss in some detail the analysisof the Noble model for the cardiac Purkinje fiber.

Although there is a logical order to the chapters in this book,certain sections are independent of one another. The reader with aserious interest in the subject can reasonably afford to omit partsof Chapters 3 and 4 on a first reading. How much of Chapter 5needs to be read depends on the reader's mathematical back-ground. (Remember that the singular perturbation theory describedin Chapter 5 is applied in Chapter 6.)

A mathematician who is simply interested in seeing the applica-tions of mathematics in nerve conduction might read Chapters 1and 2 (despite the earlier stern injunctions, not every detail ofthe derivation of the Hodgkin-Huxley equations needs to bestudied), the treatment of the FitzHugh-Nagumo equations inChapter 3, the section in Chapter 5 on singular perturbationtheory, and the discussion of the FitzHugh-Nagumo equations andthe Hodgkin-Huxley equations in Chapter 6. A physiologist who isalready acquainted with mathematical models of electrically activecells might omit most of the first four chapters and simply readChapters 5 and 6.

In writing this book, I have received essential help from manypeople. I am especially indebted to Mark Kramer, who carried outthe original calculations for Table 6.2, and to the students whoseinterest made possible the classes in which much of this materialwas developed. I am grateful to Dean John Yolton of Rutgers

Preface xi

College, who gave me the opportunity to present part of this workin undergraduate honors courses. Also, I wish to thank CarolRusnak, who typed several versions of this book with unfailinggood humor and patience.

I am happy to acknowledge support by the National ScienceFoundation Visiting Professorships for Women Program at theCourant Institute of New York University during 1984-85.

Finally, I am indebted to the staff at Cambridge UniversityPress, especially Peter-John Leone, for thorough editing that didmuch to improve this book.

Jane Cronin

Introduction

The work of Hodgkin and Huxley on nerve conduction haslong been recognized as an outstanding scientific achievement.Their papers were published in 1952 [Hodgkin and Huxley(1952a, b,c,d)] and they received a Nobel prize in physiology fortheir research in 1961. Hodgkin and Huxley's work was at once atriumphant culmination of many years of theoretical and experi-mental work by research physiologists and a pioneering effort thatset the direction and defined the goals for much of the ensuingresearch in biophysics.

The purpose of this book is, first, to provide an introductorydescription of the work of Hodgkin and Huxley and the later workthat is based on the techniques that they introduced. Our mainemphasis is on the theoretical aspect of the Hodgkin-Huxley work,that is, the derivation and analysis of their mathematical models(nonlinear ordinary and partial differential equations); the secondpurpose of this book is to describe some of the mathematics that isused to study these differential equations.

The hope is that this book will indicate to some biologists theimportance of the mathematical approach and will serve as anintroduction for mathematicians to the mathematical problems inthe field. However, this discussion is bound to be unsatisfactory tomany readers. The biologists will find the description of thephysiology simplistic, crude, if not actually misleading, and theymay also be dubious about the value of conclusions that can bedrawn from the mathematical analysis. Mathematicians who areaccustomed to the precision and stability of physics and engineer-ing, will find the inherent uncertainty of the parameters in themodels dismaying, if not disagreeable. Also, despite some verysuccessful analysis, the mathematical problems raised by thesemodels remain largely unsolved.

Introduction 2

Nevertheless, this discussion is useful because it emphasizescertain questions that must be resolved in the mathematical studyof biological problems. The reply to the biologist who doubts thevalue of mathematical techniques is that those doubts may possiblybe well founded. However, it is also true that mathematical resultshave shed significant light on certain questions in biology. Theimportant activity should consist not in expressing doubts, but inadvancing the study of the mathematical models to the point whereit can be shown clearly that they are or are not an important aspectof biological study. To the mathematician who finds the problemsdifficult or unattractive esthetically, the reply is even simpler.These problems are here, and criticizing their origin or aestheticvalue will not make them go away.

We shall assume that the reader is familiar with basic conceptsfrom electricity, that is, potential, current, resistance, the units inwhich these are measured, and Ohm's law. Since capacitance is asomewhat less elementary electrical notion and because capaci-tance plays a very important role in the derivation of theHodgkin-Huxley equations, a definition and elementary discussionof capacitance have been included in the short Appendix at the endof this book.

The problem of how a nerve impulse travels along an axon has along and interesting history. A brief summary of this history and anumber of references may be found in Scott (1975). Here we shallmerely point out a couple of the results that made the work ofHodgkin and Huxley possible. The membrane that surrounds theaxon had been discovered and its capacitance measured by Fricke(1923). Cole (1949) had pointed out that the important quantity tobe measured was the potential difference across the membrane.Equally important was the discovery by Young (1936) of the squidgiant axon. The unusually large diameter of this axon (about 0.5mm) made experimental work possible.

The work of Hodgkin and Huxley, which was a study of how anerve impulse travels along the squid giant axon, consisted of twoparts. The first was the development and application of an experi-mental technique called the voltage-clamp method, which wasinvented by Cole. By using the voltage-clamp method, Hodgkinand Huxley obtained extensive quantitative data concerning the

Introduction 3

electrical properties and activities of the axon. The second part oftheir work consisted in deriving a mathematical model (a four-dimensional system of nonlinear ordinary differential equations)that summarized the quantitative experimental data. They thencarried out a numerical analysis of the differential equations. Aswill be described later, that numerical analysis showed that thedifferential equations were remarkably successful in predicting awide variety of experimental results.

The work of Hodgkin and Huxley had the additional importancethat it set a direction for experimental and theoretical study ofother electrically active cells, that is, cells whose electrical proper-ties change during the normal functioning of the cells. Voltage-clamp methods have been developed for the study of myelinatednerve fiber (the squid axon has a particularly simple structure andis termed an unmyelinated axon), striated muscle fiber, and twokinds of cardiac fiber. For each of these, a mathematical model hasbeen derived by using basically the same approach as that used byHodgkin and Huxley in their study of the squid axon.

The mathematical analysis of the Hodgkin-Huxley equationsand the analogous models for other electrically active cells consistsof two different parts. The first part is numerical analysis, that is,computation of approximate solutions of the differential equations.Hodgkin and Huxley themselves carried out extensive numericalanalysis in their original papers and obtained the most outstandingresult of the theory: the prediction of the velocity of the nerveimpulse. However, as we shall see in Chapter 2, the equations canbe used to predict or describe many other experimental phenom-ena.

Many other numerical analyses of the Hodgkin-Huxley equa-tions have since been made, and numerical analyses, have, untilnow, been the most useful results for physiologists. There are,however, two serious drawbacks to numerical analysis. First, al-though numerical analysis can yield much useful information (as inthe example of the velocity of the nerve impulse), there are manyimportant questions that cannot be approached by use of numeri-cal analysis. Numerical analysis cannot yield an explanation ofhow the potential V and the sodium and potassium currents arerelated. As will later be shown, the simplest and crudest qualitative

Introduction 4

analysis yields far more information of this kind. Second, numeri-cal analysis requires the assignment of strict numerical values tothe parameters that occur in the differential equations. As we shallsee later, the values of the parameters, indeed the very form ofcertain of the functions, are not known with much accuracy.Consequently, it is important to study mathematically some classor family of equations to which the Hodgkin-Huxley equationsbelong, as well as to study the equations themselves.

In Chapter 2 the work of Hodgkin and Huxley is described insome detail: first the experimental work and then the derivation ofthe equations. It is important to see a fairly detailed description ofthe experimental results and their interpretation even for a readerwhose primary interest is the mathematical analysis of the equa-tions. Only a knowledge of the origin of the equations makes clearthe status of the equations and the significance to physiologists ofvarious mathematical problems concerning the equations. Chapter2 also summarizes some of the numerical analysis that was carriedout by Hodgkin and Huxley. Some of this analysis was carried outon the equatons. However, Hodgkin and Huxley also derived fromthis original system a system of nonlinear partial differential equa-tions (which we will term the full Hodgkin-Huxley equations) andthey carried out a numerical analysis to find traveling wave solu-tions of the partial differential equations. It was this analysis thatyielded the prediction of the velocity of the nerve impulse.

In Chapter 3 we describe some other mathematical models ofnerve conduction including various simplifications and modifica-tions of the Hodgkin-Huxley equations. Chapter 4 describes somemathematical models of other electrically active cells that wereobtained by using the basic techniques and ideas introduced byHodgkin and Huxley.

In Chapter 5 we turn to the problem of analyzing mathemati-cally the models that have been described. This analysis requirestwo quite distinct kinds of mathematics. First, we need materialfrom the subject of ordinary differential equations including thetheory of singularly perturbed equations. This material is sum-marized in Chapter 5. In order to study the full Hodgkin-Huxleyequations, considerable material from partial differential equations,in particular the theory of reaction-diffusion equations, is needed.

Introduction 5

Rather than attempting to present this material, we have merelydescribed it very briefly and cited a few references. There areseveral reasons for emphasizing the study of the ordinary differen-tial equations and postponing a detailed study of the partialdifferential equations. First, the two kinds of theory are essentiallyindependent of one another and represent two quite differentsubjects. Second, the ordinary differential equations are deriveddirectly from the experimental data and hence are closer to the realworld of physiology. [Very good results are obtained from studyingthe full Hodgkin-Huxley equations, but it is questionable whetherthe corresponding partial differential equations for other electri-cally active cells are realistic. See McAllister, Noble, and Tsien(1975), page 4.] Finally, it seems practical to deal with the ordinarydifferential equations in some detail first because this will helpguide future work on the partial differential equations. For exam-ple, in Chapter 6 we shall discuss the reasons why it seems strategicto regard our models as singularly perturbed systems. Moreover, byusing the singularly perturbed viewpoint we will obtain useful andenlightening information about how the electrically active cellsbehave. If more extensive research continues to show that thesingularly perturbed viewpoint is valid and informative, then it willfollow that the corresponding partial differential equations shouldbe regarded as singularly perturbed systems. But that would sug-gest the use of very specific theory of singularly perturbed partialdifferential equations rather than general reaction-diffusion theory.Thus, to some extent, the study of the partial differential equationsawaits the resolution of questions concerning the ordinary differen-tial equations.

In Chapter 6 we use the theory from Chapter 5 to study themodels derived earlier. In particular, we make a detailed study ofthe Noble model of the cardiac Purkinje fiber. Also we summarizevery briefly some of the work on traveling waves in nerve conduc-tion, that is, traveling wave solutions of the full Hodgkin-Huxleyequations.

Nerve conduction: The work of Hodgkinand Huxley

2.1 The physiological problemWe start by describing in some detail the physiological

problem to be studied. An impulse that, say, carries a commandfrom the brain to a muscle travels along a sequence of neurons thatcan be portrayed roughly as in the sketch in Fig. 2.1. When theimpulse arrives at the dendrites on the left-hand side of the neuron,the stimuli given the dendrites are integrated at the cell body toform a nerve impulse. The nerve impulse travels along the axon tothe branches of axons on the right-hand side of the neuron. Theimpulse then jumps to another set of dendrites and the process justdescribed is repeated. Neurons vary considerably in size. Thesciatic nerve of the giraffe contains an axon that may be severalmeters in length; many other axons are much shorter. The diameterof the squid axon, which has been the subject of many experimen-tal studies, is ~ 0.5 mm but it can be as much as 1 mm. Its lengthis several centimeters.

The process by which an impulse travels along a sequence ofneurons is quite complicated. For example, when the impulsejumps from one set of dendrites to another, both chemical andelectrical processes, which are not well understood even today, playan important role. The subject of Hodgkin and Huxley's work isthe process by which the impulse travels along the axon in the giantaxon of the squid. The reason that Hodgkin and Huxley studiedthe squid axon is that its diameter is unusually large and conse-quently makes experimental work possible. (The discovery of thesquid giant axon by Young in 1936 was an important step in thedevelopment of cell electrophysiology because it made a whole newdirection of experimentation possible. We note that the term "giant"axon is a relative one: It refers simply to the fact that the diameter

2.1 The physiological problem

SKETCH OF NEURON

r^,™,, .™ CELL BODYDENDRITES / DENDRITES

INPUT

Figure 2.1.

of the axon is generally ~ 0.5 mm, a much larger diameter thanthat of most axons.)

Restricting the investigation to the question of how an impulsetravels along an axon may seem to be a very narrow problem. Inactuality, it is a wide and important problem. As we shall see later,the processes by which an impulse travels along an axon arecomplicated and still not very well understood; thus the problem iscertainly deep. The techniques introduced by Hodgkin and Huxley,both experimental and mathematical, have been adapted and ex-tended so that they can be used to study other quite differentelectrically active cells. Among these are myelinated axons, striatedmuscle fibers, and cardiac fibers. All of these will be describedlater. Finally, in the study of the squid axon, the membranecovering the axon plays a very important role in the electricalprocesses. Any progress made in the study of how the nerveimpulse travels along the axon is necessarily progress in the studyof membrane biophysics. Thus the work of Hodgkin and Huxley isboth broad and profound.

In this work the axon will be regarded as having the shape of acylinder (this is a significant simplification). The axon consists ofhomogeneous aqueous matter called axoplasm, and the axoplasm issurrounded by a thin membrane (< 100 A thick) [A (angstrom) =m/1010 = mm/107]. The membrane is composed of two types ofmacromolecules: protein and lipid (fat). We assume that the squidaxon has this simple uniform structure: a cylinder of axoplasmsurrounded by a thin membrane of uniform thickness. However, invertebrates, except for the smallest axons, the structure is morecomplicated. The axon is myelinated, that is, the axon is sur-

Nerve conduction 8

rounded by a sheath of fatty material called myelin. This sheath ofmyelin is interrupted at intervals of about 1 mm by short gapscalled nodes of Ranvier. In these short gaps, the axon consists ofaxoplasm surrounded by the thin membrane previously described.The simpler axon structure, of which the squid axon is an example,is called an unmyelinated axon. Later, it will be shown that themethods of Hodgkin and Huxley, which were developed for thestudy of an unmyelinated axon (i.e., the squid axon), can beadapted to the study of a myelinated axon.

Although we will not be concerned further with the subject untilsome time later, we point out here that the speed with which anerve impulse travels along the axon depends on the diameter ofthe axon and on whether the axon is myelinated. If the axon isunmyelinated then as the diameter of the axon increases, thevelocity of the impulse increases. Indeed, the giant axon of thesquid transmits "escape signals," and thus there is good reason forrapid transmission and hence for an axon of large diameter. Butthere are practical limitations on the diameter of the axon; invertebrates, the fast transmission of nerve impulses has been ob-tained by a different evolutionary development. It turns out that ifthe axon is myelinated, the transmission of the nerve impulse ismuch faster even if the diameter of the axon remains small. For aquantitative description and analysis of how the speed of the nerveimpulse depends on the diameter of the axon and whether the axonis myelinated, see Chapter 4 and Scott (1975, p. 516).

Now we give a brief description of some of the history of thisproblem so that we can indicate the status of the problem whenHodgkin and Huxley began to work on it. Experimental studies ofthis physiological problem had been carried on for a long time. Bythe end of the eighteenth century, it had been established thatnerves could be stimulated by an electric shock and in 1850,Helmholtz measured the signal velocity on a frog's sciatic nerve.This work suggested that the basic problem of nerve conduction iselectrical in nature and one might conjecture that the axon is afairly effective biological substitute for an everyday conductor ofelectricity like copper or silver wire. That is, one might conjecturethat nerve conduction consists of an electric current. There are twocogent arguments against this idea. First, the Helmholtz experi-

2.1 The physiological problem 9

ment in 1850 showed that the signal velocity is fairly slow (27 m/s)compared to the speed of electric current in a wire, which is thespeed of light. Second, although the interior of the nerve fibercontains ions and is a fairly good conductor of electricity, thediameter of the fiber is so small that the resistance per unit lengthof a nerve fiber of average diameter is ~ 1010 fl/cm. This is anenormous resistance. It means that the electrical resistance of anerve fiber of length 1 m is about that of 1010 miles of 22 gaugecopper wire. (Copper wire, gauge 22, has a diameter of 0.7112 mmand 1010 miles is about 10 times the distance between Earth andthe planet Saturn.) By Ohm's law, we have

(potential) = (current) (resistance).

If a nonnegligible current were to be produced with such a largeresistance, a huge potential would be necessary. Because suchpotentials certainly do not occur in animals, this suggests that thenerve impulse is not carried directly by an electric current but is insome way reinforced as it travels along the nerve fiber. One of ourobjectives is to find out how this process occurs.

The objective of the work of Hodgkin and Huxley was to explainnot only the reinforcing process, but also the many phenomenaobserved in experiments conducted by physiologists over a longperiod of time. Summarizing the salient features of these experi-ments is our next step. When the giant axon is in the living squid,there is a potential difference across the membrane surface, andthis potential difference remains practically unchanged if the axonis carefully removed from the squid. The isolated axon "survives"for many hours after removal from the body of the squid in thesense that its electrical properties remain about the same. As aresult, it is possible to study the isolated axon experimentally. Thecrucial experimental observations concern how the potential dif-ference across the membrane, called the membrane potential,changes if the nerve is stimulated, and our summary consistsessentially in describing the types of changes in membrane poten-tial that occur.

If the axon is not subject to stimulation, the inside of the axon is50 to 70 mV (millivolts) less than the outside. Today the usual signconvention is to denote this by — 50 to - 70 mV (i.e., inside minus

Nerve conduction 10

outside). Hence we say that if the axon is at rest (not subject tostimulus) the membrane potential is — 50 to — 70 mV. This valuefor the membrane potential is called the resting potential. As longas the axon is not stimulated, the resting potential remains fairlyconstant. In this chapter, we will regard it as a strict constant.Later, we shall see that a more realistic view requires the introduc-tion of an element of randomness in the description of the restingpotential. Clearly a question of basic importance is that of how theresting potential is maintained. We will deal with this question alittle later.

Now suppose that a stimulus in the form of a brief current pulseconsisting of a flow of positive ions and < 1 ms in duration isapplied to the axon by an electrode inserted into the interior of theaxon. The resulting change in the membrane potential depends onthe amplitude of the stimulus in the following way.

If the stimulus is very weak, there is a temporary positive changein the membrane potential that is proportional to the amplitude ofthe stimulus, which dies away in - 1 ms and which affects themembrane potential only very near the point where the membraneis touched by the electrode providing the stimulus. The response issaid to be passive and local. If the amplitude of the stimulus isincreased, the membrane potential begins to increase faster thanthe stimulus amplitude (i.e., it is no longer simply proportional tothe stimulus amplitude), but the change in membrane potentialdoes not exceed 4-10 mV. The response is said to be active andlocal.

If the amplitude of the stimulus is large enough so that themembrane potential is raised above a critical value called thethreshold, 40 mV, then the membrane potential increasesabruptly to form a roughly triangular solitary wave ~ + 30 mV inamplitude and lasting ~ 1 ms. (This interval depends on thetemperature.) This curve is called an action potential Although itarises locally, it splits immediately into two separate waves thattravel away from the stimulating electrode in opposite directionsalong the fiber at a constant conduction velocity. The actionpotential, once it starts traveling, can be recorded along the nervefiber as it passes, but there is no way to deduce from the recordalone where the action potential originated or the amplitude of the

2.1 The physiological problem 11

stimulus that produced it. The fact that only the presence orabsence of the traveling wave can be recorded is called the all-or-nothing law in physiology. [It is interesting to note that an all-or-none mode of conduction is an efficient way to avoid certaininterference and disturbance; see Hodgkin (1971, p. 16).]

Following a first stimulus that produces an action potential thereis a time interval of about 1 ms called the absolute refractoryperiod during which no stimulus, however strong, can produce anaction potential. After the absolute refractory period there is arelative refractory period during which an action potential can beevoked but only by a stimulus of larger amplitude than the usualthreshold value. During the relative refractory, the required largeramplitude gradually decreases to the usual threshold value. As weshall soon see, the refractory period plays a crucially important rolein the propagation of the action potential.

If the amplitude of the first stimulus is below the thresholdvalue, no action potential is produced, but response to a secondstimulus is affected for an interval afterward. First there is anenhanced phase - 1-2 ms during which the threshold is lower thanthe usual value. Thus the effects of two stimuli, applied closeenough together in time, combine, and the closer together they are,the more effective. After the enhanced phase, there is a depressedphase of indefinite duration during which the threshold to thesecond stimulus is greater than the usual threshold.

If the stimulus is a very long current pulse (say 0.5 s), more thanone action potential may result. Depending on the amplitude of thestimulus, there may be as many as four or five action potentials inan interval as short as 100 ms. (Such a group of action potentials issometimes called a burst.) The action potential amplitudes decreasesomewhat during the burst.

The results that have just been described are obtained frommany difficult and delicate experiments. In order to measure themembrane potential, a microelectrode must be introduced into thenerve without disturbing or "killing" it, and then very smallpotentials must be measured. The preceding brief descriptionroughly summarizes a huge amount of hard work. [For an indica-tion of some of the difficulties in this work, see the amusingdescription given by FitzHugh (1969, p. 70).]

Nerve conduction 12

2.2 A brief summary of Hodgkin and Huxley's conclusionsThe purpose of the work of Hodgkin and Huxley was to

obtain a physical explanation of the processes that produce theexperimentally observed phenomena that have been described.Their results are a sharply quantitative description of currents thatflow across the membrane surrounding the axon and an explana-tion of how these currents produce the experimental results previ-ously described. Before starting a detailed description of the workof Hodgkin and Huxley we will give a crude brief summary inwords of the explanation proposed by Hodgkin and Huxley. Thisloosely written summary will serve as a guide in the later moredetailed account of Hodgkin and Huxley's work.

Hodgkin and Huxley's explanation is largely based on a descrip-tion of varying currents of sodium and potassium ions across themembrane that surrounds the axon, and our brief summary will beconcerned exclusively with these currents.

Suppose first that the nerve fiber is in its natural environment,that is, intercellular fluid that has a concentration of sodium (Na)ions and chloride (Cl) ions about equal to that of sea water, andsuppose that the nerve fiber is not subject to any stimulus. It isknown that the concentration of Na ions outside the axon is about10 times the concentration of Na ions inside the axon and that theconcentration of potassium (K) ions inside the axon is about 5times the concentration of K ions outside the axon. The differencein concentration across the membrane produces a chemical drivingforce on the ion that is equivalent to an electromotive force orpotential difference equal to

RT Cx

t ( )

C{ and Co are the ionic concentrations inside and outside the axon,respectively, R is the gas constant, which has the value 8.31432 X107 erg/°C (i.e., degree Centigrade) per mole, T is the absolutetemperature, and F is the Faraday number which is the product ofAvogadro's number and the charge on one electron [i.e., F= 96,500C (coulombs)]. Expression (2.1) is called the Nernst formula. For adiscussion of its derivation, see the article by Schwartz in Adelman

2.2 A brief summary 13

(1971). Thus, the electromotive force acting on the Na ions is

RT RTln(10"1)= In 10= -55 mV

and the electromotive force acting on the K ions is

RTln5 = 75mV.

The signs of these electromotive forces are chosen so as to beconsistent with the convention chosen earlier in the description ofthe resting potential. According to that convention, a positiveelectromotive force (emf) is an emf that "pushes" positive ions(electrons) in the direction across the membrane from the inside(outside) toward the outside (inside) of the axon. Similarly, anegative emf "pushes" positive ions from the outside toward theinside of the axon.

Besides the chemical driving force on the Na and K ions, there isalso the force of the resting potential, which, as mentioned earlier,has the value —70 mV. We postpone the question of how theresting potential is maintained until after further discussion ofthe currents of Na and K ions. For the present we concentrate onthe question of how ionic currents are produced by the restingpotential and the emfs that act on Na and K ions.

First, the emf equivalent to the chemical driving force acting onK ions is 75 mV, which is slightly larger than the absolute value ofthe resting potential and is opposite in sign from the restingpotential. Hence if the membrane is permeable to K ions, i.e.,allows K ions to pass through it, then we would expect that therewould be either no flow or a small outward flow of K ions. This is,in fact, what occurs. On the other hand, the emf acting on the Naions is — 55 mV, which has the same sign as the resting potential.Thus the total emf acting to push Na ions across the membranefrom the outside toward the inside of the axon is

- 55 - 70 = -125 mV.Hence we would expect (if the membrane were permeable to Naions) that when the nerve fiber is in its natural environment andnot subject to stimulus, there would be a significant inward current

Nerve conduction 14

of Na ions. Now, in fact, there is no such inward flow of Na ions;so we conclude that the membrane is impermeable to Na ionsunder the conditions described. It turns out that the permeabilityof the membrane to Na and K ions is dependent on the potentialdifference across the membrane. As we shall now see, this fact is ofcrucial importance in explaining how an impulse travels along thenerve axon. That is, we will now describe the ionic currents thattake place when some stimulus is applied to the axon so that thepotential difference across the membrane is changed.

If the membrane potential is made more negative (this is calledhyperpolarization), the membrane permeability to ionic currentsremains unchanged. Hence, there is only a very small outward flowof K ions (or none) and no flow of Na ions. However, if themembrane potential is increased (this is called depolarization) to acertain critical potential, the threshold potential (about -40 mV),the permeability of the membrane to Na ions increases, Na ionsenter the nerve fiber at an increasing rate, and the membranepotential moves rapidly toward the equilibrium potential of the Naion, that is, the potential that just balances the equivalent emfproduced by the chemical driving force acting on the Na ions. Asour earlier discussion showed, this value is 55 mV. Depolarizationalso causes an increase in the conductance of the membrane to Kions, although this increase is somewhat slower than the increase ofconductance of Na ions. The threshold is the membrane potentialat which the inward Na ion current just balances the outward Kcurrent.

For a subthreshold depolarization, the outward K current islarger than the inward Na current and the membrane potentialreturns to the resting potential. For a superthreshold depolariza-tion, the inward Na current exceeds the outward K current andcontinues to increase until the membrane potential reaches theequilibrium potential of the Na. As the equilibrium potential of theNa ion is reached, the permeability of the membrane to Na ionsdecreases fairly rapidly, but the permeability to K decreases muchmore slowly. Since the K current is an outward ionic current, themembrane potential then returns to the resting potential. (It shouldbe emphasized that the description that has just been given is veryrough. Neither the magnitude nor the timing of the events are

2.2 A brief summary 15

indicated precisely. Later we will consider experimental data andthe mathematical model that summarize the data. These will pro-vide us with a more precise description.) Meanwhile, the actionpotential is propagated along the nerve fiber because points on thefiber near the point under observation have their membrane poten-tials depolarized to the threshold value. It is important to observethe crucial role that the refractory period plays at this stage. If theaction potential has reached a certain point, then nearby points towhich the action potential has not yet arrived (say points on theright in the accompanying diagram) will be depolarized to thresholdvalue. But nearby points through which the action potential hasjust passed will be in the refractory period and hence will not bedepolarized to threshold value. Thus the action potential is propa-gated unilaterally, which is the desired objective.

-> direction of propagation

nearby points in nearby points not inrefractory period refractory period

The net result of the formation of the action potential is that thenerve fiber gains some Na ions and loses some K ions. In theformation of a single action potential, these gains and losses arevery small. However, they are cumulative and it is natural to lookfor some process that serves to offset them. Physiological studiessuggest that there are mechanisms driven by metabolic energy thatpump out Na ions and pump in K ions. For a brief description ofthese pumps, see Luria (1975, pp. 334-335). However, there areexperimental anomalies that cannot be explained by such mecha-nisms [see Beall (1981, p. 8)].

Now we return to the question of how the resting potential ismaintained. When the membrane is at rest (i.e., not subject to anystimulus), it is highly permeable to K ions, that is, K ions flowthrough the membrane easily, but the membrane is only slightlypermeable to Na ions. Because the concentration of K is muchhigher inside the axon than outside and because the K ions floweasily through the membrane, there is a leakage of K ions from theinside to the outside of the axon. Thus, the inside of the axonbecomes electrically negative with respect to the outside. That is,

Nerve conduction 16

an electrical field is set up that tends to push positively chargedparticles from the outside to the inside. The outward flow of K ionscontinues until this electrical field becomes strong enough tobalance the chemical driving force that acts on the K potassiumions and that is equivalent to the electromotive force given by theNernst formula. The potential to which the electric field gives riseis the resting potential.

2.3 The work of Hodgkin and Huxley2.3.1 The experimental results

So far we have stated the physiological problem thatHodgkin and Huxley worked on and we have described briefly andloosely the qualitative aspect of Hodgkin and Huxley's solution ofthe problem. Now we must go back and describe in detail howHodgkin and Huxley arrived at their answer to the problem. If wecompare the study of nerve conduction to a murder mystery, wemight say that we have presented the first and last chapters of thestory. We know that "the butler did it", but now we must gothrough the painstaking reasoning by which Hercule Poirot arrivedat the identity of the murderer. We want also to obtain thequantitative version of the answer. That is, we want to replacestatements such as, "this increase [of conductance of potassiumions] is somewhat slower than the increase of conductance ofsodium ions," with precise quantitative statements.

Our analysis consists of two parts. First we describe the experi-mental results and the analysis of these results. Next, on the basisof these results, we derive the Hodgkin-Huxley differential equa-tions that can be regarded as a quantitative summary of theexperimental data.

The loose description of Hodgkin and Huxley's conclusions thatwe have already given suggests the nature of the experiments thatthey carried out. It seems reasonable to assume that Hodgkin andHuxley decided somehow that Na ion current and K ion currentplayed an important role in the formation of the action potential.Then they measured the time-varying flows of Na and K current ata fixed point on the surface of the axon as an action potentialpassed by. In theory, this might be a good approach to use. In fact,it is not possible to develop techniques for directly measuring the

2.3 The work of Hodgkin and Huxley 17

flows of different ionic currents. Consequently, indirect approachesmust be used and often considerable ingenuity and effort is re-quired in order to interpret the experimental data. Moreover, inorder to interpret the data, it is frequently necessary to makecertain assumptions. These assumptions are justified later in thesense that the conclusions based on them agree with experimentalresults and can be used to predict other results. (We will presentlysee examples of these assumptions and their justifications.)

It is crucially important to have a clear picture of this structureof experimental data, interpretation of the data, and the assump-tions upon which the interpretation rests because this structure isthe foundation of the mathematical model and is a measure of thestatus of the model. A good judgment of what is a sensiblemathematical problem upon which to work must be based in parton a understanding and knowledge of the status of the mathemati-cal model (the Hodgkin-Huxley equations). It is for this reasonthat we will describe the experimental results in more detail thanmay at first seem necessary in a discussion of the mathematics ofnerve conduction.

We will describe in some detail the experiments in the first twopapers in the series on nerve conduction, that is, Hodgkin, Huxley,and Katz (1952) and Hodgkin and Huxley (1952a). In the paper byHodgkin, Huxley, and Katz (hereafter to be referred to as HHK)there are two sets of experiments. The first experiments are those inwhich a brief stimulus is applied to the nerve. (Experiments of thiskind have already been discussed Section 2.1.) Experiments of thistype had been performed in earlier work, but it was necessary tocarry them out at the start of the study in order to establish thatthe axons studied were capable of giving action potentials [seeHHK (1952, p. 433)].

Although we will describe the results of the experiments in somedetail, we will indicate the experimental procedure only briefly.There are several reasons for this omission. Most important is thefact that it is not immediately relevant to our goal. Second, theelectrical circuitry used is fairly complicated [see HHK (1952, p.430) for a diagram of the feedback amplifier used to maintain themembrane potential at a fixed value]. Finally, this circuitry is nowcompletely outdated (radio tubes, i.e., " valves" in the terminology

Nerve conduction 18

used by Hodgkin and Huxley, have long since been replaced inelectrical circuits by transistors). The brief treatment of experimen-tal procedure is misleading in one important respect. The readermay fail to realize the hardships and complications that are en-countered in carrying out these experiments. It is worthwhile toread the description of the experimental procedure given by HHK(1952, pp. 426^32) just to get an idea of the pitfalls and diffi-culties that beset the experimenter.

Before proceeding to the description of the experiments, wemake one remark concerning quantitative biological data. The"constants" that occur in biological data are not constant values inthe same sense as constants in physics, such as the gravitationalconstant. For example, we have already introduced the concept ofresting potential and used as its value - 70 mV. In fact, the valueof the resting potential varies from axon to axon and is alsodependent on the condition of the axon. Indeed, as we will seelater, the resting potential is more accurately represented by inclu-sion of a stochastic term and this fact may be important in certainanalyses. The point to be emphasized here is that measurementsvary from experiment to experiment and, moreover, we mustexpect this to occur.

In the first set of experiments described in HHK, a fine silverwire electrode is inserted into the axon, a brief pulse of current orshock is applied to the electrode, and the changes in the membranepotential are recorded. (Remember that the preceding sentence is asimplistic description of the experiment!) Note that in this experi-ment, the quantities considered are uniform along the length of theaxon. That is, the charge is uniformly distributed along the elec-trode and the potential difference across the membrane is the sameat all points along the axon. Such an experiment is sometimescalled a space-clamp experiment.

Note also that the events that occur during a space-clampexperiment are artificial. That is, such events do not occur in thenormal functioning of the axon. In a space-clamp experiment, themembrane potential has, at any given time, the same value of allpoints on the axon. As time elapses, the membrane potentialchanges, but it is constant along the axon. More precisely, themembrane potential is a function of time but is independent of


position on the axon. By contrast, in the normal functioning of theaxon, the membrane potential has the same value, the resting value,all along the axon when the axon is not subject to any stimulus, butif the axon is stimulated, say by a pulse of current, then themembrane potential changes significantly near the point where thestimulus is applied. If the stimulus is large enough, an actionpotential travels along the axon. In these circumstances, the mem-brane potential at a given time, regarded as a function of positionalong the axon, is certainly not a constant function.

The magnitude and duration of current pulse in the experimentjust described are measured and consequently the total electriccharge in the shock can be measured. Measurement of the mem-brane potential shows that the initial change in the membranepotential is proportional to the total electric charge in the shock.The proportionality constant thus computed is the capacitance perunit area of the membrane. This capacitance, which has the valueof - 0.9 ju,F/cm2, plays an important role in the later analysis aswe shall see.

The changes that take place in the membrane potential followingapplication of a shock are indicated for some typical cases in Fig.2.2, where the membrane potential is graphed against time. Thegraphs obtained depend both quantitatively and qualitatively onthe magnitude and sign of the shock.

Let us consider first the case in which the shock consists of asurge of electrons into the electrode. Then the membrane potential,measured in accordance with the previously described conventionfirst decreases (this is called hyperpolarization) and then as timepasses, increases monotonically to the resting value. Only twographs of this kind are sketched (graphs of membrane potentialsthat were reduced from the resting value by - 5 and -39 mV).However, the behavior of the membrane potential in these twocases is typical of experiments in which hyperpolarization occurs.Now consider the case in which the shock consists of a surge ofelectrons away from the electrode (a surge of positively chargedparticles into the electrode). Then the membrane potential in-creases (this is called depolarization) and the graph of the mem-brane potential as a function of time depends very strongly on themagnitude of the shock. If the depolarization is less than 15 mV,

Nerve conduction 20

301I2C

j120 mV

1120

j120 mV

Figure 2.2. Time course of the membrane potential followinga short shock at 6°C. Depolarization shown upward. Numbersattached to curves give the strength of the shock inm/x coulomb/cm2.

the membrane potential returns monotonically to the resting poten-tial, that is, there is a subthreshold response. (No graph of this isshown in Fig. 2.2) If the depolarization is more than 12 or 15 mV,then, as the graph suggests, an action potential occurs. Notice thatif the depolarization is between 12 and 15 mV, then a state ofunstable equilibrium occurs and the graph shows that it may lastfor some time before an action potential is produced or there is asubthreshold response.

It should be pointed out at this stage that the sign conventionused by Hodgkin and Huxley in measuring the membrane potentialis exactly the opposite in sign from the convention used today andpreviously described. Consequently in Hodgkin and Huxley's con-vention the membrane potential increases with hyperpolarizationand decreases with depolarization. However, the graphs in Fig. 2.2,which are taken from HHK, represent depolarization as an increaseand hyperpolarization as a decrease. That is, for these graphs,HHK departed from their own convention. In doing so, they used aconvention that coincides with the standard convention today.

The shocks that are applied in the experiments just describedhave duration of about 8 /is. It is observed experimentally that the


total current / becomes negligible (can be considered zero) follow-ing an interval of 200 /xs after the shock. Consequently, at thattime, the equation

dVI'CM-+I,

(see Appendix) becomes

~dth=-cM—.

We know the value of CM and if a fixed time t > 200 /xs is chosen,the value of dV/dt can be measured geometrically on the earliergraphs (Fig. 2.2). The value dV/dt is simply the slope of the graphat t = t0. Thus the ionic current It can be computed at a fixed timet > 200 /xs. In Fig. 2.3, It at 290 /AS is plotted against the initialdisplacement of the membrane potential. Following the conventionused by HHK, we regard the initial displacement V to be positiveif hyperpolarization occurs and regard V as negative if depolariza-tion occurs. The resulting graph in Fig. 2.3 summarizes a lot ofexperimental data and gives us insight into the relation betweencurrent and initial displacement of the membrane potential. It alsoaffords us our first example of how further information can beextracted from the explicitly given experimental data.

Figure 2.3. Graph of ionic current density against displacementof membrane potential.

Nerve conduction 22

The next experiments are the first voltage-clamp experiments. Avoltage clamp is a sudden displacement of the membrane potentialfrom its resting value to a new level at which it is held constant byelectronic feedback. In the voltage-clamp experiments, a voltageclamp is established and the membrane current is measured duringa subsequent interval of time. Voltage-clamp experiments weremade possible by the earlier work of a number of physiologists,especially Cole (1949), who introduced the technique. The work ofHodgkin and Huxley consists of a painstaking and brilliant devel-opment of this experimental technique and an original, highlysuccessful theoretical analysis of the resulting experimental data. Ameasure of the power and influence of their accomplishments is thefact that since their work, most experimental and theoretical stud-ies of electrically excitable cells have been based on the techniquesthey developed.

We will describe the results of a series of voltage-clamp experi-ments in which various values are taken for the voltage clamp (thedisplacement of the membrane potential) and in which the axon isimmersed in fluids with various Na ion concentrations. The result-ing experimental data, when analyzed with sufficient care, yield atremendous amount of information. However, it is important tomake clear that a voltage-clamp experiment consists of imposingentirely artificial conditions on the axon. In the earlier experiments(space-clamp experiments) described, the axon is stimulated by apulse of current applied to the interior of the axon. If the pulse islarge enough, the changes in potential that then occur are the samekind of changes that occur at a given point on the axon when anerve impulse passes by. But in a voltage-clamp experiment, theconditions are entirely artificial in that they do not even remotelyresemble events in the axon in its normal functioning. In avoltage-clamp experiment, the membrane potential has, at anygiven time, the same value at all points on the axon. (Thus thevoltage-clamp experiment is also a space-clamp experiment.) But,more importantly, the membrane potential remains fixed as timepasses (except at the initial moment of the experiment when thevoltage clamp is exposed). Thus the potential is independent ofposition along the axon and independent of time. In the normalcondition of the axon, the membrane potential has the same value,


the resting value, all along the axon when the axon is not subject toany stimulus. But if the axon is stimulated, say by a pulse ofcurrent, then the membrane potential changes significantly near thepoint where the stimulus is applied. If the stimulus is large enough,an action potential travels along the axon. In these circumstances,the membrane potential varies both with position along the axonand with time.

Let E denote the membrane potential and Er the resting poten-tial, and let V=E — Er. The main reason that voltage-clampexperiments yield so much useful information lies in the basicelectrical equation described in the Appendix:

dV

In a voltage-clamp experiment, there is a sudden displacement orchange of potential, that is, V is changed suddenly, and then V isheld constant. Hence there is initially a surge of capacitancecurrent, contributed by the term CM(dV/dt), but after that initialsurge, V is constant and, hence, dV/dt is zero. Hence the capaci-tance current is zero and so the total / across the membraneconsists of ionic current /,-, that is,

But / can be measured and hence in a voltage-clamp experiment,the ionic current It can be measured.

Now we describe a few typical results of voltage-clamp experi-ments performed on axons immersed in sea water, which is ap-proximately the natural environment of the axon.

If the membrane potential is decreased from the resting value— 70 to —135 mV (i.e., if V is -65 mV), then, as shown in Fig.2.4, there is a brief surge of capacitance current followed by a fairlyconstant inward current (of positive ions) of about 30 /A A/cm2.

If V has any other negative value or if V has a positive value< 10 mV, similar results are obtained; there is a brief surge ofcapacitance current followed by a constant (with time) current / ofpositive ions. If V is negative, the current is inward; if V ispositive, the current is outward.

If V is negative, there is a linear relation between V and /. Thusthe results if V is negative resemble what would be expected from

Nerve conduction 24

-65

+ 104 j2.OmA/cm2

+ 130

0 2 4 6 8 10 12msec

Figure 2.4. Record of membrane current under voltage clamp.Displacement of the membrane potential in mV is given by thenumber attached to each record. [The potential values are givenin terms of the present conventions, i.e., inside minus outside,which is the opposite of the convention used by Hodgkin, Huxley,and Katz (1952).] Inward current is upward deflection.

considerations in physics if the membrane were regarded as aconductor with constant resistance per unit area, that is, we obtaina kind of Ohm's law. (This is, of course, a rough statement becausein Ohm's law, V is the potential difference whereas here V is thedeviation of the membrane potential from the resting potential andwe assume that, for whatever reason, there is no significant flow ofcurrent across the membrane when the membrane potential isequal to the resting value.)

However, if V is positive and > 10 mV, then the ionic currenthas very different kind of behavior. This is indicated in Fig. 2.4.Unlike the results when V is negative, a version of Ohm's law nolonger provides any guidance. When V is positive and > 10 mVthe initial flow of ionic current (after the surge of capacitancecurrent) is an inward flow of positive ions. But after a shortinterval of time, the flow of ionic current reverses its direction.


Because all this time the membrane potential is constant, it is clearthat, in these conditions, we cannot think of the membrane ashaving a constant resistance. Indeed as will be gradually revealedthere is a complicated quantitative relationship among the ioniccurrents and V. (None of this is very surprising in view of ourearlier discussion in Section 2.1 because if V> 10 mV, the mem-brane potential is approximately equal to or greater than thethreshold value. Consequently, we expect nonlinear relationships tooccur.)

We emphasize once again that our descriptions of the experi-ments are simplistic. Moreover, besides omitting technical details,we are also omitting certain of the scientific results. For example,all the results we have described are affected by temperature and acomplete quantitative account requires taking the temperature intoaccount. Because our purpose is primarily to indicate the stepstaken to obtain the mathematical description rather than to give afully detailed account, we will omit all references to temperature.

In the previous experiments, the ionic current under variousvoltage clamps has been studied. The next step is to identify thevarious ions that make up the ionic current. There was earlierevidence [see Hodgkin (1951)] to indicate that the initial "hump"of inward ionic current that occurs if V has value like 20 mV iscomposed largely of sodium. In order to investigate this, Hodgkinand Huxley (1952a) conducted voltage-clamp experiments similarto the ones already described in which the axon was immersed insea water. The only difference was that the axon was immersed invarious fluids with a lower concentration of Na ions than theconcentration in sea water. From the results of these experiments,fairly direct inferences can be made concerning the presence orabsence of Na ions in the ionic current. These experiments aredescribed in Hodgkin and Huxley (1952a), which is the mostoutstanding of the experimental papers both in originality andaccomplishment. The voltage-clamp experiments described in thispaper were performed on:

(i) Axons immersed in sea water.(ii) Axons immersed in choline sea water (i.e., sea water in

which the Na ions have been replaced by choline). Cholineis an inert biochemical ion with properties such that if Na

Nerve conduction

Axon insea water

26

Axon in Axon in seacholine water againsea water

- 2 8 -

IimA/cm2

Figure 2.5. Membrane current during voltage clamps when theaxon is immersed in sea water and in choline sea water.

ions are completely replaced by choline, then the squidaxon becomes completely inexcitable, but the restingpotential is about the same as if the axon were in seawater.

