Claremont Colleges Scholarship @ Claremont

WM Keck Science Faculty Papers W.M. Keck Science Department

3-1-1995 Dynamical Disease: Identification, Temporal Aspects and Treatment Strategies for Human Illness Jacques Bélair University of Montreal

Leon Glass McGill University

Uwe an der Heiden Witten/Herdecke University

John Milton Claremont McKenna College; Pitzer College; Scripps College

Recommended Citation Bélair, Jacques, Leon Glass, Uwe an der Heiden, and John Milton. "Dynamical Disease: Identification, Temporal Aspects and Treatment Strategies of Human Illness." Chaos 5.1 (1995): 1-7. DOI: 10.1063/1.166069

This Article is brought to you for free and open access by the W.M. Keck Science Department at Scholarship @ Claremont. It has been accepted for inclusion in WM Keck Science Faculty Papers by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Dynamical disease: Identification, temporal aspects and treatment strategies of human illness Jacques Belair Centrefor Nonlinear Dynamics in Physiology and Medicine, McGill University; J6,55 Drummond Street, Montreal, Quebec H3G lY6, Canada; and Departement de Moshematiques et de Statistique, Centre de Recherches Mathematique, Universiie de Montreal, Montreal, Quebec H3C 3J7, Canada Leon Glass Centre for Nonlinear Dynamics in Physiology and Medici~e, McGill University, 3655 Drummond Street, Montreal, Quebec H$G 1Y6, Canada; and Department of Physiology, McGill University, 3655 DrummondStreet, Montreal, Quebec H3G 1Y6, Canada Uwe an der Heiden Departm~nt ofMathematics, University of Witten/Herdecke, StockumerStrasse 10, D 5810 WZ'tten, Germany John Milton Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, 3655 DrummondStreet, Montreal, Quebec H3G lY6, Canada: -and Department of Neurology and Committee. on Neurobiology, The Hospitals, 5841 South Maryland Avenue, Room B 206, Chicago, Illinois 60637 (Received 14 December 1994; accepted for publication 14 December 1994) Dynamical diseases are characterized .by sudden changes in the qualitative dynamics of physiological processes, leading to abnormal dynamics and disease. Thus, there is a natural matching between the mathematical field of nonlinear dynamics and medicine. This paper summarizes advances in the study of dynamical disease with emphasis on a NATO Advanced Research Worshop held in Mont Tremblant, Quebec, Canada in February 1994. We describe the international effort currently underway to identify dynamical diseases and to study these diseases from a perspectiveof nonlinear dynamics. Linearandnonlinear time series analysis combined with analysis of bifurcations in dynamics are being used to help understand mechanisms of pathological rhythms and offer the promise for better diagnostic and therapeutic techniques. © 1995 American Institute ofPhysics.

4 I. INTRODUCTION Mackey and Glass.3• Dynamical diseases arise because of abnormalities in the underlying physiological control mecha­ Physicians have long recognized the importance of con­ nisms. The hallmark of a dynamical disease is a sudden, sidering the temporal dimension of illness for arriving at a qualitative change in the temporal pattern of physiological diagnosis and deriving treatment strategies. Diseases can be distinguished by their onset (acute versus sub-acute) and by variables. Such changes are often associated with changes in their subsequent clinical course (self-limiting, relapsing­ physiological parameters or anatomical structures. The sig­ remitting, cyclic, chronic progressive). Recognition of the nificance of identifying a dynamical disease is that it should temporal rhythms, combined with the knowledge that certain be possible to develop therapeutic strategies based on our temporal rhythms respond best to certain treatment strate­ understanding of dynamics combined with manipulation of gies, often provides a basis for therapy. the physiological parameters back. into normal ranges. The recentanddramatic advancesin molecular medicine Progress in this direction requires the careful documentation have shifted the attention of the medical community away of the temporal aspects of disease combined with suitable from considerations of temporal phenomena. At the current theoretical analyses. Several books, conferences and reviews S 1O time, a strong impetus for studying disease dynamics is com­ have focused on nonlinear dynamics and human disease - ing from the mathematics and physics communities. Ad­ A workshop held in Mont Tremblant, Quebec, Canada, vances in our· understanding about the of oscillatory February 1994 provided an opportunity to review recent ad­ phenomena in the discipline of nonlinear dynamics offer the vances in the study of dynamical disease. The meeting at­ possibility of applying this new knowledge to gain insights tracted a multidisciplinary audience of physicians, physiolo­ into the etiology and treatment of diseases. The mathematical gists, mathematicians, and physicists. Topics included the methods for understanding the origin, stability, and bifurca­ identification of dynamical diseases, methods for analyzing tions of dynamical behavior in nonlinear equations appear to the temporal aspects of diseases, onset and offset of rhythms, be ideally suited for the analysis of complex rhythms con­ mechanisms of oscillatory phenomena, and the design of fronted by the physician on a daily basis. treatment strategies. In this article we provide an overview of Although the origins for the application of nonlinear dy­ our current understanding of dynamical disease. Although namics to the study of human disease can be traced back to we emphasize material covered at the workshop, we also the analyses of certain cardiac arrhythmias in the 1920s,I.2 mention other work in order to provide a more balanced the concept of a dynamical disease was first proposed by overview of the field.

