Intermittency and Di usion in the Hodgkin-Huxley Model...Gaspar Filipe Santos Magalh~aes Gomes Cano...

Intermittency and Diffusion

in the Hodgkin-Huxley Model

Gaspar Filipe Santos Magalhães Gomes Cano

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor: Prof. Rui Manuel Agostinho Dilão

Examination Committee

Chairperson: Prof. Lúıs Filipe Moreira Mendes

Supervisor: Prof. Rui Manuel Agostinho Dilão

Member of the Committee: Prof. Maria Teresa Ferreira Marques Pinheiro

April 2016

Acknowledgments

I would like to thank my supervisor, Prof. Rui Dilão. Without his ideas, insight, support and

thorough supervision this work would not have been possible.

I would also like to thank my family and all those who gave me their friendship and support while

I worked on this thesis.

i

Abstract

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I inter-

mittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. The

restriction of the Hodgkin-Huxley equations to the center manifolds associated with the Hopf bifurca-

tion points, subcritical and supercritical, can be described by the two-dimensional normal form of the

codimension 2 Bautin bifurcation. For the Hodgkin-Huxley spatially extended model, describing the

propagation of action potentials along axons, we show the existence of space propagating regular and

chaotic diffusion waves, as well as type I spatial intermittency and a new type of chaotic intermittency.

Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the

transmembrane potential and is characterised by the existence of a sequence of action potential spikes

occurring at irregular time intervals.

Keywords

Hodgkin-Huxley model, type I intermittency, Bautin bifurcation, diffusion waves, chaotic intermit-

tency.

iii

Resumo

Mostra-se que os potenciais de acção no modelo neuronal de Hodgkin-Huxley resultam de um

fenómeno de intermitência do tipo I, na vizinhança de uma bifurcação sela-nó de ciclos limite. A

restrição das equações de Hodgkin-Huxley às variedades centrais associadas aos pontos de bifurcação

de Hopf, subcŕıtico e supercŕıtico, pode ser descrita pela forma normal bi-dimensional da bifurcação de

Bautin de codimensão 2. Para o modelo de Hodgkin-Huxley com componente espacial, que descreve

a propagação de potenciais de acção ao longo de axónios, mostra-se que existem ondas de difusão

regulares e caóticas, bem como intermitência espacial do tipo I e um novo tipo de intermitência caótica.

A intermitência caótica ocorre na transição de um regime turbulento para um regime de repouso do

potencial transmembranar e é caracterisada pela existência de uma sequêcia de potenciais de acção

separados por intervalos de tempo irregulares.

Palavras Chave

Modelo de Hodgkin-Huxley, intermitência do tipo I, bifurcação de Bautin, ondas de difusão, inter-

mitência caótica.

v

Contents

1 Introduction 1

1.1 The Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Hodgkin-Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 The voltage-gated channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The Action Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 The Diffusion free

Hodgkin-Huxley model 13

2.1 Analysis of dynamical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 The unstable knee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The Bautin bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Type I intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Diffusion in the

Hodgkin-Huxley model 23

3.1 Propagation of action potentials along the axon . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Dynamical stability and bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Wave velocity and dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Type I spatial intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Chaotic Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Conclusions 31

Bibliography 33

Appendix A Intermittency near a saddle-node limit cycle bifurcation A-1

Appendix B Stability of the steady state of the Hodgkin-Huxley equations B-1

vii

List of Figures

1.1 Schematic of a regular neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Cover of the 1963 Nobel Prize Programme, [1]. . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Electrical circuit analogous to a cell membrane permeable to two different ionic species.

Voltages E1 and E2, imposed by the batteries, result from the active transport done by

ionic pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Electrical circuit analogous to the Hodgkin-Huxley model, with channels for Na+ and

K+, a leakage channel RL for Cl− and an input transmembrane current density i.

Resistances RNa and RK are variable, since they represent the voltage-gated channels

for sodium and potassium, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Electrical circuit showing the spatial component of the Hodgkin-Huxley model, where

x is the direction of the axon length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Left: membrane potential as a function of time. Right: two-dimensional (V, n) cut of

the (V, n,m, h) phase-space. a) The perturbation felt by the neuron was not enough

to generate an action potential, and it quickly falls back to its rest state. b) The

perturbation was enough to produce an action potential response; the neuron fires. . . 9

1.7 Rate of open n, m and h gates, ionic current densities and membrane potential during

an action potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 Property 1) Voltage (a) and phase-space (b) response of the Hodgkin-Huxley (HH)

equations (1.16) as a function of time, for the stimulation signal i(t) = 5 µA/cm2 > Itr,

during t = 35 ms, and i(t) = 0 otherwise. The impulse is above threshold. For i(t) =

5 µA/cm2, ∆ttr is 1.5 ms; for t1 = 35 ms, Itr is 2.25 µA/cm2. . . . . . . . . . . . . . . 11

1.9 Property 2) We perturbed the cell with a square wave, with a time difference between

spikes ∆t = 17.5 ms. For these parameter values, Itr = 2.25 µA/cm2, ∆ttr ' 1.5 ms

and the refractory interval is ∆tr ' 17.5 ms. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 a) Bifurcation diagram for the diffusion free HH equations (2.1) at T = 6.3 ◦C as a func-

tion of the transmembrane current density parameter i. saddle-node limit cycle (SNLC)

bifurcation occurs for i = I0 = 6.26 µA/cm2, the subcritical Hopf bifurcation for i =

I1 = 9.77 µA/cm2 and the supercritical Hopf bifurcation for i = I2 = 154.52 µA/cm

2.

b) Variation of Hopf bifurcation points I1 and I2 with temperature. Calculated with

XPPAUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

ix

2.2 Two-dimensional cuts of the orbits of the limit cycles for a) i = 7 µA/cm2 and b)

i = 50 µA/cm2 in equations (2.1). Full lines are stable cycles, dotted lines are unstable

ones. The full dot in a) represents a stable fixed point, and the open dot in b) an

unstable one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 a) Enlargement of the region in the neighbourhood of the “knee” that occurs in the

interval [I3, I4] = [7.84, 7.92] µA/cm2. b) Two-dimensional cut of the limit cycle orbits

in phase-space for i = 7.88 µA/cm2. Calculated with XPPAUT. . . . . . . . . . . . . . 16

2.4 Disappearence of the one-to-many feature of the unstable limit cycle at T = 7.72 ◦C.

Calculated with XPPAUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Normal Bautin bifurcation parameters in cartesian (β1,β2) and polar (ρ,θ) coordinates,

and the three different regions of equations (2.2). Border 1-2: θ = arccos

(√4+ρ2−2ρ

)- SNLC bifurcation; border 2-3: subcritical Hopf bifurcation (H+); border 3-1: super-

critical Hopf bifurcation (H−). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Bifurcation diagram for equations (2.2) as a function of the parameter θ = arctan(−β2/β1).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Limit cycles and fixed points (red) in the three regions shown in figure 2.5, and trajec-

tories (black) followed by the solutions of equations (2.2). Dashed circles are unstable

limit cycles, while full circles are stable ones; full dots are stable fixed points, while

hollow dots are unstable ones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 Membrane action potential response of the HH equations (2.1), for i = 6.20 µA/cm2 < I0

and initial conditions p0 = (V∗(0) + ∆V0, n

∗(0),m∗(0), h∗(0)). a) ∆V0 = 1.6 mV, b)

∆V0 = 1.7 mV, c) ∆V0 = 1.9 mV, and d) ∆V0 = 5.5 mV. The SNLC bifurcation occurs

at i = I0 = 6.26 µA/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 SNLC intermittency for the HH model. a) Logarithm of the number of spikes N of

the action potential in the left vicinity of the SNLC bifurcation, as a function of the

logarithm of ε = I0−i. The slope of the fitted line is s = −0.505, in agreement with (2.3).

b) Next amplitude map for the HH model, for the parameter value i = 6.259 µA/cm2,

or ε = 0.001. VN is the maximum value of the action potential spike number N . At the

SNLC bifurcation, ε = 0, the parabolic profile shown touches the dotted line VN+1 = VN . 21

3.1 Spatial solutions of the HH equations (3.1), for D/∆x2 = 20 ms−1 at time t = 100 ms,

for two different values of the transmembrane current density. In a), i0 = 50 µA/cm2

and the transmembrane potential converges to the steady state p∗(0) along the axon. In

b), i0 = 100 µA/cm2 and the transmembrane potential develops a propagating periodic

action potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Bifurcation diagram of the HH extended model (3.1), as a function of the transmembrane

current density i0 at the axon boundary and of the diffusion coefficient D/∆x2. We show

two different types of intermittency, oscillations and spatial chaos (dark grey regions and

dotted lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

x

3.3 a) Period (and time intervals) of the oscillatory solutions of the HH extended model (3.1),

as a function of the current density i0, for the diffusion coefficient D/∆x2 = 20 ms−1.