(iii) Axons immersed in 30% Na sea water (i.e., sea water inwhich 70% of the Na ions have been replaced by choline).

Typical experimental results are indicated in Figs. 2.5 and 2.6. The

Axon in30% sodiumsea water

Axon insea water

Axon againin 30%sodium sea water

| 1 m A / c m 2

0 2 4

msec0 2 4

msec

- I O I

0 2 4

msec

Figure 2.6. Membrane current during voltage clamps when theaxon is immersed in sea water and in 30% sodium sea water.


most important features shown in Fig. 2.5 are the following:

1. If the external Na concentration is zero, that is, if the axonis immersed in choline sea water, the initial inward currentis zero and there is an early increase in the outwardcurrent.

2. When the axon is immersed in choline sea water, the lateroutward current is only slightly altered. It is ~ 15-20%less than the later outward current that occurs if the axonis immersed in sea water.

With different values for the voltage clamp, similar records areobtained. These results give qualitative support to the hypothesisthat the early inward current is carried by Na ions. These experi-mental results can also be used to give quantitative support to thishypothesis. To show this, we introduce the notion of sodiumpotential. According to the Nernst formula, the chemical drivingforce pushing the Na ions inward is equivalent to the emf equal to

RT [Na]s

where [Na]; and [Na]o are the Na ion concentrations in the interiorand the exterior of the axon, respectively. If the membrane poten-tial equals

RT {K\ [ l

F nF l n [Na]o F n [Na], '

then the emfs just balance one another and so the net flow ofNa ions across the membrane is zero. The number (RT/F)ln^NajytNaJi) is called the sodium equilibrium potential and isdenoted by ENsi. The value of the sodium potential when the axonis immersed in sea water or 30% sea water can be estimated byexamining the graphs that are the result of voltage-clamp experi-ments. If we assume that the initial ionic current flow is a flow ofNa ions (we have already seen qualitative evidence to support thishypothesis), then if the value V of the voltage clamp is such thatthere is no initial ionic flow, then V + Er, where Er is the restingpotential, must be the sodium potential. Inspection of Figure 2.6shows that if the axon is in 30% sea water, the sodium potential is

Nerve conduction 28

very close to 79 mV, and if the axon is in sea water, the sodiumpotential is slightly less than 108 mV. That is, the voltage-clampexperiments yield fairly accurate estimates of the sodium potential.Now we use these estimates of the sodium potential to obtainquantitative support for the hypothesis that the initial ionic currentflow in a voltage-clamp experiment is a flow of Na ions.

To do this, the reasonable procedure would seem to be tocompute £ N a by using the Nernst formula and to compare theresult with the observed experimental value (close to 79 mV) justdescribed. The difficulty in such a direct procedure is that the valueof [Na]j is needed, but [Na]j is not known with any accuracy forthe axon being studied experimentally. (Indeed, to measure [Na];would require destruction of the axon.) Consequently, the follow-ing more indirect procedure is used.

Let [Na]o and [Na]'o denote the Na ion concentration in seawater and 30% sea water, respectively. Then the sodium potentialsin sea water and 30% sea water are

RT [Na]o

and

and we have

RT [Na];SNa-â—ln^. (2-2)

But

£ N a = F N a + £r, (2.3)

where Er is the resting potential and FNa is the value of the voltageclamp at which there is no initial ionic current flow in the case ofthe axon immersed in sea water. Similarly

£ N a = n . a + £ ; , (2.4)

where E'r is the resting potential and Fâ is the value of the voltageclamp at which there is no initial ionic current flow in the case ofthe axon immersed in 30% sea water. From (2.3) and (2.4), it


follows that

Now (2.2) and (2.5) give independent estimates for Eâ — £N a . Theexpression in (2.2) is a theoretical prediction based on the use ofthe Nernst formula; the expression in (2.5) is obtained fromexperimental work. That is, the values Fâ, FNa and E'r, Er can allbe determined experimentally. The expression in (2.5) is obtainedby assuming that (2.3) and (2.4) hold, but they hold if we assumethat the initial ionic current is composed of Na ions. Thus, if itturns out that the right-hand sides of (2.2) and (2.5) have the samevalue, we will have obtained substantial quantitative evidence thatthe initial ionic current is composed of Na ions. In fact the valuesobtained for the right-hand sides of (2.2) and (2.5) agree very well.For a comparison of the two values in a number of experiments,see Hodgkin and Huxley (1952a, p. 454).

This leaves us with the question of the composition of the laterionic current. If the voltage is clamped at a value V> 12 mV, thenthe later ionic current is actually in the opposite direction from theinitial ionic current (see Fig. 2.4). This suggests that the later ioniccurrent flow consists of ions different from sodium. It turns outthat there are cogent qualitative arguments to support the hypothe-sis that this current consists of K ions. See the discussion inHodgkin and Huxley (1952a, pp. 455-457).

Since we have discussed the Na ion current in some detail andsince our purpose is to convey a general picture rather than acomplete set of arguments, we will simply accept the conclusionthat the later ionic current is composed mainly of K ions and moveto the next step, that is, to obtain a quantitative description of theflow of Na ions and quantitative description of the flow of K ions.We assume that the initial ionic current consists of Na ions andthat the later ionic current consists of K ions, but that leaves openthe question of the composition of the ionic current at the in-between period during which, presumably, the Na ion flow isdecreasing and the K ion flow is increasing. Answering this ques-tion presents serious difficulties because there are no experimentaltechniques for measuring the Na ion current and the K ion currentseparately. The experimental results yield simply the magnitude

Nerve conduction 30

and the direction of flow of the ionic current. In order to obtain aquantitative description of the constituents of the ionic current,that is, to obtain graphs of the Na and K ion currents as functionsof time, we will make some assumptions about the nature of theion currents. First we will state these assumptions and then explaintheir status and how they can be justified. Let / K ( 0 and / N a ( 0denote the Na and K ion currents as functions of time, and, asbefore, let the prime sign denote the 30% sea water case.

Assumption 1. The function / K ( 0 is independent of [Na]o, theexterior concentration of Na ions.

Assumption 2. If / N a (0 and 1^(0 denote the Na ion currents asfunctions of time and if the exterior concentrations of Na ions hasthe values [Na]o and [Na]'o, then there exists a constant k such thatfor all t,

WO[We have stated this assumption in mathematical form because thatis the form in which it will be applied a little later. The originalform of the assumption, as stated by Hodgkin and Huxley (1952a,p. 457), is more tentative. Depending on the point of view, one maysay it is vaguer or less rigid.]

Assumption 3. Let [to,to+ T] denote the interval during which/N a ( / ) increases to its maximum value. We assume that if t e [/0, t0

+ T/3] or t0 < t < t0 + r / 3 , then

dIK

At this stage these assumptions can be advanced on the follow-ing basis. The first two assumptions are very simple and they donot conflict with any experimental results. The third assumption isstrongly suggested by the experimental data when the value V ofthe voltage clamp is such that V + Er is near the sodium potential,that is, if V has such a value (and hence the initial Na ion currentis negligible), the total current remains very close to zero for aninterval of time. Thus, we have reasons for trying these assump-


tions. As it turns out, they lead to a consistent theoretical structureand quantitative description that yield good agreement with otherexperimental evidence. Thus, we may say that these assumptionsare justified by the results that they yield.

Now we are ready to determine the functions / N a ( 0 and IK(t).In order to do this, we use two voltage-clamp experiments:

(i) A voltage-clamp experiment in which an axon is immersedin sea water and the voltage is clamped at a value V.

(ii) A voltage-clamp experiment in which the same axon isimmersed in 30% sea water and the voltage is clamped atthe same value V.

Allowance is made for the surge of capacitance current, the de-terioration of the axon as a result of having experiments inflictedupon it, and the difference in resting potential of the axon in thetwo different fluids in which it is immersed. How these allowancesare made is discussed in Hodgkin and Huxley (1952a, p. 458). Weobtain two functions, It(t) and //(*)» t n e ionic currents, as func-tions of time, across the membrane when the axon is immersed insea water and 30% sea water, respectively. We assume that each ofthese ionic currents is the sum of a sodium current and a potassiumcurrent, that is, we assume that

IM = IVa(t) + IK(t) (2-6)

and

i;(t) = rNa(t) + rK(t), (2.7)

where /Na(/)> ^Na(0 denote sodium currents and IK(t), I'K(t) de-note potassium currents. Our objective is to use Assumptions 1-3in order to determine /Na(0> ^Na(0> ^K(0> an<3 ^k(0- Moreprecisely, we will solve for these functions in terms of It(t) andI-(t). By Assumption 1, /K(/) = I'K(t), and by Assumption 2,

where A: is a constant that has not yet been determined. Hence,(2.7) becomes

Nerve conduction 32

Subtracting (2.8) from (2.6) we obtain

or

W , ) - ^ ^ . (2.,,

Since /,(/) and //(/) are known, it remains to determine theconstant k. In order to do this, the slopes dlt/dt and dlj/dtare measured at t = 0, that is, at the beginning of the voltageclamp (remember that the surge of capacitance current has alreadybeen allowed for). Let c be the constant defined by c =di;/dt(0) dlj/dtiO). Differentiating (2.6) and (2.7), we obtain

and

dt dt

di; drNa

dt dt

By Assumption 1,

and by

Hence,

dt dt

Assumption

dIK

we obtain

+

+

3,

~dT

drK

dt

(2,0)

(2.11)

Dividing (2.11) by (2.10) and using the definition of c, we obtain

di;/dt(o) di'c

But by Assumption 2, the ratio [/Na(01/[^Na(0] 1S a constant k


and hence that constant is

Substituting this value in (2.9), we obtain an explicit solution for/Na(0> and since

we obtain the following expression for I^

Finally by (2.6) and the condition / K ( 0 = / k ( 0 Q-Q-> Assumption1), we have

/ ; -

Thus, with the use of Assumptions 1-3, the functions /Na(0>IK(t), /Na(O> a n d ^k(0 ôr a fixed voltage clamp can be de-termined. Examples of these functions are shown in Fig. 2.7.

Knowledge of the sodium and potassium currents [i.e., / N a ( 0and / K (0 ] is sufficient to explain qualitatively how an actionpotential propagates; that is, the experimental results we have justdescribed are sufficient to suggest the explanation given in Section2.2 for how an action potential propagates.

As we have seen from the voltage-clamp experiments, if themembrane is depolarized above the threshold value (i.e., if V>12-15 mV) then Na current begins to flow. Since [Na]o>[Na] i,this Na current is directed inward and the membrane is depolarizedfurther until the membrane potential becomes positive and ap-proaches the sodium equilibrium potential. Then, as the experi-

Nerve conduction 34

Ionic current ifaxon is in sea waterand in 10% sodiumsea water

-56 mV

Sodium current insea water and 10%sodium sea water

I 1 mA/cm2

0 2 4msec

Potassium current insea water and 10%sodium sea water

Figure 2.7. Separation of ionic current into 7Na and / K .

ments show, the permeability of the membrane to Na currentbegins to decrease and the permeability of the membrane to Kcurrent begins to increase. Since [K]t> [K]o, the K current is anoutward flow of K ions. This K current brings the membranepotential back to the resting value at which level the ionic currentbecomes insignificant. Meanwhile the action potential is propa-gated along the nerve fiber because points on the fiber near thepoint under observation have their membrane potentials de-polarized to the threshold value.

Our next objective is to obtain a quantitative version of thedescription just given. In order to do this, a number of pre-liminaries must be carried out.

First, we want to establish an independence principle. Thisprinciple states that the probability that any individual ion willcross the membrane in a specified interval of time is independentof any other (chemically different) ions that are present. We willestablish the principle by using it to derive an equation thatpredicts the effect of sodium concentration on sodium current. Itcan be shown that the predictions of this equation agree withexperimental results. This establishes or certainly supports thevalidity of the independence principle.

If we assume that the probability that any individual ion willcross the membrane in a specified interval of time is independentof other ions that are present, then the inward flux MY of aparticular ion species will be proportional to the concentration cx


of that ion in the external bath. Hence, we may write

M^kfr, (2.12)

where kx is a constant that depends on the condition of themembrane and on the potential difference E across the membrane.We will take E as fixed. Similarly if M2 is the outward flux,

M2 = k2c2, (2.13)

where c2 is the ion concentration in the interior of the axon and k2

is a constant. Dividing (2.12) by (2.13), we obtain

Mi

2c2

The condition for equilibrium is Mx = M2. Substituting in (2.14),we obtain a condition for equilibrium:

r*

where c2 is a fixed internal concentration and c* is the externalconcentration that would be in equilibrium with c2 under the valueE of the membrane potential. Thus by the Nernst formula,

RT *- In —

or

EF

llT— = exp - (2.15)

Now if £* is the equilibrium potential for the ion and if cY is theexternal concentration, then (again by the Nernst formula)

RT c2 RT cxE* = — In-*--— ln^

F F

or

(2M)

Nerve conduction 36

Dividing (2.16) by (2.15), we obtain

cx \ (E*-E)F

RT

Substituting from (2.14') into (2.14) and applying (2.16), we have

Ml Cl \ (E*-E)F(2.17)

Now we compare the Na currents if the axon is immersed first insea water with sodium concentration [Na]o and then in a low-sodium solution with sodium concentration [Na]'o. We assume thatthe membrane potential has the value E in both cases. Let MNa x

and MN a 2 be the inward and outward fluxes, respectively, ofsodium when the axon is immersed in sea water. Fluxes M'Nal and^Na,2 a r e similarly defined if the axon is immersed in a solutionwith sodium concentration [Na]'o. Then

Na Na.l Na,2 /~ -, r»\= • . (2.18)

Since

and

If I ^^Q INa 1 1L J o '

then

^ i = I S r ; (219)i W Na, l Li>iaJo

but

MN a ? 2 = /

and

and thus

^N..2 = ^N.,2- (2-20)


Substituting from (2.19), (2.20), and (2.17) into (2.18), we obtain

/N._M N a i l [Na] ; / [Na] o -Af N a , 2

^Na M Na , l ~ ^Na ,2

^ {[Na];/[Na]o(MNa4/MNa,2) - l}MNa,2

{(MN a > 1/MN a i 2)- l}MN a,2

_exp([-(E*-E)F]/RT)-l

_ {[Na];/[Na]exp([-(£Na-£)F]/J?r)} - 1exp([-{Etfa-E)F]/RT)-l

(2.21)

Equation (2.21) shows how the sodium concentration affects thesodium current. It turns out that the theoretical prediction made by(2.21) agrees very well with experimental results. For a detaileddiscussion of this, see Hodgkin and Huxley (1952a, pp. 468^471).Thus the independence principle has been established.

Next it turns out that instead of dealing with the ionic currents/ N a and 7K themselves, it is considerably more advantageous toconsider the permeability of the membrane to Na and K ions. Inorder to clarify this point we first give a formal definition ofpermeability.

Definition. The permeability of the membrane to Na ions is

As previously defined,

RT [Na]o RT [Na],

*"'" ~F-XaWh=" ~F~ln¥*l-

£ N a is called the sodium equilibrium potential; - £ N a is the emf

Nerve conduction 38

equivalent to the chemical driving force acting on the Na ions.Thus, the effective external emf acting on the Na ions is E — £ N a .

The permeability is denoted by gNa. It has the dimension ofconductance (current divided by potential difference) and gNa willfrequently be referred to as the sodium conductance. Similarly, thepermeability of the membrane to K ions is IK/(E — EK) and isdenoted by gK .

Since permeability has the dimension of conductance, which isthe reciprocal of resistance, the units of measure of permeabilityare the reciprocals of units of resistance. The basic unit is the mho,which is the reciprocal of the ohm. That is, 1 fi"1 is the conduc-tance that exists if 1 A of current flows between two points whosepotential difference is 1 V.

The quantities £ N a , E, and / N a have already been described andwe know how to determine them from experimental data. Hencethe function g N a ( 0 can be determined and graphed. The value ofEK can also be determined experimentally [see Hodgkin and Huxley(1952b)] and / K has already been described. Hence the functiongK(t) can also be determined and graphed.

There are two important reasons for introducing the concept ofpermeability. First, it can be shown that gNa, gK are independentof the magnitudes of the corresponding driving forces E — ENsi,E — EK under which gNa, gK are measured. [This is shown byHodgkin and Huxley (1952a).] Thus gNa, gK measure real proper-ties of the membrane just as resistance measures a real property of,say, a length of copper wire. Second, at the sodium potential thecurrent curves reverse their signs, whereas the conductance curvesundergo no marked change. Also, the function g N a ( 0 is betterbehaved, mathematically. Indeed, g N a (0 is continuous, whereas7Na(/) is discontinuous. [See Hodgkin and Huxley (1952a, p. 477).]Consequently, gNa and gK play primary roles in the Hodgkin-Huxley equations that will presently be derived.

In order to obtain the desired quantitative description of how theaction potential is propagated, considerably more quantitative datafrom voltage-clamp experiments is needed. The voltage-clamp ex-periments described so far concern conditions in which the mem-brane potential is clamped or fixed at a value different from the


resting value. That is, we obtained a description of the currents^Na(O a nd ^ K ( 0 across the membrane when V^ 0. To obtain thequantitative description of how the action potential is propagated,we need descriptions of the currents / N a (0 and /K(/) and thecorresponding permeabilities gNa(0 and gK(t) in other voltage-clamp conditions: First the case in the membrane potential isclamped at a value different from the resting value (i.e., V=£ 0) andthen suddenly restored to the resting value (V= 0). Also necessaryare certain voltage-clamp experiments in which the membranepotential is clamped at a value different from the resting value (i.e.,V= Vl # 0) and then suddenly changed to a new value differentfrom the resting value (i.e., V=V2 and V2=t0 and V2i=V1).Experiments of this kind are described in Hodgkin and Huxley(1952b). Finally a more detailed quantitative experimental study isneeded on how the sodium conductance decreases after reaching itsinitial maximum. Such experiments are described in Hodgkin andHuxley (1952c). Since our purpose is to sketch the experimentalprocedures and results rather than to give a detailed account, wewill not describe the experiments in Hodgkin and Huxley (1952b,1952c). However, we will feel free to invoke the results of theseexperiments when we derive the Hodgkin-Huxley equations. Forthe present we will just enumerate a few of the results obtained inHodgkin and Huxley (1952b).

First the potassium equilibrium potential EK is determined.Second, the permeabilities gNa(0> £ K ( 0 a r e shown to be indepen-dent of the driving force. Finally, besides the two main ioniccurrents / N a (0 and /K(/), there is an appreciable "leakage" cur-rent due mainly to a flow of chloride ions, which is discussed andestimated quantitatively in Hodgkin and Huxley (1952b).

2.3.2 Derivation of the differential equationsIf the experimental data are compared with the previous

description of the propagation of the action potential, it becomesclear that in obtaining this description we have not made full use ofthe experimental data. That is, the experimental data consist ofspecific numbers, whereas the description of the action potential isqualitative, not quantitative. The description contains such phrases

Nerve conduction 40

as "begins to flow," "depolarize further," and "begins to decrease."But these are not numbers in the description. Clearly then weshould look for a quantitative description, that is, a descriptionthat uses the full force of the quantitative experimental results.Moreover, our quantitative description should not just specifyquantities but should be expected also to yield quantitative ex-planations: for example, we would expect to obtain a mathematicalrelationship among the membrane potential, the sodium conduc-tance, and the potassium conductance. Also we need a consider-ably more detailed description of the action potential than the onealready given. (For example, so far we have no explanation of howthe absolute and relative refractory periods occur.) This is the kindof description that Hodgkin and Huxley (1952c) obtained in theirpaper. They used the experimental results to derive a system ofdifferential equations in which the dependent variables or un-knowns are the potential difference across the membrane as afunction of time and a set of variables that describe the sodium andpotassium conductances as functions of time.

Our next step is to derive these Hodgkin-Huxley differentialequations. Before starting the actual derivation, however, it isimportant to clarify the status of these equations. This is especiallyimportant to the reader who is accustomed to differential equationsin physics, for example, mechanics or electrical circuit theory. Inderiving a differential equation or system of differential equationsin physics, one usually starts from first principles such as Newton'slaws in mechanics or Kirchhoff's laws in electrical circuit theory.The concepts in the general principles are expressed in appropriatequantitative terms. For example, in linear motion, the accelerationis represented by d2x/dt2. Finally, the particular given conditionsare expressed in mathematical terms. A familiar example is thederivation of the equation of a simple pendulum. One starts fromNewton's Second Law: Force equals mass times acceleration. If mdenotes the mass of the bob, / the length of the pendulum, and 0the angle of displacement, then

d2emass X acceleration = ml—r-.

dt2

The force that acts on the bob is a component of the gravitational


-mg

Figure 2.8.

force — mg and, as indicated in Fig. 2.8, that force is — mg sin 0.Thus the differential equation for the simple pendulum is

d2eml—Y + wg sin 0 = 0.

The derivation of the Hodgkin-Huxley equations is in starkcontrast to this familiar procedure. The chief reason for this is thatthere are no basic laws or first principles that we can follow, that is,there are no Newton's laws or Kirchhoff's laws from which to start.Experimental work suggests that the membrane potential and thesodium current and the potassium current are the most importantvariables and we would expect, therefore, that the equations mighttake the form:

dV

~di =

dt

dt

and the derivation of the equations would consist in determiningthe functions F, G, and H. [As we will see later, we do not dealdirectly with gNa and gK. Instead special forms are chosen for thefunctions gN a(0 and gK(t).] But we have no theoretical guidanceor first principles to help us determine the functions F, G, and H.Instead, the form of the differential equations is determined en-

Nerve conduction 42

tirely from the experimental data. The differential equations areempirical or ad hoc descriptions. Having indicated the nature ofthe equations, we list several points that should be emphasized.First and most important is the fact that we are forced to considerad hoc equations because there are no first principles to guide us.The quantitative aspects of physiology have not been developed tothe stage where such first principles exist; and, indeed, there is noparticular reason to believe that a set of first principles likeNewton's laws or Kirchhoff s laws will be developed. The futuredevelopment of quantitative biology remains veiled. All we know atpresent is that there is a significant quantitative aspect of biology.Second, the Hodgkin-Huxley equations turn out to be (as we willshow later) a very good summary or fusion of quantitative experi-mental results. Their empirical quality makes them analogous toKepler's laws. To some extent this analogy suggests the status ofmathematical biology at the present. Since extensive experimentaldata can be represented in compact mathematical form, that is, asystem of differential equations, it seems clear that furthermathematical developments can be expected. On the other hand,the analogy of the Hodgkin-Huxley equations with Kepler's lawscannot be pushed too far. There is no reason to conclude from theanalogy that a theoretical biologist will presently develop somebasic principles that will explain the origin of the Hodgkin-Huxleyequations just as Newton's laws can be used to derive or explainKepler's laws. (It is possibly worth noting that the fact that themathematical description given by Hodgkin and Huxley is in theform of differential equations is misleading to a reader who isfamiliar with mathematical physics. With a background of physics,one tends automatically to regard a differential equation as arepresentation or realization of basic physical laws. The Hodgkin-Huxley equations have no such properties, but correspond insteadto Kepler's laws, which, of course, are not differential equations.)

Third, it is enlightening to emphasize that Hodgkin and Huxleywere fully aware of the ad hoc nature of their differential equa-tions. As physiologists, their objective was to determine how theaction potential occurs and is propagated. (As we have pointed outbefore, even today the basic question of how the flows of Na and Kions are regulated remains unanswered.) When Hodgkin and Huxley


initiated their work, they were not looking for some differentialequations. They were concerned with studying the flow of electriccurrent through the membrane surface and finding an explanationfor how these currents occur. As it turned out, the hypothesis thatthey started out to test in their experiments proved to be incorrect;even the hope of formulating another hypothesis that would ex-plain the nature of the flow of current failed. It had seemedreasonable that with the large amount of experimental data, itwould be possible to develop such a molecular hypothesis, butHodgkin and Huxley were forced to conclude that their experimen-tal data would "yield only very general information about the classof systems likely to be involved" [Hodgkin (1977, p. 19)]. Thus they"settled for the more pedestrian aim of finding a simple set ofmathematical equations which might plausibly represent the move-ment of electrically charged gating particles" [Hodgkin (1977, p.19)].

Finally, it is important to understand the status of theHodgkin-Huxley equations in the view of physiologists. This statusis clearly described in the following excerpt from Hodgkin andHuxley (1952d, p. 541):

The agreement must not be taken as evidence that our equations areanything more than an empirical description of the time-course of thechanges in permeability to sodium and potassium. An equally satisfactorydescription of the voltage clamp data could no doubt have been achievedwith equations of very different form, which would probably have beenequally successful in predicting the electrical behavior of the membrane.It was pointed out in Part II of this paper that certain features of ourequations were capable of a physical interpretation, but the success of theequations is no evidence in favour of the mechanism of permeabilitychange that we tentatively had in mind when formulating them.

The point that we do consider to be established is that fairly simplepermeability changes in response to alterations in membrane potential, ofthe kind deduced from the voltage clamp results, are a sufficient explana-tion of the wide range of phenomena that have been fitted by solutions ofthe equations.

The Hodgkin-Huxley equations come in part from the statementconcerning current that has already been invoked several times,that is, the equation

dVI=CM—+I^ (2.22)

Nerve conduction 44

which says that the total current is the sum of the capacitancecurrent and the ionic currents. The capacitance of the membraneper unit area (i.e., CM) has already been determined. There arethree ionic currents: The two most important are /Na and /K, thesodium and potassium currents that we have described in somedetail, and a third "leakage" current, denoted by Ih consistinglargely of chloride ions, which was already mentioned briefly. Theleakage current is discussed in Hodgkin and Huxley (1952c). Thus

W N . + ' K + '/ (2-23)

and, substituting (2.23) into (2.22), we have

/ = C M — + /Na + /K + /;. (2.24)

Using the independence principle discussed earlier, we regard /Na,/K, and ^ as having no influence on one another. Hence we maydeal with the sodium and potassium currents separately and write

It turns out that the leakage current has an especially simpledescription

/ , « £ , ( £ - £ , ) ,

where g, is a positive constant and E{ is the equilibrium potentialfor the ions (mostly chloride) that constitute the leakage current.Thus (2.24) becomes

dVM dt N a a K

(2.25)

or

dV 1

dt CM N a N a

- g K ( £ - £ K ) - ! , ( £ - £ , ) ] . (2.26)

SinceV=E-ER,


it is convenient to rewrite E - £Na, E - EK, and E — Et in termsof V. To do thus, let

Then

E-EK=V-VK,

E-E,= V- V,,

and (2.26) becomes

(2.27)

Equation (2.27) is a version of the first of the Hodgkin-Huxleyequations. It would be reasonable that the remaining equationswould be of the form

- ^ — = G(/,F, gN a ,gK) ,

and our remaining problem would be to determine the functions Gand H from the experimental data. In fact, the problem is by nomeans that straightforward. Hodgkin and Huxley decided insteadto introduce other variables. The choice of these variables cannotbe entirely explained on a logical basis. All we can do is point outreasons that suggest these choices for the variables. The decision touse these variables was a brilliant intuitive step. A measure of theextraordinary accomplishment that this step represents is the factthat the mathematical formulation given by this choice of variableshas remained essentially unchanged in the 30 years of work onneurons and other electrically excitable cells that have followed.

We look first at the simpler case, that is, the potassium conduc-tance gK, and we examine how gK(t) changes when V is changed

Nerve conduction 46

8. 6

m mho/cm2 420

b)

4 0msec

(a) Rise of potassium conductance (b) Fall of potassium conductanceassociated with depolarization associated with repolarizationof 25 mV to the resting potential

Figure 2.9. Graphs of potassium conductance.

from 0 to 25 mV and then from 25 mV back to 0. This is displayedin Fig. 2.9. One of the complications that is shown by Fig. 2.9 isthe fact that if V is changed from 0 to 25 mV, then gK(t) riseswith a marked inflection. But if V is changed from 25 mV to 0,then gK(t) decreases in a simple exponential way. Consequently,finding a differential equation whose solutions would fit the twoparts of the curve in Fig. 2.9 is complicated. In order to obtain avalid description of this kind of curve, Hodgkin and Huxleyintroduced a new dimensionless variable n. They proposed toexpress the function gK(t) as

8A')=gAn(t)]\ (2.28)

where gK is a positive constant whose value will be obtained fromthe experimental data, and to require that the function n(t) satisfya differential equation of the form

dn— = aH(l-n)-PHn, (2.29)

where an and /?„ are nonnegative functions of V. The best justifica-tion for this proposal and a similar, somewhat more complicated,proposal made for gNa(0 is the fact that it works. An argumentthat suggests introducing the description of gK(t) given by (2.28)and (2.29) is the following. If an and /?„ are constants, then by anelementary method (variables separable) (2.29) can be solved andwe obtain, for the general solution,

- (a n + / O ' ] , (2.30)


where K is a positive constant. If for simplicity we take K = 1, thenif the solution n(t) is close to zero, it can be approximated bytaking fin = 0 in (2.30) and we obtain

n(t)= -e-'+l.

A simple calculation shows that the function

[n(t)]4=[l-e-']A

has an inflection point just as appears in Fig. 2.9(a). Similarly if\n(t)\ has a value close to 1, (2.30) can be approximated by takingan = 0 and K = 1, fin = 1. We obtain

n(t)=-e~\

and the function

describes a simple exponential decrease just as appears in Fig.2.9(6).

Assuming that (2.28) and (2.29) give a satisfactory description of£K(0>

w e must proceed to determine an and jin as functions of V.In order to do this, we must, of course, utilize the experimentalresults.

Let n0 denote the resting value of n(t), that is, n0 is the numbersuch that

is the potassium conductance if V= 0 (i.e., the potassium conduc-tance of the membrane if the membrane potential is equal to theresting potential). Note that since we have not specified gK (gK

will be determined later from the experimental data), we do nothave an explicit value for n0. But since gK can be determinedindependently from the experimental data, n0 has a definite valuethat we can obtain from the experimental data. Now suppose avoltage-clamp experiment is performed in which V is changedsuddenly from 0 to a nonzero value. Then an, fin are changed tovalues that correspond to the new nonzero value of V. With an, jin

at these new fixed values, (2.29) can be solved and we obtain

= -(an + Pn)t+C9 (2.31)

Nerve conduction 48

where C is a constant of integration. Let

Note that if n = nx, then by (2.29), dn/dt = 0. Hence n^ is suchthat gKn^ is the value of the potassium conductance ultimatelyobtained after V is clamped at the nonzero value. (Remember thatthe experimental data show that following imposition of a voltageclamp, the potassium conductance increases to a certain levelwhere it then remains fixed.) Let us denote this final value ofpotassium conductance by gK and observed that it is a value thatcan be obtained from this experimental data. Substituting from(2.32) into (2.30) and simplifying, we obtain

If we impose the initial condition

/i(0)=/!0,

we obtain

Introducing the notation rn = l/(an + /?w), we obtain for the solu-tion

-

Thus

Since g ^ = gKm, then

(2.33)

But gK«o is by definition the potassium conductance when V= 0and consequently the value of gK«o is known from the experimen-


tal data; so if we denote gKwo by gKo> then we may rewrite (2.33)

as4

(2.34)

Since gK and gK are known from the experimental data, then ifa fixed value is chosen for rn, the right-hand side of (2.34) is aspecific function of t. On the other hand, for fixed V, the functiong K ( 0 has been determined experimentally. Consequently in orderto obtain the correct value for rn9 we simply select the value of rn

so that the graph of the resulting gK(0> which is defined by (2.34),agrees with or coincides with the experimental graph for gK(t).Having thus determined TW, we have an equation relating an andA,, that is,

1

or«Sn + fon = l. (2.35)

Next we consider the problem of estimating gK. Experimentalresults show that if V has a large value (i.e., a value > 110 mV),then the corresponding value of gK takes values ~ 20-50% higherthan the value gK corresponding to V = 100 mV. For definitenessin calculation, we assume that the values are 20% higher, and weassume that n = 1 at those values. Since gK = 20 m.Ql~

l/cvc^ atV= 100 mV, then gK is chosen to be 24 mfl 'Vcm2 . (There is, ofcourse, an arbitrariness in this choice of gK. This arbitrariness isjustified in the same way that some of the earlier assumptions arejustified: They contribute to the development of a quantitativedescription that yields good agreement with other experimentalresults.)

Now as remarked after (2.32), if V is fixed and sufficient timehas elapsed afterward, we have

But gK has been determined and, if V is fixed, then gK can bedetermined experimentally as was shown earlier. Hence for a fixedvalue of V, the quantity n^ can be determined. From its definition

Nerve conduction 50

we have

^ T A - " - <2J6)

Now solving (2.35) and (2.36) simultaneously for an and /?„, weobtain

Thus we have computed the values of an and fin associated with afixed value of V. Values of an and (in that are associated withvarious values of V are tabulated in Hodgkin and Huxley (1952d,p. 509). It remains to choose functions ocn(V) and Pn(V),the graphs of which fit the tabulated values. The functions thatHodgkin and Huxley chose are

0.0l(K+10)

exp[(F+10)/10]-r= 0.125 exp(F/80).

Even when variations with temperature and variation of restingpotential with [Na]0 are taken into account, there is still somearbitrariness left in the selection of the functions an(V) and Pn(V).For a discussion of these points, see Hodgkin and Huxley (1952d,p. 510).

As an indication of the arbitrariness in the equation for n thathas been introduced, we point out a remark made by Hodgkin andHuxley in connection with determining the value of rn when V isgiven.

It will be seen that there is reasonable agreement between theoretical andexperimental curves [the graph of the function gK(t) defined by (2.34)and the experimentally obtained graph of potassium conductance (Fig.2.9)] except that the latter show more initial delay. Better agreementmight have been obtained with a fifth or sixth power [of n] but theimprovement was not considered to be worth the additional complication.[Italics added.]

The "additional complications" to which Hodgkin and Huxleyrefer consist mainly in the complications encountered in solving theresulting set of differential equations. It must be remembered that


Hodgkin and Huxley were writing at a time when numericalanalysis of a system of differential equations was carried out bylengthy labors with a hand-operated calculating machine [seeHodgkin (1977, p. 19)]. One may well ask whether Hodgkin andHuxley would have proposed a different equation for n{t) if theyhad had access to a modern computer.

The study of sodium conductance, that is, the function gNa(0> isanalogous to the study of potassium conductance except that it issomewhat more complicated. In particular if V is fixed at a valueabove the threshold value, then the sodium conductance first risesto a maximum value and then decreases. By considerations similarto those used to arrive at the description of potassium conductancegiven by (2.28) and (2.29), Hodgkin and Huxley decided to de-scribe the sodium conductance by means of the following equa-tions:

gNa = m3/*gNa, (2.37)

dm— =am(l-m)-Pmm, (2.38)

dh— =ah(l-h)-pkh9 (2.39)

where gNa is a positive constant and am, f3m,ah, flh are certainnonnegative functions of V. The dependent variables m and h arecalled the activation and inactivation variables, respectively. Theycan be regarded as measures of the way by which the membranepermits Na ions to pass through.

By techniques analogous to those used to study an and /?„, thecoefficients in (2.29), the coefficients am,f}m,ah, fih are determinedto be

(0.1)(F+25)

exp[(F+25)/10] - 1 '

= 0.07exp| —

exp[(K+30)/10] + 1 '

Nerve conduction 52

Substituting from (2.28) and (2.37) into (2.27), we thus obtain forthe Hodgkin-Huxley equations,

dV 1

at cM

dm— = am(l-m)-Pmm9

dh— = ah(l-h)-phh9

dn— = an(l-n)-Pnn,

where am, /?m, ah, fih, an, fin are the functions of V alreadydescribed. The preceding equations are the formulation of theHodgkin-Huxley equations that we will study. For completeness,however, it should be pointed out these equations are valid at thestandard temperature of 6.3°C. It has already been mentioned thattemperature affects the electrical activity of the axon. A moregeneral mathematical description that includes the effects of tem-perature T (in degrees Centigrade) is given by

dV 1

ai ^M

dm r [ / r - 6 . 3— [ ( ) 8 ] | ( ) |

dn

It (In 3)r-6.3

10

That is, the right-hand sides of the equations for the activation andinactivation variables are multiplied by

2.5 Mathematical status 53

2.4 What the Hodgkin-Huxley equations describeBefore going ahead with mathematical analysis of the

Hodgkin-Huxley equations, we emphasize what the equations de-scribe. The Hodgkin-Huxley equations can be regarded as describ-ing the variations of membrane potential and ion conductancesthat occur "naturally" at a fixed point on the axon. (For example,as an action potential passes a given point on the axon, certainchanges take place in the membrane potential and the ion conduc-tances.) The equations are obtained from data given by voltage-clamp experiments and are primarily a quantitative description ofthe results of experiments, that is, the voltage-clamp experiments,which are entirely artificial. That is, in voltage-clamp experimentsthe membrane potential and the ionic conductances are indepen-dent of position on the axon, the membrane potential is constant,and the ionic conductances depend only on time. These are condi-tions that do not occur in "real life" or the normal functioning ofthe axon. Later we will see how to extend the description given bythe Hodgkin-Huxley equations so that it will include the possibilityof variation of membrane potential and ionic conductance withposition along the axon. This extended description will thus bebroad enough to include description of the propagation of anaction potential along the axon.

2.5 Mathematical status of the Hodgkin-Huxley equations2.5.1 History

As a mathematical entity, the Hodgkin-Huxley equationshave a curious history. They form a significant part of the dis-tinguished physiological work of Hodgkin and Huxley. The impor-tance of this physiological work was recognized immediately; otherexperimental studies along similar lines were soon carried out byother research scientists, and Hodgkin and Huxley received aNobel prize for their work only 11 years after the papers appeared.(By the standards of physics and chemistry, this is a fairly longinterval, but compared to the intervals for many Nobel prizes inphysiology and medicine, 11 years is a very short time.) Hodgkinand Huxley included in their original papers considerable numeri-cal analysis of the equations and in the ensuing years, a number ofphysiologists have made further numerical analysis of the equa-

Nerve conduction 54

tions. Even today it seems safe to say that the majority of thephysiologically significant mathematical analysis of the equationsthat has been carried out is numerical analysis.

Nevertheless, from the point of view of physiology a morequalitative analysis of the solutions of the equations is also highlydesirable. Throughout our discussion of the derivation of theHodgkin-Huxley equations we have emphasized the uncertaintiesin numerical values and the tentativeness of the assumptions thatmust be made. Consequently it is fairly clear that a study of howsolutions of equations of this kind behave in general is just asimportant as a numerical study of solutions of a particular set ofequations. This last observation might be regarded as originatingwith a mathematician who is looking for work. But in fact the valueof qualitative analysis of classes of equations has been emphasizedby physiologists [Jack, Noble, and Tsien (1975)] and physicists[Scott (1975)].

In view of these facts, it might be expected that theHodgkin-Huxley equations would have been the subject of consid-erable attention by applied mathematicians. This is, however, notthe case. Since we are not primarily concerned here with the historyof science, we will not speculate too much about why theHodgkin-Huxley equations were neglected by mathematicians.Among the possible reasons may be the timing; they appeared at atime when nonlinear ordinary differential equations were not at thezenith of their popularity with mathematicians. Moreover at thattime (1952), there was a tendency to study very general classes ofequations rather than specific systems of equations. (The greatinterest in recent years in the Lorenz equations shows how valueshave changed since then.) Perhaps the most important reason isthat the Hodgkin-Huxley equations are difficult to study. Thisdifficulty has two aspects. First, although the functions that appearin the equations are sufficiently well behaved (in fact, they areanalytic, i.e., can be represented by power series) so that standardexistence theorems can be applied to them, the equations arenevertheless a four-dimensional system that is strongly nonlinear.Previous work in nonlinear differential equations, ever sincePoincare, suggests that analysis of these equations would be dif-ficult and that suggestion is entirely correct. There is, however,


another aspect of the difficulty with the Hodgkin-Huxley equa-tions that is even more serious, but it is also a difficulty that shouldmake the Hodgkin-Huxley equations of great interest to mathema-ticians, both pure and applied. The difficulty is that the problemsarising with the Hodgkin-Huxley equations that are of physiologi-cal interest represent a new direction in the study of nonlineardifferential equations. The classical problems in nonlinear ordinarydifferential equations that arose in celestial mechanics and later inelectrical circuit theory are concerned with equilibrium points andthen stability, periodic solutions, and almost periodic solutions.[Besides his interest in periodic solutions for problems in celestialmechanics, Poincare regarded the study of periodic solutions as acrucial first step in the general study of solutions of nonlinearordinary differential equations. See Poincare, (1892-99, Vol. I, pp.81-82).] Efforts to deal with these questions have led to thedevelopment of a large amount of mathematical theory.

The problems that arise in nerve conduction theory lie in adifferent and novel direction. From the point of view of thepreceding classical problems, the Hodgkin-Huxley equations arenot very interesting. The underlying physiology shows that thereshould be a unique set of equilibrium values V = 0 (which corre-sponds to the resting potential), m^, h^, and n^. Moreover, if oneor more of the variables V,m,h,n is displaced from the equi-librium value, then, again from the physiology, we know that aftersome time has elapsed, the values of V,m,h,n will return to theequilibrium values. In strict mathematical language we say that if(F, m, ~h, n) is a point in R4 such that

and if (K(/) , m{t\ h(t\ n{t)) is the solution of the Hodgkin-Huxley equations such that

V(0) = V,

then the solution (V(t\ m(t\ h{t), n{t)) is defined for all f > 0

Nerve conduction 56

and

lim (V(t), m(t), h(t), n(t)) = (0, m^ h^ nj.

From the physiological viewpoint, there is a limitation on theinitial value (V, m, /*, n). First, as pointed out earlier, we areinterested only in values of m,h,n that are in the interval [0,1].Second, we are interested only in values of V that do not damagethe axon. If the axon is "fried" by a large value of F, then it wouldno longer be described by the Hodgkin-Huxley equations. In alittle less precise language, all physiologically significant solutionsapproach the equilibrium point (0, m^, / i^, n^). Using conven-tional mathematical language, we would say that the problem is toprove that there exist two numbers Vv V2 such that Vx < V2 andthat the equilibrium point (0, m^, h^, n^) has a region of globalasymptotic stability of the form

{(V,m9h,n)/Vl<V<V2,O<m<l,O<h<l,O<n<l}.

This seems like a reasonable preliminary result to establish, and itis phrased in the language of classical theory of nonlinear ordinarydifferential equations. Now let us suppose that this result has beenestablished. It has two drawbacks. First, it suggests that from theviewpoint of classical theory, the Hodgkin-Huxley equations arenot very interesting. All the solutions approach an equilibriumpoint. From the classical viewpoint there seems little more to sayunless we make a finer investigation and study, for example, therate at which the solutions approach the equilibrium point. But theresult has a more serious drawback: Although it seems to be acomplete result, it tells us nothing about whether the solutionsdescribe such phenomena as the refractory period and thresholdbehavior. We need to use mathematical concepts that correspondto the physiological notions of refractory period and threshold, andwe need to establish that the solutions of the Hodgkin-Huxleyequations behave in accordance with these mathematical concepts.In other words, we need to develop another mathematical ap-proach. When the problems in nonlinear differential equations thatarise in nonlinear electrical circuit theory were first studied inten-sively in the 1920s, it was realized after a while that much of the


theory of ordinary differential equations that had been developedin response to problems in celestial mechanics (for example, thePoincare-Bendixson theory) could be utilized to study the electricalcircuit theory problems. But we have now seen there is no suchlucky circumstance when we look at problems in nerve conduction.We cannot simply fall into the familiar language and concepts.From the point of view of the mathematician, the attempt to studyquantitatively the mechanism of nerve conduction opens the possi-bility of a novel study of that most classical and conventional ofmathematical objects, the ordinary differential equation.

