Dynamical disease: Challenges for nonlinear dynamics and medicineLeon Glass Citation: Chaos 25, 097603 (2015); doi: 10.1063/1.4915529 View online: http://dx.doi.org/10.1063/1.4915529 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/25/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Introduction to Focus Issue: Rhythms and Dynamic Transitions in Neurological Disease: Modeling, Computation,and Experiment Chaos 23, 046001 (2013); 10.1063/1.4856276 Modeling oscillatory dynamics in brain microcircuits as a way to help uncover neurological disease mechanisms:A proposal Chaos 23, 046108 (2013); 10.1063/1.4829620 Randomness switches the dynamics in a biophysical model for Parkinson Disease AIP Conf. Proc. 1479, 1434 (2012); 10.1063/1.4756429 Nonlinear dynamics of the membrane potential of a bursting pacemaker cell Chaos 22, 013123 (2012); 10.1063/1.3687017 Flow characteristics in a canine aneurysm model: A comparison of 4D accelerated phase-contrast MRmeasurements and computational fluid dynamics simulations Med. Phys. 38, 6300 (2011); 10.1118/1.3652917

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Dynamical disease: Challenges for nonlinear dynamics and medicine

Leon Glassa)

Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal,Quebec H3G 1Y6, Canada

(Received 26 January 2015; accepted 2 March 2015; published online 24 March 2015)

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of

some bodily function. There is a large literature in mathematics and physics which proposes

mathematical models for the physiological systems and carries out analyses of the properties of

these models using nonlinear dynamics concepts involving analyses of the stability and

bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4915529]

Human disease is often characterized by striking changes

in bodily rhythms. This article describes how a mathe-

matical analysis of these changes may be useful for doc-

tors in developing new methods to diagnose and treat

illnesses. Practical applications developed so far include

methods to automatically assess and analyze the severity

of some abnormal cardiac rhythms; predict weight loss

using computer programs; and deliver treatments for dis-

eases such as HIV and hepatitis. Potential future applica-

tions include predicting the risk for sudden cardiac and

epileptic seizures; developing early diagnostic warning

for the onset of serious diseases such as Parkinson’s dis-

ease; developing means to improve physiological stability

to avert falls; preventing sudden infant death; developing

personalized models that can be used to predict interac-

tions of the body’s control systems with drugs adminis-

tered to boost blood cell production and regularize blood

sugar; and developing closed loop medical devices that

will automatically detect abnormal dynamics in neural

systems and automatically respond to reestablish dynam-

ics in a normal range. The large increases in data col-

lected by individuals combined with powerful methods of

data analysis available in current portable devices will

facilitate development of these approaches. All these

advances will require close collaborations between basic

scientists, physicians, and engineers.

I. INTRODUCTION

The concept of dynamical disease was developed to

emphasize the common underpinnings of nonlinear dynam-

ics and medicine.1–3 In medicine, there are sometimes strik-

ing changes of bodily dynamics that are associated with

disease: for example, systems that have irregular or constant

rhythms can develop regular oscillations or systems that

oscillate could start oscillating in new and unexpected ways

or stop oscillating. It is natural to associate these qualitative

changes in dynamics with changes in dynamics (called bifur-

cations) in appropriate mathematical models of the physio-

logical system. Mackey and I surveyed dynamical diseases

and also presented many basic models that had been pro-

posed to study disease.4 In subsequent years, detailed knowl-

edge about the biochemical and anatomical components of

physiological systems has led to the development of nonlin-

ear mathematical models that in some cases provide remark-

able ability to simulate dynamical behavior that appears to

be in good agreement with experimental and clinical data.

There are many references that provide background for this

area.5–8 Yet, it seems to me that the practical applications

have lagged behind the theoretical advances. The point of

this essay is to summarize important advances and potential

directions for practical applications. I first summarize what I

consider to be the most important theoretical approaches to

study biological dynamics. Then, I consider several physio-

logical systems that are important medically and also display

dynamics that are interesting from both a mathematical and

clinical perspective. As a challenge to readers, I specifically

list 24 practical problems related to medical treatment where

I believe ideas derived from nonlinear dynamics should help

improve diagnosis or therapy. I conclude with some general

remarks about challenges ahead.

II. THEORETICAL APPROACHES TO BIOLOGICALDYNAMICS

Biological function in animals is based on an intricate

web of nonlinear oscillations and feedbacks. These occur in

the three-dimensional anatomy of the body. There are large

numbers of oscillators (e.g., cardiac, neural, respiratory, en-

docrine) all of which interact with one another. Some bodily

functions require coordinated movements in space (as in the

heart, the gut, the lungs, the musculoskelal system, the uro-

genital system), whereas in others information is transmitted

by nerves or through the circulatory system. I discuss mathe-

matical approaches for nonlinear oscillations, nonlinear

wave propagation, feedback systems, and time series

analysis.

A. Nonlinear oscillations

Oscillations are ubiquitous in biological systems.9 Such

oscillations must be robust to variations in the structures of

the systems as well as to perturbations of the system. From aa)glass@cnd.mcgill.ca

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CHAOS 25, 097603 (2015)

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mathematical perspective, stable biological oscillations are

described by equations that display stable limit cycle oscilla-

tions. Following a perturbation to the equation that is not too

large, the stable limit cycle is re-established. Moreover,

changes in the equations themselves will typically leave the

oscillation intact provided the changes are not too big.