In the regions [I∗2 , I∗3 ] (b) and [I

∗4 , I

∗5 ] (c), the solutions of the HH extended model show

chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 a) Wave vlocity of the action potential for D/∆x2 = 20 ms−1, for different values of i0

within the parameter region [I∗1 , I∗5 ]. b) Wave velocity as a function of parameter i0 for

five different diffusion coefficients of the HH equations. . . . . . . . . . . . . . . . . . . . 28

3.5 Dispersion relation of action potential waves of the HH model, for five different values

of the diffusion coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Type I intermittency for the HH extended model (3.1), with D/∆x2 = 20 cm−1. a)

Logarithm of the number of spikes N of the action potential in the left vicinity of I∗1 =

56.012 µA/cm2. The slope of the fitted dotted line is s = −0.506, in good agreement

with (2.3). b) Next amplitude map for the parameter value i0 = 56.010 µA/cm2 or

ε = 0.002. VN (x = L/2) is the maximum value of the action potential spike number N ,

measured at the middle of the spatial domain [0, L]. The dotted line is the graph of the

equation VN+1 = VN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Chaotic intermittency for the HH extended model (3.1), with D/∆x2 = 20 cm−1. a)

Logarithm of the number of spikes N of the action potential in the right vicinity of I∗5 ,

for ε = i0 − I∗5 ∈ [0.001, 0.140]. The dotted line has slope s = −0.5. b) Next amplitude

map for the parameter value i0 = 339.369 µA/cm2 or ε = 0.010. VN (x = L/2) is the

maximum value of the action potential spike number N , measured at the middle of the

spatial domain [0, L]. Chaotic intermittency does not show any scaling behaviour on

the number of spikes as a function of ε. The dotted line is the graph of the equation

VN+1 = VN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xi

Abbreviations

HH Hodgkin-Huxley

SNLC saddle-node limit cycle

xiii

1Introduction

Contents1.1 The Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Hodgkin-Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The Action Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1

1.1 The Neuron

In the nervous system, neurons are the excitable cells that are responsible for receiving stimuli and

conducting impulses to other cells. They are able to transmit these signals over long distances across

the body, and each can form up to tens of thousands of connections [2]. These connections are called

synapses, and they can be both electrical and chemical [3]. Neurons work as the electrical wiring of

the body, forming intricate and complex networks, responsible for our every thought and decision,

movement and feeling. Just a regular human brain has dozens of billions of neurons [4, 5].

Figure 1.1: Schematic of a regular neuron.

The diagram in figure 1.1 shows the three different parts that form a neuron. From left to right: the

dendrites, which receive synaptic signals from other neurons or cells; the soma, the neuron body that

contains the nucleus, responsible for the vitality of the cell; and the axon, which is the thinner cable-

like structure, that extends from the soma, and is responsible for transmitting the signal generated

by the neuron, and ends in the tree-shaped axon terminal, which transmits the propagated signal to

other cells through synapses. Axons can have up to thousands of times in length the diameter of the

soma. An electrical signal that propagates along the axon is called an action potential [2].

Each neuron behaves like a complex dynamical system, which can produce a variety of outcomes,

according to the biochemical conditions it is subjected to, the signals it receives and the connections

it makes. Neuronal networks can have computational properties, allowing for all kinds of calculations

and outcomes, which is a field of study in itself (see, for example, [6]). In this thesis we shall focus

merely on the dynamical properties of a single neuron-like cell.

All cells have a potential difference between their inside and outside, known as the membrane

potential. This potential difference arises from the difference in ionic concentrations on both sides of

the membrane. Since cell membranes are lipid bilayers which are by themselves impermeable to ions,

membranes possess multiple specialised channels and pumps made of proteins that are responsible for

the transport of specific ions across the membrane. Pumps are responsible for moving ions against their

concentration gradient, and for this they need to consume energy – active transport. On the other

hand, channels may open and close, but if open they simply let ions pass through them according

to the concentration gradient, and therefore do not consume any energy – passive transport. When

2

ions are exchanged across the cellular membrane there are two competing effects taking place: the

difference in concentration in the two regions and the electrostatic forces between all the charged ions.

This means that there is a complex system at work, regulating how the ions flow across the membrane

while the channels are open. Channels can also be divided in two different categories – gated and

non-gated. Non-gated channels are always open, while gated channels open and close as a function of

some variable (e.g. the membrane potential itself) [7].

Since each channel only lets one specific ionic species through, a cell membrane will be as permeable

to some ion as the number of currently open channels for that ion. Each ion has a specific equilibrium

state for inside and outside concentrations, which is given by its Nernst potential

Eion =RT

zFln

[ion]i[ion]o

, (1.1)

where Eion is the Nernst potential for some ion, R is the gas constant, T is the absolute temperature,

F is the Faraday constant, z is the ion valence and [ion] represents the ionic concentration [4, 7]. In a

neuron, the main ions responsible for the membrane potential are potassium (K+), sodium (Na+) and

chloride (Cl−). Considering the electric field is constant across the membrane, and that all ions move

independently, the Nernst equation can be generalized for the three most relevant ionic species: K+,

Na+ and Cl−, giving the Goldman-Hodgkin-Katz (GHK) equation, [8]:

V =RT

FlnPK[K

+]i + PNa[Na+]i + PCl[Cl

−]o

PK[K+]o + PNa[Na

+]o + PCl[Cl−]i

, (1.2)

where V is the potential difference between the outside and inside of the membrane and P is the

permeability for each ionic species1. Through this, one can calculate the membrane potential, just by

knowing the ionic concentrations and their permeabilities.

The stable membrane potential experienced when all the different ionic fluxes are in balance is

called the resting potential. Since, at rest, the concentration of K+ is much higher inside than outside

the cell, while Na+ and Cl− have higher concentrations on the outside, the tendency will be for K+ to

leave the cell and for Na+ and Cl− to enter it. However, if no pumps existed, and ion channels were

the only way ions could travel across the membrane, K+ would tend to disappear from the cell and

Na+ and Cl− would continuously flood its inside. Eventually there would be a loss in the gradients of

ionic concentration that is necessary for the cell to maintain its resting potential. This is where the

ion pumps come in, keeping the balance of ionic concentrations across the membrane in check. The

main example of this is the Na-K pump that exchanges three Na+ ions from the inside for two K+

ions from the outside, being one of the most important ion transporters in cell membranes [7].

As we have just seen, the membrane potential in a neuron will depend on the ionic concentrations

inside and outside the cell. However, these concentrations are dependent on the membrane potential

itself, since the channels through which they pass open and close as a function of it. This means that

the system is highly complex and one must take all of these factors into account, in order to describe

it mathematically. It is in this description that the model developed by Hodgkin and Huxley was

a pioneer, since it was the first to successfully describe and predict the behaviour of the membrane

potential. In the following section we explain the Hodgkin-Huxley model in detail.

1Membrane permeability is P = D/L, where L is the membrane width and D the diffusion coefficient for the species.

3

1.2 The Hodgkin-Huxley Model

In 1952, Alan Lloyd Hodgkin and Andrew Huxley published a set of papers [9–12] where they

explained in detail how they arrived at their successful description of the membrane potential be-

haviour in the giant axon of the squid Loligo. This revolutionary work earned them the Nobel Prize

in Physiology or Medicine in 1963 (see figure 1.2).

Figure 1.2: Cover of the 1963 Nobel Prize Programme, [1].

Hodgkin and Huxley were the first to measure the membrane potential in an axon. Since the squid

giant axon can have up to one meter in length and one milimeter in diameter [13], they were able to

apply a voltage-clamp technique which allowed them to successfully perform these measurements.

In [9–12], they establish an analytical model that would describe how and why the potential behaved

in the observed way. For this, they modelled the cell membrane as an electrical circuit. This circuit has

three kinds of components: the ion channels, which act as resistors; the ion concentration gradients,

acting as batteries; and the impermeable membrane, acting as a capacitor.

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•

•

•

•

R1 R2 Cm

E1 E2

inside

outside

i2i1 iC

Figure 1.3: Electrical circuit analogous to a cell membrane permeable to two different ionic species. VoltagesE1 and E2, imposed by the batteries, result from the active transport done by ionic pumps.

4

In figure 1.3, we show a schematic of an electrical circuit, with two different ionic channels, R1 and

R2, for ionic species with Nernst potentials E1 and E2, and a capacitor Cm, representing the membrane

impermeability. Due to the effect of the ionic pumps that perform active transport across the membrane

(like the Na-K pump), we can consider that the ionic gradients and the Nernst potential always remain

the same. This is why we can represent them as a battery that imposes a constant voltage drop across

the membrane. From figure 1.3 and Kirchhoff’s circuit laws we get

iC + i1 + i2 = 0,

−i1R1 − E1 + (Vo − Vi) = 0,

−i2R2 − E2 + (Vo − Vi) = 0.