CHAOS 5 (1),1995 1054-1500/95/5(1)/1/7/$6.00 © 1995 American Institute of Physics 1 Downloaded 30 Oct 2008 to 134.173.131 .214. Redistribution subject to AI P license or copyright; see http://chaos.aip.org/chaos/copyright.jsp 2 Belair et st: Dynamical disease

II. DYNAMICAL DISEASES mortality andlor morbidity for which there is an adequate therapy. Collection of long time series often involves incon­ . In a typical medic al clinic, a significant proportion of venience and extra cost, and may be of little immediate ben­ patients present abnormalities in the dynamics of a physi­ efit to the patient based on current knowledge. Studies of ological process. Easily identified arrhythm ias are well dynamical disease are notable for the variety of lime series known in , respirology, and neurology. Arrhyth­ assembled, for the ingenuity of methodology used 10 collect mias may be significant causes of patient mortality, as in them, and the wide range of different methods that are being cardiac arrest. Symptoms which recur frequ enlly, such ob­ as developed to analyze them. structive sleep apnea or epileptic seizures, are important The techniques used to collect data include a variety of causes of patient morbidity. noninvas ive and inva sive techniques. Information about dy­ One obstacle confronted by non-physicians studying dy­ namical disease is deri ved from the following sources: n ~mi c a i disease is that they have litlle idea of the extraordi­ (1) Noninvasive methods . nary range of dynamics that can be found in the hunian body. patient diaries ;II The papers in this volume prov ide a glimpse into what is • mechanical displacements of respiratory going on. In an attem pt to docume nt dynamical disease in the structures; 12 nervous syste m, Milton and Black " carried out a literature • air flow velocities and concentrations of oxygen and survey that identified at least 32 diseases with interesting carbon dioxide in expired gases:" time courses in Which . symptoms andlor signs recur. Al­ motion of extremities using lasers and though many of these disorders are relatively commo n, for accelerometers; 13.14.22.23 example. see the discussions in this volume of hiccupS.1 2 • electromyographic (EMG) activity associated with Parkiasonism.P tardive dyskinesia.l" respiratory arrest,IS muscle;12.15,23 there are also many obscure disorders. For example, Milton • sound waves generated by the voice;24 and Black" identify 8 different disorders associated with changes in pressure on a pressure plate associated control of eye position and pupillary size. Even though the with changes in posture;2S. abnormal dynamics in these disorders are easy to observe magnetic fields from the brain;26 and can be recorded noninvasively, there has been little • blood pressure and electrical activity associated analysis to date of the rhythms from the perspective of basic with the heart beat.~7-3 o science and mathematics. (2) Invasive measurements One area in which dynamical aspects are significant, but • concentration of hormones in the blood; 17-19 still are not well understood is endocrinology. The mamma­ core body tempe rarurer" lian ovulation cycle is perhaps the most carefully studied, but • concentration of blood cells in the blood;>f there is still scant theoretical analy sis." Moreover, it is be­ intracardiac electrical activity, blood pressure and coming clear that, in general. hormone secretion displays wall motion?' complex pulsatile secretion pattern s. Since hormones are ac­ tive in such small quantities. and assays are expensive, sys­ The lime scales of measurements range fro m seconds to tematic studies are extremely difficult 10 carry out.. Notable days. Noninvasive measureme nts are capable of collecting exceptions are provided by analyses of blood concentrations vast amounts of data but limitations arise because of the of parathyroid hormone,'? insulin, IS and growth hormone.l" necessity of appropriate methods of data storage, retrieval The gaps in our understanding ' of the control of hormone and analysis. secretion are still huge. For exampl e, although it is know n that menopausal hot flashes are associated with changes in IV. TIME SERIES ANALYSIS ' hormone secretion occurring at menopause.i" the exact con­ trol mechanisms regulating the hot flashes are unknown. Typically times series of physiological data display sig­ Similarly, there are many hlood disorders characterized by nificant irregularities. Various issues arise. Are the irregulari­ striking dynamics, but the control circuits regulating these ties associated with noise or a deterministic process or some disorders are difficult to isolate and study.3.6.7.21 combination of the two? Independent of any understanding of underlying dynamic mechanism, is there any way to ana­ lyze the data to dislinguish between subgrou ps of subjects III. COLLECTION OF DATA (e.g, healthy versus diseased)? To what extent can an analy­ The first step towards obtaining a dynamical perspecti ve sis of the time series give hints about mechanisms? of disease is the careful measurement of the phenomena as a The si mplest way to analyze a lime series is 10 graph it function of lime. Such measure ments take the form of a time and examine the lime series by eye. For example, Whitelaw series, i.e. the tabulati on of the phenomena as a function of et al.' 2 illustrated this approach in their discussion of hic­ time. The analysis of such time series may provide important cups. They were able to conclude that the timing of hiccups insights into the etiology of the phenomena. is controlled by a central neural oscillator which is influ­ ]0 medicin e, collection and interpretation of time series enced by both the cardiac and respiratory cycles. . always involves difficulties. Since physicians interact with A slightly more sophisticated method of data analysis is patients, collection of data almost always requires the active to form a histogram from the time series, i.e. a graph of the collab oration of a physician. The physician's main interest is frequency at which the variable attains a certain 'value. These in identi fying disorders that are associated with significant histograms can be described mathematically in terms of a