It is rather surprising that mathematicians have not paid muchattention to this possibility. Indeed it was pointed out originally bya physiologist, FitzHugh (1969), rather than a mathematician. Ofcourse, as we shall see, there is considerable literature concerninganalysis of the Hodgkin-Huxley equations, and there are effectivemathematical tools, such as singular perturbation theory, that canbe used to approach this study. But a complete rigorous qualitativeanalysis has not yet been achieved.

2.5.2 Some successful numerical analysis of theHodgkin-Huxley equations

2.5.2.1 Analysis of the original Hodgkin-Huxley equationsHodgkin and Huxley, in their original paper (1952d),

undertook extensive numerical analysis of the equations they hadderived. Their results were largely successful in that the (approxi-mate) numerical solutions of the equations agreed very well withexperimental results. Some of these agreements are to be expected.For example, since the equations were derived from data obtainedfrom voltage-clamp experiments, the solutions of the equationsmight well be expected to give agreements with the results ofvoltage-clamp experiments. Some of the theoretical agreements orpredictions are striking (e.g., agreement with the results of current-clamp experiments, which will be described later) and one of thepredictions, the prediction of the velocity of the action potential, istruly spectacular. However, none of the theoretical studies makes atrue "prediction" in the sense of giving us new information thathad not been previously obtained from experiment or giving a new

Nerve conduction 58

viewpoint that is suggestive of new directions for experimentalwork. There are some exceptions to this statement, for example,Guttman, Lewis, and Rinzel (1980).

Because the agreement or predictions given by numerical analy-sis of the Hodgkin-Huxley equations is such an important measureof the success of the equations, we describe these agreements nowin some detail.

First and most prosaic is the correct prediction of the totalcurrent during a voltage clamp. During a voltage clamp, V isconstant. Consequently am, ftm,ah, fih, ccn, Pn are constants and,using the notation introduced in Section 2.3, we have

- (am + Pm)t],

" ( 0 = "oo - ("oo - «o)exp[ - (an + pn)t],

and consequently we can compute the currents

Since V is a given constant, the leakage current

h = s,(v-v,)can also be computed, and thus the total membrane current

can be computed theoretically and compared with the experimen-tally observed current. It turns out that the theoretical and ex-perimental^ results agree well unless F = -115 mV (the sodiumequilibrium potential); then the theoretical current has too short adelay, which is caused by the insufficient delay in the theoreticaldescription of the rise in the potassium conductance (see Fig. 2.9).

Second, a satisfactory agreement is obtained between the experi-mental results when a shock is applied to an electrode in the axon,and the theoretical results calculated by using the Hodgkin-Huxleyequations. The experimental methods were described earlier andthe experimental results are summarized in Fig. 2.2. The theoreticalresult is calculated as follows. Since the potential is constant alongthe axon, there is no current in the longitudinal direction (i.e., in


the direction of the axis of the cylinder); hence the net membranecurrent / is zero except during the shock. The form of the actionpotential (we assume that the shock is large enough so that V israised above the threshold value) can be determined by solving theHodgkin-Huxley equations with / = 0, that is,

dV 1-T=--7r{gN3Lat cM

dm

dh— = ah(l-h)=lihh9

dn— = an(l-n) = PHn,

for the solution that satisfies the initial conditions

where Vo is the initial change in the membrane potential that iscaused by the shock and m0, h0, n0 are the resting values, that is,the values of m(t), h(t), n{t), respectively, if V is the restingpotential — 70 mV. The agreement obtained is not entirely satisfac-tory. Certain discrepancies are discussed by Hodgkin and Huxley(1952d, pp. 525-526). In Hodgkin and Huxley (1952d, pp. 542-543)the extent to which these discrepancies can be attributed to knownshortcomings in the equations (e.g., the too short delay in the riseof gK) is discussed.

The numerical analysis reveals a good agreement between theo-retical predictions and experimental observations of the absoluteand relative refractory periods. There is good agreement for boththe duration of the absolute refractory period and changes in V{t)as the membrane returns from the refractory condition to thenormal resting condition.

Nerve conduction 60

Another example of agreement between theoretical and experi-mental results is anode break excitation. In an experiment demon-strating anode break excitation, a current of negatively chargedparticles is made to flow inward through the membrane in such away that the membrane potential is reduced from the resting value( — 70 mV) to a value such as —100 mV. This current is continuedfor an interval of time significantly larger than rm, rh, or rn and themembrane potential is maintained at —100 mV during this inter-val. Then the current is suddenly stopped. The observed ex-perimental result is that an action potential occurs. To obtain atheoretical description of anode break excitation, the Hodgkin-Huxley equations with / = 0 are solved for the solution with initialvalue:

F(0)= -100,

Good agreement is obtained between the experimental and theoret-ical description. The explanation of anode break excitation is thatthe decrease of the membrane potential decreases the potassiumconductance and decreases sodium inactivation. Both of theseeffects persist when the current is stopped. Hence when the currentis stopped and the membrane potential returns to the resting value,there is a reduced outward potassium current and an increasedinward sodium current. The net flow of positive ions is inward andis sufficiently large so that depolarization occurs.

A striking example of agreement between theoretical and experi-mental results occurs in work with current clamps. A small con-stant current is passed through the membrane and the resultingchanges of membrane potential measured. The current is suppliedby an internal electrode so that the membrane is subject to auniform current density. (Hence the name "current clamp.") Sincethe current is small, the changes that take place in V,m,h,n aresmall enough so that only the linear parts of the Hodgkin-Huxleyequations need be considered. (The assumption that the linearizedequations give a valid description is supported by experimentalresult that if the magnitude of the current is kept the same, but the


direction or sign of the current changed, then the membranepotential also has the same magnitude but changes its sign.) Theappropriate solutions of the linearized Hodgkin-Huxley equationsagree very well with the data obtained from current-clamp ex-periments. This is a striking confirmation of the validity of theHodgkin-Huxley equations because current-clamp experiments arevery different from the voltage-clamp experiments that give rise tothe Hodgkin-Huxley equations.

2.5.2.2 Analysis of the full Hodgkin-Huxley equationsSo far we have been looking at examples of agreement

between theory and experiment in cases where the potential isuniform on the axon, that is, the potential depends only on timeand is independent of position on the axon. Now we wish to studysituations in which the potential varies with position on the axon aswell as time. This is what occurs in the normal physiologicalfunctioning of the nerve axon. The Hodgkin-Huxley equations aspresented here cannot be used to study such situations because theequations describe conditions in which the membrane potentialand the currents depend only on time. However, minor considera-tions from electricity theory make it possible to extend the equa-tions so that they can be used to describe the situation in which thepotential and the ionic conductances are functions of distancealong the axon as well as time.

Following Jack, Noble, and Tsien (1975, p. 25ff) we obtain theextended model by combining the model of the space-clamp dataand voltage-clamp data (i.e., the Hodgkin-Huxley equations) witha few statements from standard electrical theory.

First, from Ohm's law, it follows that if V is the membranepotential, R is the resistance per unit length of the interior of thenerve, ia denotes flow of current in the axon, and x measuresdistance along the axon, then

dV— =-Ria. (2.40)ox

Then if im denotes the total current across the membrane,

Nerve conduction 62

Differentiating (2.40) with respect to x and substituting from(2.41), we obtain

d2V2=Rim- (2.42)

[For a discussion of (2.40)-(2.42) see Jack, Noble, and Tsien (1975,p. 25ff).] But in the case of nerve conduction, we have, according tothe Hodgkin-Huxley equations, that

dVtn c\ J i

dv

+ gKn4(V-VK)S + gl(V-V,)S, (2.43)

where C is the total capacitance of an axon membrane of unitlength and S denotes the area of the axon membrane of unitlength. Substituting from (2.43) into (2.42) we obtain the followingequations that describe the changes in membrane potential andflow of ionic current across the membrane:

1 32V dV

+ {gK"4(V-VK)+gl(V-Vl)}2vr,

dm— =am{\-m)-[imm,

8h— =ah(l-h)-llhh,

dn— =an(l-n)-pnn,

where r is the radius of the axon. We call these equations the fullHodgkin-Huxley equations. A solution of this system of partialdifferential equations consists of four functions V(x, t), m(x,t),h(x, t\ and n(x, t) that satisfy the four partial differential equa-tions. Now the general problem of solving the system (3fc-3tf?) isextremely difficult. However, we are not interested in finding allthe solutions of (Jfc-JP) but only those that would describe an

2.5 Mathematical status

V(x,t0)

63

Figure 2.10.

action potential. An action potential is described by a functionV(x, t), which has the following property: At a given value t = tQ,the graph of V(x, t0) has the appearance shown in Fig. 2.10. At alater time t = tl > t0, the graph of V(x, tx) has the same form asthat of V{x, /0) except that it is moved to the right (see Fig. 2.11).Such a function can be written as

where W is a function of one variable and 0 is a positive constantthat is the velocity with which the configuration moves to the right.This suggests that in order to obtain a description of the actionpotential we look for a solution of (JP-Jif) of the form

(V(x - Ot), m(x - 6t), h(x - Ot), n(x - 6t)).

Such a solution is called a traveling wave solution and has longbeen a familiar concept in mathematical physics. The search for atraveling wave solution is much simpler than the search for ageneral solution of (J$£-J{?) because it can be reduced to the studyof an ordinary differential equation. In order to carry out thisreduction, we introduce the variable £ = x - Ot. Then by the familiar

Figure 2.11.

Nerve conduction 64

chain rule from

32V

dx2

dv

dm

17 =

dh

calculus, we have

d2V

d? 'dv

dm

~6~dJ'dh

dn dn

and (J^-Jf?) becomes

1 d2V dV

R d£2

dm 1 _•^=-^[««(l-*)-j8mni], (2.44)

dh 1 r

-=--[ah(l-h)-Phh],

dn 1 r

[ ( ) ]

Thus the problem is reduced to the question of solving a system ofordinary differential equations: one second-order equation andthree first-order equations. Usually, in order to solve (2.44) wemust specify a value for 6, the velocity of the action potential. Alsowe are not searching for an arbitrary solution of (2.44). We arelooking for a solution with the property that F(£) has a configura-tion something like the graph in Fig. 2.10. Actually we impose amilder condition on F(£): We require only that l i m ^ ^ F ^ ) = 0.

Now we incorporate these two considerations into one problem,the solution of which will yield a theoretical estimate for the


velocity of the action potential, that is, the following problem isstudied: how to determine a value of 0 for which the solutions(K(O,w«),A(O,»(O) of (2.44) are such that l i m ^ F t f ) = 0.Calculations carried out by Hodgkin and Huxley (1952d, pp.522-523) show that for a particular axon, the required conductionvelocity was ^ = 18.8 m/s. The experimentally observed velocityfor the same axon was 21.2 m/s. Thus the Hodgkin-Huxleyequations yield a quite accurate estimate for the velocity of theaction potential. This estimate is probably the most spectaculartheoretical prediction of the Hodgkin-Huxley theory.

By solving (2.44) numerically with 0 = 18.8, the sodium andpotassium currents during a propagated action potential can becomputed and consequently the total addition of sodium and lossof potassium can be determined by integrating the correspondingionic currents over the whole impulse. These theoretical results arein good agreement with experimentally determined values [seeHodgkin and Huxley (1952d, pp. 531-532)].

2.5.3 Drawbacks of the Hodgkin-Huxley equationsWe have seen that numerical analysis of the Hodgkin-

Huxley equations yields excellent agreement with a wide assort-ment of experimental results. Indeed when the arbitrariness ofsome of the steps in the derivation of the Hodgkin-Huxley equa-tions is taken into account, the accurate theoretical predictions thatcome from analyzing them seem extraordinary.

Nevertheless, the Hodgkin-Huxley equations are not entirelyaccurate, and we describe now a couple of deficiencies or draw-backs, that is, we will describe two predictions made by thesolutions of the equations that differ significantly from the experi-mental observations.

First, if / is fixed, then except for a narrow range of values of /,there is a periodic solution of the Hodgkin-Huxley equations[FitzHugh (1961)]. (Since all the solutions discussed here are ap-proximate solutions obtained by numerical analysis, a "periodicsolution" means an approximate solution that looks as if it isperiodic.) But in the laboratory, stimulation of the squid giant axonby a step current produces only a finite train of up to four impulses[see Hagiwara and Oomura (1958) and Jakobsson and Guttman

Nerve conduction 66

(1980)]. Also in the laboratory, if / increases linearly with time,then if the rate of increase is below a certain threshold value, noaction potential occurs. If the rate of increase of / is above thatthreshold value, one action potential occurs. But a numericalanalysis of the Hodgkin-Huxley equations yields a periodic solu-tion for all values of / except those in the interval mentionedbefore. If the phenomenon of accommodation is taken into account,it is possible to modify the Hodgkin-Huxley equations so that acloser agreement between experiment and theory can be obtained.Accommodation is a physiological process that produces a slowdecay in a train of nerve impulses that result from a constantstimulation [see FitzHugh (1969)]. It can be regarded as an increasein the threshold of an excitable membrane when the membrane issubjected to a sustained sub threshold depolarizing stimulus or astimulus that increases very slowly. A modification of theHodgkin-Huxley equations that takes accommodation into accounthas been proposed by Adelman and FitzHugh (1975), and numeri-cal analysis of the modified equations yields results that are incloser agreement with experiments using constant current stimulus.

It should be pointed out that the discussion of accommodationin Hodgkin and Huxley (1952d, pp. 537-538) is misleading becauseit suggests that the Hodgkin-Huxley equations can be used todescribe accommodation. For a careful discussion of this point, seeJakobsson and Guttman (1980).

Nerve conduction:Other mathematical models

3.1 Earlier modelsWe have presented the Hodgkin-Huxley equations first

and in great detail because they constitute the most importantmodel of nerve conduction. There are, however, other models thatshould be discussed. Here we will describe these models, theirsignificance, and their status. However, as with the Hodgkin-Huxleyequations, a detailed mathematical analysis will be postponed untilChapter 5.

In trying to understand the development of mathematical elec-trophysiology, it is enlightening to realize that a number ofmathematical models, each consisting of a two-dimensional systemof ordinary differential equations, were introduced in the 1930s. Inthese models, one of the dependent variables can be identified asV(t), where V(t) measures the potential difference across themembrane and the second variable, say U(t), can be regarded as arecovery variable that tends to eliminate the excitability of themodel after excitation has occurred and to bring about the end ofthe impulse. The models have the form

dV

dU

where F and G are well-behaved functions. It is no longer ofinterest to analyze these models in detail, but it is important torealize that such models exist and that part of Hodgkin andHuxley's accomplishment consisted in replacing rather generalvariables, such as the U(t) in the description just given, with theconcrete variables m(t), h{t), n(t) that describe the sodium andpotassium conductances. For an account of some of these models,see FitzHugh (1969).

Nerve conduction: Other mathematical models 68

3.2 The FitzHugh-Nagumo modelA second and very different model was introduced after

the Hodgkin-Huxley equations. We have seen already that numeri-cal analysis of the Hodgkin-Huxley equations showed that thesolutions of the equations agreed, in the main, very well withexperimental results. However, attempts to study the qualitativeproperties of solutions of the Hodgkin-Huxley equations provedfruitless partly because the equations are fundamentally nonlinear,partly because they are a four-dimensional system (whereas muchof the known qualitative theory is applicable only to two-dimen-sional systems), and partly because the qualitative problems thatarise in nerve conduction had not been studied earlier.

The general qualitative theory of differential equations originatedfrom struggles to study the nonlinear differential equations thatarise in celestial mechanics. It required the genius of Poincare toshift the emphasis away from trying to obtain closed solutions (i.e.,explicit formulas for solutions) and toward studying the general orqualitative behavior of solutions. For example, if it is known thatthe system of differential equations has a periodic solution and thatall other nontrivial solutions approach this periodic solution, then afairly complete understanding of the solutions of the system hasbeen obtained even if no solution of the system has been explicitlycomputed. For mathematical models describing the axon, it isdesirable to understand the qualitative behavior of solutions thatreflects the threshold phenomenon and refractory periods. TheHodgkin-Huxley equations themselves, when first introduced, weretoo complicated to admit a qualitative analysis like this. Conse-quently, after the Hodgkin-Huxley equations had been introducedand studied numerically, efforts were made to obtain simplersystems of differential equations that would preserve the essentialqualitative properties of the Hodgkin-Huxley equations. To do thissuccessfully required an ingenious combination of mathematicaland physiological reasoning. Notice that it could not be donesimply on a mathematical basis because the essential qualitativeproperties of the Hodgkin-Huxley equations were not known. Infact, from the strictly mathematical viewpoint, the problem isimpossible: One searches for a simpler system of differential equa-tions that has the same essential qualitative properties as theHodgkin-Huxley equations, but the qualitative properties of the

3.3 The Zeeman model 69

F(V)

Figure 3.1.

Hodgkin-Huxley equations are not at all well understood, andindeed the purpose of determining the simpler system is to aid inunderstanding the properties of the Hodgkin-Huxley equations!Thus from the purely logical viewpoint, finding such a simplersystem is impossible. However, by taking into account the physio-logical background, FitzHugh (1961), and, independently, Nagumo,Arimoto, and Yoshizawa (1962), derived a two-dimensional systemthat is a desired simplification of the Hodgkin-Huxley equations.This system, usually called the FitzHugh-Nagumo equations, is thefollowing two-dimensional system:

dV V3

dW t

=<j>(V+a-bW),dt

where a, b,<j> are positive constants, / denotes the membranecurrent, which is defined as any arbitrary function of time, V is themembrane potential, and W is a recovery variable. In a moregeneral version of the FitzHugh-Nagumo equations, -V+ V3/3is replaced by a function F(V) that has the form indicated in Fig.3.1. [see Rinzel (1976)].

3.3 The Zeeman modelThe FitzHugh-Nagumo equations are a two-dimensional

simplification of the four-dimensional Hodgkin-Huxley equations.A three-dimensional model has been proposed by Zeeman (1972,1973, 1977). Zeeman derives his model by consideration ofcatastrophe theory and the use of the experimental data of Hodgkinand Huxley. Zeeman's model has serious deficiencies [these have


been discussed in detail by Cronin (1981)] and it has no novel oruseful properties.

3.4 Modifications of the Hodgkin-Huxley equationsIt is clear from our description of the derivation of the

Hodgkin-Huxley equations that there is a significant element ofarbitrariness in the derivation. Indeed, as we have pointed out,Hodgkin and Huxley themselves make this very clear. Also it hasbeen shown in Chapter 2 that the predictions made by the solu-tions of the Hodgkin-Huxley equations do not entirely agree withthe experimental results. Consequently, it is reasonable to try tomodify the Hodgkin-Huxley equations with a view to improvingtheir powers of prediction and to making them more tractable tomathematical analysis. A number of modifications have beenproposed, but no particular one has been widely accepted. TheHodgkin-Huxley equations themselves are still regarded as themost satisfactory mathematical model. We will indicate briefly afew of the modified models that have been proposed in order toindicate directions that future work may take; our descriptions arenot to be regarded as an all-inclusive account. It should be pointedout that none of these modifications has withstood as thorough anexamination as the Hodgkin-Huxley equations. In particular, nomathematical analysis of any of these models has ever been carriedout to determine what action potential velocity they predict.

3.4.1 Modifications in the description of potassium andsodium conductancesHodgkin and Huxley themselves point out a possible mod-

ification that would use a higher power of n in the expression forpotassium conductance:

(see Chapter 2). Cole and Moore (1961) have proposed that forvoltage clamps near the value of sodium equilibrium, a moreaccurate value for the exponent would be 25, although Cole (1975)later termed this a "tongue in cheek" suggestion. Other descrip-tions of the potassium conductances have been proposed by Tille(1965), Hoyt (1963), and FitzHugh (1965).

Modifications of the description of sodium conductance havealso been suggested. Hoyt (1963, 1968) has proposed the use of a

3.4 Modifications of the equations 71

single second-order equation rather than a pair of coupled first-order equations. [Hodgkin and Huxley (1952d, p. 512) state thatthey chose to use a pair of coupled first-order equations becausesuch equations were easier to apply to experimental results.] Hoytand Adelman (1970) have shown that Hoyt's model gives a betterdescription than the Hodgkin-Huxley equations of the experimen-tal results for early changes of gNa when an action potentialoccurs. Otherwise, the predictions made by the Hodgkin-Huxleymodel and the Hoyt model agree closely. Hoyt and Adelman drawfrom Hoyt's model conclusions concerning the physical mecha-nism. For a discussion of the validity of such conclusions, seeJakobsson (1973). Some other modifications are described brieflyby Scott (1975, p. 505). Of particular interest is the brief discussionby Cole, FitzHugh, and Hoyt that follows the paper by FitzHugh(1965) in which all are in agreement that at that time there was no"best" model or set of equations.

3.4.2 The FitzHugh-Adelman modelIn a more recent paper, Adelman and FitzHugh (1975)

propose a modification of the Hodgkin-Huxley equations in whichthey take into account the potassium concentration in the peri-axonal region and obtain, among other results, a more accuratedescription of accommodation during constant current stimulation.That is, the unrealistic periodic solutions described in Chapter 2 donot occur. Another modification has been proposed by Hodgkin(1975) and studied by Adrian (1975) to investigate the relationbetween sodium conductance and velocity of the nerve impulse.Huxley (1959) proposed a modification of the Hodgkin-Huxleyequations that takes into account the effects of varying the calcium(Ca) ion concentration in the bath in which the squid axon isimmersed. (Changes in the Ca ion concentration produce markedchanges in the oscillatory properties of the membrane potential.)Huxley's model has been studied qualitatively by McDonough(1979).

3.4.3 The Hunter-McNaughton-Noble modelsA very different line of study has been initiated by Hunter,

McNaughton, and Noble (1975). One of their objectives is todevelop models that closely mimic the Hodgkin-Huxley equations


(or the properties of real axons) but that are simple enough so thatclosed form solutions that describe the propagation process can beobtained. Having closed form solutions is obviously advantageous.There is another advantage in studying models that are simplerthan the Hodgkin-Huxley equations. The full voltage-clamp analy-sis that is the basis for the derivation of the Hodgkin-Huxleyequations is only possible for certain excitable cells or for a limitedrange of conditions. For example, as we will see later, experimentalstudy of cardiac muscle is far more difficult than the experimentalstudy of the squid axon. In such cases, it becomes necessary toresort to simpler models in order to study some excitation andconduction processes in a quantitative manner.

The work of Hunter et al. (1975) also includes a more generalapproach to the study of sodium conductivity. This more generalapproach is an extension of the approach used in Hoyt's model(1963), but it is based on later experimental results. Experimentalwork by Goldman and Schauf (1972) suggests that there may be arelationship between activation and inactivation. This, in turn,suggests that the sodium conductance gNa should be describeddifferently. In the Hodgkin-Huxley theory,

where m and h are activation and inactivation variables governedby the equations

Hunter et al. (1975) propose to describe gNa in terms of a linearn th-order equation

dt

~ ~~ V&Na/oo>\ a 1 a 2 , . . . , « B / dt

where ax, a2,..., an and (gNa) are functions of V.

5.5 The Lecar-Nossal stochastic model 73

3.5 The Lecar-Nossal stochastic modelFinally, we describe briefly a very important modification

of the Hodgkin-Huxley theory, which will be studied in moredetail in Chapter 6.

Since the functions that appear in the Hodgkin-Huxley equa-tions are continuously differentiable functions, the standard ex-istence and uniqueness theorems for solutions of the differentialequations are applicable. Hence a solution that satisfies the initialconditions V(0) = Fo, m = moo(0), h = hQO(0), and « = «oo(0) isunique. Thus if Vo is the threshold value, the prediction given bythe differential equation is unique: An action potential alwaysoccurs, that is, the nerve always fires. But this prediction does notagree with experimental observations. In experiments, if V(0) is setat the threshold value, the axon does not always respond by firing.Firing occurs only for a certain proportion of the number of timesthat stimulus occurs. The reason for the varying responses of thereal axon to threshold stimulus is thought to be the randomfluctuations or "noise" in the resting potential. This suggests thatthe Hodgkin-Huxley equations would give a more realistic descrip-tion of the behavior of the axon if they contained a random orstochastic term. Also the study of the Hodgkin-Huxley equationsmodified by the addition of such a stochastic term would make atheoretical connection between probabilistic firing of axons andelectrical noise generated across the nerve membrane possible. Itshould be emphasized at this point that the addition of a stochasticterm is not just for the purpose of constructing a novel mathemati-cal exercise. Random firing of axons exposed to near-thresholdstimuli has been studied by physiologists for many years. It is aserious subject in physiology and is deserving of a quantitativedescription.

A study in this direction has been carried out by Lecar andNossal (1971). They used the FitzHugh-Nagumo equations andincluded a description of the noisy nerve by adding a two-dimen-sional Brownian motion. The description given by this model is notonly more realistic in that it includes a description of the noise, butit turns out also to give a better description of the threshold thanthe FitzHugh-Nagumo equations give. We will discuss these ad-vantages in Chapter 6.

Models of other electrically excitable cells

4.1 IntroductionThe techniques developed by Hodgkin and Huxley for the

study of the squid axon have been applied in the ensuing years bymany other researchers. In a certain sense, the techniques remainthe same, that is, voltage-clamp experiments followed by a quanti-tative analysis in which activation and inactivation variables areintroduced and then described by differential equations. But thereare many obstacles to these applications of Hodgkin and Huxley'stechniques: The analysis of the ionic current becomes much morecomplicated because the descriptions of some of the currents aremore intricate and because the number of distinct components ofthe current is in some cases much larger than for the squid axon.(Later we shall describe a mathematical model for the cardiacPurkinje fiber in which the ionic current has nine components.)Also the use of voltage-clamp methods is much more difficult insome cases. For example, voltage-clamp techniques were usedsuccessfully in the study of cardiac fibers for the first time in 1964,and the voltage-clamp technique used in the study of striatedmuscle fibers was not developed until the late 1960s.

The purpose of this chapter is to describe mathematical models(systems of nonlinear ordinary differential equations) of a numberof electrically excitable cells that can be investigated by usingHodgkin-Huxley techniques. Since our primary concern is thederivation and study of these mathematical models that stem fromthe experimental studies, it is easy to forget or lose sight of theextensive and taxing work that goes into successful experiments. Inall of this discussion, it should be kept in mind that underlyingthese theoretical considerations is a great deal of original, inge-nious, and difficult experimental work.

4.2 The myelinated nerve fiber 75

Besides their intrinsic interest, the models in this chapter aretestimony to the remarkable insight and accomplishments ofHodgkin and Huxley. As we shall see, these models and thephysiological systems they describe differ in many ways from theHodgkin-Huxley equations and the squid axon that they describe.Nevertheless these models are based on the same viewpoint as thatdeveloped by Hodgkin and Huxley; part of their accomplishmentwas the development of this remarkably flexible mode of descrip-tion that can be adapted to many different physiological systems.

4.2 The myelinated nerve fiberFrom the biological viewpoint, it is clear that the speed

with which a nerve impulse travels is crucially important to theanimal, and we have already seen in Chapter 2 how a theoreticalestimate of the speed of the nerve impulse was obtained from theHodgkin-Huxley equations. The theoretical speed depends on thesquare root of the radius r of the cross section of the axon. In orderto see this, we return to the first of the equations in system {JF-J?)in Chapter 2. If we consider only solutions of the form V(x — 0/),then we have at once

d2V d2V_ /3-2

J^0 ~Jand the first equation in system (Jf-Jf?) becomes

1 32V dV

where R is the resistance per unit length of the axon, 0 is the speedof the traveling wave that describes the nerve impulse, C is thecapacitance of the membrane enclosing a unit length of axon, andIj is the ionic current flow per unit area through the membrane.Now if R; is the resistance per unit length of an axon of unit crosssection (the specific resistivity of the axoplasm), then

mrIf CM is the capacitance per unit area of the membrane, then

C=2irrCM.

Models of other electrically excitable cells 76

Then (4.1) becomes

mr1 82V 8V

or

r 32V 3V

iRfi2 dt2 M dt r

Since Rt, CM, and It are independent of r, and since dV/dt andd2V/dt2 are independent of r, then r/02 must be independent of rand hence

where A: is a proportionality constant. This result suggests that in anerve axon with the structure of the squid axon, it would benecessary to quadruple the radius of the cross section of the axonin order to double the speed of the nerve impulse.

As pointed out earlier, the radius of the squid axon is unusuallylarge. Probably the chief reason for this is that the impulse thattravels along the squid giant axon is an impulse that orders, "Flee,"and consequently speed of transmission is particularly important.The unusually large radius of the squid axon permits a compara-tively high speed of transmission.

However, for most animal species, the direction of evolution hasbeen not toward nerve axons of large radius, but instead towardthe development of myelinated axons in which a different kind oftransmission of impulse takes place. As described in Chapter 2, themyelinated axon is surrounded by a sheath of fatty material calledmyelin. The sheath is interrupted at intervals of about 1 mm byshort gaps called nodes of Ranvier. The myelin layer is a fairlygood insulator and little current crosses the membrane except atthe nodes, where excitation occurs. During impulse conduction, theexcitation jumps from node to node. This mode of transmissionpermits much higher speed of transmission even though the radiusof the axon remains small. [For a detailed analysis of this, see Scott(1975), especially p. 516.] Many nerve axons are myelinated; forexample, all the nerve fibers, except the smallest ones, in vertebratesare myelinated.


The experimental study of the myelinated axon consists incarrying out voltage-clamp experiments on the nodal membrane,that is, a section of the axon on which there is no sheath of myelin.A long series of experiments on a particular myelinated nerve axon,the sciatic nerve of the clawed toad {Xenopus laevis), was carriedout by Frankenhaeuser and several co-workers. They performedvoltage-clamp experiments and derived a mathematical model anal-ogous to the Hodgkin-Huxley equations. Finally, Frankenhaeuserand Huxley (1964) used numerical methods to obtain approximatesolutions of this mathematical model and compared these numeri-cal results with experimentally recorded action potentials. Com-plete references to the earlier work are given by Frankenhaeuserand Huxley (1964).

The results show that there are marked resemblances betweenthe ionic currents that are obtained when a voltage clamp isapplied to a squid axon and when a voltage clamp is applied to thenodal membrane of the myelinated axon, but there are also im-portant differences: for example, the relationship between ioniccurrent and membrane potential is more complicated for the nodalmembrane than for the squid axon. Hence the simple description of/ N a used by Hodgkin and Huxley, that is, the equation

cannot be used. Instead, the description of / N a is based upon theconstant field equation due to Goldman (1943/44) and describedby Hodgkin and Katz (1949). The constant field equation takesinto account the effect of the ion concentrations in the membrane.(The name "constant field equation" stems from the fact that, inthe derivation of the equation, one of the basic assumptions is thatthe electric field can be regarded as constant throughout themembrane.) From the constant field equation, the following ex-pression is obtained for /N a :

(4.2)

'Na - ^Na RJ, t ^ J o cxp[EF/RT] - 1

where E is the value of the membrane potential at the time ofmeasurement of current, £"Na is the sodium equilibrium potential,


[Na]o is the external sodium concentration, R the gas constant, Tthe absolute temperature, F the Faraday constant, and PNa is thesodium permeability, which in this work plays a role similar to gNa

in the Hodgkin-Huxley equations.Also, while the ionic current in the squid axon has three compo-

nent s-the sodium current /Na, the potassium current /K, and theleakage current IL (observe the change in notation from theHodgkin-Huxley notation for leakage current)-the ionic currentacross the nodal membrane has four components: an initial current/N a carried mainly by sodium ions; a delayed current /K carried bypotassium ions; a second "nonspecific" delayed current IP whichis carried to a large extent by sodium ions, but possibly also byother ions such as calcium; and a leakage current IL carried byboth sodium and potassium. The currents are described as follows:

where gL= 30.3 mR'Vcm2, Vdenotes displacement of the mem-brane potential from the resting potential (i.e., V= E — En whereEr is the resting potential that is taken to be -70 mV), andVL = 0.026 mV. The numerical value VL is chosen so that V = 0 ifno external current is applied to the axon. It is necessary to allowthis adjustment for VL because the data for /Na, /K, and IP, whichwe will shortly write, are made up entirely from experimental data;at V=0, it is necessary that

Now according to the Nernst formula

RT [Na]0

£ l

or

[Na]o= - ln-

RT [Na]i

or

[Na]iexp RT I [Na]o-


Substituting in (4.2), we have

RT i^lo exp(EF/RT) -

F2E [Na] o- [Na]{exp(EF/RT)

T l-QXp(EF/RT) '

In this case, the outside sodium concentration [Na]o is 114.5 mM,the inside sodium concentration [Na]f is 13.74 mM, and

PN a = PNam2/*,

where PNa is the sodium permeability constant that has the value8x lO~ 3 cm/s, and m and h are the sodium activation andinactivation constants, respectively, that satisfy the equations

dm

dh— = ah(l-m)-phh.

(See Table 4.1 for the rate coefficients ctm, ah, /?m, fih. The con-stants A, B,C that appear in Table 4.1 are listed in Table 4.2.)Similarly,

EF2[K]o-[K]{exp(EF/RT)K K RT l-Qxp(EF/RT)

where [K]o is the outside potassium concentration, which is 2.5mM, [K]; is the inside potassium concentration, which is 120 mM,and

where P^ is the potassium permeability constant, which is 1.2 X10 ~3 cm/s, and

dn— = an(l-n)-Pnn.

(See Table 4.1 for an and /?„ and Table 4.2 for the correspondingconstants A, B, C.) Finally

EF2 r [Na]o - [Na]iexp(EF/RT) 1

' PRT[ l-exp(EF/RT) J'


Table 4.1. Rate coefficient functions

A(B-V) A(V-B)

C

A(B-V)

l-ex ,

A(V-B) A(V-B)

A(B-V) A(B-V)B\\ PP(v) =

( / V-B

C

where

and Pp is the nonspecific permeability constant and

d{=ap(l-P)-PpP.(See Table 4.1 for ap and pp.)

Table 4.2. Values of constants A, B, C associated with a and i

« m

Pm« p

Ppy

A(ms-1)

o.r4.50.360.40.0060.090.020.05

B(mV)

- 1 0 "+ 454-224-13+ 40- 2 5+ 35+ 10

c(mV)

6a

103

2010201010

Reference

Frankenhaeuser (1960)Frankenhaeuser (1963b)Frankenhaeuser (1960)

Frankenhaeuser (1963a)

"Values modified to give a single curve for experimental results in Fran-kenhaeuser (1960, Fig. 4).

4.3 Striated muscle fiber 81

The equation for dV/dt is

dV 1

dt Cm

where Cm is the membrane capacitance and / is the total current.Thus the mathematical model for the myelinated nerve fiber hasthe form

_ = _ — ^ ,^ 2 [ [Na] o - [NaL^mk RT[ 1-cMEF/RT) \

2EF2 [K]o-[K]iexp(£F/^r)K" RT l-Qxp(EF/RT)

-P p 2 — - ^ — - — — - — --zr(V-V/pF RT l-Qxp(EF/RT) 5 z A L

dy— = a (l — y) — fivy, y = m, h, n, p .dt y y

It is shown by Frankenhaeuser and Huxley (1964) that a numeri-cal solution of this system that describes an action potential agreessatisfactorily with experimentally recorded action potentials. De-tails of the comparison between the theoretical prediction and theexperimental results are given in Frankenhaeuser and Huxley(1964).

The mathematical problems that must be studied for this modelare basically the same problems that must be studied for theHodgkin-Huxley equations. Again it seems reasonable to expectthat there should be a unique equilibrium point that is globallyasymptotically stable. Also, it would be desirable to obtain aqualitative analysis that shows that the solutions display thresholdphenomena and a refractory period.

4.3 Striated muscle fiberTechniques for making experimental voltage-clamp studies

of a striated muscle fiber were not developed until the late 1960sand a mathematical model was derived in 1970 by Adrian, Chandler,and Hodgkin (1970). In certain respects, the mathematical modelderived is similar to the Hodgkin-Huxley equations. There are


three components of the ionic current, all governed quantitativelyin a way similar to the three components in the Hodgkin-Huxleymodel. However, the variation of V^ (the equilibrium potentialcorresponding to the delayed potassium current) with externalpotassium concentration suggests that an equivalent circuit descrip-tion of the membrane more complicated than the Hodgkin-Huxleymembrane description should be used. As a result, the descriptionincludes the variation of two potential differences.

This model is not entirely satisfactory and we include a descrip-tion of it to indicate a direction of study in physiology rather thanfor the value of the model itself. An improvement of this modelthat uses a more elaborate equivalent circuit description of themembrane has been given by Adrian and Peachey (1973).

As with the squid axon, there are three components of the ioniccurrent: sodium current /N a , potassium current 7K, and leakagecurrent IL. These ionic currents are described by

where

Na '

RTIn

[Na]o

in which [Na]o, [K]o, [Na]i9 and [K]{ have the usual meanings (wewill not give the numerical values here) and the constant b has thevalue 1/12. The reason that the usual sodium equilibrium potentialF N a is not used is that the muscle fiber is immersed in a fluid(Ringer fluid containing 350 mM sucrose), the purpose of which isto reduce the mechanical contractions that would normally beperformed by a muscle fiber in the range of depolarizations consid-ered in the experiments. The value used for F^ a is 50 mV. V^ isdefined in a similar way and the value used for V'K is - 70 mV. Theother constants are VL = -100 mV, gL = 0.55 m^l~l/cm2, gNa = 61mB~1 /cm2 , and gK = 9.8 m.Qt~

l/cv^. The activation and inactiva-tion variables m, h, and n each satisfy a differential equation of

4.3 Striated muscle fiber 83

Table 4.3. The coefficients av and f$y

Muscle Squid giant axon

a (V—V } a ( V — V }

1 -

ahc

1 -

R P

- exp

1^xp

- exp -

10

18

14.7

P 7.6

7

(y-ym1 - exp - 10

, = A«exp -

t = «/,exp -

18

= >8AI 1 + e x p -

20

10

«*(y-K)_(y-K)

1 - exp — 10

80

the form

dy— =ay(l-y)-/iyyy y = m,h,n,

where the equation for m is the same as that in the Hodgkin-Huxleyequations for the squid axon and the equations for h and n havesimilar forms. The coefficients ay and fiy (y = m, h,n) are given inTable 4.3. In the table

5 , = 0.07, J8A = 1,

5w = 0.01, ^ = 0.125.

The equivalent circuit chosen to describe an element of membranein the striated muscle fiber is sketched in the accompanyingdiagram.


The values chosen are C'm = 1 juF/cm2, C r = 4 /xF/cm2, and Rs =150 Q/cm2. The value of Rm depends on the external solution.Values are given by Adrian et al. (1970, p. 615). Let VT denote thepotential across capacitance CT. The remaining differential equa-tions are

dV It V-VT

dt C'M RSC'M

or

dV 1 .

[

VL)\--£-£T (4-3)

and

dVT V- VT

-r^-^-- (4.4)sit H (^

Thus the entire mathematical model consists of (4.3), (4.4) and theequations

dy— = aA\ — V ) — )8,; v, y = m,h,n,dt y y

which were described earlier.On the basis of these equations, propagated action potentials

and conduction velocities were computed by Adrian et al. Thecomputed results were found to be in "reasonable agreement" withexperimentally observed values.

A mathematical study of this model or a refinement of it wouldconsist of studying the same kind of problems that are important in

4.4 Cardiac fibers 85

the study of the model of the squid axon and the model of themyelinated nerve.

4.4 Cardiac fibersWe shall describe mathematical models for two kinds of

electrically excitable cardiac cells: the cardiac Purkinje fiber andthe ventricular myocardial fiber. Developing voltage-clamp meth-ods for the study of cardiac fibers proved to be a difficult task andmodels based on experimental results were not developed until the1970s, and consequently realistic mathematical models appearedonly fairly recently.

4.4.1 The cardiac Purkinje fiber4.4.1.1 Physiological functions of the Purkinje fiber

Our first step is to briefly describe the physiological func-tions of the cardiac Purkinje fiber. The initiation of the electricalactivity that governs the heartbeat is carried out by a pacemakermechanism that is located at a position in the heart called thesino-atrial node (SA node). (The SA node is not the only region ofthe heat that possesses a pacemaker mechanism, but its rate ofbeating is higher than that of any other pacemaker region and so itsets the pace of the heart as a whole.) The electrical impulsesgenerated by the pacemaker mechanism are conducted to thevarious parts of the heart, first to the atria, then through a smallstrip of tissue called the atrioventricular (AV) node, and finally tothe ventricles. The ventricles are the two large chambers of theheart into which the blood is brought from the two smaller thin-walled chambers called the atria and from which the blood ispumped to lung circulation and systemic circulation. The electricalimpulse is conducted comparatively slowly through the AV node.Then it is conducted more rapidly to all of the walls of theventricles. The electrical impulse causes the muscles in the ventricu-lar walls to contract, thus producing the heartbeat and pumpingthe blood into the circulatory system. Consequently, it is importantthat the muscles in the ventricular walls contract almost simulta-neously so that the blood is forced into the arteries at highpressure. Thus, in turn, it is important that the electrical impulse beconducted very rapidly from the AV node to the ventricular walls.


The electrical impulses are conducted from the AV node to theventricular walls by the cardiac Purkinje fibers and their large sizeinsures that the conduction is rapid.

Conduction of electrical impulses that originate in the pace-maker region is the primary function of the cardiac Purkinje fibers,but they also have a secondary function. The Purkinje fibers firespontaneously at a regular rate. This regular rate is slower than therate of firing of the pacemaker mechanism and, consequently,under normal conditions (i.e., when the Purkinje fiber conducts anelectrical impulse that originates from the pacemaker mechanism)the Purkinje fiber does not fire spontaneously. However, if theelectrical impulses from the pacemaker mechanism do not reachthe Purkinje fibers (this happens, for example, in a clinical condi-tion called AV block), then the spontaneous firing of the Purkinjefibers plays a major role. The Purkinje fibers then act as apacemaker and the heart beats at the rate of the spontaneous firingof the Purkinje fibers. Thus a secondary function of the Purkinjefibers is to act as a "backup" pacemaker.

4.4.1.2 The Noble model of the cardiac Purkinje fiberBefore voltage-clamp techniques were completely devel-

oped for the study of cardiac Purkinje fibers, Noble (1962) pro-posed a modification of the Hodgkin-Huxley equations that can beused to describe the action potentials and pacemaker potentials ofcardiac Purkinje fibers. That is, solutions of the system of differen-tial equations proposed by Noble describe fairly accurately theseaction potentials and pacemaker potentials. More realistic models,based on voltage-clamp data were derived later by McAllister,Noble, and Tsien (1975) and Di Francesco and Noble (1985), butthe earliest model developed by Noble is worth studying firstbecause, like the Hodgkin-Huxley model, it is a four-dimensionalsystem and there is the possibility of using analysis that wasdeveloped to study the Hodgkin-Huxley equations to analyze thisnew model. (The McAllister-Noble-Tsien mode is a 10-dimen-sional model and, at present, no mathematical analysis beyondnumerical analysis has been made of that model.)