(1.3)

where: the first equation comes from applying the junction rule to the top (outside) node; the second

equation from applying the loop rule to the branches of R1 and Cm; and the third from applying the

loop rule to the branches of R2 and Cm. Considering the membrane potential to be V = Vo − Vi,

where the subscripts o and i distinguish the outside and inside of the cell membrane, respectively, we

have iC = CmdVdt , and equations (1.3) become

CmdV

dt= −g1(V − E1)− g2(V − E2), (1.4)

where g = 1/R is the electrical conductance. Through this, it is possible to know how the membrane

potential varies with time, just by knowing the membrane capacitance, the conductances of the two

ionic channels, and the Nernst potentials of the ions passing through them.

The actual circuit considered by Hodgkin and Huxley was a bit more complex than this one.

They considered the membrane was permeable to three ionic currents: one for sodium, another for

potassium and a leak current, mostly representing the flow of chloride ions. During their measurements

of membrane potential response, they also introduced an external current into the membrane through

a patch-clamp-like technique, to see how the potential would respond to certain stimuli.

Cm

i

ENa

RNa

EK

RK

EL

RL•

•

•

•

•

•

•

•

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•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

imem

imem

inside

outside

Figure 1.4: Electrical circuit analogous to the Hodgkin-Huxley model, with channels for Na+ and K+, aleakage channel RL for Cl

− and an input transmembrane current density i. Resistances RNa and RK arevariable, since they represent the voltage-gated channels for sodium and potassium, respectively.

5

In figure 1.4, we show what the actual circuit looks like. The resistors for sodium and potassium

are voltage-gated, so they are represented with an arrow across them. This means their resistance will

vary as a function of the membrane potential itself. In section 1.2.1 we explore this in detail. RL and

EL represent the resistance and Nernst potential for the mentioned leak current, while i is the current

density injected across the membrane (the one Hodgkin and Huxley were able to control during their

experiments), and imem is the total current density crossing the membrane. Proceeding in the same

way we did to get equation (1.4), we arrive at

CmdV

dt= imem − ḡNa(V − ENa)− ḡK(V − EK)− gL(V − EL)− i, (1.5)

where the variable conductances corresponding to the voltage-gated channels are represented by ḡ.

If one wants to study electrical signal propagation in an axon, the spatial component also needs

to be taken into account. For this, the neuron is considered as an electrical cable (as detailed, for

example, in [14]) with one spacial dimension x, as seen in Figure 1.5.

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Rout Rout

Rin Rin

imem

imem

iin

iout

x x+dx

outside the membrane (Vo)

inside the membrane (Vi)Figure 1.5: Electrical circuit showing the spatial component of the Hodgkin-Huxley model, where x is thedirection of the axon length.

Following the current densities in figure 1.5 (analogous to what is done in [15]), if the ionic current

densities inside and outside the cell membrane are considered to be ohmic, it arises by Ohm’s law that

Vo(x+ dx, t)− Vo(x, t) = −ioutRout,

Vi(x+ dx, t)− Vi(x, t) = −iinRin,(1.6)

where Rout and Rin are the electrical resistances of the extra- and intracellular mediums. Taking the

continuous limit, equations (1.6) become

∂Vo∂x

= −ioutRout,

∂Vi∂x

= −iinRin.(1.7)

For any circuit node at position x, Kirchhoff’s law gives

iout(x, t)− iout(x+ dx, t)− imem = 0,

iin(x, t)− iin(x+ dx, t) + imem = 0,(1.8)

which, again at the continuous limit, becomes

∂iout∂x

= −imem,

∂iin∂x

= imem.

(1.9)

6

Taking the derivative of (1.7) together with equations (1.9), we get

∂2Vo∂x

= imemRout,

∂2Vi∂x

= −imemRin,(1.10)

and, since V = Vo − Vi, (1.10) gives

D̃∂2V

∂x2= imem, (1.11)

where D̃ is the diffusion coefficient, 1Rout+Rin. We can now replace this in equation (1.5), which

becomes

CmdV

dt= D̃

∂2V

∂x2− ḡNa(V − ENa)− ḡK(V − EK)− gL(V − EL)− i. (1.12)

1.2.1 The voltage-gated channels

We can now look at the ion channel conductances and how they depend on the membrane potential.

Like we said above, the channels for sodium and potassium are voltage-gated, which means they have

gates that will open or close, depending on the membrane potential they experience. Following [16],

this model can be described by the diagram

Cα(V)

�β(V)

O, (1.13)

where C and O are the closed and open “states” of the gate, and α(V ) and β(V ) are voltage dependent

rates at which the gate goes from one state to another. If we consider the fraction of open gates to be

z, then 1− z is the fraction of closed gates, and from mass action law we get

dz

dt= α(V )(1− z)− β(V )z. (1.14)

There are two types of gates at work – activation gates and inactivation gates. Their biochemical

differences shall not concern us here, as we are merely interested in whether they are open or closed,

and for this we only need to know their α(V ) and β(V ) functions. For the K+ and Na+ conductances,

Hodgkin and Huxley proposed thatḡK = gKn

4,

ḡNa = gNam3h,

(1.15)

where n represents the probability of an activation gate for potassium to be open, while m and h

represent the probability of an activation and inactivation gate for sodium to be open, respectively.

The constants gK and gNa represent the maximum conductances for K+ and Na+. For the ions to

pass through a channel, all gates in that channel have to be open. Therefore, we can interpret (1.15)

in the following way: if potassium channels have four activation gates, and n is the probability of

one of them being open, then the total fraction of open potassium channels is n4; if sodium channels

have three activation gates, each with probability m of being open, and one inactivation gate with

probability h of being open, then the total fraction of open sodium channels is m3h. Multiplying the

fraction of open channels by the maximal conductance g, gives the conductance ḡ seen in (1.15).

7

From (1.12), (1.14) and (1.15) we arrive at the complete set of four coupled differential equations

that make the Hodgkin-Huxley (HH) model:

Cm∂V

∂t= D̃

∂2V

∂x2− gNam

3h(V − ENa)− gKn4(V − EK)− gL(V − EL)− i,

∂n

∂t= αn(V )(1− n)− βn(V )n,

∂m

∂t= αm(V )(1−m)− βm(V )m,

∂h

∂t= αh(V )(1− h)− βh(V )h.

(1.16)

Hodgkin and Huxley were then left with the problem of modelling the functions α and β for n,

m and h. They arrived at them heuristically, so that they would almost perfectly adjust to the

experimental data:

αn = 0.01φV + 10

e(V+10)/10 − 1, βn = 0.125φe

V/80,

αm = 0.1φV + 25

e(V+25)/10 − 1, βm = 4φe

V/18,

αh = 0.07φeV/20, βh = φ

1

e(V+30)/10 − 1,

φ = 3(T−6.3)/10,

(1.17)

where φ is the temperature coefficient that scales α and β for different temperature values. In these

equations, the transmembrane potential V has units of mV, the transmembrane current density i that

is applied to the cell is given in µA/cm2 and time is measured in ms. In accordance with figures 1.4 and

1.5, positive values of i correspond to currents flowing from the outside into the cell. The model has

been calibrated for the squid giant axon at the temperature T = 6.3 ◦C, so that the resting potential

when no current is applied (i = 0) is V = 0. The values of the constants are: Cm = 1 µF/cm2,

gNa = 120 mS/cm2, gK = 36 mS/cm

2 and gL = 0.3 mS/cm2, where S=Ω−1 (siemens) is the unit

of conductance. For this calibration, the Nernst equilibrium potentials are ENa = −115 mV, EK =

12 mV and EL = −10.613 mV, [12].

For D̃ = 0, the HH equations describe the potential drop across the walls of a globular cell and the

ionic gradients inside the cell are negligible. Hodgkin and Huxley have estimated that the transmem-

brane diffusion coefficient is D̃ = a/(2R2), where a is the radius of the axon (considered as a cylinder)

and R2 is the specific resistivity along the interior of the axon. For the case of the squid giant axon,

a = 238 µm, R2 = 35.4 Ω cm and D̃ = 3.4× 10−4 S, [12].

The electrophysiological state of any cell can be described by a HH type model, provided its electric

state is controlled by the opening and closing of voltage sensitive channels.

1.3 The Action Potential

The success of the HH model in describing the dynamics of certain types of neurons relies on its

accuracy in reproducing experimental facts of patch-clamp experiments, including action potentials

and their threshold properties, which we will now explain in detail.