CHAOS , Vol. 5,No.1., 1995 Downloaded 30 Oct 2008 to 134.173.131.214. Redistribution subject to AlP license or copyright; see http://chaos.aip .org/chaos/copyright.jsp Belair et al.: Dynamical disease 3

density function. A clinically familiar density function is the troIs. Bezerianoset at. 30 discuss the practical limitations dur­ "bell-shaped" normal distribution. Density functions are ing the application of the correlation dimension to heart rate very useful shorthand descriptions of a time series since they time series. They find that heavy smoking and mild exercise can be used to calculate measurable quantities such as the do not alter the complexity of EKG rhythms. mean value, the probability that the next value will be greater Determination of the entropy or one of its variants pro­ than a certain value, and so on. Demongeot et at.32 suggest vides an alternative methodto characterize the "complexity" that it may be easier to analyze mathematical models de­ of a time series.37 Using estimates of the dimension and en­ scribing neurophysiological networks from the point of view tropy, Lipsitz/" found that there is a decrease of complexity of density functions and their associated invariant measures. 2 7 Thus the measurement of a density function provides one of heart dynamics associated with aging. Peng et al have method to directly compare the predictions of a mathemati­ examined long range fluctuations using the method of de­ cal model to clinical observations. trended fluctuation analysis. There appear to be significant Most research involving data analysis requires comput­ differences between thefluctuations of heart rate in normal ers. We briefly review several of the computer-based meth­ individuals and high risk cardiac patients." Finally, Ding ods of data analysis. et al.38 suggested that analysis of dwell times may be better Some of the methods involve traditional time series than other measures of complexity for studying the dynamics methods such as autocorrelation and power spectrum. For of auditory perception. 2 2 example, Deuschl et a/ used this approach to obtain a If used only as a diagnostic tool, there is no need to have novel classification of tremor. They observed that in order to an interpretation of the dimension, approximate entropy, or distinguish between the various types of tremors observed in other statistical measure of a time series. However, there is clinical practice it was necessary to use methods based on currently significant interest in using methods of nonlinear waveform as well as frequency and amplitude of the tremor. dynamics to control complex rhythms. In this enterprise, it The role of noise in the generation of complex dynamics may be useful (or essential) to understand more about the is being attacked at a variety of levels. At the most basic level, it is still not known whether the opening and closing of underlying dynamics. In particular, it may be important to ion channels is associated with noise Or a deterministic pro­ know if the time series is generated by a deterministic or cess. Guevara and Lewis33 assume that thereis a stochastic chaotic process. If a time series is generated by a determin­ element in the opening and closing of ion channneis and istic process, then it should be possible to predict the dynam­ investigate the effect this has on the timing of interbear vari­ ics (at least for short times into the future). Chang et al.'9 and ability in a theoretical model of cardiac rhythm