Analytic insight and classification of the different ways that

oscillations can start and stop are provided by mathematical

theory including strong and weak Hopf bifurcations and the

saddle node on invariant circle bifurcation.10 Oscillations

can be reset by a single stimulus.9,11 Provided the oscillation

relaxes to the stable oscillation rapidly following a stimulus,

the effects of periodic stimulation can be analyzed by one

dimensional circle maps.4,12 However, there can be long

transient effects on the oscillation. If additional stimuli arrive

before the transients have dissipated, the problems require

higher dimensional maps to account for a variety of memory

effects.13 Mathematical analyses of the bifurcations found

from periodic stimulation of oscillations in higher dimen-

sions deserve further exploration. In the body, oscillations

interact with one another. Synchronization results from the

interaction of periodic oscillations as well as chaotic

systems.14

B. Feedback systems

As anyone who has ever had a blood test knows, values

of electrolytes, circulating blood cells, hormones, and metab-

olites and nutrients have normal ranges. Values outside those

ranges may be indicative of disease. In order to regulate

those values, the body has intricate feedback systems that

activate processes to maintain values in a normal range. For

example, in negative feedback, if a variable is too low, proc-

esses are activated to increase it, whereas if it is too high

processes are activated to decrease it. Study of control and

feedback falls cleanly under the rubric of engineering and a

great deal of research has been carried out from a systems

engineering perspective.15–17 This approach emphasizes

steady states and the ways feedback can be used to maintain

them. But nonlinear dynamics clearly plays a role.

Feedbacks are rarely instantaneous and it is essential to

incorporate factors to account for time delays in the feedback

circuits, which can lead to time delay nonlinear differential

equations. Simple nonlinear feedback systems will either

approach a stable steady state or a stable limit cycle oscilla-

tion.18 Interactions of multiple nonlinear feedbacks regulat-

ing a single variable can lead to steady states, simple or

complex oscillations, quasiperiodicity, or chaos.19,20

Changing parameters describing the feedbacks or the interac-

tions of feedback circuits lead to bifurcations in the dynam-

ics. The exploration of dynamics in systems with multiple

interacting nonlinear feedbacks is an area for further exami-

nation. Feedback functions can be non-monotonic and this

can also lead to chaotic dynamics in time delay equations.1

C. Nonlinear wave propagation

Muscles and nerves support nonlinear wave propaga-

tion. In the heart, these waves are associated with contraction

of the heart and subsequent pumping of blood to the body. In

nerves, the waves carry signals from one part of the body to

another. The waves are associated with changes in the per-

meability of specialized protein molecules, called ion chan-

nels, which are embedded in the membrane. Mathematical

models of these membrane currents range from simplified

models such as the FitzHugh-Nagumo equation to realistic

ionic models.6–8 Mathematical analyses have been carried

out for 1-dimensional geometries for nerves and 1, 2, or 3

dimensional geometries for cardiac and other tissues. The

defining properties of these excitable systems are (i) that you

need a sufficiently large (greater than some threshold) stimu-

lus to generate a large excursion (an action potential) from a

resting state; (ii) following an action potential there is a pe-

riod called a refractory time when you cannot have another

excitation; and (iii) two waves colliding head on will annihi-

late. In 2 dimensions, excitable media support many geome-

tries including plane waves, spiral waves with one or more

arms, and irregularly propagating and continually interacting

spiral geometries. In 3-dimensions, there are possibilities for

a scroll wave, which is a spiral wave that is translated along

an axis perpendicular to the plane of the spiral. But the scroll

waves can also be twisted and knotted leading to a zoo of

complex geometries.9,21 The underlying equations need to

reproduce several important physiological properties. The

duration of an excitation depends on the recovery time since

the preceding excitation. The longer the duration of the re-

covery time the longer the excitation (restitution property).

The speed of an excitation wave also depends on the duration

of the recovery time—the longer the recovery time the faster

the propagation. Bifurcations in the dynamics occur as the

restitution properties change.22–27

D. Time series analysis

The body is continually generating fluctuating temporal

signals. Dynamic signals reflecting cardiac activity (the elec-

trocardiogram—ECG) and the brain’s activity (the electroen-

cephalogram—EEG) are important markers of bodily

function and are frequently monitored, sometimes for long

time periods of hours or days, to help in the diagnosis and

treatment of disease. Other signals, for example, the fluctuat-

ing levels of blood sugar and insulin, may also be important

to assess the health and guide treatment for some individuals,

but they are difficult to monitor continuously.