If we take D̃ = 0 in equations (1.16), assuming the neuron is a globular cell without diffusion, its

electrophysiological steady state is described by the vector quantity p∗(i) = (V ∗(i), n∗(i),m∗(i), h∗(i)),

8

where the input transmembrane current density i is considered as an external parameter. When the

neuron is at rest (i.e. i = 0, no transmembrane current is being received) its steady state p∗(0) is stable

and is the unique limit set of the HH equations (1.16). These simple facts are well known and discussed

in the literature, [17], [4] and [7], among others, and we explain them in chapter 2. Due to this stability

in the vicinity of i = 0, small variations in the parameter i for a short period of time produce the same

dynamic effects as changes in the initial conditions. In both cases, the electrophysiological state is

displaced from rest, and we say the neuron has been perturbed. Its response to this perturbation can

either be a small oscillation in phase-space that soon goes back to the rest state (figure 1.6a), or it can

be a very characteristic large excursion through the phase-space (figure 1.6b). (Here, we are focusing

exclusively on the simple (globular) cell with D̃ = 0 in equations (1.16). In chapter 3 we look at how

action potentials diffuse along the axon).

0 5 10 15 20 25 30 35

0

20

40

60

80

100

t (ms)

-V(mV)

0.1 0.3 0.5 0.7 0.9

0

20

40

60

80

100

n

a)

0 5 10 15 20 25 30 35

0

20

40

60

80

100

t (ms)

-V(mV)

0.1 0.3 0.5 0.7 0.9

0

20

40

60

80

100

n

b)

Figure 1.6: Left: membrane potential as a function of time. Right: two-dimensional (V, n) cut of the(V, n,m, h) phase-space. a) The perturbation felt by the neuron was not enough to generate an action potential,and it quickly falls back to its rest state. b) The perturbation was enough to produce an action potentialresponse; the neuron fires.

This large excursion through the phase-space is called an action potential. When it is produced,

the neuron is said to have fired. The distinction between both cases is quite clear, and there is a hard

threshold between them.

In figure 1.7 we look at what happens to the gates in the channels for sodium and potassium and

how the ionic current densities behave during this action potential response. At t = 0, we have rest

initial conditions p0 = p∗(0). There are many more potassium channels open than sodium channels

(m30h0

020406080100

-V (mV)

-3-2-101

ioniccurrent

(mA/cm

2 )Na+

K+

0 5 10 15 20 25 30

0.20.40.60.8

t (ms)

rateof

opengates

n

m

h

Figure 1.7: Rate of open n, m and h gates, ionic current densities and membrane potential during an actionpotential.

potential for potassium (EK = 12 mV) and chloride (EL = −10.613 mV). Once the cell is perturbed

(at t = 0, with a 5 µA/cm2 current density i, lasting 5 ms), the membrane potential V is depolarized

(becoming more negative), which causes the m gates to open very rapidly, bringing a large influx of

sodium ions into the cell, and the membrane potential to become even more depolarized in a rapid

fashion, almost reaching the Nernst potential for sodium (ENa = −112 mV). Simultaneously to this,

the inactivation gates h for sodium have been closing during this depolarization, and the potassium

activation gates n have been opening, expelling potassium ions from the cell. Both the rise of n and

the decrease in h counter-balance the influence of the now almost completely open activation m gates.

As V reaches its peak negative value, the m gates close very rapidly, making the potential rise very

abruptly and an overshoot happens, bringing the potential close to the value of EK, higher than its

rest state. During the time it takes for the potassium n gates and sodium h inactivation gates to

recover their rest value, the cell is in its relaxation phase, in which it gradually recovers its rest value.

During this recovery time, as we shall see ahead, the cell is much less prone to firing again. The

ionic current densities in figure 1.7 are given by iion = gion(V − Eion). It is important to note that,

according to the direction of the arrows in figure 1.3, these currents have been defined to be positive

when they flow from the outside to the inside of the membrane. However, Na+ and K+ have a positive

charge, and a positive inward current implies that positive ions are leaving the cell, while a negative

inward current implies that positive ions are entering the cell. Thus, it follows that, during an action

potential response, sodium ions enter the cell, while potassium ions leave it, which is consistent with

10

the description we made in section 1.1.

The two main properties about action potentials and their associated threshold effect are the

following:

1) Consider a cell or neuron at the steady state p∗(0). Imposing a current density to the cell during

some short time t1, there exists a time interval ∆ttr and a threshold value Itr such that, if i(t) ≥

Itr, for ∆ttr ≤ t ≤ m∆ttr, with finite real m > 1, and i(t) = 0 otherwise, the system develops a

spike in its voltage response V (t) — the action potential. Then, the voltage attenuates in time

and the system returns to the stable steady state p∗(0). If i(t) < Itr, for ∆ttr ≤ t ≤ m∆ttr,

the voltage response V (t) attenuates in time to the stable steady state p∗(0) and the action

potential does not develop - Figure 1.8 shows this action potential response for i(t) = 5 µA/cm2

for t1 = 35 ms. For this value of i, ∆ttr is 1.5 ms; while for this t1, Itr = 2.25 µA/cm2.

V*(0) V*(5)

0

20

40

60

80

100

-V(mV)

a)

0 10 20 30 40 50 600

5

10

t (ms)

i(μA/cm2

)

b)

x*(0)

x*(5)

-20 0 20 40 60 80 100 1200.00.2

0.4

0.6

0.8

1.0

-V (mV)

n

Figure 1.8: Property 1) Voltage (a) and phase-space (b) response of the HH equations (1.16) as a function oftime, for the stimulation signal i(t) = 5 µA/cm2 > Itr, during t = 35 ms, and i(t) = 0 otherwise. The impulseis above threshold. For i(t) = 5 µA/cm2, ∆ttr is 1.5 ms; for t1 = 35 ms, Itr is 2.25 µA/cm

2.

2) Perturbing the HH equations (1.16) with a sequence of current density rectangular pulses above

threshold Itr, and if the temporal differences between pulses are above some time interval ∆tr,

then the solutions of the HH equations develop the same number of pulses as in the current density

exciting signal. We call ∆tr the refractory interval - Figure 1.9. If we perturb continuously the

HH model with a current density made of a sequence of rectangular pulses above threshold, and

the temporal distance between pulses is below the refractory interval ∆tr, then the number of

pulses in the voltage response is smaller than the number of pulses of the exciting signal.

In figures 1.8 and 1.9, we show the voltage response in a globular neuron (D̃ = 0) that was at

rest (p0 = p∗(0)) to a long current perturbation and to a square wave type perturbation. These two

solutions illustrate the properties of the transient action potential behaviour as explained in the two

cases described above – the threshold property and the refractory property.

11

0

20

40

60

80

100-V(m

V)a)

0 20 40 60 800

5

10

t (ms)

i(μA/cm2

)b)

x*(0)

-20 0 20 40 60 80 100 1200.00.2

0.4

0.6

0.8

1.0

-V (mV)

n

Figure 1.9: Property 2) We perturbed the cell with a square wave, with a time difference between spikes∆t = 17.5 ms. For these parameter values, Itr = 2.25 µA/cm

2, ∆ttr ' 1.5 ms and the refractory interval is∆tr ' 17.5 ms.

1.4 Thesis Outline

One of the main goals of this thesis is to explain the origin of the threshold effect and the appearance

of action potential spikes in the HH model equations and to characterise it dynamically. In chapter

2, we summarise the main dynamical properties of the solutions of the diffusion free (D̃ = 0) HH

equations (1.16). It is shown that, for the parameter values described above and bifurcation parameter

i, the asymptotic states of the HH four-dimensional equations (1.16) can be described by a codimension

2 Bautin bifurcation scenario. Upon variation of the bifurcation parameter i, this bifurcation scenario

has a subcritical and a supercritical Hopf bifurcation and a global saddle-node bifurcation of limit

cycles. We extend the concept of type I intermittency associated to saddle-node bifurcations of fixed

points to intermittency of saddle-node bifurcation of limit cycles. We derive the scaling properties of

this new type of intermittency. We show that action potentials, as observed in the HH equation, are

originated by this new type I intermittency.

In chapter 3, we analyse the spatially extended HH model (D̃ > 0) and show that the type

I intermittency is responsible for the propagation of the action potentials along the axon. In the

presence of diffusion, we find a new type I spatial intermittency, sustained oscillations develop, as

well as chaotic or turbulent action potential propagation. For high values of transmembrane current

densities, turbulent action potentials disappear and the convergence to a stable steady state occurs

mixed with intermittent action potential chaotic spikes. We call chaotic intermittency to this new

kind of intermittency of the HH extended model. The results presented in chapters 2 and 3 have been

submitted for publication [18].

Finally, in chapter 4, we summarise the main conclusions of this work.

12

2The Diffusion free

Hodgkin-Huxley model

Contents2.1 Analysis of dynamical stability . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 The Bautin bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Type I intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

13

In this chapter we study Hodgkin-Huxley model without diffusion (D̃ = 0). We review its dynamical

properties, and make a comparison with the normal form of the Bautin bifurcation. We show that the

model has type I intermittency in the vicinity of a saddle-node limit cycle (SNLC) bifurcation, which

results in an action potential response.