The Purkinje fiber differs from the squid axon in that depolariza-tion decreases the potassium permeability of the membrane. Dur-ing large depolarizations, part of this decrease seems to be transient


and the potassium permeability then slowly increases. To accountfor this, Noble assumed that the potassium ions move through twotypes of channel in the membrane. [This assumption was based onexperimental measurements made by Noble and his co-workers;see Noble (1979, p. 21).] In one channel, the potassium conduc-tance (gK ) is assumed to be an instantaneous function of themembrane potential and decreases when the membrane is de-polarized. In the other channel, the conductance (gK ) slowly riseswhen the membrane is depolarized. The empirical expression for

£ + 90

60

where potential E is the membrane potential, that is, "insideminus outside." (Note that the Hodgkin-Huxley equations concernthe quantity V, i.e., the deviation of the membrane potential fromthe resting potential, whereas in this work, the quantity £, i.e., themembrane potential itself, is studied.) The term gKi is described by

where

dn— = an(l-n)-/inn,

0.0001(-£-50)a " ~ exp[(-£-50)/10] - 1 '

- £ - 9 0B= 0.002 exp -Hn F \ 80

T h e sodium conductance g N a is described by equations similar tothose used to describe the sodium current in the squid axon. Thatis,

where gNa is a positive constant and m, h satisfy the equations

dm— = am(l-m)-Pmm,

dh— =ak(l-h)-phh9


where

0.1(-£-48)

exp[( -£-48) /15] - 1 '

exp[(£ + 8)/5] - 1 '

(-£-90)ah = 0.17 exp

20

In addition to the sodium and potassium currents, a leakagecurrent, consisting at least in part of chloride ions, is assumed toexist. Denoting this current by / ^ (An stands for anion) anddenoting the anion equilibrium potential and the anion conduc-tance by gAn, we have

E~EAn

(gAn is regarded as a constant, but various values of gAn are usedin calculations in order to reproduce the effects of anions ofdifferent permeabilities.) Thus the total membrane current is

dEm dt N a K A n

or

dEm dt N a a

+ gK(E-EK)+gAn(E-EAn),

where Cm is the membrane capacitance and is taken to be 12/xF/cm2 and gK

= £K + gK • As in the Hodgkin-Huxley descrip-tion of the squid axon, this description is space-clamped, that is, Eat a given time is assumed to have the same value at all pointsalong the fiber. Consequently there is no current flow along thefiber and, therefore, unless a current is applied, the total membrane


current is 0. Thus, for the complete system of equations, we obtain

-r = ~ — {m 3 / ,g N a (£ -£ N a ) + (gK[ + 1.2«4)m

x(E-EK) + gAn(E-EAn)}, (Nl)

^• = am(l-m)-fimm, (N2)dh

~dt— = ah(l-h)-phh, (N3)

(N4)

There is a detailed discussion in Noble (1962) of the similaritiesand discrepancies between the experimental results and the numeri-cal solutions of the system consisting of (N1)-(N4). In certainrespects these equations closely resemble the Hodgkin-Huxleyequations, but the solutions can be expected to exhibit quitedifferent behavior. For example, as pointed out in Chapter 2,solutions of the Hodgkin-Huxley equations would generally beexpected to approach an equilibrium point because after an actionpotential is formed, the membrane potential returns to the restingvalue. But the Purkinje fibers exhibit rhythmic electrical behavior.Hence, it is reasonable to expect to find periodic solutions ofNoble's system of equations, and numerical studies suggest [see,e.g., Noble (1962, p. 329)] that there exist such periodic solutions.

4.4.1.3 The McAllister-Noble-Tsien model of the cardiacPurkinje fiberA mathematical model of the cardiac Purkinje fiber based

on voltage-clamp experiments was derived by McAllister, Noble,and Tsien (1975). This model is a more realistic and accuratedescription of the cardiac Purkinje fiber than the Noble model. (Adetailed comparison of the two models is given by McAllister et al.)But it does have certain drawbacks. First, it is based on a "mosaicof experimental results" [McAllister et al. (1975, page 1)], that is,unlike the Hodgkin-Huxley model of the squid, the required infor-


mation for the model cannot be obtained from voltage-clampanalysis of a single fiber. Also, the experimental results (especiallythe voltage-clamp data) are incomplete in some ways, and themodel has one important deficiency: the sodium current is inade-quate to fully account for the upstroke velocity without makingfurther assumptions [see McAllister et al. (1975, pp. 53-54)]. Never-theless, it is useful and enlightening to study this model: Howevertentative it is in the view of physiologists, it indicates an importantdirection of work in the mathematics of physiology, and it conveysto the mathematician some idea of the complexity and difficulty ofthe problems in this subject.

The McAllister-Noble-Tsien (MNT) model is like the Noblemodel in that it is basically a quantitative description of ioniccurrents across the membrane. But later experimental results havemade possible a more detailed analysis of these currents, and so themodel is considerably more complicated.

First we describe the various components of the ionic current. Asin the Noble model, we let E denote the membrane potential and£ N a denote the sodium equilibrium potential, and use similarnotation for the other equilibrium potentials. There are two time-dependent inward currents: z'Na and isi. (The term "time-depen-dent" means that the current /Na is not simply a function of E.Later the term "time-independent" will be used. A time-indepen-dent current is a current that depends only on E. Since E itselfmay be a function of time, this terminology is not entirely logical.)The current iNa resembles the squid sodium current and can berepresented as

and

where gNa is a positive constant whose value remains uncertain[a discussion of upper and lower bounds for gNa is given byMcAllister et al. (1975, pp. 10-11)] and £N a = 40 mV. Also, theactivation variable m and the inactivation variable h satisfy the

4 A Cardiac fibers 91

equationsdm—

where the functions am, /?m, aA, /?,, are given in Table 4.4.The inward current isi, which has slower kinetics than zNa and is

carried at least partly by calcium ions, is governed by considera-tions somewhat more complicated than Hodgkin-Huxley for-malism. A sizeable fraction of isi is not inactivated even duringprolonged depolarization. The existence of residual isi is expressedby McAllister et al. by the following description of isi:

isi = gsi(E- Esi)df+ g~l(E - E,,)d',

where

g;f. = 0.04 mO" Van2,

Esi= 4-70mV,

df = {1 4- exp[ -0.15(£ 4- 40)] } ~\

and d and / are activation and inactivation variables, respectively,that satisfy the usual Hodgkin-Huxley equations with ad, fid af, fif

as given in Table 4.4.Next, there are three time-dependent outward potassium cur-

rents that will be denoted by z'K2, iXi, iXi. None of them resemblesthe squid potassium current from a quantitative point of view.However, like the squid potassium current, each of these currentscan be described with an activation variable, but without aninactivation variable. The current /Kz, which is called the pace-maker current, is described by

where

. 2.8{exp[0.04(£ + 110)] - 1 }

exp[0.08(£ 4- 60)] 4- exp[0.04(£ 4- 60)]

Table 4.4.

Current components that show inactivation

ComponentP , E 8 , £rev(ms"1) (mV) (mG~ycm2) (mV)

\1-exp

ad(E-Ed)

1 - exp -10

a/=5/exp

(E-E')

17.

P

(E-11.

(

(E-

(

1exp - -

h)

12.2 J

26

_ 1 L 4 9

16

11.49

1.13 XlO" 7 2.5 - 1 0

0002 ° 0 2 -40

0.00253 0.02 - 2 6

0.008 0.08 0

208 x 10 0 0 ° ~26

1.93 X10"4 0.033 -30

2.5

+70

- 7 0

Source: R. E. McAllister, D. Noble, and R. W. Tsien, Reconstruction of the electrical activity of cardiac Purkinje fibres, / . Physiol.251:7 (1975). Reproduced with permission of Cambridge University Press.


andds

— =as(l-s)-pss9

where as and fis are given in Table 4.5. The currents ix and ix ,called the plateau currents, are governed by the following equa-tions:

(1.2)(exp[0.04(£1x1 ~ exp[0.04(£ + 45)]

where

iX2 = 25 + 0.385£,

and

where aXi, fiXi, aX2, flX2 are given in Table 4.5. There is also atransient outward current iqr carried by chloride ions that isdescribed by

where

g^ = 2.5 mST Van2,

Ecl= -70mV,

and q and r are activation and inactivation variables, respectively,that satisfy the equations

dq

dt q q

jt=ar{\-r)-[Sr{r),

and aq, fiq, ar, fir are given in Table 4.4.

Table 4.5.

Component

Slow outward K currents

« P E g £rev

(ms^1) (ms"1) (mV) (m!2"Vcm ) (m V)

1 - exp(E-E,)

E + 50

1 + exp—TT50

17.5

[1 + exp

E + 191 -

£+20

16.67

0.001 5.0X10-5 - 52 (Rectifying) -110

5 X 1 0 " 4 0 0 0 1 3 ~ (Rectifying) -951 + e x p -

25£+20

1.27 X10-4 3X10- 0.385 -65

Source: R. E. McAllister, D. Noble, and R. W. Tsien, Reconstruction of the electrical activity of cardiac Purkinje fibres, / . Physiol. 251:8(1975). Reproduced with permission of Cambridge University Press.


Finally there are several background currents or leak currentsthat are time-independent. So far these have not been analyzedvery well by experimental means, but McAllister et al. propose thefollowing tentative description. There is an outward backgroundcurrent denoted by /K that is carried mainly by potassium ionsand described by

zKl = YI + (0.2)(£ + 30){l -exp[-0 .04(£ + 30)] J"1,

where iKi is as defined earlier. There is an inward backgroundcurrent z'Na b that is probably carried largely by sodium ions anddescribed by

where

gNa,, = 0.105,

£N a = 40 mV.

Finally there is a background current zcl b that is carried bychloride ions and described by

'ci,* = gci,*(£-£ci)>

where the following, somewhat arbitrary, value is assigned to gcl b:

gcl z, = O.OlmS2"1/cm2

and

Ecl = 0.70 mV.

The equation relating membrane potential E and current ix acrossthe membrane is

dE ix

~di = ~C'

where C is the capacitance of the Purkinje fiber and is - 12/iF/cm2. Since

h = 'Na + hi + ''K2 + iX2 + tqr + fK, + ^Na,/> + ^ , 6 »

then using the expressions we have given for these currents, we


obtain for the MNT model the following system of 10 equations:

dE 1

+ g'si(E-Esi)d'+{IK2(E)}s

where iXi, iX2, zK2, /Ki, /NaZ,, iC\,b are> a s indicated, all functions ofE, and

where y = m, d, s, xv JC2, #, /*, / , r and the functions ay(E\ fiy(E)are given in Tables 4.4 and 4.5.

These equations give a quantitative description of the ioniccurrents, that is, they describe quantitatively the sequence of eventsthat can be described roughly in words as follows: When themembrane potential is at a suitable level (when depolarization tothe sodium threshold has occurred), the inward current /Na startsto increase. The inward current isi increases more slowly and theoutward current iqr develops. The /Na decreases and there arequick variations in the membrane potential (termed a spike and anotch) followed by a longer period during which the membranepotential stays fairly level on a so-called plateau. Repolarization orhyperpolarization is triggered by the onset of the current ix

followed by the onset of the currents ix and /K . Then the currentz'K decays and this allows the inward background current to causedepolarization again to the sodium threshold. (That is, the decreasein current iKi permits depolarization to the sodium threshold. It isfor this reason that /K2 is called the pacemaker current.)

As has already been emphasized, the MNT model is tentativebecause it does not have as firm an experimental basis as theHodgkin-Huxley model. A fair judgment of the model requires astudy of the discussion given in McAllister et al. (1975). Neverthe-less, the model does summarize a considerable amount of quantita-tive data with accuracy. Moreover, it is very important for a


mathematician who is interested in cell electrophysiology to lookclosely at the MNT model. In the study of strongly nonlinearordinary differential equations, the difficulties increase very rapidlyas the dimension of the system increases. (Even the transition fromtwo- to three-dimensional systems offers very serious obstacles.)Consequently, mathematicians tend, where possible, to deal withsystems of low dimension. Models of dimension two can be usefulin some qualitative studies, as has already been indicated in thediscussion of the two-dimensional models in Chapter 3. But theMNT model, which has dimension 10, indicates clearly that morerealistic models in physiology require working with high-dimen-sional systems.

4.4.1.4 Mathematical analysis of models of the cardiacPurkinje fiber

We have seen that the physiologically significant mathe-matical problems that need to be studied for the models of thesquid axon, the myelinated axon, and the striated muscle fiber areall about the same. In each of these models, we need to study suchphenomena as threshold behavior and refractory period. But thephysiological functions of the cardiac Purkinje fiber are very differ-ent from those of the squid axon, the myelinated axon, and thestriated muscle fiber, and as a result, the physiologically significantmathematical problems that need to be studied are very different.

The numerical analysis of the MNT equations carried out in theoriginal paper [McAllister et al. (1975)] suggests the existence of anoscillatory solution. Indeed, it seems reasonable that the regularspontaneous firing of the Purkinje fibers would be described by aperiodic solution of the MNT equations. We would expect, more-over, that this periodic solution would be asymptotically stablewith a fairly large region of stability. Experimental evidence actu-ally suggests that there are two distinct oscillations [see Hauswirth,Noble, and Tsien (1969)]. We shall return to a discussion of thisproblem in Chapter 6.

Thus, the first mathematical problem to be studied for the MNTequations is a very classical question: How do we establish theexistence of an asymptotically stable periodic solution? Let usassume that this problem has been solved. Next we consider the


mathematical description of the primary function of the Purkinjefibers, that is, the transmission of regular electrical impulses thatare initiated in the pacemaker region (at the SA node) and thatoccur with a higher frequency than the frequency of the sponta-neous firings of the Purkinje fiber. In order to obtain such amathematical description, we must add to the MNT equations aterm describing the periodic electrical impulses that originate in thepacemaker. If we write the MNT equations in vector form as

x = P(x) (4.5)

and if T is the period of the asymptotically stable periodic solutionof the MNT equations, the existence of which is assumed to havebeen established, then the pacemaker impulses can be described byadding to (4.5) a term of the form F(t), where F(t) has period7\ < T. That is, F(t) is an analytic description of the pacemakerimpulse. Thus to obtain a description of the occurrence of regularor periodic impulses, we must search for a solution of

* = />(*) + /•(*), (4.6)

which has period 7\. Now this problem is the mathematical formu-lation of a well-known problem in mechanics and electric circuittheory: the problem of entrainment of frequency. The standardtechnique for dealing with the problem was introduced by Poincareand consists of replacing (4.6) with the equation

jc = P(x) + G(/,e), (4.7)

where e is a nonnegative parameter, G(t,e) has the period T(e) asa function of t, and G(t,0) = 0. Under rather general conditions, itcan then be proved that for all e sufficiently small, (4.7) has anontrivial (i.e., nonconstant) solution of period T(e).

In Chapter 5, we will discuss in detail both the problem ofshowing that (4.5) has an asymptotically stable periodic solutionand the problem of entrainment of frequency.

4.4.2 The Beeler-Reuter model of ventricular myocardial fiberA kind of companion piece to the paper of McAllister

et al. (1975) is the model of electrical activity in mammalianventricular myocardial fibers proposed by Beeler and Reuter (1977).


This model, like the models previously described, is based on dataobtained from voltage-clamp experiments.

Following Beeler and Reuter, we let Vm denote the membranepotential. The mathematical model consists of the following set ofequations, the first of which is

dVm 1

O ) ( )where Cm = 1, and

cxp[0.04(Fm_ exp[0.08(Fm + 53)]+exp[0.04(Fm

+ 0.2- ( ^ + 2 3 )

l - exp [ -0 .04 (F m

0.08(exp[0.04(Fm + 77)] - l)

exp[0.04(Fm+35)]

and

in which

gNaC = 0.003

£ N a = 50 mV.

The current carried mainly by Ca ions, denoted by is, is

h = Ssdf{V+ 82.3 + 13.0287 ln[Ca]i),

where g5 = 0.09 mfi~Vc m 2 a n d [Ca]i is in moles and refers tocalcium ion concentration in the interior of the fiber. The term zext

is a given function of time that depends on the experiment beingdescribed. See Beeler and Reuter (1977, p. 180).

The second equation in the system is

| - [ C 4 ) (4.9)


Table 4.6. C, defining function and values for rate constants (a or ft)a = (C,exp[C2(Km + C,)] + C4(Vm + C5))/exp[Q(Km + C3)] + C7).

100

Rateconstant(ms-1)

ax

P l

£

ftad

&« /

ft

(ms )

0.00050.00130

400.1261.70.0550.30.0950.070.0120.0065

(mV"1)

0.083-0.06

0-0.056-0.25

0-0.25

0-0.01-0.017-0.008-0.02

Q(mV)

502047727722.57832

- 5442830

Q[(mVms)"1]

00

- 1000000000

Q(mV)

00

47000000000

Q(mV-1)

0.057-0.04- 0 . 1

00

-0.082-0 .2-0 .1-0.072

0.050.15

-0 .2

c7

11

- 1001111111

Source: G. W. Beeler, and H. Reuter, Reconstruction of the actionpotential of ventricular myocardial fibers, / . Physiol. 268:181 (1977).Reproduced with permission of Cambridge University Press.

and the remaining six equations are

dy

dt= aJl-y)-f}Jy),

and fiy arewhere y = xv m, h, j , d, and / , and the functions ay

listed in Table 4.6.Beeler and Reuter (1977) compare this model with the MNT

model and from their discussion, it follows that the types ofmathematical problems that arise in the study of this model arevery similar to those that arise in the study of the cardiac Purkinjefiber models we have described. Here we merely point out theexistence of oscillatory phenomena [Beeler and Reuter (1977, p.200ff)] and the consequent question of the existence of periodicsolutions.

Mathematical theory

5.1 IntroductionSo far we have described a number of mathematical mod-

els of electrically excitable cells and, at least for some of themodels, we have indicated the kind of mathematical questions andanalysis that arise in work on these models. The next major step isto describe in detail how the mathematical analysis can be carriedout. In order to obtain this description, we will first give sometheory of differential equations. By doing this, we will introducethe mathematical language that is appropriate for discussing theproblems that are of concern to us. We also describe somemathematical techniques and results that will be useful in the studyof the models.

5.2 Basic theoryIn this section we describe some basic properties of solu-

tions of differential equations. These are used very frequently inthe analysis of all the physiological models.

5.2.1 Existence theorems and extension theoremsIt is reasonable that our first concern should be for the

existence of solutions. To a reader whose experience with differen-tial equations is limited to an introductory course following calcu-lus, such a concern may seem unnecessarily fussy. In a first coursein differential equations various techniques for computing solu-tions are described and it might be expected that our chief concernwould be to summarize such computational techniques; but all thephysiological models in which we are interested are nonlinearsystems, and, consequently, it is usually impossible to get explicitexpressions for the solutions (i.e., closed solutions). Frequently wemust settle for an approximation to the solution. However, in order

Mathematical theory 102

to deal with an approximation, we must know that a solutionexists. Thus we are seriously concerned with the question ofexistence, and our first step will be to state a version of the basicexistence theorem. Let x denote a real n vector, that is,

X = \X1>- • •> Xn)'>

where xl9...,xn are real numbers and let |JC| be defined by

Similarly, let

where each fj(t,x) (with j = 1,.. . ,«) is a continuous real-valuedfunction that is defined on the set

«={( / , x)/\t - to\ < a, \x - xo\ < b},

where t0, x0 are fixed and a and b are fixed positive numbers. LetM = max(,yX)^R\f(t, x)\ on R and let a be the minimum of b/Mand a. We also assume that / satisfies a Lipschitz condition withrespect to x in R; that is, we assume that there is a positive numberk such that if (t, x(1)) and (/, x(2)) are points in R, then

(It is easy to show, by using the mean value theorem, that if thepartial derivatives dft/dxp where /, y = l , . . . , « , exist and arebounded on R, then / satisfies a Lipschitz condition with respectto x in R.

Existence theorem. The n-dimensional system

dxx

^=f(t,x) or ^=f2{t,xx,...,xn), (5.1)

dxn

{with the components written out) has a unique solution x(t, t0, x0)such that

^1^0' 0» X0) = X0'

5.2 Basic theory 103

The solution x(t, t0, x0) is defined for all t^[t0- a, t0 + a] and

\x(t,to,xo)-xo\<Ma

forallte[t0-a,t0 + a].

[Following a usual convention we will refer to (5.1) sometimes asan equation and sometimes as a system of equations.]

This theorem, although useful, leaves open a number of ques-tions. Apart from the fact that, as stated, the theorem gives noinformation about how to compute the solution, there is thequestion of how large the domain of the solution is. The set Rdescribed before the statement of the theorem is not unique. Forexample, another set R with larger a and b and larger a mightexist. Thus, the solution would be extended so that it was definedover a larger interval [t0 — a,to + a]. On the other hand, we cannotbe certain of how large a domain can be found. The followingsimple example shows that the domain may be severely limited.Consider the scalar equation

dx

~It=% 'A solution of this equation that satisfies the condition to = 0,x0 = 1 is given by x(t) = — l/(t — 1). As / increases from 0 toward1, this solution decreases without bound and the solution is notdefined at t — \.

Now for convenience, we introduce a precise definition ofboundedness of solutions.

Definition. A s o l u t i o n x ( t ) of (5.1) is bounded for t > t 0 if t h e r eexists a positive number M such that

\x(t)\<M

for all t > t0 for which x(t) is defined.

It seems reasonable to expect that if a differential equation is aphysiological model, then its solutions should be defined for all /greater than some fixed value, that is, for all time later than acertain instant, and that its solutions should remain bounded. (If asolution suggests that membrane potential increases without boundas / -> 5 or even as t increases without bound, the validity of the


model is indeed questionable!) These two rather natural require-ments on the solutions, that they are defined for all t greater than afixed value and that they are bounded, are closely connected. It canbe proved that if a solution remains bounded for all t past acertain value for which the solution is defined, then the solution isdefined for all / past that certain value. In order to state this resultprecisely, we need one definition. This definition is just a formaldescription of the property that a solution has been extended as faras possible or has the largest possible domain.

Definition. If x(t) is a solution on (a, b) of dx/dt=f(t, x) andy(t) is a solution on (a, /?) of the same equation, and (a, /?) c (a, Z?)and if, for t e (a, ft), y(t) = x(t\ then y(t) is an extension of x(t).(Note: a is a real number or - oo and b is a real number or + oo.Similar conditions hold on a and /?.)

Definition. If x(t) is a solution on (a, b) of dx/dt =f(t, x) andif x(t) is such that any extension y(t) of x(t\ where y(t) is asolution on (a, /?) of dx/df =/(f, x), has the property that (a, /?)= (a, b), then x(r) is a maximal solution of dx/tffr =/(f, x).

Extension theorem. Suppose that x(t) is a maximal solution of

/ w continuous for all real t, all x e /?", #nd / satisfies aLipschitz condition in x in each bounded set E in RX Rn, that is,each set E for which there is a positive number A such that if(t, x) e £, then

\t\ + \x\<A.

Suppose further that (a,b) is the domain of x(t) and that thereexists a number c^(a,b) and a positive number B such that for all/e(c,6),

\x{i)\<B.

Then b = + oo.

Thus the problem of showing that the solutions to be studiedhave the desirable properties of being defined for all time past acertain instant and of being bounded is reduced to the single


problem of showing that the solutions are bounded. As we shall seelater in Chapter 6, when mathematical analyses of various physio-logical models are made, proving boundedness of solutions is oftendone quite easily. It is often easy to prove that all the physiologi-cally significant solutions enter and remain in a particular boundedset. Then solution x(t) is defined for all t> a, that is, b = + oo.

5.2.2 Autonomous systemsAn important special case of the differential equation

dx

is the equation

dx

that is, an equation in which the right-hand side is not a functionof the independent variable t. Such an equation is called anautonomous equation. Autonomous equations occur often in me-chanics and electrical circuit theory; all the physiological modelswe have considered so far are autonomous equations. A cruciallyimportant property of autonomous equations is that their solutionscan be studied geometrically in a particularly nice way. In order todescribe this property informally, we consider the two-dimensionalautonomous system

dx

<>

A solution of this system is a pair of differentiable functions(x(t), y(t)). The pair of functions x(t) and y(t) describe a curvein the xy plane. It is easy to show [see, e.g., Cronin (1980b)] that ifc is a constant, then (x(t + c), y(t + c)) is also a solution of thesystem that describes the same curve. Also (x(t + c), y(t + c)) isthe only kind of solution that describes that same curve. [Here weare using the term "curve" in the intuitive sense, that is, to refer tothe point set or geometric configuration. If we were to use the term


"curve" in the rigorous mathematical sense, then we would say that(x(t + c), y(t + c)) is an element of the equivalence class thatconstitutes the curve.] From the uniqueness condition in the con-clusion of the basic existence theorem, it follows that no solutioncurve crosses itself and that the intersection of two distinct solutioncurves is the null set. (It is easy to show with examples that theseresults do not hold if the system is not autonomous.) As a result, itis often enlightening to study the curves instead of the solutionsthemselves. The xy plane is often called the phase plane and thegraphs of the curves are termed the phase plane portrait. Later wewill see how important this phase plane technique and its «-dimen-sional extension are for the study of physiological models.

5.2.3 Equilibrium points5.2.3.1 Definition of equilibrium point

The simplest class of solutions of a system of ordinarydifferential equations is the constant solutions. If we consider an^-dimensional autonomous system

dXl / / \

dx2

— =/2(x1,...,xJ,

and if xv..., xn are real constants such that for j = 1 , . . . , « ,

/y(x1,...,xJ = 0,

then {xx(t\..., xn(t)\ where Xj(t) is defined by

Xj(t) = Xj for all/,

is a solution of the system. It is, of course, a very special solution,each of whose components is a constant. It is called a critical point,a singular point, or an equilibrium point. Since the terms "critical"


and "singular" have other meanings in mathematics, the termequilibrium point will be used here.

(We have defined the notion of equilibrium point only forautonomous systems. The definition can be extended to includenonautonomous systems: A point x is an equilibrium point of thesystem dx/dt=f(t,x) if / ( / , JC) = O for all /. However, in ourwork, we will be concerned only with equilibrium points of autono-mous systems.)

5.2.3.2 The two-dimensional caseAlthough equilibrium points seem like highly specialized

solutions, their locations and the behavior of solutions near themgive very useful information about the general behavior of solu-tions of the system. Our next step is to summarize briefly somewell-known results concerning equilibrium points. We begin withthe two-dimensional case for which fairly complete results can beobtained.

Linear homogeneous systems. Consider first the linear homoge-neous system

dx— = ax + by,

(5.3)dy

+ d

where a, b,c, d are real constants and it is assumed that

]=ad-bc*0.d\

Since this determinant is nonzero, then (0,0) is the only equi-librium point. We assume further that the matrix

M=\a b.Yc d

is in canonical form. Then analysis of an elementary nature [see,e.g., Cronin (1980b)] yields easily the phase plane portraits. LetXl9 X 2 be the eigenvalues of the matrix M. Then the phase portraitsdepend on the relationship between Xx and X2. Sketches of the

Mathematical theory

phase portraits and their names are as follows:

Case I. A1? A2 are real, unequal, and have the same sign.

108

y. \

stable nodeunstable node

Case II. A1? A 2 are real and equal and M = f ^ 1 .L 0 AJ

V\ \

X< 0stable node

X > ounstable node

Case HI. Xl9 A2 are real and equal and M = \^ \].L 0 A J

K < ostable node

X > ounstable node

5.2 Basic theory

Case IV. \2<0<\v

109

rsaddle point

Case V. \l9 A2 a r e complex conjugate numbers, that is, Xx = a 4-

center or vortex

Aj = a -F i 8 , a > 0 , B > 0

unstable s p i r a l point or

unstable focus

Xl = a + i/J, a < 0, j8 > 0

stable spiral point or

stable focus

If matrix M is not in canonical form, then further elementaryanalysis shows that the phase portraits are the same as those justdescribed except that they are distorted by a transformation that isa nonsingular linear transformation.


Nonlinear systems. It can be shown that if nonlinear, higher-orderterms are added to the right-hand side of (5.3), then in a neighbor-hood of the origin, the phase portrait remains essentially the sameexcept in the case of a center where the addition of nonlinear termsmay result in a center, a stable spiral point, an unstable spiralpoint, or a more complicated phase portrait [see Hurewicz (1958)].

Finally, we consider the general case

dx— =P(x,y),*dy

(5.4)

where P(xy y) and Q(X, Y) are power series in x and y and (0,0)is an isolated equilibrium point, that is, there is a neighborhood of(0,0) such that the only equilibrium point in the neighborhood is(0,0). We have just seen what the phase portrait looks like if thecoefficient matrix of the linear terms in P(x, y) and Q(x, y) isnonsingular. But so far we have no information about the behaviorof the orbits if P or Q is a series that starts with terms of degreehigher than 1. There is a remarkably simple answer to this ques-tion, which can be described roughly as follows. If (0,0) is anisolated equilibrium point of (5.4), then either (0,0) is a spiral [i.e.,the orbits spiral toward (0,0) as t increases (decreases)] or a center[there is a neighborhood N of (0,0) such that every orbit that has anonempty intersection with N is a simple closed curve] or aneighborhood of (0,0) can be divided into a finite number of"sectors" of the following types:

Type I. Fan or parabolic sector.

(0,0) (0,0)


Type II. Hyperbolic sector.

Type III. Elliptic sector.

In each of these sketches, the arrows indicate the direction ofincreasing time. Note that the only essentially new configurationthat arises when nonlinear equilibrium points are studied is theelliptic sector.

For a precise statement of this result and a proof, see Lefschetz(1963, Chap. X). For a similar discussion see Nemytskii andStepanov (1960, Chap. II).

5.2.3.3 The index of an equilibrium pointA beautiful and sometimes useful concept that is intro-

duced in the study of equilibrium points is the index of theequilibrium point. We describe first an intuitive version of thedefinition of index for equilibrium points in the two-dimensionalcase.

Definition. Let P(JC, y), Q(x, y) be real-valued continuous func-tions on an open set D in the xy plane. Let V(x, y) be the vectorfield

V(x,y) = (P(x,y)9Q(x9y))9

that is, V is a function with domain D and range contained in R2.If (x, y) e £>, then V(x, y) is the point or vector (P(x9 y), Q(x, y)).


An equilibrium point of V(x, y) is a point (JC0, y0) e D such that

Let C be a simple closed curve in D such that no point of C is anequilibrium point of V, and consider how the vector associatedwith a point (xl9 yx) e C changes as (xl9 yx) is moved counter-clockwise around C. The vector is rotated through an angle j(2ir),where j is an integer, positive or negative or zero. The number j isthe index of C with respect to V(x9 y).

It is easy to see that this "definition" is not precisely formulatedbecause several of the terms used have only an intuitive meaning.For example, what does it mean to "move counterclockwise" on C?As long as C is an easily visualized curve like a circle or ellipse, it iseasy to describe precisely what we mean by moving counterclock-wise. But if C is a sufficiently "messy" curve, then it is not at allclear which is the counterclockwise direction on the curve. Second,although for easily visualized curves C it is intuitively clear how tomeasure the angle through which the vector is rotated, no generaland precise method for describing how to measure the angle hasbeen given. Thus our "definition" is far from rigorous. An efficientway to give a rigorous definition is to formulate the definition interms of topological degree, a procedure that we will shortlydescribe.

Having obtained the definition of the index of C with respect toV(x9 y), one can then prove that if C is continuously moved ordeformed in such a way that it does not cross any equilibriumpoint of V during the continuous deformation, then the index jremains constant during the deformation.

Now suppose that (x09 yQ) is an isolated equilibrium point of V9

that is, suppose that there exists a circular neighborhood TV of(x0 , y0) such that there are no equilibrium points of V in N except(x0 , y0). Let Cl9 C2 be simple closed curves in N such that (x0, y0)is in the interior of Cx and is the interior of C2. It is not difficult toshow that C\ can be continuously deformed into C2 withoutcrossing any equilibrium point of V. Hence the index of CY equalsthe index of C2, and indeed the index j is independent of thesimple closed curve C provided C is contained in Af and (x0, y0) is


in the interior of C. Hence we may formulate the followingdefinition.

Definition. The index j is the index of the isolated equilibriumpoint O 0 , y0).

We have already seen that this definition is not based onrigorous considerations. Also it is desirable to have a definitionthat is valid in the ^-dimensional case. Both of these difficultiescan be dealt with by defining the index of an equilibrium point interms of topological degree or Brouwer degree. Next we describevery briefly how this is done. Our description is far from completebecause no attempt is made to define Brouwer degree. [For adescription of the Brouwer degree, see Krasnosel'skii (1964), Lloyd(1978), or Cronin (1980b).]

Definition. If p is an isolated equilibrium point, the index of p isdegB[/,Z?",0],

where deg5 is the Brouwer degree, and Bn is a ball with center psuch that BnaN.

If n = 2, this definition agrees with the informally describeddefinition previously given. For arbitrary «, if the matrix

is nonsingular, it follows that the index of p is

signdej^)].By using this topological version of the definition of index of an

equilibrium point, one can obtain the following existence theorem,which is useful in the study of physiological models.

Theorem 1. Given the autonomous system

-^-=fi(xl9...,xn)

: (5.5)


where f has continuous first derivatives in Rn, suppose that U is theclosure of an open ball U= {P/\p - po\< r) with center p0 andradius r, and that U is such that if a solution of system (5.5) passesthrough a point in U9 then for all later time t, the solution stays in U.Suppose also that there are no equilibrium points in U — U. Then(5.5) has at least one equilibrium point in U.

Proof See Cronin (1980b, Appendix: Corollary 6.1).

Theorem 2. If, besides the hypotheses in Theorem 1, it isassumed that the indices of the equilibrium points in U all have thesame sign, then (5.5) has exactly one equilibrium point in U.

Proof See Cronin (1980b, Appendix: Property 3).

5.2.4 Stability and asymptotic stability of solutions5.2.4.1 Introduction

In studying ordinary differential equations that describebiological systems, it is important to determine whether the solu-tions are stable. Roughly speaking, a solution x(t) is stable if,whenever another solution gets close to x(t), this other solutionthen stays close to x(t) for all later time. A solution x(t) isasymptotically stable if, whenever another solution gets close tox(t), this other solution then gets closer and closer to x(t) as timeincreases.

Such a solution can be expected to predict actual behavior of thebiological system. On the other hand, a solution that is not stablewould probably not give a good prediction of the behavior of thebiological system. The reason for this is the presence of smalldisturbances that are not described or taken into account in thedifferential equations. It is highly likely that such small dis-turbances frequently occur in physiological systems, and conse-quently, if the physiological system is described by a particularsolution of the differential equation, we may think of the system as"moving along" the solution until a small disturbance occurs,which "kicks" the system onto a nearby solution. Now if theoriginal solution is stable in the sense that nearby solutions ap-proach it, then the system will "move back" toward the originalsolution. Thus the original solution will yield a reasonably goodprediction of the behavior of the system. However, if the original


solution is not stable and if the system is kicked onto a nearbysolution by a small disturbance, the nearby solution may moveaway from the original solution and, as a result, the system willmove away from the original solution. Consequently the originalsolution will give a poor prediction of the behavior of the system.

The notion of stability, which is intuitively very reasonable,requires considerable care if we are to formulate a rigorous defini-tion. Moreover, it is usually quite difficult to determine if asolution is stable and an entire mathematical theory of stability hasbeen developed to deal with this question. The theory is notcomplete and, from the point of view of applications, the theory isnot entirely satisfactory. For the present, we shall simply stateformal definitions of stability and indicate very briefly techniquesthat have been developed to determine whether a given solution isstable.

5.2.4.2 Asymptotic stabilityWe restrict ourselves to considering just one stability con-

cept, that is, asymptotic stability, which seems to be particularlyappropriate for applications to differential equations that describebiological systems. [For a detailed discussion of the significance ofstability of solutions of differential equations, see Cronin (1980b).]

Although it is logically not necessary to give a separate definitionof asymptotic stability for the special case of a solution that is anequilibrium point, we give first such a separate definition. Thereare two reasons for doing so: first, the asymptotic stability ofequilibrium points is, in practice, a very important special case;second, the definition of asymptotic stability for an equilibriumpoint is intuitively clear and serves as a guide to understanding thedefinition of asymptotic stability for an arbitrary solution.

We consider an autonomous system

dxx

(5.6)

where fj has continuous first derivatives in Rn (y = 1 , . . . , « ) .


Definition. An equilibrium point p = (xl9..., xn) of system (5.6)is asymptotically stable if: given e > 0, then there exists 8 > 0 suchthat if x(t) is a solution of (5.6) and |x(f0) -p\<8, then for allt > t0, solution x(t) is defined and \x(t) -p\ < e and

lim x(t) =/?.t~* 00

In words, the equilibrium point is asymptotically stable if eachsolution that gets close enough to p approaches p as a limit ast -> oo (see the accompanying sketch).

This basic definition of asymptotic stability of an equilibriumpoint illustrates the fact that the mathematical theory of stability isnot entirely realistic. If an equilibrium point is asymptoticallystable and if for some values of e the corresponding 8 = 8(e) has avery small value, say 8 = (10~30)(e), then for practical purposes,the equilibrium point is unstable. On the other hand, suppose weconsider an equilibrium point of a two-dimensional system forwhich the orbits near the equilibrium point behave as sketched:


That is, there is a periodic solution (represented by the closedcurve) such that all solutions except the equilibrium point papproach the periodic solution. Then the equilibrium point iscertainly not asymptotically stable. On the other hand, if the closedcurve that represents the periodic solution is contained in a smallneighborhood of the equilibrium point, then for practical purposes,the equilibrium point may be regarded as stable.

A fundamental question in stability theory is to obtain criteriafor determining if a solution is stable or asymptotically stable. Thefollowing theorem is a classical result that is often applied.

Theorem 3. If the real parts of the eigenvalues of the matrix

are all negative, then the equilibrium point p is asymptotically stable.If M has an eigenvalue with positive real part, then p is notasymptotically stable.

Proof See LaSalle and Lefschetz (1961) or Cronin (1980).

If n > 2, the question of determining whether the eigenvalues ofthe matrix M have negative real parts is itself a serious problemespecially as n gets larger. One important method for solving thisproblem is the Routh-Hurwitz criterion. We will not describe thiscriterion in detail. We merely point out that it is described in mosttexts and that a detailed account including generalizations may befound in Marden (1966).

Theorem 3 leaves unresolved the question of whether p isasymptotically stable if the matrix M is nonsingular and all itseigenvalues either have negative real parts or are pure imaginary.These cases are not just "degenerate cases." They often occur inapplications and their study requires a more delicate investigationof the higher-order terms. Sometimes this is done by straightfor-ward (but usually rather laborious) computation. For some classesof equations, the use of Lyapunov functions is an effective method,which largely avoids computation. A lucid and beautiful introduc-tion to Lyapunov functions is given by LaSalle and Lefschetz


(1961). A complete technical account may be found in Cesari(1971).

Now we turn to the definition of asymptotic stability for anarbitrary solution. We consider an ^-dimensional system

dxx

(5-7)dxn

where fj has continuous first derivatives in RX Rn (j = 1,.. . ,«).In the following definition x(t, i, x) denotes the solution of (5.7)

such that x(i, t, x) = x.

Definition. Solution x(t) of (5.7) is stable if there is a value tQ

such that the following conditions hold.

1. There is a number b > 0 such that if

\xl-x(t0)\<b9

then x(t, t0, xl) is defined for all t > t0.

2. Given e > 0, then there exists 8 e (0, b) such that if

\xl-x(t0)\<8,

then for all t > t09

\x(t,to,xl)-x(t)\<e.

The solution JC(/) is asymptotically stable if conditions 1 and 2 holdand if also the following condition holds:

3. There exists 8 e (0, b] such that if

then

lim \x(t910, xl) - x(t9109 x°) | = 0.

From the mean value theorem, we have

u)-f(t,x) = v u + R(t, x, w),


where R is a remainder term such that

lim

and a straightforward calculation yields:

Theorem 4. Solution x(t) is asymptotically stable if and only if 0is an asymptotically stable equilibrium point of

Application of the Floquet theory [see Cesari (1971) or Cronin(1980b)] then yields:

Theorem 5. If f has period T as a function of t, if x(t) is a solutionof (5.7), and if x(t) has period T, let \v...,\n be the characteristicmultipliers and pv...9pm the corresponding characteristic exponentsof the system

£-[£"•*» u.

V l \ l < 1 i°r equivalently, R(pt) < 0], for i = 1,..., n, then x(t) isasymptotically stable. If there exists j such that \Xj\ > 1 [equivalentlyR(Pj) > 0], then x(t) is not stable.

Unfortunately, if n > 2, this theorem is more elegant than practi-cal because the problem of computing the characteristic multipliersbecomes very difficult unless / and the periodic solution x(t) aregiven very explicitly.

The following theorem lacks the elegance of the preceding theo-rem but in some cases, it may be more practical.

Theorem 6. Suppose that the equation

^=f(t,x) (5.8)

satisfies the following conditions'.

1. Each component off(t, x) has continuous third derivatives int and xl9...9 ;cn, the components of x, at each point of Rn+l.

2. There exists a bounded open set U <z Rn such that eachsolution of (5.8) that intersects dU, the boundary of U, is


"headed into" the interior of U, that is, if x(t) is a solutionof (5.8) and x(t0) e dU, then there is a 8 > 0 such that if

then x(t)^ U, and if

to-S<t<to,

then x(t)^ Rn — U. [This condition implies that no solutionof (5.8) escapes U. Also since U is bounded, it follows that ifx(t0) e U, then solution x(t) is defined for all t > t0.]

3. Let Pt y denote a matrix such that \Pt y\ = 1 and

where J is the real canonical form of fx(t, y). [For a descrip-tion of the real canonical form, see Cronin (1980).] Thereexists M > 0 such that for all y e U,

lub \P-]\<M.

4. For each t e R, y G [/, the eigenvalues of the matrix fx(t, y)satisfy the following hypotheses'.(i) There exists r>0 such that all of the eigenvalues offx(t, y) have real parts that are less than or equal to —r. Ifthere exist y e U and t > t0 such that one of the eigenvaluesof fx(t9 y) has a nonsimply elementary divisor then r> M,the constant in condition 3.(ii) There exist positive numbers 6, c such that

r — M — cr- M- c> 0 and b <

Mand such that for all y G U and t > /, the imaginary part ofeach eigenvalue of fx(t, y) has absolute value less than b. Inthe special case that the eigenvalues of the matrices fx(t, y)(where yÛ and t>t0) all have simple elementary di-visors, we require only that there exist positive numbers iand c such that

r — cr — c>0 and b <

mand such that for all y e U and t>t0, the imaginary part of


each eigenvalue of fx(t, y) has absolute value less than J).[Computable criteria for these hypotheses are given by Harden(1966).](Hi) Let —qi,...,—qn denote the real part of the eigenval-ues offx(t, y). Then for j = 2 , . . . , « , \qx- qj\ < c /8.

Conclusion: If x(t) is a solution of (5.8) such that x(t0)^ U forsome t0, then x(t) is asymptotically stable.

Proof See Cronin (1980a).

5.2.4.3 Phase asymptotic stabilityIt is natural to assume that in studying autonomous equa-

tions we should use the same definition of asymptotic stability thatwas used in the discussion of nonautonomous equations. However,it turns out that this is not possible for the most important case,periodic solutions. That is, we have the following theorem [for theproof, see Cronin (1980b)]:

Theorem 7. Ifx(t) is a nontrivialperiodic solution (i.e., aperiodicsolution that is not an equilibrium point) of the autonomous equation

dx / x

then x(t) is not asymptotically stable.

It is rather reasonable to raise, at this point, the question ofusing Theorem 5 to determine if a periodic solution is asymptoti-cally stable. Actually there is no possibility of applying Theorem 5because if x(t) is a periodic solution of

dx t x

then

Thus, dx/dt is a periodic solution of the equation


It follows that fx[x(t)] has at least one characteristic multiplierequal to 1 [see Cronin (1980b)] and hence the hypothesis ofTheorem 5 is violated.