generation. Sauer''" discuss prediction methods for time series analysis; They conclude that a significant degree of irregularity (about If one can predict the future better for a given time series 3%) can arise solely as a consequence of the underlying vari­ than for a suitable stochastic surrogate, then this provides ability in the ion channel dynamics. On a more phenomeno­ good evidence that there is nonlinear determinism that can 2 3 logical level, Boose et al modeled EMG behavior associ: potentially be exploited in applications. For example, predic­ ated with tremor as sinnsoidally modulated white noise. tion of when clinically significant events will occur in the 25 Collins and DeLuca considered postural control during future would be of obvious benefit for the design of thera­ quiet standing from the point of view of a random walk. peutic strategies for the management of sudden cardiac death They concluded that both open and closed loop control is syndrome and epilepsy. Prank et a/. I 7 conclude from their utilized by the nervous system in regulating postural sway. Finally, Paydarfar and Buerkel'? include noise as a mecha­ computations of dimension of hormone levels that a nonlin­ nism for switching between respiratory arrest and a normal ear deterministic process is underlying the dynamics in this respiratory rhythm. Analysis of experimental data in which system. Based on theoretical modeling, Chang et al.'9 find there is a combination of both noise and deterministic chaos that prediction is a better method to distinguish determinism is a difficult issue that is treated by Schreiber and Kantz.34 from noise than otherquantitative measures. "Nonlinear" methods for time series analysis developed Overall, it is difficult at this time to conclude that one in the last decade are being exercised against physiological method or style of data analysis is better than another. There time series. There are still significant methodological issues. are heated debates between proponents of the various meth­ Practical problems arise because of the shortness of the time ods. Each method has its advantages and disadvantages. It is series, lack of stationarity of time series, and the influence of of pressing importance that the different groups make both noise on the measurements. their data and their algorithms available to facilitate conver­ Probably the most commonly used nonlinear statistic for gence to common methodologies. At this stage, humility in data analysis is the correlation dimension, originally used by the face of complexity is essential. Dynamics in physiologi­ Grassberger and Procaccia.v However, Kantz and cal systems reflect random and deterministic processes at a Schreiber'6 describe the difficulties in applying the Grassberger-Procaccia correlation dimension algorithm to variety of levels ranging from the molecular level, to the experimental data and present useful advice for novices. system level, to the environment. It is impossible to control Newell et al.14 use a measurement of the correlation dimen­ all random influences and the problem of interpretation is sion to demonstrate the finger tremor in patients with tardive formidable. Facile generalizations and easy answers should dyskinesia is different from that measured in healthy con- be viewed with skepticism.

Downloaded 30 Oct 2008 to 134.173.131 .214. RedistribUtio R ~~9~ct~~ ' A\ ~ PceHs~9~ ? copyright; see http://chaos.aip.org/chaos/copyright.jsp 4 Belair et et.: Dynamical disease