2.1 Analysis of dynamical stability

As seen in equations (1.16), the diffusion free (D̃ = 0) HH equations take the form

Cm∂V

∂t= −gNam

3h(V − ENa)− gKn4(V − EK)− gL(V − EL)− i,

∂n

∂t= αn(V )(1− n)− βn(V )n,

∂m

∂t= αm(V )(1−m)− βm(V )m,

∂h

∂t= αh(V )(1− h)− βh(V )h.

(2.1)

All of the variables and constants have been described in detail in section 1.2. For the analysis about

to be made here, all the parameters are kept fixed with the exception of the current density i, which

will be considered as the free parameter of the model, and the temperature T , which is always 6.3 ◦C,

except when indicated otherwise. The basic bifurcation analysis as a function of i, including the

existence of two Hopf bifurcations of fixed points, one subcritical and another supercritical, has been

exhaustively analysed in [19] and [17]. For a review, recent references are [4] and [7].

The majority of authors have done the numerical analysis of the HH equations (1.16) with the

bifurcation analysis software AUTO, [20], and XPPAUT, [21], which incorporates the former. All

bifurcation diagrams shown here have been calculated with XPPAUT.

I0 I1

I2

V*

LCLC

0 50 100 1500

20

40

60

80

100

i (μA/cm2)

-V(mV)

a)

0 50 100 1500

5

10

15

20

25

30

35

i (μA/cm2)

T(ºC) I1 I2

b)

Figure 2.1: a) Bifurcation diagram for the diffusion free HH equations (2.1) at T = 6.3 ◦C as a function of thetransmembrane current density parameter i. SNLC bifurcation occurs for i = I0 = 6.26 µA/cm

2, the subcriticalHopf bifurcation for i = I1 = 9.77 µA/cm

2 and the supercritical Hopf bifurcation for i = I2 = 154.52 µA/cm2.

b) Variation of Hopf bifurcation points I1 and I2 with temperature. Calculated with XPPAUT.

In figure 2.1a, we show the bifurcation diagram for the diffusion free HH equations (2.1) at T =

6.3 ◦C, as a function of the transmembrane current density parameter i. In it, we represent the stability

of the coordinate V ∗(i) of the fixed point p∗(i) and the maximum values of the V (i)-coordinate of the

limit cycles (LC) associated with the Hopf bifurcations. Dotted lines correspond to unstable states

14

and continuous lines to stable ones. The SNLC bifurcation occurs for i = I0 = 6.26 µA/cm2, the

subcritical Hopf bifurcation for i = I1 = 9.77 µA/cm2 and the supercritical Hopf bifurcation for

i = I2 = 154.52 µA/cm2. In figure 2.1b, we see how the position of the Hopf bifurcation points I1 and

I2 varies with temperature. The two points coalesce at T = 28.85◦C, after which the fixed point p∗(i)

remains stable for all values of i and no limit cycles appear. For temperatures below this point, the

bifurcations and limit cycles are analogous to what is seen in 2.1a.

The bifurcation diagram depicted in figure 2.1 summarises the main characteristics of the asymp-

totic solutions of the diffusion free (D̃ = 0) HH equations (2.1). These equations have a unique fixed

point with coordinates p∗(i) = (V ∗(i), n∗(i),m∗(i), h∗(i)), whose position in phase space depends on

i (the other parameters in (2.1) are kept fixed). The fixed point p∗(i) has two Hopf bifurcations, one

subcritical for i = I1, and another supercritical for i = I2. For i > I2 and 0 ≤ i < I1, the fixed point

p∗(i) is stable. The local bifurcation analysis shows that for i < I1 and I1− i sufficiently small, the HH

equations have at least two limit cycles, one stable and another unstable, [19] and [17]. The unstable

limit cycle is created at I1, for decreasing values of i, and the stable one is created at I2, also for

decreasing values of i. These two limit cycles collide at i = I0 < I1 and, for i < I0 they do not exist.

At i = I0, the HH equation has a SNLC bifurcation, [22].

In figure 2.2, we show what the orbits of the limit cycles look like in the regions [I0, I1], where two

limit cycles exist and [I1, I2], where only one exists. These are two-dimensional cuts of trajectories in

the four-dimensional phase space (V, n,m, h).

a)

-20 0 20 40 60 80 1000.30.4

0.5

0.6

0.7

0.8

-V (mV)

n

b)

-20 0 20 40 60 80 1000.30.4

0.5

0.6

0.7

0.8

-V (mV)

n

Figure 2.2: Two-dimensional cuts of the orbits of the limit cycles for a) i = 7 µA/cm2 and b) i = 50 µA/cm2

in equations (2.1). Full lines are stable cycles, dotted lines are unstable ones. The full dot in a) represents astable fixed point, and the open dot in b) an unstable one.

The fixed point p∗(i) is unstable for i in the interior of the interval [I1, I2] and stable outside. Away

from the bifurcation points, the fixed point p∗(i) is hyperbolic. The overall behaviour of the bifurcation

diagram in figure 2.1a can be understood as a codimension 2 Bautin or generalised Hopf bifurcation,

[22] (in section 2.2 we cover this analogy in detail).

2.1.1 The unstable knee

At both I3 = 7.84 µA/cm2 and I4 = 7.92 µA/cm

2 a saddle-saddle bifurcation of limit cycles

appears, which produces a “knee” in the parameter region [I3, I4] in the unstable limit cycle. In figure

15

2.3a, we zoom in to this region to better show what it looks like in the bifurcation diagram, and in

2.3b we show the limit cycle orbits in a two-dimensional cut of the phase space inside [I3, I4]. We see

I0 I1

V*

LC

LC

I3

I4

6 7 8 9 10 110

20

40

60

80

100

i (μA/cm2)

-V(mV)

a) b)

-20 0 20 40 60 80 1000.30.4

0.5

0.6

0.7

0.8

-V (mV)

n

Figure 2.3: a) Enlargement of the region in the neighbourhood of the “knee” that occurs in the in-terval [I3, I4] = [7.84, 7.92] µA/cm

2. b) Two-dimensional cut of the limit cycle orbits in phase-space fori = 7.88 µA/cm2. Calculated with XPPAUT.

that we can have up to four limit cycles – three unstable and a stable one.

Strange phenomena have been reported in this very small region, such as the existence of chaotic

behaviour ([23]), as well as a period doubling effect on the period of the limit cycles, (reported in [19]

and [17]), however this effect is not a period doubling codimension 1 bifurcation.

We can safely assume this does not affect the overall behaviour of the solutions of the HH equations,

since it is restricted to the unstable limit cycle, for a very small region of i, and for T below 7.72 ◦C.

As we increase the temperature for values higher than this, the dotted limit cycle curve in figure 2.3a

loses its one-to-many feature.

7.7 7.8 7.9 8.0 8.10

2

4

6

8

i (μA/cm2)

T(ºC)

I3 I4

Figure 2.4: Disappearence of the one-to-many feature of the unstable limit cycle at T = 7.72 ◦C. Calculatedwith XPPAUT.

In figure 2.4, we can see the two points I3 and I4 coalescing and disappearing at T = 7.72◦C. For

temperatures higher than this, the unstable limit cycle behaves smoothly without further bifurcations.

16

2.2 The Bautin bifurcation

The analysis done so far has shown that the Hodgkin-Huxley model has a generalized Hopf bifurca-

tion, also called a Bautin bifurcation. We will now look at the Bautin bifurcation alone in more detail.

Since equations 2.1 are so complex, it is easier to look at the normal form of the bifurcation, which is

much simpler, and produces the same behaviour in only two dimensions (unlike the four needed in the

HH model) and two parameters.

The normal form of the Bautin bifurcation in cartesian coordinates can easily be obtained from its

expression in polar coordinates (seen in Appendix A, equations (A.1)), and is{ẋ = β1x− y + β2x

(x2 + y2

)− x

(x2 + y2

)2,

ẏ = x+ β1y + β2y(x2 + y2

)− y

(x2 + y2

)2,

(2.2)

where β1 and β2 are real parameters. Figure 2.5 shows an analysis of these parameters and for which

values the different stages of the Bautin bifurcation occur.

0

θ

ρ

H+

H-

SNLC

1

2

3

-β1

β2

Figure 2.5: Normal Bautin bifurcation parameters in cartesian (β1,β2) and polar (ρ,θ) coordinates, and the

three different regions of equations (2.2). Border 1-2: θ = arccos

(√4+ρ2−2ρ

)- SNLC bifurcation; border 2-3:

subcritical Hopf bifurcation (H+); border 3-1: supercritical Hopf bifurcation (H−).

The horizontal axis of figure 2.5 represents the symmetric of the parameter β1, and the vertical axis

represents β2. In order to better characterise the Bautin bifurcation, it is more intuitive to describe

these parameters in terms of their polar equivalent coordinates ρ and θ, that, as seen in the figure,

have been chosen to be ρ =√β21 + β

22 and θ = arctan(−β2/β1). For β1 = −β22/4, which is the

same as saying, for θ = arccos

(√4+ρ2−2ρ

)there is a saddle node bifurcation in the radial variable

r =√x2 + y2, which corresponds to a SNLC bifurcation in the cartesian coordinates (x, y). At θ = π/2

there is a subcritical Hopf bifurcation, and at θ = 3π/2 there is a supercritical Hopf bifurcation.