Given the ready accessibility of the ECG, analysis of

heart rate variability is perhaps the paradigmatic biological

problem that has been studied in a time series analysis con-

text. Classic approaches measure the mean heart rate, stand-

ard deviation, power spectrum, and density distributions.28

Following the development and popularization of nonlinear

mathematics, new measures, such as detrended fluctuation

analysis and wavelet analysis, have been proposed that

reflect scaling properties, long range correlations, fractal,

and chaotic measures of variability.29,30 Another approach

develops symbolic representations of dynamics and then

assesses various measures of entropy.31–33 The goals of this

have been to identify compelling aspects of the signals, to

use the signals to improve diagnosis and prognosis, and to

use the analysis to help identify physiological

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mechanisms.34,35 Though less developed, many of these

measures have been used for the analysis of EEG data.36,37

Time series analysis have also been applied to other systems

including motor function,38,39 respiration,40 and endocrine

fluctuations.41

III. PHYSIOLOGICAL SYSTEMS WITH COMPLEXDYNAMICS

With this summary of mathematical approaches in

mind, I focus on various bodily systems. I briefly review the

physiology and then identify practical medical problems for

which deeper theoretical analysis may lead to improvement

of clinical treatments. This list is highly subjective and

reflects my judgment about problems for which mathematics

(especially nonlinear dynamics) has been used or may be

useful in the future.

A. The heart

The normal heart beat is generated by a pacemaker

called the sinus node that is located in the right atrium.

Waves of excitation originating in the sinus node pass

through the atrioventricular (AV) node and then to the ven-

tricles. The excitation of the ventricles leads to a contraction

that pumps blood through the body. The frequency is modu-

lated by sympathetic and parasympathetic nerves that

respond to bodily demands.

Abnormal cardiac rhythms are called arrhythmias. To

understand the arrhythmias, we just need to understand the

interplay of three processes: (i) pacemakers generate waves

of excitation; (ii) excitation waves circulate in the heart fol-

lowing normal or abnormal pathways; and (iii) waves can be

blocked. I now describe some selected arrhythmias in which

one or more of these factors play a role.42,43

AV heart block is perhaps the earliest arrhythmia studied

mathematically with descriptions of appropriate models dat-

ing to the early twentieth century.44 Conduction of the exci-

tation from the atria can be described by a one dimensional

map. As the frequency of the atria pacemaker increases,

there will typically be a frequency at which there is no longer

1:1 conduction through the AV node, but rather there are

other rhythms such as 3:2 heart block with atrial excitations

for each 2 ventricular complexes. The ability to predict the

sequence of different rhythms and their ordering as heart rate

increases represents a beautiful and concrete application of

mathematics.45,46 Yet, there are few cardiologists who are fa-

miliar with the mathematics. Since cardiologists do not need

the mathematics to do their job, the mathematics is not

taught to them. AV heart block is easily diagnosed from the

ECG, and if the AV block is severe enough, the patient is eli-

gible for a pacemaker.

Ventricular tachycardia47 (VT) is another arrhythmia

that has attracted a large mathematical literature. In ventricu-

lar tachycardia, there is an abnormally rapid heart rate that

originates from excitation in the ventricles. The source of the

accelerated rhythm could be an abnormal pacemaker or a cir-

culating excitation. The circulating excitation can be mod-

eled by excitation on a ring, in a sheet or shell, or in a solid

structure that may have a realistic geometry. VT can either

be monomorphic or polymorphic. Monomorphic VT (the

ECG shows a repetitive wave form with a single morphol-

ogy) often occurs in patients who have had blockage of a

coronary artery (a heart attack). Cardiologists identify the

anatomical substrate in this case as a scar with one or more

narrow isthmuses of viable tissue. Such an isthmus can form

one part of a re-entrant circuit. If the frequency of the heart

is not too high, the arrhythmia is not necessarily fatal, the

patient can make it to the cardiologist, and therapy can be

initiated. Therapies include ablating a part of the re-entrant

circuit, prescribing medications that reduce the incidence of

the arrhythmia, or implanting a medical device that can

deliver pacing that will terminate the arrhythmia using small

periodic shocks or a large shock. In polymorphic VT, there

are multiple morphologies of the ventricular complexes.

This rhythm often degenerates to ventricular fibrillation

(VF), a fatal arrhythmia. A patient who experiences poly-

morphic VT or VF will die unless they receive immediate

care. Therapies include drugs to reduce the frequency of the

arrhythmia and implantable cardioverter defibrillator (ICD)

devices that can deliver a large shock to the heart that would

usually be able. Transitions to VT are sometimes preceded

by alternation of the morphology of complexes on the ECG

and there has been development of models extending period-

doubling bifurcations to spatially distributed systems. There

is a very large nonlinear dynamics literature in this area,

much of which focuses on properties of re-entrant waves

(waves on rings, spiral waves in sheets, scroll waves, and

other waves in 3D) and instabilities that arise in these waves

as parameters or anatomical substrate are changed.22–27

Practical advances in cardiology to date have been largely

carried out by biomedical engineers working in collaboration

with cardiologists.

Atrial fibrillation (AF) is a rhythm in which the upper

chambers of the heart display irregular rhythms that many

believe are associated with multiple re-entrant rhythms cir-

culating on the heart.48,49 It is characterized by an irregular

sequence of ventricular activations. Much of the theoretical

work developed for VT and VF is also applicable to AF. AF

leads to reduced capacity for exercise, increased risk for

stroke, but is not usually fatal. Treatment for atrial fibrilla-

tion includes administration of drugs that reduce the inci-

dence of AF and ablation. Original ablation procedures

generated barriers to electrical propagation from the pulmo-

nary veins to the atria.50 The theoretical underpinning for

this procedure is the observation that rapid stimuli often em-

anate from the pulmonary veins. Another approach targets

the central region of spiral waves, determining by intracar-

diac mapping, for ablation.51 Understanding the dynamics of

the arrhythmias and the transitions between normal sinus

rhythm and the various abnormal rhythms represent key

questions in both mathematics and in clinical medicine.