V. BIFURCATIONS IN DYNAMICS AND ORIGINS OF maps." Because of the relevance of the dynamics in model DYNAMICAL DISEASE systems to human health and disease, periodic forcing is an important technique to study physiological dynamics and A striking aspect of some dynamical diseases is the sud­ control. den change in qualitative dynamics. The most dramatic of these changes lead to imminent death as in respiratory or cardiac arrest or a seizure in a patient with epilepsy. From a B. Hopf bifurcation mathematical perspective, the observations of qualitative From a mathematical perspective, one of the best known changes in dynamics may provide a clue for the initiation of and understood bifurcations is the Hopf bifurcation. In the theoretical analyses. Indeed, mathematical analyses of bifur­ Hopf bifurcation, as a parameter is changed, a stable steady cations in nonlinear equations combinedwith experimental state is destabilized and is supplanted by a stable oscillation. obsevations of dynamics in biological systems often provide This occurs in two different ways. In the supercritical Hopf an entry point for theoretical analyses. bifurcation, the amplitude of the oscillation grows gradually A. Bifurcations in maps larger, and no parameter values lead to bistability between a steady state and an oscillation. In the subcritical Hopf bifur­ One of the most extensively studied problems in nonlin­ cation, there is a range of parameter values that gives either ear dynamics of biological systems is the periodic forcing of a stable steady state or stable oscillation, depending on the oscillatory and excitable biological systems.' Understanding initial condition. A dynamical signature of a subcritical Hopf of the dynamics in these systems has implications for a wide bifurcation is a sudden onset of a high amplitude oscillation variety of phenomena ranging from generation of cardiac as a parameter changes with hysteresis effects depending on arrhythmias from the competition between competing pace­ whether the parameter is increased or decreased. makers to the entrainment of the circadian rhythm by light. ,It is known that gradual tuning of parameters in negative In the simplest cases, biological systems can be modeled by 4 1 feedback loops can give rise to a supercritical Hopf low dimensional maps. Kunysz et al described the effects bifurcation.i An intriguing but speculative possibility is that of periodic stimulation on in vitro preparations of atrioven­ dynamical diseases'with an insidious onset may'be associ­ tricular (AV) nodal tissue. This work is directly relevant to ated with supercritical Hopf bifurcations. For example, per­ functioning of AV node in normal and pathological condi­ haps the gradual development of abnormal tremor in tardive tions. Periodic stimulation of the tissue generated rhythms dyskinesia is associated with gradual modification of feed­ similar to what is observed during AV heart block. Kunysz et back loops for movement control. 14 Similarly, some ofthe al. found however that one dimensional maps are not ad­ altered dynamics in aging may occur as a consequence of equateto interpret the entirerangeof phenomenaobserved." changes in the time delays and sensitivity of physiological The effects of periodic forcing in hormonal and control feedback 100ps2 ' systems is more difficult to analyze because of the experi­ Sudden onset or offset of large amplitude oscillations is mental difficulties of measuring hormonal concentrations. observed in many circumstances. An important case is respi­ 18 However, Sturis et al. analyzed the effects of periodic glu­ r~tory arrest, ¥tudied by Paydarfar and Buerkel. 15 They pro­ cose infusion in humans and in mathematical models. They pose that this, important clinical arrhythmia may be associ­ found evidence for phase locking between the glucose infu­ ated with a bistability between a steady state and an sion rhythm and the insulin rhythm. The phenomena they oscillation where there is a perturbation that can take you observed are similar to bifurcations in low dimensional maps from one basin of attraction to another. This analysis ernpha­ 42 proposed for phase locking. Foweraker et al. modeled the sizes an important point of direct relevance for an under­ pulsatile release of luteinizing hormone releasing factor standing of human disease. In dynamical systems with mul­ (LHRH) from the hypothalamus. LHRH regulates the release tiple basins of attraction, changes in dynamics can arise as a of two hormones from the pituitary gland which are essential consequence of: changes in control parameters that destabi­ for the maintenance of normal reproductive function. In their lize the basins of attraction; changes in the boundary basins model, the external periodic forcing arises from adrenergic as a consequence of parameter modification; and perturba­ inputs to the hypothalamus and the oscillator is formed by a tions that take you from one basin to the other.' However, network composed of reciprocally connected LHRH neurons much more analysis is needed of such phenomena as parox­ and gamrna-aminobutyric acid containing neurons. To ana­ ysmal cardiac arrhythmias, hiccups, hot flashes, and respira­ lyze this they carried out extensive simulations of periodi­ tory apnea in order to identify the origin of the abnormal cally forced Fitznugh-Nagumo equations. The bifurcations rhythms. observed can be at least partially understood from bifurca­ Another important class of phenomena are those in tions of low dimensional maps. Finally, Mayer et al.21 show which there is a sudden switch between two qualitative that periodic forcing of a simple model for the immune sys­ rhythms. A striking example is provided in the study of tem can display chaotic dynamics. Beuter and Vasilakosl3 who demonstrated a sudden switch to These studies of different systems underscore the follow­ large amplitude: Parkinsonian tremor from' a tremor that ap­ ing observation: periodic forcing' of nonlinear" oscillations peared superficially to be normal. This is similar to the sub­ gives rise to a large number of different regular and irregular critical Hopf bifurcation, but there is a bistability between rhythms that can be understood based on fundamental math­ different rhythms. Switches between physiological tremor ematicalprinciplesrelated to bifurcations of lo':¥ dimensional and pathological tremor are also described by Boose et al.23