Since the behaviour of the system is similar for all values of ρ, we can fix it at ρ = 1 and study the

dynamical behaviour of the system as a function of the parameter θ – This is shown in figure 2.6.

17

LC

LC

x*

SNLC

0 π2 π 3 π2 2π0

0.5

1

θ

x

Figure 2.6: Bifurcation diagram for equations (2.2) as a function of the parameter θ = arctan(−β2/β1).

Comparing the three different regions in figure 2.5 with the bifurcation diagram of figure 2.6, we

can see that in region 1, θ ∈ [0, cos−1(√

5− 2)] ∪ [3π/2, 2π], the fixed point p∗ = (x∗, y∗) is stable and

there are no limit cycles; in region 2, θ ∈ [cos−1(√

5 − 2), π/2], the fixed point remains stable, and

there are two limit cycles, one stable and another unstable; and in region 3, θ ∈ [π/2, 3π/2], the fixed

point is unstable and there is only one limit cycle, which is stable. The unstable limit cycle originates

at the subcritical Hopf bifurcation at θ = π/2, the stable limit cycle at θ = 3π/2, and both collide

at the SNLC bifurcation, at θ = cos−1(√

4+ρ2−2ρ

). This is the exact same behaviour we observed in

figure 2.1a for the Hodgkin-Huxley model, now for a much simpler system of equations.

In figure 2.7 we show what the limit cycles (in red) in the three regions of figure 2.5 look like,

and the black arrows represent the trajectories followed by solutions of the normal form of the Bautin

bifurcation.

It has just been shown that the response of the Hodgkin-Huxley model to changes in the parameter

i is exactly the same as the response of the normal form of the Bautin bifurcation for the parameter

θ = arctan (−β2/β1).

2.3 Type I intermittency

We have just established in detail that, just like in region 1 in the normal form of the Bautin

bifurcation, for i < I0, the HH equations only have one stable fixed point, and no limit cycle exists.

However, for this region, these equations develop an action potential, resembling the limit cycle re-

sponse only found in other parameter regions. In fact, one can now realise that figures 1.6b, 1.8 and

1.9 were examples of this. It is this phenomenon that we now explain in this section.

If a cell or neuron is perturbed with some constant transmembrane current density i < I0, where

I0 is the parameter value of the SNLC bifurcation, and there is no diffusion (D̃ = 0), the asymptotic

time solutions of the HH equations converge to the stable fixed point p∗(i), for any initial condition

away from it. In this parameter region(i < I0), if the initial condition is away from the fixed point,

let us say p0 = (V0, n0,m0, h0) 6= p∗(i), then the response of the system has two possible outcomes.

18

1

-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.0

0.5

1.0

x

y

2

-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.0

0.5

1.0

x

y

3

-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.0

0.5

1.0

x

y

Figure 2.7: Limit cycles and fixed points (red) in the three regions shown in figure 2.5, and trajectories(black) followed by the solutions of equations (2.2). Dashed circles are unstable limit cycles, while full circlesare stable ones; full dots are stable fixed points, while hollow dots are unstable ones.

If p0 is close enough to p∗(i), then the asymptotic solutions of the HH equations converge to p∗(i),

without ever doing a long excursion through phase space regions away from the fixed point. On the

contrary, if p0 is sufficiently displaced form p∗(i) in the (V, n,m, h) four-dimensional phase space, the

solution of the HH equations does a large excursion in phase space, resembling, during some time, an

almost periodic orbit — action potential response. These two regions in phase space are separated by

a boundary or threshold.

As the parameter i approaches I0 and p0 is sufficiently displaced from p∗(i), the larger are the

number of transient spikes that appear in the potential V . For the same i < I0, it is possible to

produce zero, one, or more spikes, depending on how far p0 is from p∗(i). There is, however, an

upper limit for which, no matter how much we continue to displace p0 from p∗(i), no more spikes are

produced. Thus, there is a maximum number of obtainable spikes for each i < I0. Depending on

the value of p0, the number of spikes generated must be either equal to or below this maximum. In

figure 2.8, we show this transient behaviour of the solutions of equations (2.1), for several values of p0

and fixed i < I0.

As shown in figure 2.8, the solutions of the HH equations are action potential type responses and,

as we shall see now, they are the result of a type I intermittency phenomenon, [24], [25] and [26],

19

0 20 40 60 80 100-20020

40

60

80

100

120

t (ms)

-V(mV)

a)

0 20 40 60 80 100-20020

40

60

80

100

120

t (ms)

-V(mV)

b)

0 20 40 60 80 100-20020

40

60

80

100

120

t (ms)

-V(mV)

c)

0 20 40 60 80 100-20020

40

60

80

100

120

t (ms)-V(m

V)

d)

Figure 2.8: Membrane action potential response of the HH equations (2.1), for i = 6.20 µA/cm2 < I0 andinitial conditions p0 = (V

∗(0) + ∆V0, n∗(0),m∗(0), h∗(0)). a) ∆V0 = 1.6 mV, b) ∆V0 = 1.7 mV, c) ∆V0 = 1.9

mV, and d) ∆V0 = 5.5 mV. The SNLC bifurcation occurs at i = I0 = 6.26 µA/cm2.

near the codimension 2 SNLC bifurcation. For these parameter values, the unique attractor in the

4-dimensional phase space is the fixed point p∗(i).

To show that the transient time behaviour shown in figure 2.8 corresponds to type I intermittency,

we analyse the behaviour of the solutions of the HH equations near the SNLC bifurcation at i = I0

(figure 2.1a). At i = I0 and near the fixed point p∗(I0), the HH equations have a two dimensional

center manifold and a two dimensional stable manifold. By the reduction principle, [22], near the

SNLC bifurcation, the asymptotic time solutions of the HH equations are topologically equivalent to

the asymptotic time solutions of the normal form for this bifurcation.

With this simple fact, in Appendix A, we show that the intermittency characteristic of the SNLC

codimension 2 bifurcation has the same scaling behaviour in the bifurcation parameter as the type

I intermittency observed in interval maps, [24] and [25]. To be more specific, with ε = I0 − i, the

permanence time of the orbits of the HH equations (2.1) in the vicinity of the limit cycle that appears

at the SNLC bifurcation (ε = 0) is tper = c/√ε, where c is a constant. Denoting by P the period

of the shadow limit cycle responsible for the spiky action potential response and by N the number of

action potential spikes generated before the system goes to the rest steady state, we have NP = tper,

implying that

lnN = C − 12

ln ε, (2.3)

where C is a constant (Appendix A).

The first test showing that the action potential solutions of the HH equations are associated with

type I intermittency is to verify that the maximum number of action potencial spikes developed in the

20

solution of the HH equations obeys the scaling relation (2.3). A second test of type I intermittency is

to show that, close to the SNLC bifurcation, the maximum amplitude of each action potential spike,

as a function of the maximum amplitude of the previous one, has a parabolic profile. This graph will

be called the next amplitude map, [24].

In figure 2.9a, we show the number of spikes of the action potential generated by the HH equation,

in the left vicinity of the SNLC bifurcation, as a function of ε = I0 − i. For ε ∈ [0.00030, 0.07532], the

slope of the fit is s = −0.505, in agreement with the estimate (2.3). In figure 2.9b, we calculate the next

amplitude map, where we observe the typical parabolic profile associated with type I intermittency.

-8 -6 -4 -2 00

1

2

3

4

ln(ϵ)

ln(N)

a)

89 90 91 92 93 94 95 9689

90

91

92

93

94

95

96

-VN (mV)

-V N+1(mV)

b)

Figure 2.9: SNLC intermittency for the HH model. a) Logarithm of the number of spikes N of the actionpotential in the left vicinity of the SNLC bifurcation, as a function of the logarithm of ε = I0 − i. The slopeof the fitted line is s = −0.505, in agreement with (2.3). b) Next amplitude map for the HH model, for theparameter value i = 6.259 µA/cm2, or ε = 0.001. VN is the maximum value of the action potential spikenumber N . At the SNLC bifurcation, ε = 0, the parabolic profile shown touches the dotted line VN+1 = VN .

The main conclusion of this analysis is that the diffusion free (D̃ = 0) HH equations (2.1) exhibit

type I intermittency in the left vicinity of the SNLC bifurcation. This intermittency phenomenon is

responsible for the action potential spiky signals. The threshold is associated with a boundary in phase

space that separates the two possible types of transient solutions. Our numerical analysis shows that

for i ∈ [0, I0) and if the electrophysiological state of the cell is sufficiently far from the steady state, the

HH model always shows an intermittent response, with one or several action potential spikes. These

results have been submitted for publication [18].