The stability definitions that are frequently used in the study ofsolutions of autonomous equations are definitions of orbital stabil-ity. As the name suggests, these concepts arose in the study ofcelestial mechanics, and they are not entirely satisfactory for themodels that are studied here. Because the choice of a stabilitydefinition is important, it is worthwhile to explain this point indetail and we begin by stating the definitions.

Definition. Let x(/) be a solution of the n-dimensional equation

^ = / (* ) , (5-9)

where / is defined and has continuous derivatives in Rn such thatx(t) has period T. Let C be the orbit of x(t\ that is,

C= {*(/)//€= [0,r]}

or C is the underlying point set of solution x(t). If p e Rn, let

d(p,C)= ini \p-q\.C

Solution x(t) is orbitally stable iff: given e > 0, then there existsS > 0 such that for any solution x^\t) of (5.9) for which there is at0 such that

it is true that JC(1)(/) is defined for all t > t0 and

for all t > t0. Solution x(t) is asymptotically orbitally stable if x(t)is orbitally stable and there is an e0 > 0 such that for any solutionx(1)(t) for which there exists t0 with

it is true that

lim d(x(1)(t),C)=0.t-*oo

Orbital stability is a meaningful property of a solution only if theorbit is a simple configuration. In practice only the case in which


the orbit is a simple closed curve (so that the solution is periodic) isconsidered. That is why in the definition of orbital stability, weimposed the hypothesis that x(t) has period T. Orbital stabilitysays roughly that if a solution gets close to the orbit of the givensolution, it stays close. However, if y(t0) is close to C, it does notfollow that

\y(t)-x(t)\ (5.10)

remains small. If the solutions move at different speed, the expres-sion (5.10) may not stay small. In other words, a solution may beorbitally stable but not stable in the sense of the definition ofstability in the preceding section on nonautonomous equations.[For an example of a periodic solution that is orbitally stable butnot stable, see Hahn (1967, p. 172).]

For our purposes where we would expect the solutions near the"stable" periodic solution to have some kind of approximateperiod, orbital stability is not enough. Roughly speaking, we wantthe nearby solutions to have about the same speed as the "stablesolution." Now we describe an example that illustrates this desir-able behavior and after that we will introduce a definition sug-gested by this example.

Example.

dt

In polar coordinates, this system becomes

dr dx dyx2 + y2)(l — x2 — y2) = r2(\ — r2)

ui ui ai

or

jt=r(\-r2) (5.11)

and

f - l . (5.12)


Elementary computations show that the general solution of thesystem (5.11) and (5.12) is

0 = -t + C,1

r =

where C and k are arbitrary real constants. If k is negative, then ris real only if t is large enough so that

-\<ke~2'.

If C = 0, then in Cartesian coordinates, each solution is

(**(O, yk(O), where

_ _ c o s / _ _ - s i n ?

* * U J [l + * - 2 < ] 1 / 2 ' M '[l + * e ] [l + ke-2']1/2'

Note that if k = 0, then

Now, if A:o nt 0,

< a -and, thus,

\im{\xk(t)-x0(t)\ + \yk(t)-y0(t)\}=0.

But since the system of differential equations is autonomous, thenif T is a nonzero constant, the pair (5co(f), 7o(OX where

xo(t)=xo(t + r), yo(t)=yo(t + r),

is also a solution and for most values of T,

l i m { \ x k ( t ) - x o ( t ) \ + \ y k ( t ) -

Thus, if the phase of solution (xQ(t), yo(t)) is changed, the asymp-totic stability property is lost.

This example suggests the following definition, which will beuseful.


Definition. Let x(t) be a solution of the ^-dimensional equation

^ = / ( * ) , (5-13)

where / is continuous on Rn. Then x(t) is uniformly stable if thereexists a constant /: such that, given e > 0, then there exists 8 > 0such that if u(t) is a solution of (5.13) and if there exist tl912 witht2> K and such that

\u(h)-x(t2)\<8.

then for all t > 0,

\u{t^t)-x{t2 + t)\<e. (5.14)

The solution x(t) is />/*ase asymptotically stable if (5.14) is satisfiedand also there exists t3 such that

lim |w(0-*('3 + 0l = 0-t-*oo

[In other words, if u(t) gets close enough to the orbit of x(t) and ifthe phase of x(t) is suitably chosen, then the distance between thesolutions goes to 0 as / increases without bound.]

The concept of phase asymptotic stability has been known for along time in differential equations, and, as the preceding discussionindicates, it seems particularly well adapted for use in problems inphysiology and biology. Phase asymptotically stable solutions havea very important property, which is described in a theorem due toSell (1966) [see also Cronin (1980b)].

To state Sell's theorem, we need a few definitions.Suppose that we consider the n-dimensional system

^ = / (* ) • (5-15)

Definition. The point 3c is an co-limit point of a solution x(t) of(5.15) if there exists a monotonic increasing sequence {tn} such

that

lim tn — oo and lim x(tn) = x.« —• oo tn —• oo

Definition. The set of all co-limit points of x(t) is the Q-limit setof x(t) and is denoted by S[x(/)]-

Mathematical theory Y16

[Notice that if x{t) is bounded, then fi[x(O] is nonempty.]

Notation. Let O[x(t)] denote the orbit or underlying point set ofthe solution x{t), that is,

O[x(t)] = {x(t)/t in the domain of x(t)}.

5.2 A A Sell's theorem

Sell's theorem. If x(t) is a bounded phase asymptotically stablesolution of (5.13), then there exists a phase asymptotically stableperiodic solution y(t) of (5.13) such that

Roughly speaking, Sell's theorem says that a bounded phaseasymptotically stable solution approaches a phase asymptoticallystable periodic solution. (This periodic solution may be just anequilibrium point.) In view of the earlier discussion, Sell's theoremsuggests that the physiologically significant solutions all approachequilibrium points or nontrivial periodic solutions. (We use theterm "suggests" here because Sell's theorem is based on conceptsfrom the mathematical theory of stability, and that theory is notentirely realistic.)

5.2.4.5 Existence of phase asymptotically stable solutionsAlthough Sell's theorem provides a large clarification of

the picture of how phase asymptotically solutions behave, it raisesa serious problem, namely, how to determine whether there exists aphase asymptotically stable solution, especially when we are dealingwith a system in which we have very little information about theexplicit form of the solutions. Of course if the solution is anequilibrium point, phase asymptotic stability coincides withasymptotic stability and the familiar criterion for asymptotic stabil-ity (Theorem 3) is applicable. But if we want to use Sell's theoremto search for nontrivial periodic solutions, then we must establishthe existence of nontrivial phase asymptotically stable solutions. Inorder to do this, we use the same kind of technique that isdescribed in Theorem 6. That is, the eigenvalues of the matrix

are examined at all points x that are contained in an orbit or, more


practically, at all points x in a set from which a solution cannotescape. Considerations of this kind yield the following theorem.

Theorem 8. Let

^=/(*) (5-16)

be an n-dimensional autonomous system such that f has continuoussecond derivatives at each point of Rn. Suppose there exists a boundedopen set U c Rn such that the following conditions hold:

1. Each solution of (5.16) that intersects dU is headed into theinterior of U, that is, if x(t) is a solution of (5.16) andx(t0) G dU, then for all t > t0,

X ( / ) G U.

2. There is no equilibrium point of (5.16) in U.3. For all X G ( / , each component of f is a power series in

xv..., xn, the components of x.4. Suppose y E:U and Py is a matrix such that 1^1 = 1 and

where J is the real canonical form offx(y). Then there existsM > 0 such that

\\xb\p-l\<M.

5. For each y e U, the eigenvalues of the matrix

fAy)satisfy the following hypotheses'.(i) There exists r > 0 such that, for ally G U, the matrix

fAy)has (n — 1) eigenvalues that have real parts that are lessthan or equal to —r. If any of the eigenvalues of a matrixfx(y) has a nonsimple elementary divisor, then we requirethat r> M.(ii) The imaginary parts of the eigenvalues of fx(y) for ally e U have absolute value b, where


and f is a positive number if all the eigenvalues have simpleelementary divisors. If any of the eigenvalues have nonsimpleelementary divisors, then the imaginary parts of the eigen-values have absolute value less than or equal to

r — M — 7] r 7}

M = M " 1 " M 'where TJ is a positive number. [Remember that according to(/) if there exists an eigenvalue with a nonsimple elementarydivisor, then r> Mso that if i\ is sufficiently small,

r i)1 > 0 .

M MWe note that computable criteria for condition 5 are given byMarden (1966), especially on pp. 197 and 203.]

6. There exists /} > 0 such that if x G U and the nth eigenvalueof fx(x) has real part greater than —r and u(x) is aneigenvalue of unit length associated with the nth eigenvalueof fx(x), and if f^(x) is the coefficient of u(x) in theexpansion off(x) in terms of the n (generalized) eigenvec-tors of fx(x), then

7. Let —qx,...,-qk denote the real parts of the (n - 1)eigenvalues that have real part less than or equal to — r.Then for j = 2 , . . . , k,

Conclusion: If x(t) is a solution of (5.16) and there exists t0 suchthat x(t0) G U, then x(t) is phase asymptotically stable. If U isconnected there is exactly one phase asymptotically stable periodicsolution u(t) such that

O[u(t)]<zU

and if x(t) is a solution such that x(t0) G U for some t0, then

Sl[x(t)]=O[u(t)].

Proof See Cronin (1980a).

We have already seen that physiologically significant solutionsare bounded and we have indicated how boundedness of solutions


is established by showing that there is an appropriate bounded setinto which all the physiologically meaningful solutions enter andremain. In some cases, it may be possible then to apply Theorem 8or some theorem like it. This direction is still largely unexploredterritory, but there is an extreme case that should be pointed outexplicitly.

Suppose that we consider a model of a physiological situation

dx / x

and suppose there exists a set £/, the closure of a bounded open set,such that all the physiologically meaningful solutions of the modelenter U and remain thereafter in U. The next step would be tosearch for phase asymptotically stable solutions in U. Since anysuch phase asymptotically stable solution would be bounded, thenSell's theorem would be immediately applicable to it.

Now suppose that by some means we were able to show thatthere are no phase asymptotically stable solutions in the set U.Then it follows that no solution of the model can be used to predictthe behavior of the physiological system. That is, if the physiologi-cal system is "moving along" a solution and a disturbance "kicks"the system off that solution and onto another, then there is noreason for the system to return to the first solution because the firstsolution has no stability properties. As pointed out earlier, inmathematical models of physiological systems, we must assumethat small disturbances, not taken into account in the model, areconstantly impinging on the physiological system. Thus, since thesystem is kicked from one solution to another in random fashion,the solution of the model cannot be used to predict the behavior ofthe physiological system. The only prediction that can be made isthat the physiological system stays in or near the set U.

In such a case, there are two possible conclusions. One is, verysimply, that the model is a poor description of the physiologicalsystem and a better model should be sought. There is a secondpossibility, which is very serious because it brings into questionwhat can be expected of a mathematical model. Suppose, in thehypothetical case we just described, there is good reason to believethat the model is a fairly accurate description. Then the informa-


tion that the model gives us is that the physiological systembehaves in a random or unpredictable manner. Now it turns outthat such behavior or activity is actually observed in some systems.For such cases, a mathematical model that is a system of ordinarydifferential equations cannot be expected to give much information[For a detailed discussion of a problem of this kind, see Cronin(1977).]

The possibility is related to the subject of "chaotic" systems thathas been studied extensively in the last few years and in whichmany questions remain open.

5.3 Periodic solutions5.3.1 Autonomous systems

We have already seen in Sell's theorem that the study ofasymptotically stable solutions leads to the existence of periodicsolutions. The problem of determining whether there exist periodicsolutions is an important one for the study of physiological modelsbecause, as has been shown in Chapters 3 and 4, we are oftenconcerned with oscillatory phenomena in physiological systems andthese are often well described mathematically by periodic solutions.The most conventional approach to the study of periodic solutionsis to establish the existence of the periodic solution and thendetermine whether it is stable. Our next step is to list criteria forthe existence of periodic solutions. We start with consideration ofthe two-dimensional case for which there is an old and verywell-known result, the Poincare-Bendixson theorem.

5.3.1.1 Poincare-Bendixson theorem

Poincare-Bendixson theorem. Given the autonomous system

dx-r = P(x,y),d (5.17)

iwhere P, Q are continuous and satisfy a local Lipschitz condition ateach point of an open set in R2, suppose that the solution S =

), y(t)) of (5.17) is defined for all t>t0, where t0 is a fixed

5.5 Periodic solutions 131

Figure 5.1.

value, and is bounded. Suppose also that ti(S) contains no equi-librium points of (5.17). Then one of the following two alternativesholds.

1. Solution S is a periodic solution [in which case tt(S) is theorbit of S].

2. Q(S) is the orbit of a periodic solution and solution Sapproaches S2(5) spirally from the outside of spirally fromthe inside {see Fig. 5.1).

The Poincare-Bendixson theorem suffers from the same usualdrawback of pure existence theorems. That is, the existence of aperiodic solution is established, but the periodic solution may notbe unique, and the theorem gives no hint about how to computethe periodic solution. The Poincare-Bendixson theorem has anotherserious limitation. It is well known from consideration of higher-dimensional examples that the theorem is not true if the dimensionn > 2. Since most of the models we have considered have dimen-sion at least 4, we certainly need a higher-dimensional result.

5.3.1.2 Bendixson criterionA simple and often convenient test for the nonexistence of

periodic solutions in the two-dimensional case is given by thefollowing condition.

Bendixson criterion. Given the system

dx

dy


where P and Q have continuous first partial derivatives with respect tox and y at each point of the (x, y) plane, then if the function

dP dQ

dx dy

is nonzero at each point of the (x, y) plane, the system has nonontrivial periodic solutions.

Proof See Cronin (1980b) or any standard textbook on differen-tial equations.

5.3.1.3 Hopf bifurcation theoremOnce we consider dimension n > 2, there are very few

results about the existence of periodic solutions of autonomoussystems except Sell's theorem. There is, however, one well-knownresult, the Hopf bifurcation theorem, that should be described indetail.

The Hopf theorem is concerned with an autonomous system witha parameter. Very roughly speaking, it states that periodic solutionsmay bifurcate (or appear) from an equilibrium point as the param-eter is varied through a critical value. The precise statement of thetheorem is as follows.

Hopf bifurcation theorem. Consider the n-dimensional equation

-£=A(e)x+f(x,e) (5.18)

where e is a parameter, the function f has continuous first derivativesin x and e, and f is such that

r l / ( * ' e ) l nhm = 0|x|->0 \X\

uniformly in e for \e\ sufficiently small', A(e) is a differentiatematrix function of e. Let x(t, x°, e) denote the solution of (5.18)such that x(0, x°, e) = x°. Suppose that matrix A(e) has the eigen-values a(e) + ifi(e) and a(e) — //J(e), where a(e) is a real differen-tiate function such that a(0) = 0, a'(0) ¥= 0, and /?(0) # 0. Supposealso that /j3(0) is an eigenvalue of multiplicity 1 of the matrix


A(0) and suppose that A(0) has no eigenvalue of the form /[«/?(0)],where n = 0, ±2 , + 3 , . . . . Then there is an interval I = ( — r, r),where r > 0, and there exist real-valued differentiable functionse(s), h(s), c3(s),..., cn(s), all with domain / , such that

e ( 0 ) = M 0 ) = c 3 ( 0 ) = ••• = c n ( 0 ) = 0

and such that, if

c(s)= [0,s9c3(s),...9cn(s)]9

then the solution

x[t,c(s),e(s)]

of (5.18) has period

2ITVh(s)\.

The function e(s) is nonpositive or nonnegative, so the periodicsolution is obtained only for e > 0 or e < 0. Thus the periodic solution''bifurcates" from the equilibrium point 0 as e passes through thevalue 0.

It is clearly important to determine if the periodic solution thusobtained is stable, and there are explicit, that is, computable,criteria for stability. A good discussion of these is given by Poore(1976). The Hopf theorem is directly applicable only to a limitedclass of equations. Probably its primary importance lies in showingthat a certain kind of qualitative behavior (the appearance ofperiodic solutions) can occur and this kind of qualitative behavioris analogous to or serves as a model of experimentally observedbehavior of certain biological and hydrodynamical systems.

Essentially we have just two results about the existence ofperiodic solutions if n > 2: Sell's theorem and the Hopf bifurcationtheorem. The Hopf theorem is a local result: It says that if aparameter is changed slightly, then under certain circumstances asmall periodic solution appears near an equilibrium point. Sell'stheorem is an in-the-large result: It says that if there is a phaseasymptotically stable solution, that solution approaches an equi-librium point or a periodic solution.


5.3.2 Periodic solutions of equations with a periodic forcing term5.3.2.1 The need for such periodic solutions

The theorems about periodic solutions of autonomoussystems that we have described are likely to be useful in the studyof our physiological models because most of these models areautonomous systems. However, there are important modificationsof these models that are systems of differential equations that haveperiodic forcing terms. To study these modifications we needtheorems about the existence of periodic solutions of nonautono-mous systems, and this is the topic to which we turn now.

Before embarking on the mathematics, we describe briefly a fewof these modifications that are nonautonomous systems. We havealready described (in Chapter 4) the example of a periodic currentstimulus applied to a squid axon. This was studied numerically byBerkinblit et al. (1970), but a rigorous qualitative study requires theuse of theorems about periodic solutions of nonlinear nonautono-mous systems. Much more important examples are nonautonomousmodifications of the model of the Purkinje fiber. If the action of aPurkinje fiber in its natural setting is to be studied, then theinfluence of the periodic pacemaker pulse, that is, the pulseoriginating in the pacemaker region of the sino-atrial node, must betaken into account. Mathematically, this means adding a periodicforcing term to the model of the Purkinje fiber, for example, theNoble equations or the McAllister-Noble-Tsien equations.

5.3.2.2 A general theorem about existence of periodic solutionsWe consider an ^-dimensional nonautonomous system of

the form

dx

or

(5.19)

where dft/dt, dft/dxj (/, j = 1,.. . ,«) exist and are continuous in

-£ = /„('> * i , . . • , * „ ) ,


Rn+1 and f(t, x), regarded as a function of /, has period T, whereT is a positive constant; that is, for all x and all /,

f(t,x)=f(t+T,x).

The question is: Does equation (5.19) have a solution of period TlIn order to state a theorem that answers this question, we letx(t, c) denote the solution of (5.19) such that x(0, c) = c.

Theorem 9. Suppose there is a bounded open set Ua Rn such thatU is homeomorphic to a closed ball in Rn and such that U has thefollowing property: if cG U, then x(T, c) e U. {This property can bedescribed in words as: a solution that passes through a point in U isalso in U after the time interval T has elapsed.)Conclusion: There is a point c0 e U such that x(t,c0) has period T.

This theorem is proved by applying the Brouwer fixed pointtheorem to the mapping M of U into itself, defined as follows: IfCG U,

M: c->x(T,c).

The hypothesis of the theorem guarantees that mapping M takes Uinto itself and, hence, by the Brouwer fixed point theorem, map-ping M has a fixed point c0, that is,

x(T,co) = co = x(0,co).

It is easy to show that this condition implies that the solutionx(t, c0) has period T.

One drawback of this theorem is that the Brouwer fixed pointtheorem is used and consequently the proof of the theorem yieldsno information whatever about how to compute the solution unlesswe refer to a constructive proof of the Brouwer theorem. Second,since we do not known how to compute solution x(t, c0), there islittle hope of investigating its stability. Finally, the theorem justsays that at least one periodic solution exists. We have no estimateof an upper bound for the number of periodic solutions. Theremight, indeed, exist an infinite set of periodic solutions. In order tocircumvent this difficulty, we impose additional hypotheses onfj(t9xl9...,xn). Let us suppose that each fj(t, x l 5 . . . , xn) has theform

Pj(t,xl9...,xn),


where Pj(t9 xv..., xn) is a polynomial of degree qj in xl9..., xn

and the coefficients in Pj are continuous functions of / that haveperiod T except for the coefficients of terms of degree q.. Theselast coefficients are assumed to be constants. Thus, Pj(xv..., xn)may be written as

where Rj is a polynomial homogeneous of degree q} inxl9..., xn, Qj(t, xl9..., xn) is a polynomial of degree qj—l in

x! , . . . , xn, and the coefficients are continuous functions of / withperiod T. Finally we assume that the homogeneous polynomialsRl(xv..., xn)9..., Rn(xl9..., xn) have no common zeros in Rn

except the origin. [A necessary and sufficient condition that thishold is that the resultant of Rv...,Rn is nonzero; see van derWaerden (1940) or Macauley (1916).] Under these hypotheses, wehave the following result.

Theorem 10. The number m of periodic solutions of (5.19) is eitherinfinite or, if m is finite,

/ / n = 2, then m is finite and 1 < m < qxq2. For arbitrary n, if weadd an arbitrarily small constant term of / , m becomes finite and

Proof See Cronin (1980b, Appendix).

5.3.2.3 Branching of periodic solutionsJust as the Hopf bifurcation theorem shows that a certain

kind of qualitative behavior (the appearance of periodic solutions)can occur in autonomous equations, certain theorems aboutbranching of periodic solutions show that analogous qualitativebehavior occurs for nonautonomous equations. We consider theclassical problem of searching for periodic solutions of an «-dimen-sional system of the form

^=f(t,x,e), (5.20)

where e is a real parameter such that |e| is small and / has


components fj(t, xv..., xn, e), j = 1, . . . , «, where each fj hascontinuous partial derivatives of the third order with respect tot, JC19 . . . , xn, e. Further, we assume that for each fixed value of e,fj(t,xl,...,xn,e) has period T(e) where T(e) is a differentiablefunction of e. We assume that for e = 0, (5.20) has a solution JC(/)of period T(0) and we study the following problem.

Problem 1. If |e| is sufficiently small, does (5.20) have a solutionx(t, e) of period T(e) such that for each real t,

limx(t,e)=x(t)7e->0

By using a strategic change of variables and applying Floquettheory [see Cronin (1980b)], this problem can be reduced to thestudy of the equation

-£-=Ax + eF(t9x,e) + G(t), (5.21)at

where A is a constant real matrix, the functions F and G havecontinuous first derivatives in all variables, and F and G haveperiod T in t, where T denotes the number T(0). Problem 1 can berephrased as:

Problem 2. If \e\ is sufficiently small, does equation (5.21) havesolutions of period Tl

To answer this question, we have first a classical theorem due toPoincare.

Theorem 11. If the equation

dx— =Axdt

has no nontrivial solutions of period Inir/T {or, equivalentiy, ifmatrix A has no eigenvalues of the form Iniri/T, where n =0, + 1 , + 2, . . . ) , then there exist numbers i ] 1 >0, T J 2 > 0 such that foreach e with \e\ < r}l9 there is a unique vector c = c(e) such that

\c(e)-co\<-q2,

where

Co= - fTe-sAG(s) dsJo


and

x(t9e9c(e))

is a solution of (5.21) that has period T.

Now suppose that matrix A does have one or more eigenvaluesof the form 2n<iri/T. This is sometimes called the resonance case.In certain applications, the resonance case is by far the moreimportant case, but its analysis requires a more detailed study ofthe influence of the nonlinear terms. For an account of analyticalmethods of study, see Hale (1963). For a discussion of the use oftopological methods, see Cronin (1964,1980b). These mathematicalresults can be used to describe such oscillatory phenomena asentrainment of frequency and subharmonic oscillations. Topologi-cal methods can be used to obtain lower and upper bounds on thenumber of periodic solutions and information (albeit limited) aboutthe stability of these solutions.

5.4 Singularly perturbed equations5.4.1 Introduction

It has long been known that problems in electrical circuittheory often give rise to so-called singularly perturbed systems ofdifferential equations [see, e.g., Mishchenko and Rozov (1980,Chap. I)]. We consider a singularly perturbed system of the form

dxx

/ ( )

dxk

li' ' "> Xn)dt

— =fn(xl,...,xn).

That is, some (but not all) of the first derivatives are multiplied bya small parameter e. The study, both qualitative and quantitative,

5.4 Singularly perturbed equations 139

of solutions of such singularly perturbed systems is by no meanscomplete, but we shall summarize some of the important results ofthese studies because there is considerable evidence to indicate thatmany of the models of electrically active cells are singularly per-turbed systems of ordinary differential equations or can be ap-proximated by such systems. We will review this evidence at thebeginning of Chapter 6.

5.4.2 Some examples5.4.2.1 Two-dimensional examples

We begin by looking at a fairly simple two-dimensionalsystem

dy

where g(x, y) has continuous first partial derivatives with respectto x and y at each point of R2. To get an idea of how the solutionsof this system behave when e is very small and positive, we noticethat if y = JC3, then dx/dt = 0; but at any point of R2 that is noton the curve y = x3, \dx/dt\ is very large if e is sufficiently small.That means that except for points on the curve y = x3 or pointsvery close to that curve, the vector (dx/dt, dy/dt) is almost hori-zontal, pointing left or right depending on the sign of dx/dt. Thusas a rough sketch of the vector field we obtain Fig. 5.2.

In order to investigate what happens very close to the curvey = JC3, it is necessary to specify g(;c, y) and to study its values onthe curve y = x3. For the present, it is sufficient to observe that if a

y y=x3

Figure 5.2.


-3x-y =0

Figure 5.3.

solution of (5.22) is on or very near the curve y = x3, then it willstay close to the curve y = x3. The curve y = x3 is sometimes calledthe slow manifold because it is only on or very near this curve thatthe motion along the solution curves is not very fast in a nearlyhorizontal direction. Without looking more closely at this config-uration, let us consider a slightly more complicated singularlyperturbed system:

dt

(5.23)

where g(x, y) is as in (5.22). A rough sketch of the vector field isshown in Fig. 5.3.

Now further, let us suppose that in (5.22), g(x, y) is positive onthe half-plane y < 0 and that g(x, y) is negative on the half-planey > 0. This, together with Fig. 5.2, suggests that all the solutions of(5.22) approach the equilibrium point (0,0). Next suppose that in(5.23), g(x, y) is positive on the half-plane x < 0 and negative onthe half-plane JC> 0. What happens to the solutions of (5.23) is notclear. However, it seems possible that there might exist a solutionclose to the dashed curve in Fig. 5.3, that is, a periodic solutionconsisting of two "fast" path segments that alternate with two"slow" path segments. The fast and slow path segments are de-termined by "solving" the degenerate system, that is, system (5.23),with e = 0:

y = x3 — 3.x,

dy


(Notice that we multiply the first equation of (5.23) by e be-fore setting e = 0.) The "fast segments" are {j> = 2, — 1 < JC < ^3"}and {y=—2, — J3<x<l}. The slow segments are then

T/3 < J C < - 1 , y = x3-3x] and {(JC, y)/l < x < i/3,y = JC3 — 3x}. Now in fact, this sketchily stated conjecture is true,that is, it can be proved in very general circumstances that if wefind a "discontinuous solution" of the degenerate system (i.e., finda connected sequence of fast and slow segments), then for e smalland positive, the singularly perturbed system has a solution that isvery close to the discontinuous solution of the degenerate system.This solution need not, of course, be periodic. We have representedthat particular case in Figure 5.3 because it is a familiar andimportant case. Equally important for our purposes is the followingkind of example. Consider

dx

dy

1

where g(x, y) = 0 is the straight line indicated in Fig. 5.4 and thesigns of g(jc, y) are as indicated in Fig. 5.4. It is easy to see that arough sketch of the vector field is as indicated in Fig. 5.4. Thepoint E is an equilibrium point and a possible discontinuoussolution starts at the point A and is sketched with a dashed curve.Note that it approaches the equilibrium point. Also note that a

g(x,y)=O

g<0

x3-3x+y =0


solution of the degenerate system that starts at the point A and isalso sketched with a dashed curve moves much more directly to thepoint E and that the points A and A may be chosen close togethereven though the solutions that pass through them behave verydifferently.

5.4.2.2 A three-dimensional exampleWe consider one more example, a three-dimensional singu-

larly perturbed system. We give this example partly because it isenlightening to illustrate the ideas that have been advanced withanother and significantly different example and partly because thethree-dimensional example is an illustration of the elementarycatastrophe that is most widely used in applications of catastrophetheory. See Zeeman (1972, 1973, 1977). (It should be noted thatZeeman has proposed in his papers a three-dimensional model ofthe voltage-clamp behavior of the squid axon. Because Zeeman'smodel has serious deficiencies [see Cronin (1981)], we have notincluded a description of it in Chapter 3.)

We consider the singularly perturbed system

dx 1- - - - ( , 3 + ^ + , ) ,

dz

where / , g have continuous first partial derivatives at each point inthe yz plane. As in the preceding examples, we wish to obtain arough sketch of the vector field given by (5.24) if e is positive andvery small. If we choose the x axis to be the vertical axis, then if eis sufficiently small, the solutions move in a nearly vertical direc-tion except in a neighborhood of the surface Sf described by theequation

F(x, y, z) = x3 4- xy 4- z = 0.

The folds in surface Sf are characterized by the equations

dF— =3x2+y = 0 (5.25)dx


and

F(x, y, z) = JC3 + xy + z = 0, (5.26)

and the projections of the fold curves onto the (y, z) plane, whichare derived by eliminating x between (5.25) and (5.26), are de-scribed by

4 / + 27z2 = 0. (5.27)

Equation (5.27) describes a cusp # and it is easy, by elementaryarguments, to show that the surface 5? and the cusp <$ appear assketched in Fig. 5.5. (Figure 5.5 is a representation of the elemen-tary catastrophe that is most widely used in applications ofcatastrophe theory.) Let us assume that functions / and g in (5.24)are such that the dashed curve is a solution of the degeneratesystem. The same kind of arguments used for system (5.23) suggestthat system (5.24) has a solution near the discontinuous solutionsketched by a dashed curve.

5.4.3 Some theory of singularly perturbed systems5.4.3.1 Theorems of Mishchenko and Rozou

The remainder of our discussion of singularly perturbedsystems consists in describing rigorously and precisely the mathe-matics that is suggested by the preceding examples. In this, we willfollow Mishchenko and Rozov (1980), Levinson (1951), and Sibuya(1960). We will state several general theorems concerning the


existence of solutions of the singularly perturbed system that areclose to "solutions" of the degenerate system. By a "solution" ofthe degenerate system we mean a curve made up of segments thatare solutions of the degenerate system (i.e., "slow" segments)alternating with the kind of straight line segment that occurred inthe preceding examples or, more generally, a curve that is asolution of the "fast" system that will be described in detail in thefollowing. The closed curve in Fig. 5.3 is such a "solution." Such acurve (not necessarily closed, of course) is called a discontinuoussolution of the degenerate system. The word "discontinuous" refersto the behavior of the tangent of the curve. Generally the tangentchanges discontinuously at certain transition points from one curvesegment to another.

Now we summarize results of Mishchenko and Rozov (1980).Our first step is to give a formal definition in the ^-dimensionalcase of a discontinuous solution of the degenerate solution. For thispurpose we need some preliminary definitions. We consider thesystem

(5.28)

where i = 1 , . . . , k, j = 1 , . . . , / , /: + / = «, and e is a small positiveparameter. The notation

will be used; the ^-dimensional Euclidean space of points(x 1 , . . . , xk) is denoted by Xk, the /-dimensional Euclidean space ofpoints (yl,...,yl) is denoted by Yl, and the ^-dimensionalEuclidean space of points (x1 , . . . , xk, yl,..., yl\ which is thedirect sum of Xk and Yl, is denoted by Rn. Rn is the domain of thevector functions / and g, and / and g are assumed to have


continuous first derivatives in all variables at each point of R" (sothat the standard existence and uniqueness results hold) and alsoall other derivatives required in the discussion that follows.

The degenerate system corresponding to (5.28) is the systemobtained from (5.28) by setting e = 0, that is, the system

fi(x\...,xk9y\...9y

i) = 09 / = l , . . . , f c ,

dyJ (5-29)— =gJ{x\...9x

k9 / , . . . , / ) , 7 = 1 , . . . , / .

The fast-motion equation system corresponding to (5.28) is the^-dimensional system

dxl

e—=fi(x19...9x

k9y\...9y

l)9 1 = 1 , . . . , * , (5.30)

where the numbers yl9..., yl are regarded as parameters.

Assumption 1. If y1,..., yl are arbitrary fixed values, then eachsolution (xl(t),..., xk(t)) of (5.30) is such that if it has an co-limitset, the co-limit set is an equilibrium point.

Let F denote the /-dimensional surface in Rn defined by the kequations

/ ' ( x 1 , . . . , * * , / , . . . , / ) = ( ) , 1 = 1 , . . . ,* . (5.31)

[In referring to F as an /-dimensional surface, we are assuming thatthe (5.31) are such that F is an / manifold.]

Next let ^ ( J C 1 , . . . , xk, y1,..., yl) be the kxk matrix definedby

The matrix s/(xl,..., xk, yl,..., yl) is defined at all points of Rn.Let

r_ = {(x\ . . . , xk, y\..., yl) e T/all eigenvalues of

jtf(xl,..., xk9 yl

9..., y ) have negative real parts)

The set T _ is called the staWe reg/Vw of F.


Assumption 2. No point of F_ is an equilibrium point of (5.28).That is, if (3c1,..., Jc\ y\...,?') G T_, then

{f\x\...,xk,y\...,y'),...,fk{x\...,xk,y\...,y'),

g\x\...,xk,y\...,y'),...,g'(x\...,xk,y\...,y'))

Let

The points in Fo are called nonregular points. In general, the set Fo

is an ( / - l)-dimensional subset of T and

where

C={(x\...,xk,y\...,y<)

G T/det s/(xl, ...,xk,y\...,y')>0},

D = {(x\...,xk,y\...,y>)

G T/det J / ( X \ . . . , xk, y\..., y1) < OJ.

From the definition of Fo, it follows that if (xj , , . . . , XQ, y\,..., yl)G Fo, then the matrix

j y ^ 0 , . . . , J V : 0 , y 0 , . . . , y 0 )

has at least one eigenvalue that is zero.

Definition 1. A n o n r e g u l a r p o i n t S ( X Q , . . . , X Q , y ^ . . . , y^) i s ajunction point if and only if the following conditions are satisfied.

1. S is not an equilibrium point of (5.28). Sincethen f(S) = 0. Hence, this condition means that g(S) # 0.

2. Zero is an eigenvalue of algebraic multiplicity 1 of stf{S).All the other eigenvalues of sf(S) have negative real parts.

3. The fast-motion equation system with (yl, ...,yl) =(jô1,..., yl), that is, the system

e^-=f'(xl9...9x

k9yl...,y{>)9 (5.32)


has the point (X\,...,XQ) as an equilibrium point [since/ ( S ) = 0]. We require that ( X Q , . . . , X Q ) be an isolatedequilibrium point and that ( X Q , . . . , XQ) be the a-limit setof exactly one orbit of (5.32). That is, there is exactly oneorbit of (5.32), the underlying point set of a solution x(t)of (5.32), such that lim,_ - ^ ( 0 = ( 4 > . . . , x§).

4. Let A i / denote the A>dimensional hyperplane

{{x\...,xk,y\...,y')\/=yl...,y>=y<>}.

Then, if X^ y/ is translated by any sufficiently smallvector, it does not contain an equilibrium point of thefast-motion equation system near S.

Definition 2. Let (Jc1,..., xk, yl,..., yl) e Rn - T and considerthe orbit of

e-^=f'(x\...,xk,y\...,yk), (5.33)

which passes through the point (Jc1,..., x1). If the co-limit set ofthis orbit is a stable equilibrium point (jc1,..., jc*) and if

then the point (Jc1,..., Jc , y1,..., yl) is called the drop point corre-sponding to the point (Jc1,..., Jc , yl,..., yl).

Definition 3. Let S(xl0,..., XQ, yl,..., y$) be a junction point

that satisfies the following condition: The orbit of

which has (x^ , . . . , JCQ) as its a-limit set (condition 3 in Definition1), has the property that its co-limit set is a stable equilibrium point

Then P is the drop point following the junction point S.

With Assumptions 1 and 2 and Definitions 1-3, we are ready todefine a discontinuous solution of the degenerate system (5.29). Indescribing the discontinuous solution, we will use the term "phase


point" and speak of the "phase point traversing" a curve. The term"phase point" simply refers to a position on a curve and how thephase point " traverses" the curve is given by the analytic descrip-tion of the curve. A discontinuous solution of (5.29) with initialpoint Q^Rn is the continuous curve 3~c Rn obtained by applyingsuccessively the following steps.

Step L If Q(jc1,..., xk, yl9..., yl) £ T9 then the system

1,...9xk9y\...,

has a unique solution

(xl(t),...,xk(t))

such that

xi(t0)=xi9 / = l , . . . , / c .

We assume that the co-limit set of the solution

is a stable equilibrium point (x1 , . . . , xk) and that

That is, we assume that (Jc1,..., xk9 y1,..., yl) is a drop point

corresponding to Q. The first section of the continuous curve ST isthe curve

{(x\t),...,xk(t),y\...,y')\t>t0}.

The phase point traverses this section of ST instantaneously andthis is called a fast-motion part of ST.

Step 2. If Q(X\...,Zk,y\...9yl)*=T_9 then from the defini-

tion of F_ and by the implicit function theorem, the system ofequations

fi(x19...9x

k9y

19...9y

i)=09 i = l , . . . , f c ,

can be solved uniquely for xl,...9xk in terms of yl,...,yl in a

neighborhood of the point Q(xl,..., xk9 yl

9..., yl). That is, there


exist differentiate functions

such that

and such that

/'[x1(/.....A....^(/.-.A/.-.y]-o.(5.34)

Substitutingx ' = x ' ( / , . . . , y ) , i= i , . . . , * ,

into (5.29), we obtain (5.34) and

y) *vy = l , . . . , / . (5.35)

The existence theorem can be applied to (5.35) and we obtain theconclusion that there exists a solution

of (5.35) such that

and such that

This last statement follows from (5.34). Next we assume that thereexists tx > t0 such that the solution

is defined for all t e [^0î) a nd that

exists and that

S=\im{xl[y\t),...,y'(t)],...,xk[y\t),...,yl(t)],


is a junction point in Fo. In this case, the resulting section of thecurve 2T is

The time interval during which the phase point moves along thissection is tx — t0 and this section is called a slow-motion part of y .

Step 3. If Q(xl0,..., XQ, yl,..., y^) is a junction point, we assume

that there exists a drop point P following the junction point Q.Then the corresponding section of the curve $" is defined asfollows. Let the orbit of

which has (JCQ,. . . , JCQ) as its a-limit point, be described by thesolution

The corresponding sections of F is

{(x1(t),...,xk(t),yl0,...,y<)\trea\}u{P},

where

P= ]im(xl(t),...,xk(t),yl...,yfi.

The phase point traverses this section of ST instantaneously andsuch a section is called a fast-motion part of &.

We illustrate these definitions with an example (shown in Fig.5.6)

dy

5.4 Singularly perturbed equations

y

151

3-3x-y=0

Figure 5.6.

The degenerate system is

y = x3 — 3JC,

dy I 1

The fast-motion equation system is

d x < -K x

e— = -{x'-'ix-y).

For fixed y, this is a single first-order differential equation andhence Assumption 1 is automatically satisfied. The set T is thecurve

y = x3 — 3x.

The matrix J / is the 1 X 1 matrix (i.e., just a scalar)

J / = - 3 J C 2 + 3 .

The set F_ is

F _ = {(x, y)\y = x3 - 3x and x 2 > 1} .

Since £ is the only equilibrium point, Assumption 2 is certainlysatisfied. The set Fo consists of the two points S\ —1,2) andS 2( l , —2). Also each of the points Sl and S 2 is a junction point.


To prove this statement we note first that conditions 1 and 2 inDefinition 1 (the definition of junction point) are obviously satis-fied. Equation (5.32) in condition 3 becomes, for this example withthe point Sl,

e— = - (x3 - 3x - 2) = 2 + 3x - JC3.dt

Since - (JC3 - 3x - 2) = - (x - 2)(x + I)2, then dx/dt = 0 at x =— 1 and dx/dt > 0 for JC in a neighborhood of — 1 (except at — 1itself). Thus, condition 3 is satisfied. Condition 4 is satisfied byinspection. Thus, S1 is a junction point. A similar proof shows thatS2 is also a junction point. Using again the fact that

- (JC3 - 3x - 2) = - (JC - 2)(JC 4- I ) 2 ,

we see easily that P\2,2) is a drop point following S1 and P2

( — 2, —2) is a drop point following S2.Now we are ready to describe some discontinuous solutions.

Suppose first that the initial point Q is such that Q^T_; inparticular, take Q to be, say, the point (\/T,0). By Step 2 in thedefinition of discontinuous solution, the first section of the discon-tinuous solution is

Since 5 2 is a junction point, then by Step 3 in the definition ofdiscontinuous solution, the second section of the discontinuoussolution is

{{x,y)\-2<x<l,y=-2}.

The sections that follow are

{(x,y)\-2<x< - 1 , y = x3-3x),

{(x,y)\-l<x<2,y = 2],

and

{(JC,J) |V / 3"<JC<2, y = x3-3x).

The discontinuous solution is a simple closed curve.We obtain a second discontinuous solution by considering an

initial point Q such that Q £ T. In particular, take Q to be, say, thepoint (1,0). Then by using the steps in the definition of discontinu-


ous solutions, we see easily that the discontinuous solution that hasas its initial point Q = (1,0) consists of the sections

and then the same sections as for the preceding discontinuoussolution.

For a three-dimensional example, see Mishchenko and Rozov(1980, p. 19).

The first mathematical question that we consider is the follow-ing. Suppose that ^ is a discontinuous solution with initialpoint Qo = (xl

O9...,x%9y%9...,y&). Let ^ be the orbit of the

solution of (5.28) that has Qo as its initial value at / = t0. That is,STE is t h e o r b i t of t h e s o l u t i o n (x\t),..., x k ( t ) , y l ( t ) , . . . , y \ t ) )of (5.28) such that (x\t0),..., xk(t0), y\t0), • • • ,/(*<>)) =Qo(xl

o,...,x%9y£,...,yl). How far apart are ^ and ^ , espe-cially as e -> 0? The first answer to this question is given byMishchenko and Rozov (1980, p. 174, Theorem 1).

Theorem 12 (Mishchenko and Rozov). Let ^ be a finite seg-ment defined on the interval [t0, tx\ of the discontinuous solution &"0.That is, ^0 consists of the slow-motion parts of ^0 whose domainsare contained in [/0> ' J and the fast-motion parts that occur betweenthese slow-motion sections. Then if e is sufficiently small, the solution

of which 3TE is the orbit, is defined for all t e [f0, fj. / /

and if

where

d[p,^0}= glb\p-q\,

then

Roughly speaking, this theorem says that if e is small enough,then ^ is a good approximation to the orbit ^ . This is an


extremely powerful result because, as we shall see in later applica-tions, finding a discontinuous solution is, in some cases, mucheasier than directly studying the solutions of the singularly per-turbed system.

The second question that we want to consider is the following.Suppose that the discontinuous solution ^ is a closed curve.Then if e is small, is ^ a closed curve? In other words, if e issmall, does the singularly perturbed system have a periodic solutionwhose orbit is near ^ ? This question is answered by Mishchenkoand Rozov (1980, p. 204, Theorem 1).

We impose the same hypotheses as in the preceding theoremexcept that we assume also that ^ is a simple closed curve andthat STQ is isolated and stable. That ^ is isolated and stablemeans the following. If S is a junction point of ^ and if N(S) isany sufficiently small neighborhood in ro of S, let p e N(S) — S.We consider the discontinuous solution with orbit ^ whichcontains p. (It is, of course, necessary to prove that such adiscontinuous solution exists.) Assume that if we follow ^ afterp, ^0 will intersect N(S) in a point p. Define the mapping

<t>' P-*P,that is,

That ^~0 is isolated and stable means that we assume that <f> has aunique fixed point S and that the linear part of <j> is a mappingfrom N(S) into N(S) and that this mapping also has a uniquefixed point, the point 5.