CHAOS, Vol. 5, No.1, 1995 Downloaded 30 Oct 2008 to 134.173.131.214. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp Belair 9t el.: Dynamical disease 5

C. Skipped beat rhythms In this section we wish to summari ze the directions that now seem the most promising. In some biological rhythms , there appears to be a clock setting the rhythm, but there is a haphazard appearance to (1) Diagnosis. There is active work trying to utilize time expressed rhythms with a seemingly random dropping of series analysis for practical applications. These include: beats. As Longtin43 discusses, such rhythms can arise from a identification of those at high risk for cardiac death ,27 number of different mechanisms-the simplest might be a early diagnosis and classification of abnormalities of 13 1422 gradua l increase of a variable controlling a rhythm to an tremor • . and postural contro!.2S oscillating threshold combined with noise. Experimental ob­ (2) Therapy-

CHAOS, Vol. 5"No. 1, 1995 Downloaded 30 Oct 2008 to 134.173.131.214. Redistribution subject to AI r- license or copyright; see http://chaos.aip.org/chaos/copyright.jsp 6 Belair et al.: Dynamical disease

Many present-day physicians and medical researchers 7L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms ofLife are searching for the "magic bullet." They hope that by re­ (Princeton University Press, Princeton, 1988). placing molecules that are altered or made deficient by dis­ 8J. Jalife (Editor), "Mathematical approaches to cardiac arrhythmias," Ann. N. Y. Acad. Sci. 591, t-417 (1990). ease, or by delivering the right drug, disease will be cured. 9L. Glass, P. J. Hunter, and A. McCulloch (Editors), Theory of Heart: Although one cannot dispute the profound successes of this Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function strategy (in the discovery of many miracle drugs­ (Springer-Verlag, New York, 1991). 10L. Glass (Editor), Chaos Focus Issue on Nonlinear Dynamics of Physi­ antibiotics, hormones, cortisone), theoretical analyses of ological Function and Control, Chaos 1, 247-334 (1991). complex systems such as presented here shift attention to 11J. Milton and D. Black, "Dynamic diseases in neurology and psychia­ system interactions and dynamics. Dynamics can be extraor­ try.t'Chaos 5.8-13 (1995). dinarily sensitive to minute changes in drug concentration. 12w. A. Whitelaw, J-Ph. Derenne, and 1. Cabane, "Hiccups as a dynamical disease," Chaos 5,14-17 (1995). The cardiac arrhythmia suppression trial offers a striking ex­ 13A. Beuter and K. Vasilakos, "Tremor: Is Parkinson's disease a dynamical ample of increased mortality associated with increased sud­ disease?" Chaos S, 35-42 (1995). den cardiac death in patients receiving a drug that reduced 14K. M. Newell, F. Gao, and R. L. Sprague "The dynamical structure of certain types of cardiac arrhythmias." Another graphic ex­ tremor in tardive dyskinesia,"Chaos 5, 43-47 (1995). 15D. Paydarfar and D. M. Buerkel, "Dysrhythmias of the respiratory oscil­ ample is in Sacks' recounting of the difficulties encountered lator,"Chaos 5, 18-29 (1995). in titrating the concentration of dopamine in encephalitis 16L. E. Meuli, H. M. Lacker, and R. B. Thau, "Experimental evidence panents." Even small changes of delivered drug can change supporting a mathematical theory of the physiological mechanism regulat­ a potential beneficial effect to an effect that causes significant ing follicle development and ovulation number," BioI. Reprod. 37, 589­ 594 (1987). patient morbidity or even mortality. 17K. Prank, H. Harms, G. Brabant, R-D. Hesch, M. Dammig, and F. The road from the development of ideas at the benchtop Mitschke, "Nonlinear dynamics in a pulsatile secretion of parathyroid to their application at the bedside is fraught with false turns hormone in normal human subjects," Chaos S, 76-81 (1995). and hidden pitfalls. Scientific wisdom dictates that the cru­ 18!. Studs, C. Knudsen, N. O'Meara, J. S. Thomsen, E. Mosekilde, E, Van Cauter, and K. S. Polonsky, "Phase-locking regions in a forced model of cially important experiments cannot be performed without a slow insulin and glucose oscillations" Chaos 5, 193-199 (1995). theoretical knowledge of the properties of the underlying 19R. Lanzi and G. Tannenbaum, "Time course and mechanism of growth control mechanisms and their response to perturbations. hormone's negative feedback effect on its own spontaneous release," En­ docrinology 130, 780-788 (1992). Nonetheless, the use of empirical methods, such as drug tri­ 20p' Kronenberg, "Menopausal hot flashes: Randomness or rhythmicity," als involving animals and patients, is the mainstay of Chaos 1, 271-278 (1991). present-day clinical research. This meeting drew attention to 21 H. Mayer, K. S. Zaenker, and U. an der Heiden, "A basic mathematical the fundamental importance of carefully documenting the model of the immune response," Chaos 5, 155-161 (1995). 