Open questions where theoretical analysis may be useful

include

(1) Improving algorithms for anti-tachycardia pacing. The

algorithms used in these devices are now empirically

determined rather than based on dynamics of the rhythms

under periodic stimulation.52

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(2) Improving ability to carry out ablation for VT. Current

work is making use of the detailed information obtained

from mapping ECG activity on the surface of the body

and during intracardiac mapping to develop realistic

models of arrhythmias that will be sufficiently detailed

to direct the cardiologist to appropriate sites for abla-

tion.53,54 It is possible that the style of using simplified

models in simplified geometries will not be adequate.

(3) Improving ability to carry out ablation for AF. The various

approaches to ablation of AF are being carried out by physi-

cians50,51 and there is not yet consensus on what works best.

Current theoretical models are likewise attempting to de-

velop sufficiently accurate models to inform theory.55

(4) Improving defibrillation methods. Current methods for

defibrillation use large currents delivered by an ICD.

Experimental work and theoretical work are directed

towards optimizing the waveforms of the defibrillations

pulse.56 Work is also being done on the efficacy of using

smaller stimuli.57,58

(5) Predicting the risk for sudden cardiac death. ICDs can be

successful in reversing VF, thereby saving lives. However,

ICDs are expensive and they can lead to infections or other

complications. Current guidelines lead to a large number of

false negatives and false positives, so a large percentage of

the devices that are implanted, do not lead to a defibrillation

episode. Possible approaches for improving risk stratifica-

tion include carrying out time series analysis of heart rate

variability59–61 and developing new metrics based on ana-

lyzing mechanisms of arrhythmias that precede sudden car-

diac death.62 Technological developments that enable

individuals to take their own ECGs, post them in a public

space, and have their data analyzed by others offer the possi-

bility of revolutionizing our understanding of arrhythmias

and medical care. Many small companies are working in

this space, but none has emerged yet as a leader.

(6) Identifying atrial fibrillation. Algorithms that distinguish

normal sinus rhythm from atrial fibrillation can be used

clinically in implantable devices to help guide therapy

by assessing the frequency of occurrence and total occur-

rence of atrial fibrillation over extended time periods.

Algorithms emanating from a nonlinear dynamics per-

spective involve assessing the probability density distri-

butions of the change in the times between successive

beats (DRR) and return maps of that give the dependence

of the interbeat timing on the preceding beat and also the

return maps of DRR intervals.63,64 The algorithms were

used in the development of a clinical device (Medtronics

LINQ implantable recorder).

(7) Improving methods for intrapartum cardiac monitoring

during delivery. Although monitoring fetal heart rate plays

an important role in evaluating fetal stress during labor, it

is difficult to quantitate.65 Recent suggestions propose

using methods that assess complexity of fetal heart rate to

provide a new approach to fetal monitoring.66

B. Nervous system

The brain provides compelling scientific problems at all

level of organization from subcellular structures to the entire

brain. A landmark achievement in nonlinear dynamics and

neurophysiology was the development of nonlinear mathe-

matical model of the action potential in squid giant axon.67

Subsequently, dynamic fluctuations of brain activity and

behaviors controlled by brain activity have attracted wide

theoretical interest.7,8 In contrast to the heart, the functional

significance of brain rhythms is poorly understood. EEGs are

surface electrical recording of voltages associated with brain

activity.68 Electrodes are positioned on several locations on

the scalp. Each electrode reflects the activities of millions of

cells. Rhythms are classified by their spectral frequency:

alpha wave (7.5–12.5 Hz), beta wave (12.5–30 Hz), delta

wave (0–3 Hz), theta wave (4–7 Hz), and gamma wave

(32–100 Hz) (the frequency ranges differ in different sour-

ces). Neurophysiologists can also record from single cells.

The most compelling aspect of these recordings is the

changes in activity associated with some stimulus or task.