21

3Diffusion in the

Hodgkin-Huxley model

Contents3.1 Propagation of action potentials along the axon . . . . . . . . . . . . . . . 24

3.2 Dynamical stability and bifurcations . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Wave velocity and dispersion relation . . . . . . . . . . . . . . . . . . . . . 26

3.4 Type I spatial intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Chaotic Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

23

In this chapter we look at what happens when one-dimensional diffusion is considered in the

Hodgkin-Huxley model. We present the equations for the model with diffusion, where a neuron is

considered as a cable-like structure, and analyse the resulting propagating action potential waves. We

find type I spatial intermittency, as well as a strange type of spatial chaotic intermittency.

3.1 Propagation of action potentials along the axon

The HH equations (1.16) with spatial term D̃ > 0 describe the axonal propagation of the potential

function, as well as the opening and closing of ion channels. We consider a 1-dimensional domain of

length L representing the axon. In the interior of the spatial domain, there is no transmembrane current

excitation (i = 0), but at the boundary x = 0 the neuron may be excited with some (transmembrane)

current density i(t). Under these conditions, the HH equations (1.16) are rewritten in the form

∂V

∂t= D

∂2V

∂x2+ F (V, ~z)− 1

Cmi(t), for x = 0,

∂V

∂t= D

∂2V

∂x2+ F (V, ~z), for x ∈

]0, L

],

∂~z

∂t= G(V, ~z), for x ∈

[0, L

],

(3.1)

where i(t) has been set to zero on the second equation (the excitation signal i(t) only exists for x = 0),

x is measured in cm and t in ms. The three gating variables n, m and h are represented by the vector

~z, and their three equations in (1.16) have been collapsed into then final equation in (3.1). The vector

functions F and G are defined by comparison between equations (3.1) and (1.16) and D = D̃/Cm. We

further consider that the transmembrane potential and the gate variables obey Neumann or zero flux

boundary conditions∂V

∂x

∣∣∣∣x=0,L

= 0 and∂~z

∂x

∣∣∣∣x=0,L

= 0. (3.2)

In Appendix B, it is shown that the diffusion term in (3.1) does not change the stability of the

steady state p∗(0), and the linear analysis leads to the conclusion that the homogeneous steady state

of the extended HH equation is stable. However, away from the steady state, the situation can be

different. As the local dynamics of the HH model have intermittent solutions (as seen in section 2.3),

we analyse numerically how intermittency and diffusion affect the propagation of the action potential

along the axon.

To simulate numerically the reaction-diffusion equations (3.1) we have used a benchmarked numer-

ical method, [27], obeying the discrete conservation law ∆x =√

6D∆t, where ∆x and ∆t are space

and time discretisation steps. This relation between space and time steps minimizes the integration

error. We have chosen the axon length L = 50 cm, with the spatial region divided into M = 400 small

intervals of length ∆x, where L = M∆x. As D = ∆x2/(6∆t) = L2/(6M2∆t), we change ∆t in the

interval [0.003, 0.033] ms, which corresponds to variations in the diffusion coefficient in the interval

[0.23, 2.34] cm2/ms. The value suggested by Hodgkin and Huxley, [12], is D̃ = 3.4× 10−4 S, giving

D = 0.34 cm2/ms, which is within the range of our numerical analysis. To analyse the solutions of

the extended HH equations (3.1), we have imposed a constant signal i(t) = i0, for every t ≥ 0, at

x = 0. The initial condition along the axon was set to p∗(0) = (V ∗(0), n∗(0),m∗(0), h∗(0)). According

24

to the parameters described above, we have varied the diffusion coefficient in the realistic range D ∈

[0.23,2.34] cm2/ms, which corresponds to having D/∆x2 ∈ [5, 50] ms−1.

0.0 12.5 25.0 37.5 50.0

0

20

40

60

80

100

x (cm)

-V(mV)

a)i0 = 50 μA/cm2

0.0 12.5 25.0 37.5 50.0

0

20

40

60

80

100

x (cm)

-V(mV)

b)i0 = 100 μA/cm2

Figure 3.1: Spatial solutions of the HH equations (3.1), for D/∆x2 = 20 ms−1 at time t = 100 ms, for twodifferent values of the transmembrane current density. In a), i0 = 50 µA/cm

2 and the transmembrane potentialconverges to the steady state p∗(0) along the axon. In b), i0 = 100 µA/cm

2 and the transmembrane potentialdevelops a propagating periodic action potential.

In figure 3.1, we show that we can have sustained oscillations along the axon for a realistic diffusion

coefficient, even though the local dynamics at any point in the spacial domain (with the exception

of the initial axon boundary) are those of p(i = 0), where the unique limit set of the diffusion free

HH equations (2.1) is the stable steady state, as seen in chapter 2. These sustained oscillations must,

therefore, result from the intermittency effect of the diffusion free HH equations for i = 0 we saw in

section 2.3. If the input transmembrane current density at the axon boundary is low, for example

i0 = 50 µA/cm2, the resulting transmembrane potential along the axon converges to the steady state

p∗(0), in agreement with the stability analysis in Appendix B. If the input transmembrane current

density at the axon boundary is above some threshold, stable oscillations develop.

3.2 Dynamical stability and bifurcations

Further numerical simulations have shown that, for a certain range of the parameter i0, the extended

HH system (3.1) has spatial intermittency and periodic oscillations. In figure 3.2, we depict in the

(i0, D/∆x2) parameter space, the regions where both phenomena are observed. This result shows that

the extended HH system has some attractor set other than the homogeneous fixed point p∗(0).

In figure 3.2, the black lines I∗1 and I∗5 delimit the regions where equations (3.1) show solutions with

intermittency from regions with oscillatory and chaotic solutions. For parameters in the intermittency

regions, the solutions of equations (3.1) show a finite number of spikes along the spatial domain before

going to the stable steady state. The light grey region marks the solutions that are oscillatory and

propagate through the spatial region (figure 3.1b). Between the light dashed lines I∗2 and I∗3 the

time interval between successive spikes is irregular — dark grey region. The light dashed line I∗4

precedes the final line I∗5 only by a couple of decimal places and marks the beginning of a chaotic

region, characterised by the chaotic behaviour of the time interval between successive action potential

spikes (figure 3.3c). The chaotic region ends giving rise to what we call chaotic intermittency. Type I

intermittency and chaotic intermittency will be analysed in more detail in the next subsections.

25

0 100 200 300 400 500

10

20

30

40

50

i0 (μA/cm2)

D/Δx2

(ms-1 )

Oscillations

TypeI Intermittency

ChaoticIntermittency

I*1 I

*2

I*3

I*4

I*5

Figure 3.2: Bifurcation diagram of the HH extended model (3.1), as a function of the transmembrane currentdensity i0 at the axon boundary and of the diffusion coefficient D/∆x

2. We show two different types ofintermittency, oscillations and spatial chaos (dark grey regions and dotted lines).

In figure 3.3, we make a detailed bifurcation analysis along a cross section of the bifurcation diagram

in figure 3.2, for the diffusion coefficient D/∆x2 = 20 ms−1 and spanning the whole region [I∗1 , I∗5 ]. In

this figure, the period of oscillations as a function of the transmembrane current density i0 is plotted.

Figure 3.3a shows the whole region, while figures 3.3b and 3.3c are different zoomings of the bifurcation

diagram. As shown, regular single-period oscillations occur in the regions [I∗1 , I∗2 ] and [I

∗3 , I

∗4 ] and there

are complex bifurcations or chaotic regions in the regions [I∗2 , I∗3 ] and [I

∗4 , I

∗5 ], where the time intervals

are irregular with period bifurcations (figure 3.3b) and show a bifurcation pattern characteristic of

chaotic maps of the interval (figure 3.3c).

3.3 Wave velocity and dispersion relation

We also wanted to study the properties of the action potential waves that are produced and propa-

gated along the axon. For this we have calculated its velocity of propagation for different values of the

diffusion coefficient. We measured the time at which the wave passed through each of the M = 400

small intervals that discretized the 50 cm of axon length, and found that these data adjusted perfectly

to a linear fit, from which we could get the value of the velocity. We then divided the i parameter

region for which oscillations are produced in 30 parts, and thus obtained 31 different values of velocity

(one for each i), each represented by a point in figure 3.4a. In figure 3.4b we show this curve for five

different values of the diffusion coefficient.

We can see that the generic profile in figure 3.4a is kept independently of the diffusion coefficient,

and shares a similarity in shape to the time intervals between oscilaltions seen in figure 3.3a. The

oscillations begin with some velocity v, which than decreases as i0 is increased. When some value

near the end of the oscillatory region is reached, the velocity of propagation increases abruptly, and

stagnates close to its initial velocity until the end of the oscillatory region. When we increase the

diffusion coefficient, this whole picture is shifted up, meaning that, generally speaking, higher diffusion

coefficient mean a higher velocity of propagation, as seen in figure 3.4b, which, of course, makes sense.