22G. Deuschl, M. Lauk, and J. Timmer, "Tremor classification and tremor temporal aspects of disease and emphasized that "pencil and time series analysis," Chaos 5, 48-51 (1995). paper" research may have important implications for manag­ 23 A. Boose, C. Jenrgens, S. Spieker, and J. Dichgans, "Variations on tremor ing patients at the bedside. It is anticipated that forming an parameters," Chaos 5, 52-56 (1995). effective partnership between theorists studying dynamical 24H. Herzel, D. Berry, I. Titze, and I. Steinecke, "Nonlinear dynamics of the voice: Signal analysis and biomechanical modeling," Chaos 5, 30-34 disease, basic biological scientists, and clinical investigators (t995). holds great promise for treating human disease. 25J. J. Collins and C. 1. De Luca.v'Upright, correlated random walks: A statistical-biomechanics approach to the human postural control sys­ tern," Chaos 5, 57-63 (1995). ACKNOWLEDGMENTS 26 J. A. S. Kelso and A. Fuchs, "Self-organizing dynamics of the human brain: Critical instabilities and Sil'nikov chaos, "Chaos 5, 64-69 (1995). The workshop was a NATO Advanced Research Work­ 21 C. K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger, "Quantifica­ shop with additional support and funding provided by NIH, tion of scaling exponents and crossover phenomena in nonstatlonary Centre de Recherches Mathematiques (Montreal), The Fields heartbeat time series," Chaos 5, 82-87 (1995). 28L. A. Lipsitz, "Age-related changes in the 'complexity' of cardiovascular Institute for Research in Mathematical Sciences (Waterloo, dynamics: A potential marker of vulnerability to disease," Chaos 5, 102­ Ontario), Foundation International for Nonlinear Dynamics 109 (1995). (Bethesda), Medtronics (Quebec). Some of the material in 29 J. Kurths , A. Voss, A. Witt, P. Saparin, H. J. Kleiner, and N. Wessel, Sec. V on bifurcations is based on an analysis of Professor "Quantitative analysis of heart rate variability," Chaos 5, 88-94 (1995). 30 A. Bezerianos, T. Bountis, G. Papaioannau, and P. Polydoropoulos "Non­ W. F. Langford (Guelph) presented at the workshop. linear time series analysis of electrocardiograms," Chaos 5, 95-101 (t995). 31 D. P. Zipes and J. JaIife (Editors), Cardiac Electrophysiology: From Cell I W. Mobitz "Uber die unvollstandige Storung der Erregiingsuberleitung to Bedside, 2nd ed. (Saunders, Philadelphia, 1994). zwischen Vorhof und Kammer des mensclichen Herzens," Z. Gesarnte 32J. Demongeot, D. Benaouda, and C. Jezeque " 'Dynamical confinement' Exp. Med. 41, 189-237 (1924). in neural networks and cell cycle," Chaos 5, 167-173 (1995). 2B. van dcr Pol and J. van der Mark, "The heartbeat considered as a 33M. R. Guevara and T. 1. Lewis, "A minimal single-channel model for the relaxation oscillator and an electrical model of the heart," Philos. Mag. 6, regularity of beating in the sinoatrial node," Chaos S, 174-183 (1995). 763-775 (1928). 34T. Schreiber and H. Kantz. "Noise in chaotic data: Diagnosis and treat­ 3M. C. Mackey and L Glass , "Oscillation and chaos in physiological con­ ment," Chaos 5, 133-142 (1995). trol systems," Science 197, 287-289 (1977). 35p. Grassberger and I. Procaccia, "Measuring the strangeness of strange 4L. Glass and M. C. Mackey, "Pathological conditions resulting from in­ attractors," Physica D 9, 189-208 (1983). stabilities in physiological control systems," Ann. N.Y. Acad. Sci. 316, 36H. Kantz and T. Schreiber, "Dimension estimates and physiological data," 214-235 (1979). Chaos 5, 143-154 (1995), 5L Rensing, U. an der Heiden, and M. C. Mackey (Editors), Temporal 37 S. Pincus, "Approximate entropy (ApEn) as a complexity measure," Disorders in Human Oscillatory Systems (Springer-Verlag, Berlin, 1987). Chaos 5, 1IO-tI7 (1995). 6J. Milton and M. C. Mackey, "Dynamical diseases.vAnn. N.Y. Acad. Sci. 38M. Ding, R Tuner, and J. A. S. Kelso, "Characterizing the dynamics of 504,16-32 (1987). auditory perception," Chaos 5, 70-75 (1995).