We need just one more assumption, which is a nondegeneracycondition. Without loss of generality, we may assume that ^consists of four parts: two slow-motion parts and two fast-motionparts. Let Sx and S2 be the two junction points of ^0. Let Sf bethe (/ — l)-dimensional plane tangent to Fo at Sv and let S* be the(/ — l)-dimensional plane tangent to ro at S2. We make the follow-ing assumption. [Nondegeneracy assumption; see Mishchenko andRozov (1980, p. 202).]

The vector g(Sx) is transversal to Sf (i.e., if vx is a nonzerovector in Sf, then (\g(Sx) • ^ i D / d g ^ ) ! \vx\) < 1. Also the vector

is transversal to S*.


Now we have:

Theorem 13 [Mishchenko and Rozov (1980, p. 204)]. There existse0 > 0 such that if 0 < e < e0, then (5.28) has a closed trajectory ZTt,which describes a periodic solution, such that

limp(iT0,ir ) = 0.

[Whether this closed trajectory ê is unique is not known; seeMishchenko and Rozov (1980, p. 203).]

For the special case n = 2, Theorem 13 has a simpler strongerform, which we will use in the analysis of the FitzHugh-Nagumoequations.

Theorem 14 [Mishchenko and Rozov (1980, pp. 141-142)]. Giventhe singularly perturbed system

4-/<*.,).4-

where x, y are scalars, we assume that the following conditions aresatisfied:

1. / , g have continuous second derivatives of all kinds at eachpoint in the xy plane.

2. If F denotes the curve defined by the equation

f(x,y) = 0

then each point of F is an ordinary point, that is, at eachpoint of F,

[fx(x,y)Y+[fy(x,y)}2>0.3. The nonregular points of F, that is, the points at which

are isolated and each of them is nondegenerate, that is, ateach of them,

fxx(x,y)*0.


4. Let the regular part of T between nonregular points p andp be described by

yp<y<y-p.

If for all y e [yp, yp], we have

then this regular part of T is called a stable part. Weassume that g(x, y) =£ 0 at all points of the stable part andat all nonregular points of F. [In other words, we assumethat there is no equilibrium point on a stable part and thatno nonregular point is an equilibrium point of (5.36).]

5. No two nonregular points of F have the same ordinates.6. Suppose the degenerate system corresponding to (5.36),

that is, the system

0=f(x,y),dy- = g ( * , 7 ) ,

has a closed discontinuous solution ST^.

Conclusion: If e is sufficiently small, the system (5.36) has a uniquestable limit cycle STB such that

e->0

5.4.3.2 Theorems of LevinsonWe will see in Chapter 6 that the preceding theorems from

Mishchenko and Rozov are useful in the analysis of our models ofelectrically active cells. However, there is one important class ofsolutions that cannot be obtained by using the Mishchenko-Rozovtheory. If we consider the FitzHugh-Nagumo equations or theHodgkin-Huxley equations and regard them as singularly per-turbed systems, then a solution that describes an action potential isnot a periodic solution but is a solution that approaches a stableequilibrium point. This stable equilibrium point describes the rest-ing or stable quiescent condition of the unstimulated axon. Now ifsuch a solution is close to a discontinuous solution, then this mustbe a discontinuous solution that approaches a stable equilibrium


point, that is, the kind of discontinuous solution indicated in Fig.5.4. But the equilibrium point E in Fig. 5.4 is an asymptoticallystable equilibrium point. Thus, Assumption 2, which holds inTheorems 12 and 13, is violated. Also condition 4 in the statementof Theorem 14 is violated. Thus, Theorems 12, 13, and 14 are notapplicable. However, certain parts of the theory due to Levinson(1951) are applicable. Although we shall use only a two-dimen-sional version of Levinson's results, we summarize some of the^-dimensional theory so as to indicate the relationship and overlapof the Levinson theory and the Mishchenko-Rosov theory. Itshould be noted also that Levinson's approach to singularlyperturbed equations is somewhat more general than that ofMishchenko and Rozov because Levinson includes nonautonomousas well as autonomous equations, whereas Mishchenko and Rozovconsider only autonomous equations. We consider the system

dxt

d2uE y

_|_ l O" 1 '

L o \"

...,xn,u,t,e)

du

due ^ dt

= 0, (5.37)

where x x , . . . , xn9 w, /, e are real and f l 9 . . . , /„, ^ , . . . , <£>„, g, A arereal-valued continuous functions for xv..., xn, w, r real and e > 0.Most frequently we will write (5.37) in vector form as

dx du

* ^ , (5-38)d u du

The degenerate system of (5.38) is

dy dv— = / — +<?>," ' * (JJ9)dv

g - + A-o.


The degenerate system is obtained from (5.38) by setting e equal to0 and using the notation (y, v) in place of (x, u). As will be seenlater, the change of notation is a clarifying convenience.

System (5.38) includes as a special case the system

dx— =H(x,w,t,e),

(5.40)dw

£—- = G(x,W,t,£),at

where x, H are n vectors and G, w are scalars. [System (5.40) is thekind of system that occurs in mathematical modelling of electri-cally active cells.] To show that (5.40) is a special case of (5.38), weargue as follows. First, if the right-hand sides of (5.40) are linear inw, let w be such that

du— = w.dt

Then (5.40) can be written as

dx du— =f(x,t,e)— + 4>(x,t,e),

d2u dut—i + g(jc, /, e)— + h(x, t, e) = 0.

System (5.41) is a special case of (5.38). Second, in the general case,that is, the case in which at least one of the right-hand sides of(5.40) is not linear in w, by differentiating the last equation in(5.40) with respect to /, we obtain

dx— =H{x,w,t9e)9dt (5.4O0d2w dG dw dG dx dG

d d d£ dt2 ~ dw dt dx dt dt '

This last system is a special case of (5.38) with / = 0.Our next step is to define a discontinuous solution of (5.39). This

step is parallel to the similar step taken in the Mishchenko-Rozovtheory. However, in Levinson's work, the concept of discontinuoussolution differs significantly from the concept of discontinuous


solution used by Mishchenko and Rozov. At the outset there is alarge difference because system (5.39) is, in general, nonautono-mous. Hence, the discontinuous solution will be a curve in (y, v, t)space, that is, (n + 2) space. [Since Mishchenko and Rozov dealonly with an autonomous system, their discontinuous solutions arecurves in (y, v) space, i.e., (n + 1) space.]

In order to define a discontinuous solution, we shall be obligedto write a fairly lengthy account that will include making a numberof assumptions. The clearest way to present these assumptions is tostate them as they are required rather than to try to list all of themin advance. We shall indicate the introduction of an assumptionsimply by using the word "assume" in italics.

First we assume that there exists a point A = (y, v, a) in (y, v, t)space such that

g(y,v,a)>0.

Then in a neighborhood of (j>, v, a), system (5.39) becomes aconventional (n + l)-dimensional system of ordinary differentialequations

dy dv

dt J dt 9'

dv h

dt g

or

dy r h]

* - _ * (5-42)

dt g'

Consequently, it is reasonable to expect that we can solve (5.39) ina neighborhood of (y, v, a) or, more precisely, that there exists asolution (y(t), v(t)) of (5.39) that satisfies the initial value

y(a)=y,

v(a) = v.

Indeed, if / , g,h,<f> satisfy slightly stronger conditions than just


continuity, then the existence theorem stated earlier in this chapterguarantees the existence of such a solution. We shall assume thatthere exists a number TX with the following properties:

1. rx > a.2. There exists a solution (y(t), v(t)) of (5.42) such that

(y(a),v(a)) = (y,v)

and the domain of (y(t),v(t)) contains [a, TX). We willdenote this solution by So. Also lim, tT y(t) = y(rl — 0)and limnTv(t) = v(r1 - 0) exist and are finite. We willdenote y(rx - 0) and v(rx - 0) by yB, vB.

3. I f / e [ a , T l ) ,

g[y(t),v(t)9t90]>0

and

]img[y(t),v(t),t,0] =0.

Let Bx denote the point (yB, vB, TX) in (y, u, /)"sPace a n d let ^ ^ idenote the union of the orbit of solution So for a < t < rx and thepoint Bv That is,

The first arc of the discontinuous solution is defined to be ABV

The second arc of the discontinuous solution is a curve in thehyperplane t = r1 in (y, v, t) space. In order to describe this secondarc, we first make another assumption. Let

and denote I(yB>vB>Ti) ^y IB- We assume that IBJ=0. We alsoassume that h(yB, vB, T1?0) # 0. The second arc of the discontinu-ous solution is the orbit of the solution of

^ =f(y,v,r.,0) (5.43)

with initial value

y=yB= lim.y(0,

v = vB = l im v(t).


We arrive at this description of the second arc by the followingconsiderations. Since h{yB, vB, T1 ?0) # 0, then if t < TX and if TX — tis sufficiently small, it follows that

h[y(t)9v(t)9t,0]

g[y(t),v(t)9t,0] * °

and hence dv/dt±0. Thus, dt/dv is defined. Multiplying bothsides of the equations in (5.39) by dt/dv, we get

dy dt dv dt dt

dt dv dt dv dv'

or

dv%—* dt

dy

dv

g +

dt

dv

= /+*

dt

dt

dv

dt

~dv'

0.

= 0

(5.44)

If t is held constant so that dt/dv = 0, then (5.44) becomes

g[y(t),v(t),t,0}=0.

Since

then as t \ TX system (5.45) becomes

It is these considerations that suggest that the second arc of thediscontinuous solution is the orbit of the solution of

^=f(y, v,rlf0) (5.43)

with initial value


In (5.43), variable v is an independent variable. We will consider vto be increasing above vB or decreasing below uB according ash[yB,vB,rv0] is negative or positive. (Remember that we haveassumed that h[yB, vB, TÔ] =£ 0.) The reason for this considerationis that if t <rx,

dv h

dt~ g*

Hence, if rx — t is sufficiently small, then since

g[y(t)9v(t)9t90]>09

it follows that dv/dt has the sign opposite to the sign ofh[yB,vB, TV0]. For definiteness, we will assume that h[yB, vB, r l90]is negative and hence that v is increasing.

Let y(v) be the solution of (5.43) such that

By assumption

g[yB, VB9TX9O] = 0. (5.46)

But if v # vB and \v — vB\ is sufficiently small, then

d n dg dy dg

dv = i dyi dv dv

Thus

lim —j?fy(w),i;,T1,Ol= E -7

But this last expression is IB, which has been assumed to benonzero. This fact along with (5.46) implies that if \v — vB\ isnonzero but sufficiently small, it follows that

g[j>(l>),»,T1 ,0]*0.

Hence, \i[v — vB] is positive but sufficiently small,

j""g[.K0).0.Ti»O]<fc*O.


Now we assume that there is a smallest number vc such thatvc > vB, the domain of solution y(v) of (5.43) contains the interval[vB9 vcl and

Let Cx denote the point (y(v c \ vc9 TJ. Denote g[y(vc), vc9 TÔ] bygc and assume that

(This implies that / 5 , which was earlier assumed to be nonzero, isactually negative.) Since g c ^ 0 , then in a neighborhood of thepoint Cl9 the degenerate system (5.39) is a conventional (n 4-1)-dimensional system

Jy h

dt g'

Consequently it is reasonable to assume that there exists a solution(y(t)9 v(t)) such that

We assume that there exists T2 > TX such that:

1. The domain of (y(t), v(t)) contains the set [T1? T2).2. Forf€E[TlfT2),

3. l im / T T 2>;(O=j(T 2 -0) and limtUy(t) = V(T2-0) existand are finite,

4. lim,T

Let i?2 denote the point (y(r2 — 0), v(r2 - 0), T2) in (j>, i;, /)space. Next we assume that the same kind of hypotheses hold forthe point B2 that held for Bv Hence, we can then proceed in thesame way at B2 as we did at Bx with v increasing or decreasingaccording as h[y(r2 — 0), v(r2 — 0), T2 ,0] is negative or positive.


Continuing in this manner, we obtain a curve

ABXCYB2 • • • BNCNA'

(where A' is a point at which g =£ 0).

Definition. The curve

ABXCXB2 • • • BNCNA\

which will be denoted by ^ , is a discontinuous solution of (5.39).

Now we are ready to state Levinson's theorem. We make thefollowing assumptions:

Al . There exists a discontinuous solution Sf of (5.39) definedfor a < t < /?.

Al. There exists an open set R contained in (y, v, t) spacesuch that:(i) ^ c R.(ii) f(y, v, t, e), <£>, g, h and their first partial derivativeswith respect to j> l5..., yn, v, t are uniformly continuousand bounded as functions of y, v, t, e for (y, v, t) e R and0 < e < r, where r is a small positive number,(iii) g, h have continuous second partial derivatives onRX[0,r]. [Levinson (1951, p. 75) points out that thisassumption can be avoided.]

In the statement of the theorem, we use the vector norm that wasintroduced earlier in this chapter. That is, if x is the n vector(xv..., xn), then the norm of x is

Theorem 15 (Levinson's theorem). Suppose that (5.39) satisfiesthe preceding assumptions Al and A2. If e,8v82 are sufficientlysmall and positive, there exists a solution (x(t), u{t)) of (5.38) witha domain that contains [a, /$] for any set of initial values (x(a), u(a))that satisfy the inequalities

du dv


As e, Sl9 82 -> 0, the curve in (JC, U, /) space described by(x(t), u{t), t) for a<t<fi approaches Sf. In particular, for anyfixed 8 > 0, the function

\x(t)-y{t)\ + \u{t)-v(t)\

converges uniformly to zero on the set

E = [a + 8, rx - 8] U [rx + 8, r2 - 8] • • • [rN + 8, 0 ]

as £, 81? 82 -» 0. Also the functions

du dv d2u d2vand —2 2

converge uniformly to zero on the set E as e, 8V 82 -> 0.

Theorem 15 was introduced by Levinson to treat a system of theform (5.37). Actually, our models are all of the special form (5.40).Consequently when Theorem 15 is applied, the initial values arespecified values only of w(a) and x(a). The dw/dt(a) is thenspecified by the second equation in system (5.40). [If we specify avalue of dw/dt(a), then, integrating the second equation of (5.40'),we would obtain

dwe— = G(x, w, t, e) + C,

dt

where C is a constant of integration whose value would be de-termined by the specified value of dw/dt(a). Thus, unless C = 0,we would not have the system (5.40) that we are trying to study.]Thus 82, and the inequality

dw dv— (a) (odt dt

can be disregarded; that is, the version of Theorem 15 that we willuse is:

Theorem 15R (revised). Suppose that the degenerate system of(5.40) satisfies assumptions A\ and A2. Then if e and 8X aresufficiently small, and positive, there is a solution (w(t), x(t)) of(5.40) with the solution defined for a<t</$ for any set of initialvalues w(a), x(a) such that


[The discontinuous solution given by assumption Al is described by(y(t), v(t)).] Also as e,Sl tend to zero, the curve representing thesolution (w(t), x(t)) in (w, x, t) space tends to Sf. In particular,for any fixed 8 > 0,

\w(t)-v(t)\ + \x(t)-y(t)\

tends uniformly to zero over the intervals

as e, 8X —> 0.

If Theorem 15 is to be applied to system (5.40), the first step isto differentiate the equation

dw

dt

with respect to t. As a consequence of this, it turns out that thediscontinuous solution with initial point A, where dG/dvÔ,stays (for some time interval) in the manifold in (y, v, t) spacedescribed by

G(y9v9t9O) = GA9

where GA denotes the value of G at the point A. The reason forthis is that we have

dwe— = G(x, w, /, e)

dt

and, after differentiation,

d2w d

Setting e = 0, we get

- [ G ( x , w , / , 0 ) ] = 0 .

The first step in obtaining the discontinuous solution is to solve the


system

-^- = H(y,w,t,O),

d ( 5 - 4 ? )

-[G(y,w,t,O)]=O

or

or

(dG\tdv\ ldG\dy dG

\Hv)\dt) + \dy)~dt + ~dt = °

dy— = H(y,v,t,0),dt (5.48)do ldG\-lUdG\dy dG'dt \ dvj [\ dyjdt+ 8t

We seek a solution (y(t), v(t)) such that

and the point A = (yA,vA,a) is such that at A, dG/dvi=0. LetGA = G(yA,vA,a,0). Then

because

and by (5.48) or, equivalently, (5.47),

dG— [y(t),v(t),t,O]=O

so that G[y(t), v(t), /,0] is a constant. Thus the first arc of thediscontinuous solution lies in the manifold

G(y,v9t9Q) = GA.


But we seek solutions of system (5.40), the degenerate system ofwhich is

dx— = H(x,w9t90),at

Thus, we seek discontinuous solutions, the slow arcs of which arecontained in the manifold described by

G(x,w,t90) = 0.

Hence, the point A must be contained in this manifold.However, Theorem 15 can be extended to a discontinuous solu-

tion, the first arc of which is fast, by using arguments in Levinson'spaper [Levinson (1951, pp. 81-84, Proof of Theorem 1, Part 2)].We will need this extension in Chapter 6 for the study of theFitzHugh-Nagumo equations.

We illustrate the preceding theory with the example consideredearlier:

dy / 1

T \( , 3 , , ) .

To cast this system in the form of (5.37), we differentiate thesecond equation with respect to / and obtain

dy

d2x r „ , dx dy£ dt2 l •"" • ~J dt • dt

or

dt ' 2 ' ' (5.50)</2x , dx[to + j, L


This is a special case of (5.37) with y = xv x = w, and

/=o,

g=(3x2-3),

The corresponding degenerate system is

dy / 1

(5,1)

The manifold M, described by

G(x,w,f,0) = 0,

becomes, in this case,

x3-3x-y = Q.

(This manifold is, of course, the curve shown in Fig. 5.6.) Let A bea point on this manifold such that

g = 3 * 2 - 3 > 0 .

(Any point on M such that \x\ > 1 satisfies the condition.) We seeka solution of (5.51) that has the initial value A at some value t0 oft. If we apply Levinson's theory to the degenerate system (5.51),then we consider curves in (y, x, t) space, where these curves aredescribed by solutions of (5.51). However, in this case the equa-tions dealt with are autonomous. Hence, we consider the projec-tions of these curves into (y9 x) space. Theorem 15R can be used toconclude that the projections of the curves described by the discon-tinuous solution and the solution of (5.49) itself stay close together.

Referring to Fig. 5.6, let A be a point on the manifold describedby

x3-3x-y = 0

such that

g = 3 x 2 - 3 > 0 .


For example, suppose

As shown earlier, the first arc of the discontinuous solution, whichhas A as its initial point, lies in the manifold

and the arc is that part of the manifold that joins the point A =( - \/3\0) and S1 = (-1,2). The point (-1,2) is the first point atwhich

g=3x2-3

is zero. Thus Sl is the point B1 described in the definition of thediscontinuous solution. [More precisely, Sl is the projection in(y, x) space of the point Bv] Now for this example the equation(5.43) becomes

dx

because in this example the function / is identically zero. Hence,the second arc of the discontinuous solution is a segment of the line

y = 2.

The segment starts at ( — 1,2). Since h<0 at x = l, then x isincreasing and the integral

f g[y(v)9v9Tl90] dv

for this case becomes

fX (3x2 - 3) dx = [x3 - 3JC]* x = x3 - 3x + 1 - 3J-i

= x3-3x-2

= {x + \)\x-2).

Thus, the point vc is 2 and the second arc of the discontinuoussolution is the line segment


The third arc of the discontinuous solution is the subset of mani-fold

which is the point set

{(x,y)\l<x<2, y = x3-3x).

We are especially interested in the case of relaxation oscillations,that is, the case in which the discontinuous solution is a closedcurve and nearby solutions of the singularly perturbed system areperiodic. Although the results of Mishchenko and Rozov deal withthis problem, we describe Levinson's somewhat different approachto the problem because it is valuable to have as much informationas possible about the question. In order to deal with this case, wewill need the following further theorem. First, we denote by d/d adifferentiation with respect to one of the (n + 1) initial coordinates(w(a), x(a)) or with respect to one of the corresponding initialcoordinates (v(a), y(a)).

Theorem 16. With the same hypotheses as in the preceding theo-rem, we have, if e, Sv 82 -» 0,

dw

~da

d

Ja~

(0dw

It

dv

~Jad

~Ja

(')dv

It

+dx

~da

—»

da'0,

The convergence is uniform over the same set of intervals as in thepreceding theorem. Moreover, if b denotes the initial value of du/dtat a, then

and

dx

d du

~db~dt

du

~db(0 •0

uniformly over the same set of intervals as e, 8X, 82 —> 0.


Now suppose that G and H, as functions of t, have period Twhere T is a positive constant. That is, suppose that for allX, W, / , £,

Suppose further that (5.39) has a discontinuous solution withperiod T, that is, the point A1 coincides with point A and theinterval (a, /?) is (a, a + T). By using the immediately precedingtheorem and conventional arguments, we obtain:

Theorem 17. Let D(al9..., aw+1, &, e) denote the determinant

dxx

dax

dx2

dax

dw

dax

d dw

da, dt

da2

dx2

da2

dx2

dwi

d

db

db

dx2

db

dw

db

dw

dt

where the entries in the determinant are evaluated at t = fl = a+ Tand al9...9 an+l9 b denote initial values xx(a),..., xn(a), w(a),dw/dt(a). Suppose that D is continuous and nonzero for e sufficientlysmall, and for the terms \aj-yj(a)\ (j = 1 , . . . , n\ \an + l - v(a)\9

and \b - (GyH + Gt)/Gv\, where (GyH + Gt)/Gv is evaluated at thepoint A, all sufficiently small. Then for sufficiently small positive e,equation (5.37) has a unique nearby solution of period T.

If G, H are independent of t, and (5.39) has a discontinuoussolution of period T9 and if the hypotheses of the precedingtheorem are satisfied, then a similar conclusion holds except thatthe period of the periodic solution of (5.37) is 7(e), where T(e) is acontinuous function of e and lime^07X£) = T.


5.4.3.3 Theorem of SibuyaThe work of Levinson is restricted to the case in which just

one of the equations is singularly perturbed, that is, w is a scalar(or real variable). Some of Levinson's work (not the part onperiodic solutions) has been extended to the general case by Sibuya(1960). It is necessary to use Sibuya's result in order to search fornonperiodic solutions of singularly perturbed systems in whichthere are at least two equations with a factor 1/e. As we shall seelater, this is exactly the problem that must be studied when theHodgkin-Huxley equations are regarded as a singularly perturbedsystem.

Before stating Sibuya's results, we point out explicitly someconditions in the results. First we note that the vector norm usedby Sibuya is equivalent to, but not identical with, the vector normused by Levinson. Second, certain characteristic roots of coefficientmatrices that occur have value zero. These characteristic roots arerequired to be simple. (This is a kind of nondegeneracy hypothesis.)Finally, the nonzero characteristic roots are all required to bepositive.

The system of equations studied by Sibuya is the following:

d2U _ dUe—Y +A(t,U9 JC,C) + a(t,U, x,e) = 0,

dt dt (5.52)dx „ dU „— = B(t9 U9 x, e ) — + b(t9 U9 x, e),dt dt

where x and U are m- and ^-dimensional vectors, respectively, /is a real scalar, and e is a real nonnegative parameter. It is assumedthat there exists an open set R in (t9U,x) space such that thecomponents of the matrices A, B and the vectors a, b and theirfirst-order partial derivatives with respect to t and the componentsof U and x are all uniformly continuous and bounded as functionsof (r, £/, x, e) if (/, U,x)^R and e is small. Let


Then the degenerate system corresponding to (5.52) is

(5.53)

dVAo(t,V,y)—+ao(t,V,y) =

dy . dV „— = B 0 ( / ,K ,^ )— + bo(t,V,y).

Now we define a discontinuous solution of the degenerate system.Let the point Px = (sl9 Vl9 yx) be such that Px e R, and assumethat the real parts of all the characteristic roots of A^s^ Vv yx) arepositive. Let (v(t), y(t)) be a solution of (5.53) such that

Assume that there exists s2 > sx such that:

1. The solution (v(t)9 y(t)) is defined for t e (sl9 s2).2. limtts (v(t% y(t)) exists and is finite. [Denote that limit

by (v29 y2).] Assume also that (s2, v29 y2) e R.3. The real parts of the characteristic roots of A0(t9 v(t)9 y(t))

are all positive for / e [sl9 s2).4. The matrix A0(s2, v2, y2) has 0 as a simple characteristic

root, and all the other characteristic roots of A0(s2, vl9 y2)have positive real parts.

By assumption 4, we can assume that the first row and the firstcolumn of A0(s2, v2, y2) are zero. If ml — n 4-1, then U = (w, w)and V=(v9r) where u and v are scalars and w and r are nvectors, and we can write systems (5.52) and (5.53) in the followingform:

d2u du dwe-—r- + g ( r , ii, w9x9e)— + p(t9u9w9x9e) —

dt at at+ h(t9u,w,x,e) = 0,

d2u du dwe—5r + <x(t9u9w9 x9 E)—-~ + A(t9u9w, x, e)-— /* rA\

dt dt dt p.j4j

+ a{t, u, w, x, e) = 0,

dx du dw— =f(t9u9w9x9e)— + B(t9u,w9x9e) —dt dt dt

+ Z)(/, w, w, x, e)


and

dv dr) ( ^ ^ ^

dv drao(t, v, r,y)—+ A0(t, u, r,y)— + ao(t, v, r, y) = 0,

at at

(5.55)

dv dr= / 0 ( / v ry)

w h e r e

where g, A are real scalars, a and a are column n vectors, /? is arow « vector, 4 is an « X « matrix, / and Z> are column m vectors,and B is an m X « matrix. Similar formulas hold for Ao, a0, and soon.

Let

We assume that at P2, we have

We assume also that if t<s2 and if t is sufficiently close to s29

then for (/, v, r, ^) = (/, u(0, r(/), ^(0), where (v(t\ r(t)) = w(/),it is true that

Let y = <f>(t>) be the solution of the system

— =fo(s2>v>r2>y)

such that


Let v3 be the smallest value of v > v2 such that

v2

and assume that the curve

# = {(t,v,r9y)\t = s2, v2<v<v3, r = r2

is contained in R. Also we assume that on curve

and that the real parts of all the characteristic roots of the matrixAo — go/w, where In is the n X n identity matrix, are positive.

Now let y3 = <i>(v3) and let P3 = (s2, v3, r2, ^3) . Assume that allthe characteristic roots of A0(t, v, r, y) have positive real parts atP3. Let (v(t), r(t\ y(t)) be the solution of (5.55) such that

Assume that there exists s3 > s2 such that:

1. (v(t), P(t\ y(t)) is defined for s2 < t < s3 and is containedin R.

2. If Xj(t), j = 1,2, . . . ,«,« + 1, are the characteristic rootsof the matrix

A0(t,v(t),r(t),y(t)),

then

R e X y ( / ) > 0 , 7 = 1 ,2 , . . . , H, « + 1 and s2< t<s3.

Then a discontinuous solution So of the degenerate system is the

union of the curve

{(t,v,r,y)\Sl<t<s2, v = v(t), r = r(t), y=y(t)}

the curve # , and the curve

{ ( t 9 v 9 r 9 y ) \ s 2 £ t £ s 3 , v = 6 ( t ) 9 r = r ( t ) , y = y ( t ) } .

In stating Sibuya's result, we use the following definitions of thenorm of a vector and the norm of a matrix. If c = (c 1 ? . . . , cn), then


the norm of c is defined to be

|c|=max|cy.|,

and if A is an n X n matrix, then the norm of A is defined to be

Theorem 18. / / 8V 82, 83, and e are sufficiently small positivenumbers, there is a solution (u(t), w(t), x(t)) of (5.54) defined onthe interval sx< t < s3 for any initial values satisfying the conditions

\u(Sl) - v(Sl) | + | w(Sl) - r(Sl) | < 8l9

du

~dt^Sl

dv

'~~dt^Sl*+

dw

~dJ^hdr

'"dt

and as 8V 82, S3, and e approach zero, the curve Sf representing thesolution (u(t), w(t), x(t)) in (t, u,w, x) space approaches the curveSo, that is,

lim max [</(/?, ^ ) ] = 0 .O ^S

Also for any fixed 8 > 0, the terms

and

du dv

Tt^'Yr'converge uniformly to zerc

sx<r<s2- 8

and

s1 + 8<t<s2-t

+dw dr

~dl^'~dt

> on the intervals

5,

respectively. Moreover, the terms

u(t)-v(t)\ + \

x(t)-y(t)\,

w(t)-r(t)\,


and

du dv dw dr

converge uniformly to zero on the interval

5.5 Partial differential equationsAll the mathematics described so far in this chapter is

concerned with systems of ordinary differential equations and isintended to be applied to some one of the voltage-clamp modelsdescribed in Chapters 2-4. We have already seen in Chapter 2 thatif we are to obtain a mathematical description of an action poten-tial, then we must use a mathematical model that combines thevoltage-clamp model with cable theory. Since the resulting model isa system of partial differential equations, which we termed the fullHodgkin-Huxley equations, it would seem reasonable to proceednow to a description of the mathematics of partial differentialequations, that is, theory that would be applicable to the fullHodgkin-Huxley equations and analogous models for other physi-ological models.

Actually, we will not present such theory because it is in theprocess of being developed. We have already emphasized, in ourdiscussion of ordinary differential equations, that much of theknown theory of differential equations has no direct or usefulapplications in the study of the Hodgkin-Huxley equations andthat further mathematical theory needs to be developed to dealwith the problems that arise naturally in the study of physiologicalmodels, that is, voltage-clamp models. The situation with the fullHodgkin-Huxley equations is similar except that there is not verymuch theory, applicable or otherwise. The full Hodgkin-Huxleyequations are an example of a system of reaction-diffusion equa-tions. Although reaction-diffusion equations are now the subject ofextensive study, they received comparatively little attention untilabout 20 years ago. The extensive work that is now being done onreaction-diffusion equations [see Fife (1978) for a summary ofthese studies] suggests that, in time, a large body of theory will bedeveloped that will be applicable to such systems as the full

5.5 Partial differential equations 179

Hodgkin-Huxley equations. However, at present, many of themathematical problems concerned with the full Hodgkin-Huxleyequations remain partially or totally unsolved.

There is a second reason for not giving an account of themathematical theory that might be applicable to the fullHodgkin-Huxley equations. That reason lies in the fact that, aswork on the Hodgkin-Huxley equation progresses, the results mayindicate the best direction for work on the full Hodgkin-Huxleyequations. Consequently, to some extent, work on the fullHodgkin-Huxley equations should await results concerning theHodgkin-Huxley equations.

For these reasons, we will restrict ourselves merely to describingbriefly (in Chapter 6) some of the work that has been done onversions of the full Hodgkin-Huxley equations.

Mathematical analysis ofphysiological models

6.1 IntroductionWe have already described the numerical analysis of the

Hodgkin-Huxley equations (in Chapter 3) and other models (inChapter 4). In view of the high success of much of this numericalanalysis, it is natural to ask why any further mathematical analysisis required. That is, when a model of an electrically excitable cellhas been obtained, why not answer all mathematical questions thatarise by simply carrying out a numerical analysis? The answer tothis question lies partly in the nature of the models that we haveconsidered and partly in the kind of results that can be obtainedfrom numerical analysis. As we have seen, the models consideredare empirical descriptions and the constants that appear in themare, at best, reasonably good approximations. In fact, for somepurposes, it is well to regard these constants as parameters thathave various values. Moreover, the very functions that appear inthe models are, in some cases, quite tentatively proposed [seeChapter 2 and Hodgkin and Huxley (1952d, p. 510).] In carryingout a numerical analysis, an entirely specific model must be consid-ered. That is, the forms of the functions and the values of theparameters must be completely specified. Moreover, the numericalanalysis gives no information about whether solutions of similarmodels have the same or similar behavior. As a simple example,suppose that the numerical analysis of one model suggests thatthere is a unique asymptotically stable periodic solution. (Of course,numerical analysis alone cannot be used to prove the existence of aperiodic solution.) Now suppose that one of the constants or one ofthe functions in the model is changed slightly. Does the resultingmodel also have a unique asymptotically stable periodic solution?

6.1 Introduction 181

It seems reasonable to guess that there is such a periodic solution,but the question cannot actually be answered by the originalnumerical analysis. Strictly speaking, the only way to answer thequestion is to carry out a numerical analysis of the new model. Thiskind of argument suggests the importance of qualitative analysis(which does not depend on exact numerical values) of the solutionsof the models studied. However, it should be emphasized that thesearguments have been adduced by physiologists [see Jack, Noble,and Tsien (1975, p. 379)]. They are not just arguments put forth bymathematicians seeking employment!

A second drawback to the numerical analysis is that it does notgive us any understanding of the in-the-large behavior of thesolutions. For example, the numerical analysis of the Hodgkin-Huxley equations shows that approximate solutions display behav-ior that describe threshold phenomena, but one obtains no under-standing of how or why this behavior occurs. However, as we shallsee in this chapter, viewing the FitzHugh-Nagumo equations as asingularly perturbed system makes for a much clearer understand-ing of the how and why of this behavior.

Two very different kinds of mathematical analysis are needed todeal with models of electrically active cells. First, a qualitativeanalysis is required for the solutions of the models derived fromvoltage-clamp experiments: the Hodgkin-Huxley equations and themodels of other electrically active cells that are obtained fromvoltage-clamp experiments. For brevity, we shall term them volt-age-clamp models. As pointed out earlier, each of these modelsdescribes the relationships among the membrane potential andionic currents, all regarded as functions of time only. Each of thesemodels is a system of nonlinear ordinary differential equations, andthe solution of such a system can be interpreted in two ways. First,the solution can be regarded as describing the membrane potentialand the ionic currents (as functions of time) at a fixed position onthe axon when the axon is functioning naturally. Second, thesolution can be regarded as describing the membrane potential andionic currents that occur in experiments in which the membranepotential and ionic currents have at a given time the same value atall points along the axon. Such experiments include not only the

Mathematical analysis of physiological models 182

voltage-clamp experiments, but other experiments described inChapter 2 in which the membrane potential is allowed to vary [seeHodgkin, Huxley, and Katz (1952)]. If the membrane potential andthe ionic currents have the same value at all points along the axon,they are said to be uniform. The term "uniform action potential"refers to a solution of the system for which the membrane potentialand the ionic currents are uniform and the initial value of themembrane potential exceeds the threshold value. Then a large, butbrief, flow of sodium current occurs and is followed by an outwardflow of potassium current.

Since all the voltage-clamp models are systems of nonlinearordinary differential equations, then the qualitative theory of suchdifferential equations that was described in Chapter 5 can bebrought to bear on the analysis of the models. As it turns out, themost effective viewpoint is to regard the system of ordinary dif-ferential equations as a singularly perturbed system and to studythe solutions that are close to the discontinuous solutions (thetheories of Levinson and of Mishchenko and Rozov that weredescribed in Chapter 5). Since the decision to adopt this viewpointis crucially important to the later analysis, we summarize here themathematical and physiological reasons for this decision. First,numerical computations of values of the functions that appear inthe equations suggest strongly that the equations should be re-garded as singularly perturbed. We will see this later in the discus-sion of the Hodgkin-Huxley equations and the Noble equations.Next, experimental results show that the membrane potential Vand the Na+ current undergo rapid smooth changes. Moreover,numerical study of the solutions of the Hodgkin-Huxley equationsand the Noble equations show that the solution component V(t)and the solution component m(t), which is a measure of the Na+

current, display very rapid smooth changes. These smooth rapidchanges are characteristic of the solutions of a singularly perturbedsystem that are close to discontinuous solutions. Third, numer-ical studies of the Hodgkin-Huxley equations made by Cole,Antosiewicz, and Rabinowitz (1955) show that if / increases abovea particular value, a periodic solution appears, that is, bifurcationoccurs. However, as will be discussed in Section 6.2.1.3 of this

6.1 Introduction 183

chapter, the bifurcation that occurs does not look like a Hopfbifurcation, but rather like the kind of bifurcation that occurs insingularly perturbed systems. In such bifurcations, the periodicsolution that appears is near a discontinuous solution. Finally, itshould be pointed out that discontinuous solutions give quickglobal pictures of the behavior of the general solutions. This is justthe kind of information that the physiologist needs to complementnumerical studies.

In the analysis of Hodgkin-Huxley equations in Chapter 2, itwas shown that in order to obtain a quantitative description of thenatural behavior of the axon, it is necessary to combine theHodgkin-Huxley equations (i.e., the voltage-clamp equations) withcable theory to derive a system of nonlinear partial differentialequations. We referred to this system as the full Hodgkin-Huxleyequations. A solution of this system describes the membrane poten-tial and the ionic currents as functions of both time and positionalong the axon. The search for a solution that described an actionpotential was shown to be a search for a traveling wave solutionand this, in turn, becomes the study of a system of nonlinearordinary differential equations. But the problems that must bedealt with concerning these nonlinear differential equations arevery different from those that arise in the study of the voltage-clampequations. In the last part of this chapter, we will describe brieflythe kinds of problems that arise and some of the work that hasbeen done.

It should be pointed out at this stage that, while numericalanalysis has shown that the full Hodgkin-Huxley equations are avalid model of the squid axon, it does not follow that combiningvoltage-clamp equations with cable theory always yields a validmodel. In fact, such a technique does not seem to work for modelsof the Purkinje fiber [see McAllister et al. (1975, pp. 4 and 53)].

Finally, we must point out that this chapter is inevitably frag-mentary. The reason for this is the present state of our knowledgeof the subject. As we shall see, this knowledge is the sum of effortsin various directions, and many of these efforts have not beenbrought to full fruition. Indeed, in some cases, it is not clearwhether these are efforts in the right direction. Thus, this chapter is


doomed to a certain incoherence. The one virtue of our discussionis that it shows clearly that there are many open problems in thissubject.

6.2 Models derived from voltage-clamp experiments6.2.1 Nerve conduction models6.2.1.1 Some numerical analysis

Although we are primarily concerned with qualitative anal-ysis of the voltage-clamp models, we start by summarizing brieflysome numerical analysis of the Hodgkin-Huxley equations. Besidesbeing valuable per se, these numerical results are important be-cause they are suggestive of directions for qualitative study. Wehave already described in Chapter 2 the results of the numericalanalysis carried out by Hodgkin and Huxley in their original paper.Here we describe briefly some later numerical analysis that issuggestive of qualitative study.

The first numerical analysis of the Hodgkin-Huxley equationswith a modern computer was carried out by Cole et al. (1955); seealso FitzHugh and Antosiewicz (1959). They studied (i) constantcurrents that are " threshold," that is, produce an action potential,(ii) the minimal rate of increase of current required to produce anaction potential, and (iii) repetitive firing, that is, spontaneousperiodic uniform action potentials in response to a constant cur-rent. For a sufficiently large constant current /, their numericalresults suggested the existence of a periodic solution, that is, aninfinite train of uniform action potentials. (If there is a periodicsolution, the amplitude of the current stimulus does not affect thefrequency (or period) of the periodic solution. That is, there is nofrequency modulation. But many axons, especially in the sensorysystems, require this frequency modulation [see Cohen (1976, pp.101 and 121)]. This shows that the Hodgkin-Huxley model is notsuitable for sensory pulse conduction. Dodge (1972) has discussedthis point.) Studies of repetitive firing have also been carried out byCooley, Dodge, and Cohen (1965) and others [for references, seeRinzel (1979)].

If the bath in which the axon is immersed has a calciumconcentration lower than the normal value, then the axon some-times displays spontaneous periodic action potentials. This experi-

6.2 Models derived from voltage-clamp experiments 185

mental phenomenon has been studied mathematically by Huxley(1959), who proposed a modification of the Hodgkin-Huxley equa-tions to describe the axon in a reduced calcium concentration.Huxley also carried out a numerical analysis of these modifiedequations and obtained periodic solutions. [McDonough (1979),has carried out an analytic study of these modified equations.]More recently, Guttman, Lewis, and Rinzel (1980) have shownexperimentally that such repetitive firing can be annihilated by abrief pulse of appropriate magnitude and phase. Under anothersuch pulse, repetitive firing resumes. It is worth observing thatthese researchers state that their experimental work was motivatedby theoretical studies. (References for the theoretical studies aregiven in the paper; see especially page 389.) This is an example ofhow theoretical studies do have a contribution to make in physio-logical research.

Now we describe briefly some numerical results that should besupported by rigorous mathematical reasoning. First, Berkinblitet al. (1970) have carried out computer studies of the Hodgkin-Huxley (HH) equations that model certain phenomena observed inphysiological experiments, especially, summation of subthresholdresponses and periodic omission of pulses. The first of theseconsists in studying the effect of adding a periodic sequence ofsquare current pulses that have a subthreshold effect (the chargethus added raises the membrane potential, but not up to thethreshold). The second consists in studying the effect of adding aperiodic sequence of square current pulses that have a super-threshold effect. The numerical results obtained agree with thephysiological observations and thus provide additional evidence ofthe validity of the HH equations. But if the numerical results couldbe supported by rigorous mathematical reasoning, this evidencewould be stronger and also a clearer understanding might beobtained of the processes involved.

Another interesting and important numerical study that whichshould be analyzed theoretically is the numerical study of themodel due to Adelman and FitzHugh (1975). As we have pointedout earlier (in Chapter 2), one of the drawbacks of the HHequations is that they exhibit periodic solutions that do not corre-spond to physiological behavior. The existence of such periodic


solutions is suggested by numerical analysis [Cole et al. (1955)] andis proved for the FitzHugh-Nagumo equations (see Section 6.2.1.2of this chapter). In Chapter 2, this point was discussed and we sawthat this drawback of the HH equations is related to the fact thatthe HH equations do not describe the phenomenon of accommod-ation or adaptation. Mathematically, this means that there is nodependent variable that is sufficiently slow changing. In FitzHugh'sterminology, there is no variable of Type 4 [FitzHugh (1969, pp. 4and 32)]. There seems little doubt that a more accurate or morerealistic version of the HH equations would require somehowtaking adaptation into account.

Adelman and FitzHugh (1975) introduced a modification of theHH equations that is a significant step in this direction. Theirmodification is based on results of physiological experiments. First,it has been observed experimentally that K5, the concentration ofpotassium ions in the periaxonal space between the outside of themembrane and the inside of the Schwann cell sheath, playsa significant role in the electrical activity of the axon [seeFrankenhaeuser and Hodgkin (1956)]. The concentration Ks

changes continually due to the potassium ion flow through themembrane and diffusion across the Schwann cell layer [seeAdelman, Palti, and Senft (1973)]. Hence, the potassium potentialFK, which depends on Ks (remember the Nernst formula), is afunction also of time. Also, Adelman et al. (1973) showed how gK

and an and /?„ are affected by the varying of VK with Ks.Adelman and Palti (1969) showed that dh/dt, the rate of change ofsodium inactivation, depends on K5. These dependencies weretaken into account by Adelman and FitzHugh in a modified formof the HH equations. Actually, they set up three modifications ofthe HH equations: the first modification introduced K5; the secondintroduced K5, new values of gK, and new functions for an and fin\the third introduced K5, new values of gK, and new functions foran9 /?„, ah, and fih.

The third modification, called by Adelman and FitzHugh thefully modified HH equations, which includes all the changes,can be described as follows. Let K{ denote the potassium ionconcentration in the interior of the axon, Ko the potassium con-centration in the external bulk solution, and let P£ denote the


permeability to potassium of the diffusion barrier between theperiaxonal space and the external bulk solution in centimeters persecond. The modified equations are

dV - 1— = —{/-gN a

dm— = (\-m)am{V)-mfim{V),

— = (l-h)ak(V)-hfih(V),

dn— = (l-n)an(y)-nfin{y),

dKs

dt

1

e1 r-vK)

where 8 is the radial thickness of the phenomenological periaxonalspace, gK = 62.5 Q'Vcm2, and

RT (KJ1/ I i

The numerical analysis of this system of equations gives a moreaccurate representation of a number of physiological phenomenaincluding a better description of adaptation during long durationconstant current stimulation. That is, finite trains and nonrepetitiveresponses are obtained.