Identification of high frequency (>40 Hz) fluctuations of ac-

tivity in sensory regions of the brain led to an early proposal

that these rhythms help to bind or coordinate inputs from dif-

ferent cells.69

There is a current focus on determining the wiring dia-

gram of the brain (the “connectome”).70 But Kopell and col-

leagues stress that identification of the connections is not

sufficient. Since there are often significant changes in

dynamic signatures in different brain states, it is also neces-

sary to know about the dynamics (the “dynome”).71 Neural

activity in individuals, as monitored by the EEG, reflects var-

ious normal and abnormal dynamics including the large

spike and wave configuration configurations associated with

epilepsy,72,73 the changes in the composition of the power

spectrum that can identify different sleep and consciousness

states including the effects of anesthesia, Parkinson’s dis-

ease, and schizophrenia.74 There are a large number of neu-

rological disorders with striking dynamical features.75

Open questions where theoretical analysis may be useful

include

(8) Predicting the onset of epileptic seizures before they

occur. If an algorithm was available, then it might be

possible to institute therapies that would avert the sei-

zure or to change behavior to mitigate harm from the

seizure (e.g., a person could stop driving).73 The ability

to predict onset of epilepsy has been controversial but

is an important area of active research.36,37,76,77

(9) Improving algorithms for deep brain stimulation for

Parkinson’s disease. Deep brain stimulation involves

implanting electrodes deep in the brain and stimulating

at frequencies greater than 100 Hz.78 Deep brain stimu-

lation using electrodes implanted in the thalamus has

been successful for the treatment of Parkinson’s disease

and can reduce tremor and improve quality of life. The

mechanism of deep brain stimulation is controversial. It

may block nerve activity, desynchronize neural oscilla-

tions,79 or lead to the release of a chemical that induces

a Hopf bifurcation in an oscillatory pathway.80

(10) Develop closed loop stimulation for neurological disor-

ders. Deep brain stimulation is delivered at a localized

spot in either a predetermined manner or following

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activation by the patient. Several groups have proposed

a closed loop system that would locally record neural

activity and then deliver stimuli based on the activity

and algorithms built into the device. In particular,

there is a proposal to use cortical multielectrode arrays

to both monitor the activity and deliver the

stimuli.79,81,82

(11) Develop new modes of electrical stimulation for other

disorders than Parkinsonian tremor. In the heart, pace-

makers are often employed to correct pathological

rhythms. Thus, even though there may be a structural

defect, such as a damaged AV node, the pacemaker can

correct the rhythm. To the extent that abnormal neural

rhythms are associated with pathologies, it is possible

that correcting the rhythm will be medically useful

even though it may be impossible to cure the disease.

Deep brain stimulation has been proposed as a possible

therapy for many disorders including Tourette’s syn-

drome, essential tremor, and cluster headaches.71,83 In

all cases, developing a theoretical understanding of

mechanisms could lead to improved procedures for

locating electrodes and stimulating.

C. Musculoskeletal system

Under control mediated by the nervous system, the mus-

culoskeletal system supports the body and enables motor

functions posture, the use of trunk, arms, legs, and especially

locomotion. Proper motor function typically entails genera-

tion of central signals as well as monitoring of motor activity

via a variety of proprioceptive feedbacks. There is current

active research in engineering for the development of pros-

thetics that can aid individuals with motor deficits to carry

out motor tasks.84,85 Such studies necessarily involve con-

cepts of feedback, oscillation, and synchronization, but the

work to date appears to be largely carried out by the engi-

neering community, rather than the nonlinear dynamics com-

munity. In addition to the normal rhythms of the body, there

are a large variety of pathological tics and tremors,72 some

which may be associated severe neurological disorders or

may be premonitory signals that precede the development of

disabling diseases.39,86 From a practical standpoint, falls rep-

resent a major source of disability in elderly patients and the

possibility of developing strategies to reduce falls is a major

focus.

Open questions where theoretical analysis may be useful

include

(12) Develop methods to assess early onset of diseases,

drug toxicity, or chemical toxicities that lead to

impaired motor function. A variety of tasks including

standing87,88 walking,38 stick balancing,89,90 curve

tracing91 all display significant temporal fluctuations

and there have been numerous suggestions that these

fluctuations might provide an early sign for the devel-

opment of more serious disease. Understanding the

mechanisms of these fluctuations necessarily entails

analysis of multiple time delay control systems.

Further, the signals themselves show complex time

series which are being approached using a large number

of approaches including many of those mentioned in

Sec. II.39,91

(13) Develop methods to reduce falls in the elderly.

Stochastic resonance refers to the enhanced transmis-

sion of information for some optimal level of noise.

Collins and colleagues proposed that adding noise to

the insoles of shoes might improve balance and lead to

the reduction of falls.92 Recent work supports these

claims.93,94

(14) Develop methods to avert or reduce the effects of mi-

graine headaches. Migraine headaches can be disabling

and difficult to treat. Recent research has provided evi-

dence that spreading waves on the surface of the cortex

may be associated with migraine episodes. Unlike the

rapid velocity excitatory cardiac waves, waves associ-

ated with migraine are conjectured to be due to spread-

ing depression and move with a much slower velocity

of propagation.95 Given our knowledge about nonlinear

wave propagation, it may be possible to develop better

electrical or pharmacological methods for control.

D. Respiratory system

Like the heart, the respiratory system generates a rhythm

that is necessary for life. In awake conscious people past a

certain age, the respiratory rhythm can be under conscious

control. A large number of mathematical models for respira-

tory rhythmogenesis have been proposed over the years that

share a common feature of having stable limit cycle oscilla-

tions.96–98 However, the anatomical components of the mod-

els can differ, and even knowing whether the limit cycles are

generated by endogenous pacemaker cells or by networks of

cells which do not spontaneously oscillate is still controver-

sial. Just as in the heart, there are pathological respiratory

rhythms, but these are not as clinically common or important

as the cardiac arrhythmias.99 One rhythm, Cheyne-Stokes

respiration is characterized by a waxing and waning of respi-

ration with a periodicity of about 40–60 s. This rhythm is

observed in some terminally ill patients, in some patients

with neurological diseases, in obese individuals, and in nor-

mal individuals at high altitude. One approach to understand-

ing this rhythm is by considering a nonlinear time delayed

negative feedback system. Increasing the sensitivity of the

feedback control, as might occur in neurological disease, or

increasing the time delay as might occur in obese people or

people with impaired cardiac function can lead to supercriti-

cal Hopf bifurcation leading to this rhythm.1,100,101

Respiratory arrest, for example, as occurs in sudden infant

death syndrome, is another important abnormality of the re-

spiratory rhythm.