26

50 100 150 200 250 300 350

10

12

14

16

18

20

22

i0 (μA/cm2)T(ms)

I*2 I

*3 I

*4

a)

285 290 29510

12

14

16

18

20

22

i0 (μA/cm2)

T(ms)

I*2

I*3

b)

339.0 339.1 339.2 339.3 339.418.5

19.0

19.5

20.0

20.5

21.0

21.5

i0 (μA/cm2)

T(ms)

I*4 I

*5

c)

Figure 3.3: a) Period (and time intervals) of the oscillatory solutions of the HH extended model (3.1), as afunction of the current density i0, for the diffusion coefficient D/∆x

2 = 20 ms−1. In the regions [I∗2 , I∗3 ] (b)

and [I∗4 , I∗5 ] (c), the solutions of the HH extended model show chaos.

More broadly speaking, we can also see that these results are of the same order of magnitude of the

experimental velocity v = 21.2 m/s= 2.12 cm/ms measured by Hodgkin and Huxley in the giant axon

of the squid Loligo, [12].

We then proceeded to study the dispersion relation (how the angular frequency ω and the wave

number k relate to each other), for different values of i0 and D. For this we again used 31 different

values of i within the oscillatory region for each D, measuring for each of these, both the period

between oscillations and their wavelength. We neglected the chaotic regions [I∗2 , I∗3 ] and [I

∗4 , I

∗5 ], since

for these regions it wouldn’t be practical to obtain a single period and/or wavelength (which is what

makes them chaotic in the first place). In figure 3.5 we show what this dispersion relation profile looks

like.

We can see that all of the points obtained fall along a very smooth curve for every diffusion

coefficient. Even neglecting the chaotic regions, while progressing along i0 in the parameter space, each

set of (ω, k) points would fall somewhere along the smooth curves seen in figure 3.5. It is interesting

to note that even with such a strange chaotic profile in some regions of oscillation (figures 3.3b and

c), the dispersion relation behaves so smoothly, giving us no clue of the complexity of what is actually

happening.

27

50 100 150 200 250 300 350

0.95

1.00

1.05

1.10

1.15

1.20

1.25

i0 (μA/cm2)

v(cm/

ms)

a) D/Δx2(ms-1)50

40

30

20

10

0 100 200 300 400 5000.5

1.0

1.5

2.0

i0 (μA/cm2)

v(cm/

ms)

b)

Figure 3.4: a) Wave vlocity of the action potential for D/∆x2 = 20 ms−1, for different values of i0 within theparameter region [I∗1 , I

∗5 ]. b) Wave velocity as a function of parameter i0 for five different diffusion coefficients

of the HH equations.

D/Δx2(ms-1)10

20

30

40

50

0.30 0.40 0.55 0.60

10

20

30

40

50

ω (ms-1)

k(cm-

1 )

Figure 3.5: Dispersion relation of action potential waves of the HH model, for five different values of thediffusion coefficient.

3.4 Type I spatial intermittency

In the region i0 < I∗1 of figure 3.2, the HH model with diffusion has type I spatial intermittency

solutions. This means it will produce a finite number of propagating action potentials, before going to

rest. To characterise this intermittency, we have tested the parameter scaling and the next amplitude

map as in section 2.3. In the intermittency regime, we have counted the number of action potential

spikes that propagate along the domain [0, L], and have calculated the next amplitude maps.

In figure 3.6 we show the analogous of figure 2.9, now for the spatial HH equations, with ε = I∗1 − i0in the domain ε ∈ [0.001, 0.231]. The numerically determined slope of the scaling relation (2.3) is

s = −0.506, in agreement with the theoretical prediction s = −0.5. The next amplitude map has

been calculated for i0 = 56.010 µA/cm2 < I∗1 = 56.012 µA/cm

2 and D/∆x2 = 20 cm−1, showing a

parabolic profile, characteristic of type I intermittency. This shows that, not only is type I intermittency

responsible for the propagation of action potentials at the local level (with D = 0) but it appears again

in the waves produced by the spatially extended HH system.

28

-7 -6 -5 -4 -3 -2 -112

3

4

5

ln(ϵ)

ln(N)

a)

103.2 103.3 103.4 103.5 103.6103.1

103.2

103.3

103.4

103.5

103.6

-VN(x=L/2) (mV)

-V N+1(x=L

/2)(mV) b)

Figure 3.6: Type I intermittency for the HH extended model (3.1), with D/∆x2 = 20 cm−1. a) Logarithmof the number of spikes N of the action potential in the left vicinity of I∗1 = 56.012 µA/cm

2. The slope of thefitted dotted line is s = −0.506, in good agreement with (2.3). b) Next amplitude map for the parameter valuei0 = 56.010 µA/cm

2 or ε = 0.002. VN (x = L/2) is the maximum value of the action potential spike number N ,measured at the middle of the spatial domain [0, L]. The dotted line is the graph of the equation VN+1 = VN .

3.5 Chaotic Intermittency

A new intermittency of the HH extended model appears for i0 > I∗5 . However, for this region, we

have not found the same pattern we did in figures 2.9 and 3.6.

-7 -6 -5 -4 -3 -2 -11.52.0

2.5

3.0

3.5

4.0

4.5

ln(ϵ)

ln(N)

a)

103.1 103.2 103.3 103.4 103.5103.1

103.2

103.3

103.4

103.5

-VN(x=L/2) (mV)

-V N+1(x=L

/2)(mV) b)

Figure 3.7: Chaotic intermittency for the HH extended model (3.1), with D/∆x2 = 20 cm−1. a) Logarithmof the number of spikes N of the action potential in the right vicinity of I∗5 , for ε = i0 − I∗5 ∈ [0.001, 0.140].The dotted line has slope s = −0.5. b) Next amplitude map for the parameter value i0 = 339.369 µA/cm2or ε = 0.010. VN (x = L/2) is the maximum value of the action potential spike number N , measured at themiddle of the spatial domain [0, L]. Chaotic intermittency does not show any scaling behaviour on the numberof spikes as a function of ε. The dotted line is the graph of the equation VN+1 = VN .

In figure 3.7, we show the logarithm of the number of spikes as a function of the bifurcation

parameter ε = i0 − I∗5 , and the next amplitude map for the parameters i0 = 339.369 µA/cm2, with

diffusion coefficient D/∆x2 = 20 cm−1. From figure 3.7a, we conclude that the intermittent behaviour

has no apparent scaling and is different from other types of intermittency. Indeed, the number of spikes

as a function of the distance ε to the bifurcation point seems to behave randomly, without any scaling

behaviour. Furthermore, the next amplitude map shown in figure 3.7b is not characteristic of any

type of known intermittency phenomenon. Due to the irregular form of this map and the bifurcation

diagram of figure 3.3c, we have called this phenomenon chaotic intermittency. These results have been

submitted for publication [18].

29

4Conclusions

31

We have found the geometric and dynamical origins of the action potential type response of the

Hodgkin-Huxley neuron model. This peculiar response is due to type I intermittency occurring in the

vicinity of a saddle-node bifurcation of limit cycles. In this regime, neurons have a stable steady state

but for large amplitudes of excitation they develop the action potential type of response, shadowing

the existence of a stable limit cycle that appears at different parameter values. These conclusions were

obtained under zero diffusion, implying that our results remain true for any cell with an electrophysi-

ological state controlled by voltage sensitive channels.

We have extended our analysis to neurons with long axons. In this case, the diffusion coefficient

of the HH model is positive and the solutions of the HH model equations show a more complex

behaviour. For this case, we have assumed that neuron excitation is done at one boundary of the

spatial domain through a transmembrane current. We have shown that, above some transmembrane

current density threshold, action potentials spikes develop, showing type I intermittency characterised

by a finite number of action potential spikes propagating along the axon. Increasing the values of

the transmembrane current density at the boundary of the axon, periodic propagating stable diffusion

waves along the axon appear. In the parameter range of oscillations, we may have turbulent oscillations

or chaos. These chaotic oscillations appear on the irregular time interval between successive action

potential spikes and have a bifurcation diagram similar to the ones found in interval maps (figure 3.3c).

For the parameter values where oscillatory or chaotic solutions exist, the steady state of the HH

equations remains stable and is reached for small values of transmembrane current densities at the

axon boundary.

As far as we know, this is the first time that intermittency phenomena, chaotic or type I, are

reported in an electrophysiological model of a cell. However, it is a common phenomenon found in

electroencephalogram, [28], and epilepsy, [29].

Our analyses have been done for the original and calibrated HH model equations (1.16) and (3.1),

with realistic diffusion coefficients. This implies that all the phenomena described here are predictions

that can be explored in patch clamp experiments on giant axons. All the simulations are in agreement

with HH observations, including the velocity of propagation of action potentials measured along the

Loligo giant axon.

The original research in this thesis has been submitted for publication [18].

32

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