It is clear that this modified HH system is more complicatedthan the original HH equations and since qualitative analysis of theHH equations is very limited, there is little hope of making arigorous qualitative analysis of the modified HH equations. Thereremains, however, the possibility of making a qualitative compari-son of the two systems with a view to understanding why differenttypes of solutions occur.


6.2.1.2 Analysis of a two-dimensional modelThe use of qualitative analysis. In Chapter 3, we introduced

two-dimensional models that are simplifications of the HH equa-tions. As explained there, the purpose of introducing the simplifiedmodel is to make possible a qualitative analysis of the solutionsand thus obtain a clearer understanding of the processes that aretaking place. Since we are so much concerned here with qualitativeanalysis, it is worthwhile first to illustrate the advantages of quali-tative analysis with a simple familiar example: resonance in aspring-mass system. We consider a spring-mass system withoutdamping. If m is the mass and k is the spring constant, the actionof the system is described by the differential equation

mu + ku = 0.

If w = yjk/m, then elementary techniques show that the generalsolution of this equation is

u(t) = cxcoswt + c2sinwt,

where cv c2 are constants. Now suppose a periodic external forcewith period equal to the period of the preceding general solution,say A cos wt, where A is a nonzero constant, is applied to thespring-mass system. The system is now described by the equation

mu + ku = A cos wt (6.1)

and the general solution is shown by elementary methods to be

Au(t) = cxcoswt + c2sin wt + 1 sin wt. (6.2)

2* m.wRegardless of the initial conditions, the term {A/2mw)tsm wtappears in the solution. The graph of (A/2mw)tsinwt, shown inFig. 6.1 shows how the solution becomes unbounded as t -> oo. It isimportant to observe here that we have not only determined thatthe solution becomes unbounded but also that we have a clearpicture of how the solution oscillates more and more wildly. Sincewe have an explicit formula for the solution, then for given initialvalues (which will determine the constants cx and c2), we can findthe value of the solution at any fixed time /. (That is, we havecomplete quantitative information.) But the qualitative information(how the solution oscillates more and more wildly) is more im-


Figure 6.1.

portant in understanding the action of the spring-mass system.Now suppose that instead of solving (6.1) explicitly, we had carriedout a numerical analysis. A sufficiently detailed analysis wouldprobably suggest (although, of course, not prove) that the solutionoscillates with increasing wildness. But the numerical analysismight not give us as explicit qualitative information. Moreover, wewould have little or no understanding of why it occurred.

Since there is no hope of obtaining, for the differential equationsthat we study, explicit or closed solutions [i.e., solutions likesolution (6.2) of equation (6.1)] we try to search directly forqualitative information. The primary purpose of this chapter is toshow how the techniques described in Chapter 5 can be used forsuch a search.


The FitzHugh-Nagumo equations: Numerical results. Our firststep will be to look at a numerical analysis of the two-dimensionalFitzHugh-Nagumo equations. Since the system is two-dimensionaland autonomous, its solutions can be represented by curves in aplane (i.e., the phase plane). By studying the graphs of these curves,we will obtain a clear picture of how the solutions behave generallyand how this behavior corresponds quite closely to the experimen-tally observed physiological phenomena. Thus, we will see how thequalitative behavior of the solutions accounts for the physiologicalphenomena.

As introduced in Chapter 3, the FitzHugh-Nagumo equationsare

dV V3

= <f>(V+a-bW),dt

where V denotes the membrane potential, W is a recovery variable,a, b,<j> are constants, and I is the total membrane current. Thecurrent / can be an arbitrary function of time, but we consider firstthe case that describes the normal condition and functioning of thenerve, that is, 7 = 0.

Following FitzHugh (1969) we take a = 0.7, b = 0.8, and <f> = 0.08in (6.3). Then by numerical methods, approximate solutions of(6.3) can be computed. The phase plane portrait of the computedsolutions is shown in Fig. 6.2 [which is Fig. 3.2 in FitzHugh (1969)].The dashed curves are the isoclines, that is, the curves

dV V3

anddV— = V+a-bW=0.dt

The critical points or singular points or equilibrium points are thepoints of intersection of the isoclines, that is, the points (V,W)such that

V3

V W=0


I -

and

V+a-bW=O.

In this case, there is just one critical point R, which has coordi-nates

{VR,WR) = (-1.1994, -0.6243).

The solution curves, with arrowheads to indicate the direction ofincreasing /, are given by solid curves. The curve with doublearrowheads, called the threshold separatrix, separates solutioncurves that represent action potentials from those that do notrepresent action potentials. Since the results shown in Fig. 6.2summarize numerical analysis, they are not all sharply defined. Inparticular, the threshold separatrix is not well defined. As shown inFig. 6.2, all the solution curves approach critical point R as timeincreases. Now suppose that we consider a solution that has theinitial value (-1.1994+ Vo, -0.6243), that is, a solution that de-scribes what takes place if the initial value of V is VR + Vo and theinitial value of W is WR. If Vo is positive but small enough so thatthe point (VR+ VQ,WR) is on the left of the threshold separatrix,then as Fig. 6.2 shows, the solution curve moves almost directlytoward the critical point R. This is a solution that does not


describe an action potential, that is, it describes a passive localreaction. On the other hand if the initial point (VR + Vo, WR) is onthe right of the threshold separatrix, then, as the graph of thesolution curve shows, V(t) increases rapidly as time passes; thenW(t) increases, V(t) decreases, and the solution curve movestoward the critical point R. This solution describes an actionpotential.

The behavior of these solutions shows how the phase plane canbe made into a kind of map of different physiological states. Forexample, the region labeled AR (for absolutely refractory) consistsof points (V,W) such that if a shock is applied so that V isincreased to a larger value F, then no action potential will occur nomatter how large V is. That is, each point in the region correspondsto a state of the axon in which the axon is absolutely refractory.Similarly, the points in the region labeled RR (for relativelyrefractory) correspond to states of the axon in which it is relativelyrefractory. Also the points in the region labeled E (for enhanced)correspond to states of the axon in which it is enhanced, that is,will respond with an action potential even though the shockapplied is not large enough to raise the membrane potential fromVR to the usual threshold level. The regions labeled RR, AR, and Ein Fig. 6.2 are not indicated very clearly; their boundaries aresimply suggested. The reason for this lack of definiteness is that weare summarizing results of numerical analysis and, hence, theresults are not sharply delineated; rather the results suggest theexistence of such regions.

The relationships among these solutions of (6.3) that we haveseen give us an "in-the-large" understanding of the various actionsof the membrane potential that could not be achieved fromnumerical analysis of the HH equations. Of course, we have paid aprice for this qualitative understanding: Much of the realism of thedescription given by the HH equations has been sacrificed. Theactivation and inactivation variables that describe quantitativelythe currents of sodium and potassium ions in the HH equationshave been replaced by a recovery variable W that has no specificphysiological meaning. Nevertheless, the derivation and analysis ofthe FitzHugh-Nagumo equations represents a large forward step inthe quantitative analysis of nerve conduction.


W

I

Figure 6.3.

Numerical analysis of the FitzHugh-Nagumo model can also beused [FitzHugh (1961)] to explain a number of other properties ofthe nerve membrane: for example, abolition of an action potentialby anodal shock, and anodal break excitation.

Suppose that an action potential occurs and is described by thesolid line in Fig. 6.3. If an anodal shock is applied at the instantthat the action potential is described by the point P, then the phasepoint is displaced horizontally to the left of P to a new point P\and if this displacement is large enough so that P' is on the left ofthe threshold separatrix (as sketched), then the phase point con-tinues on the solid line back to the equilibrium point R. Thus, theaction potential is abolished by the anodal shock. This phenome-non has been observed experimentally: See Blair and Erlanger(1935) and Tasaki (1955).

Next suppose that a constant membrane current /, where / < 0,is applied for an interval of time of length, say, T. Then the Visocline becomes

V3

V W+I = 03

and its graph is obtained simply by lowering the V isocline in Fig.6.2, by |/ | units. Then the new equilibrium point R' is below R (see


Figure 6.4.

Fig. 6.4). If the phase point starts at R when / is applied, it willmove toward R' along the dotted line in Fig. 6.4. Now if |/ | and Tare large enough, then at the end of the time interval of length 7,the phase point will be below the threshold separatrix and so whenthe membrane current / is made zero again, an action potentialwill occur. This is the phenomenon of anodal break excitation thatwas discussed earlier (Chapter 2). In Chapter 2, we stated thatappropriate numerical solutions of the Hodgkin-Huxley equationsdescribe fairly accurately anodal break excitation. Also we pre-sented an explanation in words of why anodal break excitationoccurs. It is interesting to compare the explanations in Chapter 2with the preceding explanation, which is quantitative, not just anexplanation in words; also it relates the phenomenon of anodalbreak excitation to other phenomena.

FitzHugh also made a motion picture entitled "Impulse propa-gation in a nerve fiber," which is based on numerical study ofequation (6.3). This motion picture, which is very helpful in under-standing how an action potential occurs, is available on loan fromthe National Medical Audiovisual Center (Annex), Station K,Atlanta, Georgia 30333.


The FitzHugh-Nagumo equations'. Singular perturbation analysis.FitzHugh's work is based entirely on numerical analysis, and ournext step is to back up that numerical analysis with rigorousmathematical theory. We will use singular perturbation theory tosupport the numerical analysis and also to give a clearer under-standing of the behavior of the solutions. Further examination ofFig. 6.2 indicates a very important property of the FitzHugh-Nagumo equations: The solutions display behavior that is typicalof solutions of a singularly perturbed system. When consideredfrom this viewpoint, the behavior of the solutions of theFitzHugh-Nagumo equations is easier to understand. In particular,the origin of the threshold separatrix becomes clear. In order to seethis, we show first how to change variables so that theFitzHugh-Nagumo equations become a singularly perturbed sys-tem.

Let T = t<j>. Then (6.3) becomes

dV dV dr dV V3

dt dr dt dtY 3

or

dW~dt~~

dV

dW

~d7 =

dW~~cfr

1 /

- h

F +

dr

It

a —

dW

~~cfr

F3

T~

bW.

Since $ = 0.08, the (6.3') has the conventional form of a singularlyperturbed system of ordinary differential equations, that is, asystem of the form

dx 1

where e is small. Hence, we may apply the theorems described inSection 5.2.4 of Chapter 5.


For simplicity of notation we write t in place of r in (6.3'). Thensince a = 0.7 and b = 0.8, (6.3') becomes

T/3

(6.3")dW

= J/+0.7-0W,dt

where we have replaced <J> = 0.08 by the small positive parameter e.The graph associated with (6.3") is indicated in Fig. 6.5. Exceptvery close to the curve

V3

v- — -w=o,the directions of the solution curves are nearly horizontal and theygo to the left or the right (according as V— V3/3— W$0) asindicated by arrows in Fig. 6.5. Also, on the curve

V3

V W=0,

V + 0.7-0.8W=0

E=H.2,-0.6)

Figure 6.5.


dW/dt is positive on the right of the equilibrium point E =( — 1.2, —0.6) and negative on the left of E, as indicated in Fig. 6.5.

Notice that if we take a sufficiently large contour ^ (sketched inFig. 6.5), every solution that intersects # crosses it and goes intothe interior of #. Thus all solutions ultimately enter a bounded set,the interior of #.

Now we show that in a certain approximate sense the solutionsof (6.3") approach the equilibrium point E. First, the point E is anasymptotically stable equilibrium point of (6.3"). This is becausethe linear part of the right-hand side in a neighborhood of E is

e1 - 0 . 8 J

- [ -0 .44] - -

1 -0.8

Since the trace of this matrix is negative and the determinant ispositive, it follows that the eigenvalues of the matrix have negativereal parts. Hence, by a standard theorem, equilibrium point E isasymptotically stable. That is, if e is fixed, then ex > 0 implies thereexists 8 > 0 such that if (V(t), W(t)) is a solution of (6.3") and if(F(/o), W(t0)) e N8(E), then for all t > t0, (V(t), W(t)) is definedand

(V(t),W(t))<=Nei(E).

Actually we can say more. The 8 clearly depends on el9 but 8 isindependent of e. To see this, we simply take e = 1, conclude thatE is asymptotically stable, and then obtain the correspondingLyapunov function V. But it is easy to see, because of the simplerole that e plays in (6.3"), that Lyapunov function if "works" forall values of e. The neighborhood N8(E) can be defined in terms ofan inequality of the form

where e2 > 0-

Next, we observe that there exists a positive number T such thatif S is a discontinuous solution defined on the interval [a, a + T]with the initial point of S contained in the set Int(^), then thepoint of S corresponding to a + T is contained in N8/2(E). Thatis, all discontinuous solutions that have initial point p e Int(^)arrive in N8/2(E) after the time interval T.


Finally by Theorem 15R in Chapter 5, if q e Int(#), then if e issufficiently small, there is a solution (V(t), W(t)) of (6.3") thatis defined for [a, a + T], which is such that (V(a), W{a)) = q andis such that

Hence, the solution is defined for all t > a and for / > a + T, thesolution remains in the neighborhood N (E).

This shows that the equilibrium point has a kind of globalasymptotic stability property. Of course, this conclusion is validonly if e is sufficiently small, but since the FitzHugh-Nagumoequations are intended to give only a qualitative description, wemay choose a value of e for which the conclusion is valid; that is,there is no physiological evidence that limits us to the value

(On the other hand if we try to use Levinson's theorems or theMishchenko-Rozov theorems in analyzing models that are derivedfrom quantitative physiological data, that is, models like the HHequations, then we do not have such freedom in the choice of e.)

Now suppose we describe the situation in which a constantcurrent / , where / > 0, is impressed on the membrane. Then thesystem (6.3) is

dV V3

— = V W+I,dt 3

dW= <j>(V+a-bW)

dtand the system (6.3") becomes

dV II V3

— = -\V- — -dt e \ 3

dW= F+0.7-0.8JF,

dtwhere a, b are taken as 0.7 and 0.8, respectively. The graphassociated with this system is indicated in Fig. 6.6. Roughly speak-ing, it is obtained by "lifting" the cubic curve a vertical distance


W

Figure 6.6.

equal to / . If the cubic curve is lifted enough so that the line

intersects the cubic curve between its minimum and maximumpoints (as indicated in Fig. 6.6), then there is a discontinuoussolution of the degenerate system that is a closed curve (sketched asdashed curve).

Now we rewrite (6.3") in the notation of the Mishchenko-Rozovtheorems described in Chapter 5, and it becomes

dx

dy— =x 4-0.7 -0.8y.dt

That is,

x3

f(x,y)=x- — -y + I,

g(x, y) = x + 0.1 -0.8y.

Hypothesis 1 of the Mishchenko-Rozov theorem (Theorem 14 of


Chapter 5) is obviously satisfied. Since

hypothesis 2 is satisfied.The nonregular points of

T={(x,y)\f(x,y) = 0}

are the points that satisfy

fx(x,y) = l-x2 = 0,

that is, the points (1, § + / ) and ( - 1 , f + J). Thus the nonregularpoints are isolated. Since

then each of the nonregular points is nondegenerate. Thus, hy-pothesis 3 is satisfied.

The stable parts of T are the points (x, y) with x < -1 and thepoints with x > 1. As Fig. 6.6 indicates, the line

g(x,y) = odoes not intersect the stable part of T or the nonregular points.Thus, hypothesis 4 is satisfied.

The two nonregular points have different ordinates. Hence,hypothesis 5 is satisfied. As already indicated, there is a closeddiscontinuous solution, that is, hypothesis 6 is satisfied. Therefore,we may apply the Mishchenko-Rozov theorem (Theorem 14,Chapter 5) and conclude that there is a unique stable periodicsolution of the singularly perturbed system for each sufficientlysmall positive 6.

Finally, let us suppose that / is large enough so that the cubiccurve is lifted so much that the line

intersects the cubic curve on the right of the minimum and themaximum points (as indicated in Fig. 6.7). Then as the discontinu-ous solutions (sketched as dashed curves) indicate, the equilibriumpoint E is globally asymptotically stable.

As a second example of singular perturbation analysis, we con-sider the mathematical description of the results of experiments inwhich the nerve fiber is immersed in a solution bath of reducedcalcium concentration. It has long been known that if the calcium


W

Figure 6.7.

concentration is reduced, the axon may display spontaneous oscil-lations or repetitive activity. A quantitative study of this activitywas made by Frankenhaeuser and Hodgkin (1957) and a mathe-matical analysis of their results was carried out by Huxley (1959).

Frankenhaeuser and Hodgkin found that the main results couldbe "summarized by saying that the effects of a fivefold reductionof calcium on the system controlling Na and K permeability aresimilar to those of a depolarization of 10-15 mV." Huxley pro-posed describing the corresponding change in the mathematicalmodel, that is, the Hodgkin-Huxley equations, by replacing V inthe equations for dm/dt, dh/dt, and dn/dt by F + AF, where

[Ca]/where k = 9.32 and [Ca] is the calcium concentration and [Ca]n isthe normal calcium concentration. (Thus if the calcium concentra-tion is lower than [Ca]w, then AF is negative.) Hence, the effect inthe FitzHugh-Nagumo equations [written in the form (6.3")] wouldbe to change them to

dV \ \ F3

— = -\v- — -wdt e [ 3

dW= (j/_#) + 0.7-

dt

(6.4)


Figure 6.8.

where R = — AK (Remember that the variable W corresponds tothe variables m, h,n governing the flow of Na and K ions.)

The graph associated with system (6.4) differs from the graphassociated with the original system (6.3"), that is, Fig. 6.3, in thatthe diagonal line described by

F + 0 . 7 - 0 . 8 W = 0is replaced by the line described by

V-R + 0J-0.W=0.That is, the diagonal line is moved to the right by distance R. Thusif,

the graph is as in Fig. 6.8. Thus, a relaxation oscillation of the formsketched in Fig. 6.8 occurs. Finally, if R > 2, the graph is as in Fig.6.9. Thus, as indicated in that graph, the equilibrium point p isglobally asymptotically stable.

On the other hand, if R is negative, that is, if the calciumconcentration [Ca] is increased, then the diagonal line is moved tothe left and the graph has the form shown in Fig. 6.10. Thus, theequilibrium point is globally asymptotically stable.

A stochastic model of nerve conduction: The model of Lecar andNossal. Next we describe another two-dimensional model that isalso a modification of the HH equations, but that is a study in avery different direction-a direction that should be pursued further.In order to motivate the introduction of this model, we point out


W

if>°-l.7\. \ /

Figure 6.9.

two drawbacks of the HH equations. First, there is a significantdifference between what is observed experimentally when a stimu-lus just equal to the threshold value is applied to an axon and theprediction made by solutions of the HH equations of what willoccur. The functions that appear in the HH equations are "well-behaved" functions and hence the basic existence and uniquenesstheorems for solutions of systems of differential equations areapplicable. Because of the uniqueness of the solution, the HHequations always make the same prediction concerning the re-

Figure 6.10.


sponse of the axon to a stimulus equal to the threshold value.However, the experimental result is that an action potential occursfor only a certain fraction of stimuli of threshold value [see TenHoopen and Verveen (1963)]. Second, the all-or-none law, which isan empirical statement or summary of experimental results, statesthat the axon either does not fire or responds with a full-sizedaction potential, but computation of approximate solutions of theHH equations shows that stimuli of various magnitudes produceintermediate responses. Such intermediate responses are not ob-served experimentally. These conflicts between experimental ob-servation and theoretical prediction can be explained if we takeinto account an unjustified assumption that is used in the deriva-tion of the HH equations: the assumption that the resting potentialis a constant. Actually, experimental results show that the restingpotential is subject to small random fluctuations or noise [seeVerveen and Derksen (1968)]. The processes by which the restingpotential is maintained are not very well understood, but theoriesof how these processes occur suggest that the processes are of adiscrete nature and, thus, it is not surprising that such noise shouldexist.

These remarks suggest that the HH equations might yield morerealistic or accurate predictions if a random or stochastic term wereincluded in the equations. The model that we will describe is a firststep in this direction. It was proposed and studied by Lecar andNossal (1971). Their work included analysis of both the HHequations and the model introduced by Frankenhaeuser andHuxley (1964) for the study of the myelinated axon (see Chap. 4,Sec. 4.2). In both these models, the quantities rh(V) and rn(V) areconsiderably larger than rm(V). They are so much larger that areasonable approximation to the HH equations or the Fran-kenhaeuser-Huxley equations can be obtained by assuming thatdh/dt = 0, dn/dt = 0, and that h and n have the constant values*oo(Jo) and H^VQ), where Vo is the resting value of the membranepotential. The resulting two-dimensional system is a "reasonableapproximation" in the sense that for limited intervals of times itssolutions, together with the conditions h = / ^ ( F Q ) and n = nîVo),are near the solutions of the full four-dimensional system. Thereason that the approximation works fairly well is that V and m


vary much more rapidly than h and n. This kind of approximationwas introduced by FitzHugh (1961) and was termed the method ofreduction.

First we describe this method very briefly. A more detaileddescription is given in Section 6.2.1.3. Following any stimulus ofthe axon, the quantities V,m,h,n ultimately return to their restingvalues. Thus, we expect that the HH equations have a uniqueglobally asymptotically stable equilibrium point. The reduced sys-tem is:

dV \{

dm

where

— mrm(V) '

and it can be shown [see FitzHugh (1969)] that the reduced systemhas three equilibrium points: two asymptotically stable equilibriumpoints and a saddle point. See Fig. 6.11, where the points A, B, Care the equilibrium points. The point A corresponds to the restingstate of the system; B is the saddle point. The dashed line is thethreshold separatrix. Solutions that pass through a point on theright of the dashed line approach the equilibrium point C as

m

Figure 6.11.


t -> oo, and solutions that pass through a point on the left of thedashed line approach the equilibrium point A as t -> oo. Thus thesaddle point and the threshold separatrix provide a model of asystem with threshold behavior.

The solutions do not approach a unique globally asymptoticallystable equilibrium point as would be expected for the HH equa-tions. The reason for this is that h and n largely describe ordetermine the later behavior of the solutions of the HH equations.Since h and n are "suppressed" in the reduced system, we can nolonger expect the behavior that occurs with the HH equations.

Now the analysis carried out by Lecar and Nossal is the follow-ing. A stochastic or random term is added to each of the equationsin the reduced system. The resulting system is linearized in aneighborhood of the saddle point B and the linear system thusobtained is studied. The first step in describing these results is torewrite the reduced system in a different and more convenientnotation. The following notation is introduced:

J=ClI.Lecar and Nossal also introduce a different conductance variable athat, for the case of the Hodgkin-Huxley equations, is

a = m3.Their reason for introducing this conductance variable is that astudy of experimental data [see Lecar and Nossal (1971, p. 1053)]suggests that the steady state value of a, that is,

seems more likely to be a universal quantity (i.e., one that describesdata from a wide class of experiments) than the steady state valueof m.


With these notations, the reduced system becomes

dV

).

where, in the case of the HH equations,

[Lecar and Nossal emphasize that much of the analysis that followscan be carried out without reference to the particular form of

For the Frankenhaeuser-Huxley model, a similar reduced systemcan be derived. It has the form

dV 1

- = *(»),

where

- Vp),

a = m2,and

2a1/2

Now if noise is taken consideration, then random or stochasticterms must be added to each equation in the reduced system (6.5)or the system (6.6). Of course, once this is done, we are no longerconcerned with the problem of solving a system of differentialequations, but instead a system of stochastic differential equations.Instead of looking for a deterministic solution (V(t), o(t)) of (6.5),we must seek a solution of the system of stochastic differential


equations, (i^(t), 2(0), where for each fixed value of t, say t0, theexpressions i^(t0), 2(/0) represent random variables. That is,( ^ ( / ) , 2 (0 ) is a time-varying probability distribution for theposition of the system in the (Vo) plane.

Adding random terms to the system (6.5), we obtain

dV(Vo)-y^(V-V1) + Fv((o),t), (6.7)J

^ 0, (6.8)

where Fv and Fa are random functions of time (Langevin forces);Fv is assumed to depend on (a) , the mean value of a, and Fa isassumed to depend on (V), the mean value of V.

Now (6.7) and (6.8) are expanded in a neighborhood of theequilibrium point B = (VB, oB). Using the dimensionless variables,

e =V-VB

v,

the expanded forms of (6.7) and (6.8) become

— - + + + —dt " u V1

^ =A21e + A22(i+ • • • +FO(VB, t) + • • •, (6.10)dt

where

A12 = yT—iT(v1-vB),1 yi

\do(V)

L 9A•22 ~

The linearized version of (6.9) and (6.10) is

de 1i. " i i - "izr- rz ^V\°B'> * )> \®-*^)

(6.12)


and it is this last system that Lecar and Nossal analyze in detail.Thus their conclusions are based on the following assumptions:

1. The system (6.5) is a good approximation for the HHequations; and similarly the system (6.6) is a good ap-proximation for the Frankenhaeuser-Huxley equations (asindicated earlier, this assumption is valid if the interval issmall).

2. The linear system (6.11) and (6.12) is a good approxima-tion for the system (6.9) and (6.10).

(Since both eigenvalues of the matrix

Au Au

have nonzero real parts, then a sufficient condition that assumption2 be valid is that e and JU, be small enough.)

It is also assumed that Fv(oB, t) and Fa(VB, t) each representsthe result of many small independent random events and, hence,for each t is a Gaussian or normal random variable. (Regarding Fv

and Fa, for each /, as normal random variables is suggested by thecentral limit theorem.)

On the basis of these assumptions, Lecar and Nossal were ableto obtain several very interesting results. For example, they calcu-lated the probability of firing (occurrence of an action potential)for a myelinated axon given the initial value of V. Their resultsagree with experimental results [see Lecar and Nossal (1971, p.1058)]. Another result is the distribution of latency values. Latencyis the physiological term for the time interval from the beginning ofthe stimulus to the appearance of an action potential (measured,say, at the time at which it reaches half its full height). Experimen-tal results usually suggest that the latency is bounded, but in thesaddle point model of threshold behavior the latency is unbounded.The reason for this is that on an orbit that passes very close to thesaddle point, such as the orbit Sf in Fig. 6.11 the speed (i.e.,[(dV/dt)2 + (dm/dt)2]l/2) is very small because at the saddlepoint,

dV dm


Hence, the time required to pass on ^ from point a to point /? isvery large; but Lecar and Nossal's results show that the probabilityof large latencies is very small. Thus the model is more realistic. Inother words, the addition of random terms in the differentialequations makes the saddle point a more accurate or realisticmodel of threshold behavior.

The work of Lecar and Nossal is the only work in the directionof including a stochastic term in the mathematical model, and itraises many questions, but it slows clearly a direction of study thatshould be pursued more extensively.

6.2.1.3 Analysis of the Hodgkin-Huxley equationsBoundedness of solutions. The analysis will be described in

terms of the following version of the Hodgkin-Huxley equationsgiven by FitzHugh (1969):

dV 1

{

dm

dt

dn _

dt ~ rn(V) '

We notice first that if m = 0, dm/dt is positive and if m = 1,dm/dt is negative, and hence every physiologically significantsolution of the Hodgkin-Huxley equations stays in the set

rJV)

(V)-

(V)-

m

h

n

Similar arguments hold for h and n and we conclude that allsolutions of interest stay in the set


Now let

F=max[FNa,FK,FL],

If V> V, then dV/dt < 0 and if V< V, dV/dt > 0. Thus, if r is apositive number, it follows that all the physiologically significantsolutions stay in the closed bounded set

_ m(V,m,h,n)\V-r<V<V + r,0< h

nReduction method of FitzHugh. In view of the ease with which the

desired boundedness condition on the solutions is established, it israther a surprise to find that further analysis of the HH equation isextremely difficult. We will describe a couple of approaches thathave been used to obtain some qualitative understanding of thesolutions. Both of these approaches depend upon the fact thatrm(V) is about one-tenth the size of rh(V) and rn(V). The firstapproach, the reduction method of FitzHugh, consists in firstapproximating the HH equations by taking dh/dt = 0 and dn/dt =0, that is, by taking TH and TW to be infinite and letting h and n beequal, respectively, to /^(O) and ^(O). Then the HH equationsare reduced to the two-dimensional system

dV 1

dm m^V) - m

di= rm(V)

Numerical analysis [see FitzHugh (1969)] shows that this systemhas three equilibrium points. A sketch of these equilibrium pointsand some nearby orbits is given in Fig. 6.12. [The sketch is notnumerically accurate. For a more accurate representation, seeFitzHugh (1969).] Equilibrium points A and C are stable; equi-librium point B is a saddle point.

Mathematical analysis of physiological models

m

212

J

Figure 6.12.

Equilibrium point A is the resting point. Stimulation by acurrent shock of the correct sign causes a displacement from A tothe right on the horizontal dashed line. If the displacement is, say,to the point D, then this is a subthreshold shock and the phasepoint in the Vm plane, which describes the condition of the axon,returns to the neighborhood of point A. If the displacementreaches the point E, then the phase point proceeds to a neighbor-hood of equilibrium point C. This describes the first part of anaction potential. (The point does not return to equilibrium point Abecause the variables h and n, which describe how this return takesplace, are kept constant.)

The next step is to allow h and n to vary in the way governed bytheir differential equations

dh

dt " (V)

dn _nao(V)-n

dt rn(V) '

and to study numerically how the equilibrium points of (F, m)change as h and n are varied. The equilibrium points B and C


approach each other, merge, and disappear. The only remainingequilibrium point is A, and the phase point returns to a neighbor-hood of A. A more detailed account of these changes is given byFitzHugh (1969). Our discussion is not rigorous, and even a moredetailed description is numerical and suggestive rather than rigor-ous. However, this kind of intuitive procedure provides very usefulguidance in the study of these complicated nonlinear systems ofordinary differential equations.

The singular perturbation viewpoint. Next we reformulate the HHequations as a singularly perturbed system in a manner parallel tothe study of the FitzHugh-Nagumo equations that was describedin an earlier section of this chapter. In order to formulate the HHequations as a singularly perturbed system, let

maxrm(F) '

rh(V)

Then

dV

dm

dt

dh

dt

dn

maxTn(F)

the HH equations become

1

CmJV)

TM(V)mw

hJV)

TH(V)mzx

nJV)

i3*(K-KN . )

-m

-h

— n

~ VL)},

dt

Since rm(V) is roughly one-tenth as large as rh(V) and rn(V) [see


FitzHugh (1969), p. 25], we rewrite these equations as

dV 1{

dm _ mx(V)-m

~di~ TM{V)

= eTN{V)

where e is a small positive number (actually about one-tenth). Nowwe define r as T = et. Then dV/dt = (dV/dr)(dr/dt) = £(dV/dr)and similarly for dm/dt, dh/dt, and dn/dt. Hence, system (6.13)may be written

dV 1E~dt= ~~C

dm m

dhLdt

dnLdt

or

dV

dt ~

dm

dt

dh

dt

dn

TM(V) '

" P / \

\ TH{V) 1

\ TN{V) j

1 {" M

J 00

«co(^)-»

'-V^) + gKn4(V-VK) + gL(V-VL)},

(6.14)

dt TN(V) "


System (6.14) is a singularly perturbed system and hence thesingular perturbation theory of Chapter 5 can be applied to it.Since there are two singularly perturbed equations in system (6.14),then the theory of Sibuya and the theory of Mishchenko and Rozovare applicable. Such an application has not actually been carriedout, but seems to be a step that should be taken.

Now we turn to the question of a mathematical description ofrepetitive firing of the nerve. We have already seen, in the analysisof the FitzHugh-Nagumo equations, how the addition of a con-stant term / can cause the appearance of a periodic solution. Itmay be possible, by extending the singular perturbation analysis ofthe FitzHugh-Nagumo equations described in Section 6.2.1.2, toshow that if / is a constant in a suitable interval of values, then theresulting HH equations have a periodic solution. It should beobserved that the two-dimensional case (illustrated in Fig. 6.7)suggests that if / exceeds a fixed value, then a discontinuousperiodic solution appears and the amplitude of the discontinuoussolution at once exceeds a fixed value. That is, the amplitude doesnot increase continuously from zero as would be the case with aHopf bifurcation.

The Hopf bifurcation theorem has been used by Troy (1974/75)to investigate the appearance of periodic solutions as the current /is varied. In order to determine which of the two approaches, thesingular perturbation method or the Hopf bifurcation theorem, ismore realistic, it is enlightening to study the numerical analysis[carried out by Cole et al. (1955)].

In the graph given by Cole et al. (1955, p. 166), it is shown that ifthe current / takes larger values, first one action potential occurs,then for a larger value of /, there are three action potentials, andfinally, if / is large enough, the action potentials continue indefi-nitely, that is, an oscillatory phenomenon occurs. In terms of themathematical description, there is a periodic solution. However,one point should be emphasized about the graphs just described.All the action potentials have the same amplitude. Unlike the resultof the Hopf bifurcation theorem, where, as the parameter changes,the amplitude of the periodic solution increases monotonicallyfrom zero upward, the periodic solution, as soon as it appears, hasa fixed amplitude. Thus, the numerical experimental result is more


like the appearance of a periodic solution in a singularly perturbedsystem, as illustrated for the two-dimensional case in Fig. 6.6.

6.2.2 Analysis of the Noble model of the cardiac Purkinje fiber6.2.2.1 The Noble equations viewed as a singularly perturbed system

As the earlier discussion (Chapter 4) shows, a mathemati-cal analysis of the Noble model should account for or describe twophenomena: the transmission of impulses that originate in thepacemaker region of the heart and the regular spontaneous firingof the Purkinje fibers. The regular spontaneous firing correspondsto a stable periodic solution of the model, and the transmission ofimpulses would probably be well described by the mathematicaltheory of entrainment of frequency described in Chapter 5, as wewill see later.

However, before undertaking the search for a periodic solution,we must first choose a viewpoint of the Noble equations. Instudying the FitzHugh-Nagumo equations qualitatively, it turnedout to be very enlightening to regard them as a two-dimensionalsingularly perturbed system. We shall find that a similar viewpointis helpful in studying the Noble equations. We first look at a fewnumerical values to guide our choice of a viewpoint.

The Noble equations have the following form:

dV 1— = {(400m3/* + 0 .14) (F- 40) + ( l . 2exp [ ( - V- 90)|50]

dm

~dt

dh

~dt

dn

= «*(!

-«(1

+ 0.015 exp[(F +

-h)-fihh,

-n)-Pnn,

90) |60] + 1.2«4)(F + 100)},

(6.15)

where V denotes the membrane potential (inside potential minus


outside potential) and we are assuming that the anion current IAn

is zero. [Noble (1962, pp. 319 and 343ff) discusses the effect ofvarying /An. We choose /An = 0 because this is a value for whichNoble obtained numerical results and we can compare our qualita-tive results with the numerical results.] The other functions thatappear in the equations are defined as follows:

0 .1 ( -F -48)

exp[(-F-48)/15] - 1 '

0.12(F+8)

exp[(F+8)/5]-l'

-F-90

- F - 4 2 \ I-1

10

0.0001(-F-50)

exp[(-F-50)/10] -1 '

-F-90pn = 0.002 exp

\ 80

Following the same convention used in the study of theHodgkin-Huxley equations, we may write the last three equationsof system (6.15) as

dm m o o (F ) -m

dt

dh

dt

dn

dt

rm(V)

(V)-

( F ) -

r(V)

h

n


where

and hîV), rh(V), nx(V), and rn(V) are similarly defined.Numerical values of the functions rm(V), rh(V), and Tn(V) arelisted in Table 6.1. Notice that

minTm(F) = 0.1, minTA(K)«l, minTn(F) = 106,

Mm=

Mn=

0.206, Mh = « 8.3,

min T m_ » 0.486, min ^LJ- » 0.121,K M m v Mn

min - ^ = ^ » 0.198.M

These values suggest that T A ( F ) is between 5 and 10 times as large

Table 6.1

V

-90-80-70-60-50-40-30-20-10

010203040

rm(V)

0.098890.110250.123460.138520.155280.173020.1904760.2044990.206270.190580.1671960.145220.127460.112209

5.612988.34725.55863.0031.773991.2871.104851.037451.0131.00441.00141.00031.00005

*n(V)

481.9277520.833534.759510.204452.488377.358306.748250.62657207.90176.056151.97568132.9787118.483106.496


as Tm(V)9 and rn(V) is more than 60 times as large as rh(V) andmore than 500 times as large as rm(V). If we let

Mm = lnbrm(V),

Mh = \ubTh(V),

Mn = lubrn(V),

where the lub is taken over V in the interval [ - 90,40] (this intervalcontains the set of values actually assumed by the membranepotential), and if we define

Tm(V)=rm(V)/Mm,

= rh(V)/Mh,

= rn(V)/Mn,

then system (6.15) may be rewritten

dV—dt

dm

di = MmTm(V) '

dh_ _ hjv)-h ( 6 - 1 6 )

dt MhTh(V) '

dn _njy)-n

dt MJn{V) '

where, for convenience, we have written the right-hand side of thefirst equation in system (6.15) in abbreviated form. The functionsTm(V), Th(V), and Tn(V) are all of the same order of magnitude inthat their values are positive numbers in the interval [0.1,1]. Thenumerical values previously displayed of the functions rm(V) andrh(V) suggest that Mm is about 0.21, Mh is about 8.3, and Mn isabout 535. Thus, Mh is about 40Mm and Mn is about 2548MW.


Hence, we can rewrite (6.16) asdV— =&(V,m9h,n),at

dm ^ o o ( ^ ) ~ m

~d7= Mj

dt MMmTh{V)'

dn _ nx{V)-n

dt ~ 2548MmJn(F) •

Further, if we let e = 1/40 and TJ = 1/(2548/40) « 1/64, then wemay rewrite (6.17) as

dV— =&(V,m,h,n),dtdm mîV) — m

" MmTm(V) '

dh_= \hJV)-h] ( 6 1 8 )

dt £[ MmTh(V) \'

dt eV[ MmTn(V) \

Now we introduce a conventional change of the independentvariable; that is, we set

r = et,

where e is regarded as a small positive number. (Since e is small,this change of variable amounts to "stretching out" the time axis.)Then by the chain rule from calculus,

dV dV dr dV

dtdm

dh

~dt =

dn

~dt~

dr dtdm

dh

~d~rZ'dn


and we may rewrite (6.18) asdV

— edr

dm ôo(^) ~" m

MmTm(V) '

dhTr'

dn—dr

-£ = £MmTh(V)

MmTn(h)

or, denoting the independent variable by t so as to conform toconventional notation, we have

dt e\ MmTm(V) j '

dh (6.19)

Now e and 7} were originally equal to 1/40 and 1/64, respectively.However, if we regard e and TJ as small positive parameters, thenthe original problem of mathematically analyzing the solutions ofsystem (6.15) suggests the following problem: to analyze the solu-tions of (6.19) when e and 77 are sufficiently small positive num-bers.

Notice that we have not said that this new problem is strictlyequivalent to the original problem of studying system (6.15). Tobegin with, in deriving (6.19) we used the approximations

Mh = 40Mm,

M =

Second, even if we are able to make a rigorous analysis of (6.19) for


sufficiently small positive e and 17, it does not follow that theconclusions are applicable to (6.15) unless we can show that therigorous analysis of (6.19) is applicable to a range of values of eand -q that includes e = 1/40 and 77 = 1/64. As we have alreadyseen in Chapter 5, theorems that yield results valid for sufficientlysmall e often do not give explicit estimates on the range of validityof 6.

In spite of these qualifications or reservations, we shall go aheadto the analysis of the solutions of (6.19). There are two reasons forthis decision. First, as seen earlier in this chapter, the analysis of asimilar approximation of the FitzHugh-Nagumo equations turnedout to yield a lot of enlightening and useful information aboutbehavior of solutions. Second, there is the rather negative reasonthat a direct rigorous analysis of the solutions of (6.15) seems to bebeyond the reach of known methods, whereas we can obtain atleast some partial results concerning solutions of (6.19) if e and TJare sufficiently small positive numbers.

6.2.2.2 Analysis of the singularly perturbed systemReduction to three-dimensional system. In view of the messi-

ness of the function (V,m,h,n) in the first equation of system(6.19), one might expect that the chief difficulties that would arisein the study of (6.19) would stem from the fact that one is dealingwith complicated algebraic expressions. Actually this is not so. Itturns out that the beginnings of a qualitative analysis of (6.19) arereadily accessible. The obstacle to a complete qualitative analysislies in the absence of enough qualitative theory about singularlyperturbed systems.

Now we turn to the analysis of (6.19). First, let us comparesystem (6.19) with system (6.14), the version of the Hodgkin-Huxleyequations that is obtained in the preceding section and that is asingularly perturbed system. Systems (6.14) and (6.19) are similarin that both are four-dimensional singularly perturbed systems andeach has a two-dimensional "fast" system [the first two equationsin each of the systems (6.14) and (6.19)]. There is, however, animportant difference between (6.14) and (6.19). In (6.19), the lastequation has a small parameter TJ multiplying the right-hand side.Hence, the last equation can be regarded as a perturbation of the


equation

dn

The conventional mathematical approach to such a system is toassume that 7) = 0. Then dn/dt = 0 and n(t) must be a constant,say n0, and system (6.19) becomes

dV 1— = -&(V,m,h,n0),at e

dm 1 mîV) — m

dh _hx{V)-h

dt MmTh{V) '

The solutions of (6.20) are then studied, and it is shown that thesolutions of (6.19) with TJ nonzero but small are close to thesolutions of (6.20).

[Note that although (6.20) is a singularly perturbed system, it is athree-dimensional system. Consequently if the conventional math-ematical approach just sketched can be applied, we would expectthe analysis of (6.19) to be easier than the analysis of (6.14), thesingular perturbation version of the Hodgkin-Huxley equations,because (6.14) is a four-dimensional system.]

Our first step will consist in showing that this conventionalmathematical approach fails. The approach fails for two reasons.The first reason for failure is purely mathematical: The analysis of(6.20) shows that it has no periodic solutions. Thus, we have nohope of using this approach to find a periodic solution of (6.19).Although this is a negative result, we will carry out a detailedanalysis of (6.20) in the next section because the results will bereferred to in a later section and because it provides an applicationof the singular perturbation theory introduced in Chapter 5.

The second reason for failure is more fundamental: The assump-tion dn/dt = 0 is physiologically untenable. This point will bediscussed in more detail following the analysis of (6.20).

Demonstration that the three-dimensional system does not haveperiodic solutions. Since we are looking for a periodic solution that


would describe the regular spontaneous firing of the Purkinje fiber,a reasonable procedure is to use the singular perturbation theory toestablish a periodic solution, with orbit, say S?, of (6.20), and thenshow that if 17 is sufficiently small, system (6.19) has a periodicsolution or an approximately periodic solution whose orbit is closeto the orbit SP. However, we shall now show that this seeminglyreasonable procedure fails, that is, we will show that we do notobtain a periodic solution for system (6.20). Then we will explainwhy the physiology would lead us to expect this result.

Following the theory of singularly perturbed systems describedin Chapter 5, we seek discontinuous solutions of the degeneratesystem

MmTm{V) U'

dh _hJV)-h

dt " MmTh(V)

or

m = mJV),

dh _hJV)-hdt MmTh{V) •

A discontinuous solution is a curve consisting of two kinds ofsegments:

(i) those that are described by solutions of

dh _mJV)-h

dt MmTh(V)

and that are contained in the manifold Jt defined by

j(= {(F, m, h)/^{V, m,h,no) = 0 and m = m


m

/

/

v/m0O(V)-m>O

I i— i i i i i i i

1 O1 .V/

/0.8

0.6

0.4

0.2

-

-

-

i i i i V-80 - 60 - 4 0 - 2 0 0 20 40

THE FUNCTION m^ (V)

Figure 6.13. The function m^V).