Open questions where theoretical analysis may be useful

include

(15) Develop better methods to prevent sudden infant death

syndrome. In early work, Paydarfar considered the pos-

sibility that respiratory arrest might arise from a stimu-

lus that would shift the stable limit cycle observed in

normal breathing to a stable steady state in which there

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was not respiration.102 Based on this conceptual picture,

Paydarfar has been exploring the possibility that small

perturbations delivered to a mattress might be useful

for stabilizing the respiratory rhythm in infants.103

Preliminary results are positive, and there is a commer-

cial possibility for developing new types of mattresses

that will reduce the risk of long apneic spells in babies

and sudden infant death syndrome.

(16) Develop better methods for adjusting ventilators during

forced ventilation. There are many different modes of

mechanical ventilation. In some, the patient’s inspira-

tory effort triggers the ventilator. In other settings, there

is no feedback from patient to ventilator and the venti-

lator delivers periodic ventilation. Nonlinear interac-

tions occur between the ventilator and the intrinsic

respiratory rhythm.104–107 These phenomena are diffi-

cult to explore in the clinic. In general, adjustment of

ventilation is done empirically. Ideally, ventilators

could be adjusted based on a nonlinear analysis of the

coupling between the ventilator and the patient.

Another innovative suggestion is that irregular ventila-

tion patterns mimicking natural variability may elimi-

nate some of the practical difficulties associated with

current ventilation protocols.108

E. Immunology and hematology

Circulating blood cells play crucial roles in transport-

ing oxygen to the tissues and protecting the body from dis-

ease. Stem cells in the bone marrow are responsive to

signals from the periphery and adjust the rates of synthesis

of the various blood types. Perhaps related to this continual

turnover of the circulating cells, loss of control can lead to

various cancers.109 HIV viruses also invade some types of

white blood cells leading to reduced immune function and

development of serious diseases if the viral growth is not

treated.110 Various drug regimens, particularly chemother-

apy for cancer patients, lead to destruction of circulating

cells and stem cells and can lead to loss of immune func-

tion. An early model that incorporated non-monotonic con-

trol of blood cell production by a time-delayed feedback

proposed by Mackey and myself showed the possibility of

chaotic dynamics in control of blood cell production.1

Detailed data sets of levels of circulating blood cells at fine

time resolutions are not easy to collect and not easy to

find.

One of the triumphs of mathematical biology has

been in the development of treatment of viral diseases

such as HIV/AIDS and hepatitis.111,112 The analysis of

kinetics of reproduction and mutation rates combined with

mathematical models of virus clearance has led to the de-

velopment of the combination therapies that are currently

used today.

Open questions where theoretical analysis may be useful

include

(17) Optimize the temporal administration of drugs that

stimulate blood cell production in patients receiving

chemotherapy.113,114 Low white blood cell count is an

unwanted side effect of some aggressive chemothera-

peutic protocols and can potentially result in devastat-

ing infection. To counteract the effects of the

hormones, agents such as colony stimulating factor,

which stimulate white blood cell production, are admin-

istered. These drugs are expensive. The protocols for

administering the drugs have been worked out in clini-

cal trials, largely funded by the companies that manu-

facture the drugs. A typical protocol is administration

of the CSF by subcutaneous injection for several con-

secutive days at the same time each day. In view of the

inherent time delays in blood cell production and the

intrinsic feedback systems, the system is quite compli-

cated, and measurements of cell counts on one day do

not necessarily reflect the dynamics to be expected in

the future. There is striking need to develop better mod-

els, perhaps individualized based on measured

responses to the drug in individual patients.

(18) Develop individualized multi-scale models for the

interaction of anti-viral agents and viral diseases includ-

ing hepatitis and HIV/AIDS. Combining knowledge of

biochemical pathways with kinetic data in in vitroexperiments and clinical data is now making possible a

detailed development of nonlinear theoretical models of

virus-drug interaction.115,116 It should be possible to

use kinetic data of viral load obtained during treatment

to optimize the drug administration on an individual-

ized basis.

F. Endocrine system

Like the hematological and immune systems, it is not

straightforward to obtain accurate data for long times with a

fine temporal resolution due to the difficulty and cost of

obtaining the data. However, there are many prominent hor-

monal rhythms covering a broad range of time scales.