CHAOS, Vol. 5, No.1, 1995 Downloaded 30 Oct 2008 to 134.173.131 .214. Redistribution subject to AI P license or copyright; see http://chaos.aip.org/chaos/copyright.jsp Belair et al.: Dynamical disease 7

39T. Chang, T. Sauer. and S. J. Schiff. "Tests for nonlinearity in short sta­ 45R. Langer, "New methods of drug delivery," Science 249, 1527-1533 tionary time series," Chaos 5.118-126 (1995). (1990). 4oT. Sauer. "Interspike interval embedding of chaotic signals," Chaos 5. 46 A. E, Reinberg, G. Labrecque, and M. H. Smolensky, Chronobiologie et 127-132 (1995). Chronotherapeutique: Heure Optimaled'Administration des Medicaments 41 A. M. Kunysz, A. A. Munk, and A. Shrier, "Phase resetting and dynamics (Flammarion, Paris, 1991). 47 A. Garfinkel, M. L. Spano, W. L. Ditto, and J, N. Weiss, "Controlling in isolated atrioventricular cells," Chaos 5, 184-192 (1995). cardiac chaos," Science 257,1230-1235 (1992), 42 J. P.A. Foweraker, D. Brown. and R. W. "Discrete-rime stimula­ Marrs, 48W,L. Ditto and L. M. Pecora, "Mastering chaos" Sci, Am, 269(2), 78-84 tion of the oscillatory and excitable forms of a FitzHugh-Nagumo model (1993). applied to the pulsatile release of luteinizing hormon e releasing hor­ 49S. J. Schiff, K. Jerger, D. H. Duong, T, Chay, M, L. Spano, and W, L. mone," Chaos 5, 200-208 (1995). Ditto, "Controlling chaos in the brain," Nature 370,615-620 (1994). 43 A. Longtin, "Mechanisms of stochastic phase locking." Chaos 5. 209­ 50D, S. Echt et al., "Mortality and morbidity in patients receiving encainide, 215 (1995). flecainide, or placebo: The cardiac arrhythmia suppression trial," N. Engl. 44D. Claude, "Shift of a limit cycle in biology: From pathological to physi­ J. Moo. 324, 781-788 (1991). ological homeostasia," Chaos 5. 162-166 (1995). 51 0 . Sacks. Awakenings (Harper & Row, New York, 1974).

Downloaded 30 Oct 2008 to 134.173.131.214. RedistribUtioX~~\?J~t~P6' RI~cJ nJ~~1 copyright; see http://chaos.aip.org/chaos/copyright.jsp