(ii) those segments that are "jump arcs," that is, solutions of thefast system.

To visualize the discontinuous solution clearly, we obtain first apicture of the manifold Jt. The manifold Jl is described by thetwo equations

(6.21)

(6.22)

Equation (6.21) describes a cylindrical surface that is perpendicularto the (V, m) plane. This surface is simple to visualize because thefunction mîV) is the simple monotonic increasing functiongraphed in Fig. 6.13 [see Noble (1962, p. 326, Fig. 4)]. Equation(6.22) describes a cylindrical surface perpendicular to the (F, h)plane. Although the function J^ is complicated looking, the curvein the (F, h) plane described by (6.22) is a simple configuration.


For the case n0 = 0, (6.22) becomes

(400m3/z + 0.14)(F-40)

( ( - F - 9 0 \ / K+90U= - 1.2 e x p — - — + 0 . 0 1 5 exp——(K+100) ,50 / r\ 60 /

and solving for h as a function of V, we obtain

1h = h(V) =

400m3(F-40)

X{ - ) - 1.2exp

+ 0.015 exp

- K - 9 0

50

F+90

60(K+100) .

The graph of the function h(V) is given in Fig. 6.14. [Because ofthe scale of the drawing in Fig. 6.14, it is impossible to see how the

- 8 0 - 6 0 - 4 0 - 2 0

Equilibrium point E

[ 20 40J-.—.—.—i—. V

(V|fh,)


h (V)

Figure 6.15.

graphs of the functions h(V) and h^V) intersect. A distortedversion of these graphs, which illustrates the intersection moreclearly, is shown in Fig. 6.15.] It is essential to observe that thegraph of h^{V) intersects the graph of h(V) at the point E, whereh(V) is an increasing function. The graph of the function h(V)drawn in Fig. 6.14 is obtained by plotting the points whosecoordinates are listed in Table 6.2 and joining these points by asmooth curve. The in-the-large properties of the resulting curve,especially the property of being S-shaped, play an important rolein determining the form of the discontinuous solutions, as we willsee shortly. One might well argue that if more points on the graphof the function were plotted, then the in-the-large character of thegraph and, hence, the form of the discontinuous solutions could besignificantly changed. The answer to this is that the functions thatappear in the original Noble equations are chosen to be consistentwith a finite set of data points obtained from laboratory observa-tions. That is, the equations are empirical equations or are ap-proximate summaries of laboratory data. Hence, a small change in

228

V

-90-87-83-80-77-73-70-67-63-60-57-53-50-47-43-40-37-33-30

-27-23-20-17-13-10-7-303710131720232730333740434750

am

0.2720.31290.37580.42990.49050.58210.65970.74540.87290.97901.0951.2641.401.5511.7651.932.1172.3732.58

2.7873.0823.313.553.8764.124.3854.7365.0045.2765.6445.926.2065.5866.8747.1637.5517.8438.1378.5298.8259.1219.5179.814

A,9.849.489.008.648.287.807.447.086.606.245.885.405.044.684.203.853.493.022.672.331.891.581.2940.9490.7280.5420.3790.2430.1640.0940.0610.0380.0200.0120.00750.00380.00230.00140.000660.000390.000230.000110.00006

0.0000750.0000940.000130.000160.000190.000260.000310.000380.000490.000580.000690.000850.0010.001160.001390.001580.001790.002080.00231

0.000560.002890.003160.003430.003790.004070.004360.004740.005030.005330.005720.006010.006310.006710.007010.007310.007710.0080.00830.00870.0090.00930.00970.01

A,0.0020.001930.001830.001760.00170.001620.001560.00150.001420.001380.001320.001260.001210.001170.001110.001070.001030.000980.00095

0.000910.000870.000830.000800.000760.000740.000710.000670.000650.000630.000590.00057O.OOO550.000530.000510.000490.000460.000440.000430.000410.000390.000380.000360.00035

ah

0.170.14630.11970.10310.08870.07260.06250.05380.04410.03790.03260.02670.02300.01980.01620.01390.01200.00980.0085

0.00730.00590.00510.00440.00360.00310.00270.00220.00190.00160.00130.00110.00090.00080.00070.00060.00050.00040.000360.00030.000250.00020.000150.0001

Table 6

h0.008160.01090.01630.02190.02930.04310.05730.07590.1090.1420.1820.24970.310.37750.4750.54980.6220.71090.7685

0.81750.86980.90000.92410.94780.96080.97070.98010.98520.9890.99260.99450.99590.99730.99790.99850.99890.99930.99940.99960.99970.99980.99990.9999

.2

0.026890.03090.04000.04740.05590.06940.08140.09530.11680.13560.15690.18960.21740.24880.29560.33510.37750.44000.4911

0.54440.61940.67630.73290.80330.84980.88990.93140.95390.96990.98360.98980.99390.99690.99830.99890.99950.99970.99980.99990.99990.99990.99990.9999

0.03590.04640.06440.08170.10240.13680.16730.20210.25440.29780.34260.405

lim = 0.4520.49830.55680.59600.63450.67960.7097

0.73780.76950.79120.81030.83220.84690.86020.87550.88570.8950.90570.91300.91560.92740.93270.93750.94330.94720.95080.95510.95810.96080.96410.9664

0.95420.93060.88000.82480.75170.62750.52170.41480.28800.21070.15190.09660.06910.04980.03290.02460.01890.01360.0109

0.00880.00670.00560.00470.00380.00320.002770.002440.001920.001620.001310.00110.00090.00080.00070.00060.00050.00040.000360.00030.000250.00020.000150.0001

h(V)n = 0

-5.98-1.920.25450.62670.65190.51480.39440.290.18770.13370.09480.05990.04270.03050.01980.01450.01070.00730.0055

0.00430.00320.00260.00220.00180.00160.00150.001480.001510.001580.00170.00190.00220.00260.00310.00370.00510.00680.010.025—

-0.0267-0.0122-0.0088

h(V)n = 0.75

-2.231.3692.32.111.721.1930.8740.62830.40170.28590.20370.13030.09410.06810.04500.03360.02510.01750.0135

0.00820.00800.00670.00570.00480.00440.00420.00410.00420.00450.00490.00550.00620.00750.00880.01060.01430.01910.02810.0680—

-0.072-0.032-0.023

229

h(V)n = nx(V)

-5.984-1.9240.2550.6270.6520.51560.39560.2920.19060.13750.09950.0660.04950.03780.02740.02220.01810.01420.0119

0.01030.00850.00760.00690.00630.00620.00610.00640.00680.00750.00860.00480.01130.01390.01680.02050.02830.03820.05670.1387—

-0.1486-0.0665-0.0482


the equations may make them no less accurate, and may indeedmake them a more accurate description of the data. Consequently,if we make a small change in the equations in order to be able toapply known theory and draw significant inferences about thesolution of the equations, the equations thus modified may providean equally accurate and possibly more effective model. We shalluse this viewpoint a number of times in the analysis that follows.

The graphs in Figs. 6.13 and 6.14 show that the manifold Jt iseasily visualized. Now all the segments of a discontinuous solutionexcept the jump arcs, lie in Jt. To get a complete picture of adiscontinuous solution, it is only necessary to determine the possi-ble jump arcs. Inspection of the graph in Fig. 6.14 shows that thepoint (F, mjy\ h) = (V, m,Ji), where h = h(V) and the functionh(V) has a maximum at V= V, is a possible junction point. Then ifthere is a jump arc starting at this junction point, it is a curve in theplane h = Ti. This curve must be an orbit of a solution of the fastsystem

dV 1

at e

dm lUmJV)-m)]dt ~ e[ MmTm(V) \

Since we are looking only at orbits and since the factor 1/e appearsin both equations, then it suffices to look at the orbits of thesystem

dv

—

dm mx(V)-m ^ ^

~d7^ MmTm{V) '

To show that (V,m,Ti) is a junction point, it is necessary first toshow that there is exactly one orbit [of a solution of (6.23)] that"comes out" of (F, m). More precisely, it is necessary to prove thatthere exists a solution (V(t), m(t)) of (6.23) such that

lim (V(t),M(t)) = (V,m)t-> — oo

and that all solutions with this property have the same orbit. Let us


assume that such a solution with orbit, say, ^ \ , exists. (It isstraightforward but fairly lengthy to prove this assumption.)

As can be visualized from Fig. 6.14, system (6.23) has just twoequilibrium points: (F, m), where m = mO0(V)y and another pointthat we denote by (Vv mx), where VY is the positive solution of theequations h = h(V) and h = h(V), and m1 = mO0(Vl). [The point(Vl9 hx), where hx = hiV^, is shown in Fig. 6.14.] In order to showthat £fx is a jump arc, we must show that orbit Sfx has as itsco-limit set the point (Vv mx). In order to show this, we observethat

jF(F,m,/*,«0) = - — [400m3(V-40)][h-h(V)].Cm

In this discussion, we assume V < 40. Hence, V — 40 < 0. From thedescription of the function m^V) given by Noble (1962, seeespecially p. 326, Fig. 4) and the equation

dm M

di MmTm(V) '

it follows that if the initial value of m(t) is in the interval (0,1)(and this is the only case that is physiologically significant), thenfor all later /,

0<m(t) <l.

Also since Cm > 0, it follows that J^(F, m,Ji,nQ) has the same signas Ji — h(V), but Ji is a local maximum value of h{V). This can beseen from the graph in Fig. 6.14. (The positive direction on the haxis in that graph is downward.) Hence, it follows that

h-h(V)>0

for V< V and for Fe(F,K1). It can also be seen that

and that, if V> Vv

(V,m9h,no)<0.

We have already observed that

0<m(t) <l


and inspection of ^(V, m,7i,n0) shows that if V is a sufficientlylarge positive number, dV/dt is negative, and if V is a negativenumber of sufficiently large absolute value, then dV/dt is positive.Hence, V(t) remains bounded for all t. Hence, the solution withorbit 6^x remains bounded. Moreover, it follows from the signs of&(¥, m, h, n0) and [m00(F) - m] that the solution with orbit Srx

must have as its co-limit set an equilibrium point, that is, either(V, m) or (Vv mj. But a solution of system (6.23) that "comesout" of (V9 m) must be in the set

Since ^(V, m, Ti, n0) > 0 on the set A, then dV/dt is positive.Therefore, the orbit Sfx must approach the equilibrium point(Vl9 mx). Hence, the orbit SPX may be regarded as a jump arc. Nowthe next segment of the discontinuous solution would start at thepoint (F1? ml5 A), where Ji — h{V{) and w1 = w00(F1). Thus, thisnext segment would be a solution of the equation

dh h

dt " MmTh{V)

with initial value Ji. Also, the solution lies in the manifold de-scribed by the two equations

(6.24)

,no)=0. (6.25)

The graph of h = h(V) in Fig. 6.14 shows that near (K1; h^), we cansolve (6.25) uniquely for V as a function of h, that is, we obtain

V=G{h),

and the segment is described by the solution h(t) of the equation

dh hx[G(h)]-h

dt ~ MhTh[G{h)]

such that h(0) = h, by V= G[h(t)] and by

The projection of the segment just described on the (V,h) planeis shown in Fig. 6.14 by a jagged line. However, this solution


approaches the equilibrium point E that has coordinate V (be-tween 0 and _+10) tât is s^h that h(V) = hO0(V). [Thus E hascoordinates (F, m j F ) , hM(V)).]

The same is true for other discontinuous solutions; that is, eachdiscontinuous solution approaches the point E as r->oo. Forexample, consider the discontinuous solution that has as its initialpoint the point P in Fig. 6.15. [Actually P is the projection of thatinitial point from manifold Jt onto the (F, h) plane.] The projec-tion of this discontinuous solution on the (V, h) plane is indicatedby the dashed line.

It follows from the theory of Mishchenko and Rozov that if £ issmall, system (6.20) with no = 0 has no periodic solutions; that is,in this case, n0 = 0, we obtain no periodic solutions by using theviewpoint of singular perturbation theory. Further analysis of thesame kind for other fixed values of n0 shows that, again, noperiodic solution is obtained; that is, for fixed values of n0, thediscontinuous solutions approach an equilibrium point. Thus, thestrategy proposed earlier, that is, to find a periodic solution of(6.20) and then show that if i\ is sufficiently small, system (6.19)has an approximately periodic solution, is doomed to failure.System (6.20) does not have closed discontinuous solutions.

Physiological reasons why three-dimensional system does not haveperiodic solutions. When we consider the physiological significanceof the variable n, it becomes clear that this failure is inevitable. Thevariable n is the activation variable for the flow of potassium ions.The varying flow of potassium ions produces the pacemaker effectthat in turn makes possible the regular firing of the Purkinje fiber[see Noble (1962, p. 333)]. If the activation variable n remainsconstant, the underlying ideas in the derivation of the Nobleequations are no longer valid. [See the discussion in Noble (1962,pp. 326-327). In that discussion, the variation in n plays a crucialrole in the repetitive action in the Purkinje fiber.] Thus, there is nopossibility of reducing our problem to the study of the three-dimensional system (6.20).

Comparison between four-dimensional and three-dimensional sys-tems. As we have seen, approximating the solutions of (6.19) withsolutions of (6.20) fails because (6.20) has no periodic solutions.That is, the periodic solutions of (6.19), if there are any, elude this


approach. This is actually not surprising because in searching forperiodic solutions, we are looking at the asymptotic behavior ofsolutions, that is, the behavior of the solutions as /->oo. Thetechnique of approximating solutions of (6.19) with solutions of(6.20) is valid only for intervals on the t axis of finite length. Thisis illustrated by the following examples.

Example 1.

dx

y0' (6.26)

dt

where TJ is a small positive number. If TJ = 0, then each solution of(6.26) is a constant 2-vector. But if TJ > 0, then

where A' is a constant. Thus, if K i= 0, then

lim |t-* oo

This is certainly not the behavior of solutions of (6.26) with y = 0.

Example 2.

dx

dy

dz

•j-V.

where TJ is a small positive number. If TJ = 0, then each solution of(6.27) is a constant 3-vector. But if TJ > 0, then a fundamentalmatrix of (6.27) is

100

0COST)?

sinii/

0- sin TJ?

cos -nt

That is, each solution of (6.27) is a linear combination of the


columns of the preceding matrix. The solution

0COST]/

sinT]/

is certainly not well approximated by a constant 3-vector no matterhow small 17 is.

Example 2 shows how a periodic solution can elude us if thesolutions of the equation with T] = 0 are studied. It is also enlighten-ing to take the singular perturbation viewpoint illustrated by thefollowing example.

Example 3.

dx 1 dx— = --{x5-3x-y) or r\— = -(xi-3x-y),

dy

1t=X' <6-28)

The degenerate system corresponding to (6.28) is

x3 — 3x — y = 0,

±-dt~x-

The degenerate system is the result of setting 17 = 0, and solving thedegenerate system means solving

— = x

on the manifold defined by

y = x3 — 3x.

Thus, no periodic solution is obtained. However, the same kind ofanalysis that was used earlier in this chapter shows that (6.28) has aclosed discontinuous solution and, hence, a periodic solution.

With these examples in mind, we return to system (6.19). Let usfirst make a change of the independent variable in (6.19):

T = y\t.


Then (6.19) becomes

dV dV dt 1 1— = — — = *(V,m,h,n),dr dt dr TJ e

dm 1 1 m<X)(V)-m

~dr~ = ^~e MmTm(V) '

k ( 6 - 2 9 )

dr r, MmTh(V) '

dh _nx(V)-n

dr ~ MmTn(V) '

If we regard e simply as a fixed constant and ?) as a smallparameter, then system (6.29) may be regarded as a singularlyperturbed system in which the fast system consists of the first threeequations and the slow system is the last equation. Consequently,one segment of a discontinuous solution of (6.29) consists of anorbit of the fast system

dV 1 1— = - -dr r] e

dm 11 moo(F)-m

~^=l~e MmTm(V) ' ( 6 ' 3 0 )

dh 1 hJV)-h

dr V MmTh(V) '

where n is a constant. But the orbits of (6.30) are the orbits ofsystem (6.20) that was investigated earlier by using singular per-turbation analysis (where e is regarded as a small positive parame-ter).

Thus, we conclude that a rigorous study of (6.19) might beundertaken by using a two-stage singular perturbation analysis:First, a singular perturbation analysis of (6.30) in which e isregarded as a small positive parameter; and second, a singularperturbation analysis of (6.29) in which TJ is a small positiveparameter. The results of the analysis of (6.30) are used to carryout the analysis of (6.29).


We have already see how the orbits of (6.20) [and, hence, also of(6.30) with n = 0] approach the equilibrium point E. Such an orbitcan be regarded as a fast-motion part of a discontinuous solutionof (6.29). The fast-motion part is followed by a slow-motion partthat is a solution of the degenerate system

dn n ^

MmTn(V) '

m,h,n) = 0y *r[V9mJV),hO0(V)9n]=0,

or m = moo(F), (6.31)

This solution starts from the point

f=(f,mx{f),hoo(P),0

[Remember that the point E has coordinates

Since noo(V)>09 then it follows that dn/dr>0 at the point Pand, therefore, the solution n(r) increases. As n(r) increases, Vchanges in accordance with the equation

Note that n(r) does not "reach" n^{V)\ that is, there is nonumber T0 such that

where

[Vo,mJVo,hJVo),n(ro)}=0.The reason for this is that the point

would be an equilibrium point of (6.31) and, hence, no orbit couldcontain that point. Thus, we can only conclude

remains positive but is decreasing. To complete the analysis, itwould be necessary to determine if a junction point occurs on thisslow-motion part, then determine if there exists a drop point


(V,m,h)- space

Figure 6.16.

following this junction point, and so forth. This would require, firstof all, a fairly extensive numerical study, and we will not undertakesuch a study here. We observe only that on each slow-motion partof the discontinuous solution, the term

is positive and is decreasing as r increases. Thus, the closeddiscontinuous solution that would be sought would have the formindicated in Fig. 6.16. [In Fig. 6.16, the (K, m, h) space is repre-sented by the (V, m) plane.]

A modification of the three-dimensional system. We have alreadyindicated the complications that arise in the study of system (6.19).In this section, we describe a three-dimensional system that is akind of simplification of (6.19), that is, the following modificationof system (6.20):

d v l r[dt

dm

Itdh

~dt

MmTm(V)(6.32)

hJV)-kMmTh{V) •

[Formally, this modification of (6.20) consists in replacing n0 byin the first equation.]


There is no rigorous justification for making this modification.Indeed, the geometric behavior of the solutions of system (6.32) isdifferent from the behavior of the kind of periodic solutions thatwe would expect to get for system (6.19). The fast-motion part ofthe discontinuous solution approaches E as t_^> oo. Thus, for tsufficiently large, V is approximately equal to V. Thus, the fourthequation in system (6.29) [which is system (6.19) with a simplechange of the independent variable]) becomes

dn _nK{f)-ndt MmTn(¥) '

and the solution n(r) of this equation with «(0) = 0 is easily seento be

1 - expMmTn(v)

Thus, «(T)->«00(F)asT-ôo.[In actuality, as n(r) increases, thevalue of F, which is governed by the equation

also changes.]By considering the system (6.32), we are assuming in advance

that n = nO0(V). The graph of a solution of (6.32) would berepresented in Fig. 6.16 by a curve in the surface

in (V, m, h, n) space.By applying the Mishchenko-Rozov theory, a detailed analysis

of system (6.32) can be made, and existence of a periodic solutioncan be established. The analysis turns out to be fairly lengthy, andthis suggests that the analysis of more realistic models of electri-cally active cells (such as the 10-dimensional McAllister-Noble-Tsien model) may present formidable obstacles.

6.2.2.3 Using entrainment of frequency to describe the primaryfunction of the Purkinje fibersIn Chapter 4, we described the primary and secondary

functions of the Purkinje fiber. The primary function is conductionof the electrical impulses that originate in the pacemaker region


and the secondary function is to act as a "backup" pacemaker by"firing" spontaneously and regularly. The periodic solution ofsystem (6.32) that was obtained in the previous section can beregarded as a mathematical description of the secondary function.

Now we turn to a mathematical description of the primaryfunction, that is, a description of the conduction of the electricalimpulses.

In view of the lengthy effort that was required to obtain thedescription of the secondary function, we might expect that themathematical description of the primary function would requireconsiderable work. Actually the description is derived very easilyby applying the theory of entrainment of frequency, and we beginby summarizing this theory briefly.

The physical phenomenon of entrainment of frequency can bedescribed as follows. Suppose a physical system (e.g., a tuning fork)has a natural frequency of oscillation, say coo, and suppose anoutside force with frequency co is impressed on the system. If co isnot close to coo, that is, if |co - coo| is large enough, the twofrequencies together produce a beat frequency. This can be ob-served in the simple experiment of striking two tuning forks, onewith frequency <oo and one with frequency co. The oscillations ofthe nearby air are produced by the sum of the two oscillations, andthe experimenter hears the beat frequency. But if the frequency cois close enough to coo, that is, if |co - coo| is small enough, then thephysical system oscillates with frequency co. The "free" or naturalfrequency coo is said to have been "entrained" by the outside orextraneous frequency co.

Examples of entrainment of frequency have long been observedby scientists. In the seventeenth century, Huyghens observed amechanical example: Two clocks slightly out of synchronizationbecome synchronized when fixed on a thin wooden board. Anexample that was once of great practical importance and thatmotivated the mathematical analysis carried out by van der Pol,was the three-element radio tube (or electron tube) in which thefrequency of oscillation of the tube is entrained by the frequencyimpressed on the grid. Since radio tubes have long since beensupplanted by transistors, it is not purposeful to describe thissystem in detail. [For a mathematical discussion, see Stoker (1950,


p. 147ff) and Minorsky (1962, p. 429ff).] The importance of thisexample today is that it gave rise to a mathematical theory that hasan independent identity and that can be applied to entirely differ-ent physical systems such as the Purkinje fiber.

The mathematical description of entrainment of frequency con-sists, first, of an autonomous system of nonlinear ordinary dif-ferential equations that describe the physical system that has anoscillation of frequency <o0. Let us write this system as

^ = / ( * ) , (6.33)

where x and / are n vectors. Then the oscillation of frequency coo

is a solution x(t) of (6.33) such that x(t) has period l/coo. Thepostulated outside periodic force is represented by a term that hasfrequency <o or period l/<o. If we denote this term by a functiong(t), then the description of the system and the impressed forcingterm is given by

-£=f(x)+g(t). (6.34)

A mathematical description of entrainment of frequency would bea solution of (6.34) that has period 1/co and that is "close" (insome sense) to the given solution x(t). But the question of whethersuch a solution exists is a special case of the theory discussed inChapter 5, where Theorem 11 gives sufficient conditions for theexistence. At this stage, we simply emphasize that the entrainmentof frequency solution exists independent of the magnitude of g(t).Thus, a forcing term of very small magnitude may produce anoscillation of frequency co that has a large magnitude.

Now we have already shown that the Purkinje fiber can beregarded as a system with a natural frequency (the frequency of thespontaneous firings) and if we describe the electrical impulses thatarrive regularly at any given position on the Purkinje fiber by aperiodic forcing term added to the system (6.15), then, in general,the theory of entrainment of frequency will guarantee that thePurkinje fiber will oscillate or fire with a frequency equal to thefrequency of the forcing term.


Now we describe this application of entrainment of frequency tothe system (6.19)

dV

~dt =

dm

dt

dh

dt

dn

dt

1

e

1

£

{ ,m, , n),

mx(V)-m'

[ MmTm(V) \

MmTh{V) '

- vnJV)-n'

MmTn(V) JLet us assume that if e and y are small enough positive numbers,system (6.19) has a nontrivial periodic solution of period, say To.Denote this periodic solution by (V(t), m(t), Ti(t),n(t)) and writethe forcing term that describes the influence of the electricalimpulses that arrive regularly from the pacemaker region byUi(t,p\f2(t,p\f,(t,p\Ut)\ where for / = 1,2,3,4, the func-tion ft(t9 p) has continuous first partial derivatives in t and /?, andp is a small positive parameter. We assume that ft(t, p) has periodT(p) in t, where T(p) is a continuous real-valued function of psuch that lim^^Q^^) exists and

r, p) = 0. We consider the ques-

lim T(p) = T0.p->0

Finally, we assume that limtion of whether the system

dV 1


has a solution of period T(p) such that, as p -> 0, this solutionapproaches the periodic solution (V(t), m(t), /*(/), n(t)). But thisis a special case of the question answered by the theorem ofPoincare (Chap. 5, Theorem 11). Poincare's theorem states that if acertain hypothesis is satisfied, then such a periodic solution exists.The hypothesis concerns the nonexistence of periodic solutions ofthe linearized equation. This hypothesis can be expressed as thecondition that a certain determinant is nonzero, that is, a conditionof genericity is satisfied. Since the terms in system (6.32) are, atbest, good approximations, we may certainly assume that such acondition is satisfied. Hence, the periodic solution exists and itdescribes the oscillatory behavior of the Purkinje fiber at any givenpoint along the fiber where the period of the oscillation is theperiod of the regular behavior of the electrical impulses arrivingfrom the pacemaker region.

Of course our mathematical analysis is far from complete. Themost serious gap is the absence of a stability analysis. The periodicsolution (V(t), m(t),Ji(t)) should be globally asymptotically stableor at least have a good-sized region of asymptotic stability. Alsothe periodic solution given by Theorem 11 of Chapter 5 shouldhave a good-sized region of asymptotic stability. Finally, it isimportant to show that the entrainment of frequency takes placeeven if T(p) decreases very rapidly, that is, even if dT/dp at p = 0is very negative. The reason for this is that the natural frequency offiring of the Purkinje fiber is often much slower than the pace-maker frequency.

6.2.2.4 Further directions for study of the Noble model and othercardiac modelsIn the preceding sections, we have carried out a detailed

analysis of one of the basic mathematical problems that arises inthe study of the Noble model. In this section we will describebriefly several other mathematical problems that arise naturally inthe study of cardiac models. Since little or no work has been doneon most of these problems, we will merely state the problems and,in a few places, indicate possible approaches to the solution of theproblems. It should be emphasized that the brevity of these de-


scriptions is a poor measure of the importance or difficulty of theproblems.

Computation of period of periodic solution. The natural frequencyof the spontaneous oscillations of the Purkinje fiber is known fromlaboratory observations. Consequently, if a periodic solution of amodel such as the Noble model is determined, the next step is todetermine the period of the solution and compare this result withthe experimental result. Suppose the existence of a periodic solu-tion is established as in the earlier discussion, that is, by finding aclosed discontinuous solution and then proving that there is aperiodic solution near the closed discontinuous solution. Then onepossible way to estimate the periods of the solution is to estimatethe time interval required to "travel around" the discontinuoussolution. In making this estimate, we assume that travel along thesegments of the discontinuous interval that are described by thefast system is instantaneous. Thus to estimate the time intervalrequired to travel around the discontinuous solution is to estimatethe time required to travel along the segments of the discontinuoussolution that are governed by the slow system. Since the slowsystem is explicitly known, such an estimate can be carried out bystraightforward methods.

Description of the influence of drugs. As pointed out earlier, oneof the main purposes in studying mathematical models of electri-cally active cells is to obtain a clearer and more accurate picture ofhow the cell functions. In the earlier discussion, we were referringto the normal functioning of the nerve cell or Purkinje fiber. Butthe study of abnormal functioning, owing, for example, to theinfluence of drugs, is also very important, especially in cardiacmodels.

The first step in such a study of abnormal functioning is toderive a mathematical model that takes into account the presenceof agents such as adrenaline, tetrodotoxin, or manganese ions.There are many experimental studies in this direction. For work upto 1979, see Noble (1979), especially Chapter 8.

Description of other oscillations in cardiac models. In the earlierdiscussion, we considered only the normal spontaneous oscillationsof the Purkinje fiber. There is another oscillation of the Purkinje


fiber that occurs at low membrane potential. This oscillation hasbeen studied experimentally by Hauswirth, Noble, and Tsien (1969).In a sufficiently realistic mathematical model of the Purkinje fiber,a stable periodic solution that describes this oscillation shouldoccur.

Beeler and Reuter (1977) describe experimental conditions underwhich the myocardial fiber displays oscillatory behavior. Beelerand Reuter show by numerical analysis that their mathematicalmodel of the myocardial fiber (described in detail in Chapter 4 ofthis book) has oscillatory solutions if terms describing these experi-mental conditions are added to the original mathematical model.Qualitative analyses of these modifications of the model mightshow how these oscillations arise and are maintained.

6.2.2.5 Description of traveling waves in nerve conductionIn Chapter 2, Section 2.5, we derived the full Hodgkin-

Huxley equations, which can be used to describe the normalphysiological functioning of the nerve axon. The fullHodgkin-Huxley equations are the partial differential equations

1 d2V dV—+g

\V- VK)

dm— -am(l-m)-Pmm,

dh— ~ah(l-h)-phh,

dn— = a n ( l - n ) - & n .

As pointed out earlier, the general problem of solving the system(JP-Jf) is extremely difficult (see the discussion at the end ofChapter 5). However, we are interested only in solutions of (Jfi-Jf)that would describe an action potential. As shown in Chapter 2,these are traveling wave solutions, and to find such solutions, we


need to solve the system of ordinary differential equations:

1 d2V dV

+ {gKn4(V- VK) + gl(V- V

dm 1 r

— =--[am(l-m)-Pmm], (2.44)

(ft 1 ,

dn 1 r

— = - - [ a B ( l - n ) - j 8 B / i ] .

In these equations, 0 is a parameter that represents the velocity ofthe action potential.

As discussed in Chapter 2, the mathematical problem of findinga traveling wave solution becomes the question of how to de-termine a value of 0 for which there are solutions (F(£), m(£),/*(£), «(£)) of (2.44) such that l im^^Kd) = 0. Since system (2.44)is a nonlinear system, this question is an example of a nonlineareigenvalue problem. Numerous studies of this nonlinear eigenvalueproblem have been carried out and we will restrict ourselves here toa very brief listing of some of this work.

Before listing the work, however, it seems purposeful to indicatea little of the physiological significance of this nonlinear eigenvalueproblem. First of all, as discussed in Chapter 2, Hodgkin andHuxley made a numerical study of the problem in their originalwork. They found numerically a solution of the problem, that is,they found numerically a value of 0 and a corresponding solution(K(g), m(£), A(O, «(£)) of (2.44) such that l i m ^ K t f ) = 0. Thevalue of 0 that Hodgkin and Huxley determined represented anestimate of the velocity of the nerve impulse. This velocity had longsince been determined in the laboratory. As we pointed out earlier,the comparatively close agreement of 0 (=18.8 m/s) and theexperimentally observed velocity of 21.2 m/s is one of the most


spectacular triumphs of the Hodgkin-Huxley theory. Thus, wehave an example of the crucial physiological importance of thenonlinear eigenvalue problem.

On the other hand, in deriving the full Hodgkin-Huxley equa-tions, we take a step away from the reality of experimental data. Asit turns out, this step is not significant in the Hodgkin-Huxleywork. This is shown, of course, by the successful estimation of thevelocity of the nerve impulse. But that same step away from thereality of the laboratory may cause serious difficulties in the studyof other electrically active cells. For example, McAllister, Noble,and Tsien (1975) point out that there seem to be serious difficultiesin using their model of the cardiac Purkinje fiber to study conduc-tion of impulses along the fiber.

The decision to search for traveling wave solutions may also bean obstacle to realistic study of the action potential. The assump-tion that the action potential can be described by a traveling wavesolution seems eminently reasonable and it is well justified in thecase of the nerve axon (again by the successful estimate of thevelocity of the impulse). Whether this assumption holds for otherelectrically active cells remains an open question.

As our discussion indicates, the search for traveling wave solu-tions of the full Hodgkin-Huxley equations has considerable physi-ological importance. Now we turn to a brief listing of some of thetheoretical and qualitative results in this direction.

Summaries of work on traveling wave solutions of the fullHodgkin-Huxley equations may be found in the papers by Scott(1975) and Cohen (1976). Scott also describes results on travelingwave solutions of other models, for example, the full FitzHugh-Nagumo equations. Especially interesting is the study by Rinzeland Keller (1973) of a simplified FitzHugh-Nagumo equation. Thisis also discussed by Rinzel (1976), and further results, both qualita-tive and numerical, have been obtained by Rinzel and Terman(1982). Hastings (1974, 1976, 1975/76) has obtained results forclasses of equations like the full FitzHugh-Nagumo equations andlike the full Hodgkin-Huxley equations. A particularly interestingtechnique has been used by Carpenter (1977a, b) to study travelingwave solutions of the FitzHugh-Nagumo equations and the


Hodgkin-Huxley equations. Carpenter regards the subject as aproblem in singularly perturbed equations and uses the method ofisolating blocks.

So far, we have enumerated only studies concerning the existenceof traveling wave. Naturally, the question of stability properties ofthe traveling wave solution arises. Suppose (F(£), w(£), /*(£), «(£))is a suitable solution of (2.44). Then the actual traveling wavesolution of (JfZ-Jff) is (v(x - 0t\ m(x - 0t), h(x - 0t\ n(x - Ot)).To study the stability of this solution, we must compare, in somesense, to nearby solutions of {Jft-Jtf). That is, we study the stabilityof the solution of a partial differential equation. The stabilitytheory in Chapter 5 is not applicable. In the case just described, wedo not study the stability of the solution (v(i-), m(£), /*(£), w(£))°fsystem (2.44). For an introduction to this subject, see Scott (1975).For a detailed discussion, see Evans (1972a-c, 1975).

Appendix

In order to understand the experiments of Hodgkin andHuxley, a little background in electricity is needed. We will assumethat the reader is familiar with the notions of electrical charge,potential difference, current, resistance, and the units of measure ofthese. We will also need a basic relation among these, that is,Ohm's law, which is the statement

V=IR,

where V is the potential difference, / the current, and R theresistance.

In addition, we need to use the notion of capacitance. Sincecapacitance is a less familiar electrical concept and because it playsa crucial role in electrophysiology, we describe it in some detail.

Let A, B be conducting plates made, for example, of copper. LetA be connected to G, a gold leaf electroscope. If a charge is put onA by touching A with a charged rod, then, because the chargedistributes itself uniformly in the gold leaf, the two parts of thegold leaf have the same charge. Thus, they are repelled by oneanother and hence the leaf is deflected as indicated in Fig. A.I. Theamount of deflection measures the potential of A.

Now if plate 2?, which has no charge, is brought near A asindicated in Fig. A.I, there is a decrease in the deflection of the leafin G. This is because a rearrangement of electrons and ions in Boccurs and this rearrangement neutralizes part of the electriccharge on A. For example, if there is a positive charge uniformlydistributed on A and G, then electrons will move on B so that theyare as close as possible to A. Thus, some of the positive ions on Aand G will be bound to A. The result is that the potential on A isdecreased even though the electric charge on A remains constant.

Appendix

G ^

250

AFigure A.I.

Definition. The quantity of electricity (i.e., electrical charge)required to impart unit potential to a conductor is the capacitanceof the conductor. If q units of electricity raise the potential of theconductor by V units, the capacitance is given by

In the experiment just described, the capacitance of A is in-creased by bringing B closer to A. If an insulating material isplaced between A and B, the deflection of the gold leaf is furtherreduced, that is, the capacitance of A is further increased.

Definition. A pair of conductors carrying equal and oppositequantities of electricity and separated by insulating material is acapacitor or condenser. The conductors are called the plates and theinsulating material is called the dielectric. The capacitance C of acapacitor is given by

c

where Q is the quantity of electricity on either plate and V is thepotential difference between the plates.

Capacitance is measured in farads [F = 1 C/V (coulomb/Volt)].In other words, a capacitor has a capacitance of 1 F if the potentialdifference between the conductors is changed by 1 V if 1 C ofelectric charge is transferred from one of the conductors of thecapacitor to the other.

The capacitance of a capacitor depends upon: (i) size of theplates (capacitance increases with size of the plates); (ii) distance

Appendix 251

apart of the plates (capacitance increases when plates are broughtcloser together); (iii) intervening dielectric. [The influence of thedielectric can be visualized in terms of dipole molecules. Whenthe molecules are oriented properly, they neutralize the charges onthe plates, thus decreasing the potential and increasing the capaci-tance. The measure of the influence of the dielectric is the permit-tivity or dielectric constant. If a given capacitor with free space (orair) between the plates has capacitance C and the capacitanceincreases to C" when some other dielectric is substituted, thepermittivity of that dielectric is C/C\

Now we consider a simple electrical circuit (Fig. A.2) in whichthere is a capacitor C, a battery B, and a switch S. If the switch isclosed, there is a surge of electrons toward one plate of thecapacitor and a surge of electrons away from the other plate andtoward the battery terminal. This is not a current in the sense of acontinuous flow of electrons or ions through a conductor, but it is acurrent in the sense that there is movement of electrons. It issometimes called a displacement current.

If a displacement current ic occurs during a time interval A/(during which the switch is closed), an electric charge *'C(A/) isadded to one plate and removed from the other. A potentialdifference between the plates, denoted by AF, is developed and AFis proportional to the charge Q = ic(ht); that is,

where C is the capacitance of the capacitor. Then

AV

—I i-C

Figure A.2.

Appendix 252

Letting Af -> 0, we obtain

dV

Displacement currents are crucially important in our study be-cause it turns out that the membrane that surrounds the axonbehaves as a capacitor. Hence, the total current / across themembrane is

dVI=CM—+Ii9 (A.I)

where Ii is the current caused by the flow of ions across themembrane, CM(dV/dt) is the displacement current, and CM is thecapacitance per unit area of the membrane.

If we speak more precisely, we should say that it is assumed thatequation (A.I) is a correct description of the current /. Thisassumption is one of the basic assumptions in the work of Hodgkinand Huxley.

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INDEX

accommodation, 66action potential, 10

burst of, 11activation variable, 51Adelman, 12Adelman and FitzHugh, 66, 71Adelman and Palti, 186Adelman, Palti, and Senft, 186Adrian, 71Adrian, Chandler, and Hodgkin, 81Adrian and Peachey, 82all-or-nothing law, 11angstrom, 7anode break excitation, 60asymptotically orbitally stable

solution, 122asymptotically stable solution, 118autonomous equation, 105axoplasm, 7

Beeler and Reuter, 98-100, 245Bendixson criterion, 131Berkinblit, Dudzyavichus, Kovalev,

Fomin, Kholopov, andChailakhyan, 134, 185

bounded solution, 103

capacitance, 19, 149, 150membrane, 19

capacitor (or condenser), 250cardiac Purkinje fiber, 85-98

functions, 85-86Noble model, 86-89McAllister-Noble-Tsien model,

89-97Carpenter, 247catastrophe theory, 142Cesari, 118, 119choline, 25Cohen, 184Cole, 22

Cole, Antosiewicz, and Rabinowitz,182, 184

Cole and Moore, 70constant field equation, 77Cooley, Dodge, and Cohen, 184Cronin, 70, 107, 113, 117, 119, 121,

125, 128, 136, 137, 142current clamps, 60

degenerate system, 145dendrite, 6, 7depolarization, 14, 19dielectric, 250

dielectric constant, 251discontinuous solution of Levinson,

158-164discontinuous solution of Mishchenko

and Rozov, 144, 148displacement current, 251Dodge, 184drop point, 147

entrainment of frequency, 98, 239-243equilibrium point, 106-111

asymptotically stable, 116index of, 111-113

Evans, 248existence theorems (for differential

equations), 101-104extension of solution, 104extension theorem, 104-105

fast-motion equation system, 145FitzHugh, 11, 57, 65, 67, 68, 69, 70,

186, 190, 193, 210FitzHugh and Antosiewicz, 184FitzHugh method of reduction, 205,

211FitzHugh-Nagumo equations, 69, 190

singular perturbation analysis,195-202

Index 260

Frankenhaeuser and Hodgkin, 186,201

Frankenhaeuser and Huxley, 77-81,204

Goldman, 77Goldman and Schauf, 72Guttman, Lewis, and Rinzel, 185

Hagiwara and Oomura, 65Hahn, 123Hale, 138Hastings, 247Hauswirth, Noble, and Tsien, 97, 245Hodgkin, 11, 25, 43, 51, 71Hodgkin and Huxley, 1,12,16,17, 25,

29, 38, 39, 43, 44, 45, 46, 50, 51,59, 71, 73, 75, 180

Hodgkin and Huxley Nobel prize (in1961), 1, 53

Hodgkin-Huxley equations, 55, 56, 58,61, 62, 65, 66, 67, 68, 70, 75, 77,82, 181

derivation, 43-53full Hodgkin-Huxley equations, 62,

178, 183influence of temperature, 52mathematical status, 53-57scientific status, 40-43

Hodgkin, Huxley, and Katz, 17, 20,21, 24, 182

Hodgkin and Katz, 77Hopf bifurcation theorem, 132-133Hoyt, 70, 71Hoyt and Adelman, 71Hunter, McNaughton, and Noble, 71,

72Huxley, 71, 185, 201hyperpolarization, 14, 19

inactivation variable, 51independence principle, 34

application of, 44ionic current, composition of, 25-34

Jack, Noble, and Tsien, 54, 61, 62,181Jakobsson and Guttman, 65, 66junction point, 146

Krasnosel'skii, 113

LaSalle and Lefschetz, 117leakage current, 44Lecar and Nossal, 73, 204, 206, 209,

210

Lecar-Nossal stochastic model, 73,202

Lefschetz, 111Levinson, 143, 156,164,165,166,168,

169Lipschitz condition, 102Lloyd, 113

maximal solution, 104McAllister, Noble, and Tsien, 89-97,

183McDonough, 71, 185membrane potential, 9Minorsky, 241Mishchenko and Rozov, 138,143,144,

153, 154, 155myelinated axon, 7, 75, 76

sciatic nerve of clawed toad, 77

Nagumo, Arimoto, and Yoshizawa, 69Nemytskii and Stepanov, 111Nernst formula, 12neuron, 6, 7Noble, 217, 244Noble model of cardiac Purkinje fiber,

216-239nonregular point, 146

Ohm's law, 9, 24co-limit point, 125fi-limit set, 125orbitally stable solution, 122

permeability of membraneto Na ions, 37to K ions, 38unit of, 38

permittivity, 251phase asymptotically stable solution,

125Poincare, 54, 55, 68, 98, 243Poincare-Bendixson theorem, 130potassium current, 30Purkinje fiber (see cardiac Purkinje

fiber)

qualitative analysis, 188

Ranvier, nodes of, 76refractory period, 11

absolute, 11relative, 11role of, 15

Index 261

response asymptotically orbitally, 122active and local, 10 orbitally, 122passive and local, 10 phase asymptotically, 125

resting potential, 10 uniformly, 125maintenance of, 15 Stoker, 240

Rinzel, 184, 247 striated muscle fiber, 81-85Rinzel and Terman, 247

Ten Hoopen and Verveen, 204threshold potential, 10

Scott, 8, 54, 76 Tille, 70Sell, 125 traveling wave solution (of fullSell's theorem, 126 Hodgkin-Huxley equations), 63Sibuya, 143, 173-178 Troy, 215space-clamp experiment, 18sodium current, 30 uniformly stable solution, 125sodium equilibrium potential, 27speed of nerve impulse, 8, 65 velocity of action potential, 65

relation to radius of axon, 76 ventricular myocardial fiber,squid giant axon, 6 Beeler-Reuter model of, 98

diameter, 6 Verveen and Derksen, 204discovery (in 1936), 6 voltage-clamp experiment, 22, 74length, 6structure, 7 Young, 6

stable solution, 118asymptotically, 118 Zeeman, 69, 142

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