Circadian (about 24 h) fluctuations occur for several hor-

mones including cortisol, ACTH, and growth hormone.117

However, superimposed on the 24 h rhythm can sometimes

be pulsatility with a shorter cycle, as occurs with growth hor-

mone.118 Hormones, such as insulin and other hormones that

respond to circulating levels of nutrients, fluctuate following

meals.119 On the other hand, female reproductive hormones

have a periodicity of about 28 days.120 The development of

many synthetic hormones including insulin, growth hor-

mone, thyroxin, estrogen, progestin, testosterone, cortisone

represents one of the major successful themes of 20th cen-

tury medicine that has had profound medical implications

that has positively affected huge numbers of lives. In view of

the complex mechanisms of action of hormones, the intrinsic

feedback loops regulating their production, and the intrinsic

rhythmicity of many hormones, it is striking that medicine

has progressed so far with little input from the nonlinear dy-

namics community.

Open questions where theoretical analysis may be useful

include

(19) Develop better methods for administering growth hor-

mone. Growth hormone has normal pulsatility of

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several hours. When growth hormone is administered

therapeutically, it is typically administered on a daily

basis.121 This mode of administration does not corre-

spond to the normal pulsatility. It would be useful to de-

velop a better understanding of the normal

physiological mechanisms for release and function of

growth hormone with a goal of optimizing treatment

schedules.

(20) Develop individualized models for glucose metabolism

in diabetics and use this to optimize insulin administra-

tion. Diabetics attempt to control blood sugar within a

normal physiological range. When blood sugar is too

low, there can be loss of consciousness and even death.

Blood sugar that is too high leads to complications

including heart disease and stroke. Diabetics measure

their blood sugar levels several times per day and titrate

their eating or administration of insulin depending on

the results. A closed loop artificial pancreas would mea-

sure blood sugar, compute the expected time course fol-

lowing insulin administration, and administer the

optimal dose. Current research is working to develop

personalized models for the artificial pancreas where

parameters could be set based on measured

responses.122,123

(21) Develop personalized programs to predict weight loss

for obese patients. Development of mathematical mod-

els for weight loss based on food intake, metabolism,

and activity are now capable of predicting weight

change on an individualized basis.124 The further

refinement and testing of these models provide another

clinical application of potential major importance in

view of the obesity epidemic in some western

countries.

(22) Develop better methods to mitigate the effects of jet

lag. Because of the importance of the circadian rhythm

and its physiological significance, it has been a long-

standing topic for mathematical analysis and model-

ing.9,125–127 The endogenous circadian rhythm is

normally entrained to the 24 h light dark cycle, and fol-

lowing transfer from one time zone to another, the en-

dogenous rhythm takes time to readjust leading to sleep

disturbances and reduced function. Finding an optimal

stimulus or stimuli that will rapidly and reliably reset

the circadian rhythm has been one of the holy grails of

this field with enormous practical significance. Despite

a large number of proposals concerning the best way to

reset the circadian rhythm using light, eating habits,

and drugs, there is not now a consensus of the optimal

procedures.

G. Miscellaneous

Since organ systems interact with one another, the above

discussion based on classifying disorders based on the main

organ involved is necessarily open to criticism and revision.

For example, sympathetic and parasympathetic nerve activ-

ities play important roles in the onset and continuation of

cardiac arrhythmias, respiratory and musculo-skeletal

rhythms are generated by the nervous system, many

hormones are secreted by or under the control of neural ac-

tivity, levels of some circulating hormones that regulate

blood pressure and heart rate, respond to cardiovascular ac-

tivity, and so forth. So far I have also not mentioned diseases

involving the kidneys, the liver, or the gut and there are

many disorders of a dynamic nature in these organs.

However, there are a few additional suggestions that involve

techniques that are not specifically directed towards single

organs.

Open questions where theoretical analysis may be useful

include

(23) Develop better methods to predict multi-organ failure

in the critical care setting. Patients in intensive care

units in hospitals are subjected to continual monitoring

of multiple vital functions. The amount of data recorded

for each subject can be massive. Some functions, such

as the heartbeat are easy to monitor and an alarm will

sound if the heart rate falls too far outside of a normal

range. But a major problem is to identify the small sub-

set of patients who will go on to develop multi-organ

failure—an occurrence that will lead to death if not

remediated rapidly. There is now technical feasibility to

do large scale data analysis in real time in the inten-

sive.35,128,129 There remains a problem of identifying

algorithms that will be effective. There have been

recent reports of successful use of time series analysis

for improving treatment of very low birth weight

babies.130

(24) Utilize the techniques introduced by synthetic biology

to develop new classes of medical treatment. Synthetic

biology refers to the design and implementation of

genetic circuits to carry out some function or dynamics

that would not normally take place in the target cell.

This young and rapidly evolving discipline has pro-

found possible technical applications to health includ-

ing new strategies for treating cancer and

reprogramming cells to regenerate.131 Because of the

difficulties involved in designing and synthesizing

genetic networks, nonlinear mathematical modeling

will play an important role. Recent work suggests that

E. coli bacteria could have engineered circuits that

would function to detect a specific pathogen and secrete

a particular therapy.132

IV. CONCLUSIONS

The complex dynamics of normal bodily function has

provided a steady stream of effects and questions that pro-

vide a challenge to basic scientists. Mathematical analysis of

nonlinear oscillation, nonlinear wave propagation, nonlinear

feedback, and time delay equations is of intrinsic interest,

completely apart from the potential practical applications to

basic science and clinical medicine. However, here I have

focused on the possibility of specific applications to medi-

cine. Although there are only a small number of concrete

applications to date, at the current time, large numbers of ba-

sic scientists, engineers, and start-up companies are explor-

ing future opportunities. Based on the current review, I make

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a number of generalizations and conclusions and identify

emerging themes and approaches.

A. Personalized medicine

Although personalized medicine is often associated with

developing medications specifically chosen based on the

genetic profile of an individual patient, there is an important

dynamic aspect that is crucial. Dynamic models based on

clinical data that may include genetic information are cur-

rently being developed for the ablation of anatomical sub-

strates for cardiac arrhythmias and the administration of

medications including insulin and growth colony stimulating

factors. Wearable portable devices can now collect and ana-

lyze data in real time for cardiac function, blood sugar, tem-

perature, and activity.41,133 Future challenges include

developing means for data to be standardized and uploaded

by individuals into public data banks. Mining such data has

the potential to provide profound new insights into the dy-

namics of individuals.

B. Large data

Related to the development of personalized medicine is

the collection of huge amounts of data. Although much of

the data is static (genomes of individuals, some types of

imaging, medications administered, and outcomes), the vari-

ous applications mentioned in the body of this paper may all

potentially involve collection of large amounts of data.

Physionet has pioneered the collection of large open data

sets for research and algorithm development.134 The level of

interest in this area is made clear by consideration of the

large literature dealing with time series analysis.41,60,71,77 As

the collection of data proceeds, a major bottleneck will lie in

developing appropriate algorithms to interpret and utilize

this data.

C. Closed loop systems

The development of closed loop systems is already

developed in cardiology with a variety of devices such as

anti-tachycardia pacemakers and ICDs that are able to detect

arrhythmias and administer stimuli to convert the rhythm to

a normal sinus rhythm. As noted earlier, there is likewise in-

terest in developing closed-loop systems for neural stimula-

tion to alleviate Parkinsonian tremor or avert an epileptic

seizure.82 The development of closed loop systems for the

administration of drugs and hormones represents another

direction of current research.122

D. Critical transitions

In ecology and natural science, work over the last fifteen

years has focused on the application of theory of bifurcations

(often called “tipping points” in the popular press and by

some scientists) to study large transitions due to a saddle

node transition from one basin of attraction to a sec-

ond.135,136 The current work has a clear precedent emerging

from discussions of transitions and multi-stability in ecosys-

tems.137 As well, the ideas were put forward by proponents

of catastrophe theory and attracted a great deal of interest

and criticism.138 The current resurgence in these areas ech-

oes some of the early themes, but focuses on the possibility

of predicting change based on an analysis of fluctuations in

the neighborhood of bifurcations points. Although the claims

of universality generate broad interest, sparse data associated

with climate change and ecological transitions make caution

essential.139–141 Yet, recent experimental work on popula-

tions in the laboratory has shown increased fluctuations pre-

ceding population collapse.142 A still unpublished

manuscript motivated by Scheffer and colleagues discusses

the possibility of extending these ideas to predict transitions

in medicine.143 With increased attention to the study of tran-

sitions in medicine, useful practical applications may emerge

involving predicting transitions such as cardiac arrhythmias

leading to sudden cardiac death or epileptic seizure.

E. Easy or hard mathematics

The study of nonlinear dynamical systems has led to the

discovery of a host of interesting properties concerning dy-

namics and bifurcations in mathematical models. Yet many

applications that have had an impact on current practice do

not involve the exotic phenomena that attract many to non-

linear dynamics. Examples mentioned earlier include linear

models to predict drug effects on HIV,111 the use of density

histograms and return maps to diagnose atrial fibrilla-

tion,63,64 and nonlinear models with a single stable fixed

point to predict weight loss during dieting.124 In contrast, the

analyses and development of realistic models in cardiac elec-

trophysiology require large research teams and a very high

level of technical excellence.53,54 In seeking and developing

applications of mathematics to medicine, we must recognize

that important advances may not depend on a mathematical

breakthrough, but may rather emerge from appropriate use

of well known concepts to vital problems. Independent of

whether the underlying mathematics is easy or hard, in order

for mathematical advances to be implemented, it is essential

that they are developed to the point where implementation of

them is transparent and easy to use. From a practical per-

spective, implementation of new techniques is often cata-

lyzed by adequate remuneration to the health care providers.

In the course of preparing this article, I have been in

contact with many working on problems related to dynami-

cal disease. They share a common recognition of the com-

plex dynamics manifest by the human body. There is a

strong sense that the increased understanding is leading to

new approaches and will lead to the development of useful

diagnostic and therapeutic procedures. Hopefully, the current

review will help spur the progress.

ACKNOWLEDGMENTS

I thank NSERC and the Canadian Heart and Stroke

Foundation for financial support over many years. I have

benefited from a Lady Davis Visiting Professorship to visit

The Racah Institute of Physics at Hebrew University in

Jerusalem during the preparation of this manuscript. I thank

many colleagues for useful suggestions including Michael

Mackey, Ary Goldberger, James Collins, Nancy Kopell,

David Paydarfar, Alan Perelson, and Jack Feldman.

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