Read an Excerpt
Dynamical Chaos
Proceedings of a Royal Society Discussion Meeting Held on 4 and 5 February 1987
By Michael V. Berry, Ian C. Percival, Nigel Oscar Weiss PRINCETON UNIVERSITY PRESS
Copyright © 1987 The Royal Society of London and authors of the individual papers
All rights reserved.
ISBN: 978-0-85403-333-1
CHAPTER 1
Diagnosis of dynamical systems with fluctuating parameters
By D. Ruelle
Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
Many time evolutions occurring in Nature may be considered as non-autonomous, but dependent on parameters that vary slowly with time. It is argued here that some, but not all, of the tools used to understand chaotic dynamics remain useful in this situation.
In recent years it has been ascertained that many time evolutions observed in Nature exhibit the features of chaos. This means that they are deterministic time evolutions involving only a finite number of degrees of freedom, but that a complicated non-periodic behaviour is observed, due to sensitive dependence on initial condition. Mathematically, the deterministic time evolution corresponds to an autonomous differentiable dynamical system, sensitive dependence on initial condition means that a small perturbation of the initial condition will grow exponentially with time (as long as it does not become too large). The asymptotic evolution of the system takes place on a (usually) complicated set in phase space called a strange attractor.
A fundamental finding is that hydrodynamic turbulence is chaotic, and described by strange attractors. Many other examples of chaos have been demonstrated clearly in various areas of physics and chemistry, and less clearly in biology and economics. Investigations of experimental chaos have been mostly of either geometric or ergodic nature. (The study of chaotic power spectra, broad band spectra, has also played an important role historically, but does not at this time yield the sort of detailed information that is provided by other techniques.) The geometric approach visualizes reconstructions of attractors and of bifurcations and is limited to weakly excited systems ('onset of turbulence'). The ergodic approach determines information dimensions, characteristic (i.e. Liapunov) exponents and entropies, and is applicable to moderately excited systems.
The analysis of relatively modest data can provide a usable power spectrum, or an estimate of the information dimension by the Grassberger–Procaccia algorithm. In general, however, the detailed diagnosis of chaotic dynamical systems requires long time series of high quality (stability of parameters of the system and precision of experimental measurements). One faces then the problem that the systems for which our techniques work best are not those in which we are mostly interested. Among the latter we may quote pulsating variable stars, electroencephalograms and time series of economics. One can of course dismiss at least the last two examples by arguing (reasonably) that the electrical activity of the brain, and the stock market, are not autonomous dynamical systems with few degrees of freedom.
Against this reasonable view let me remark that information dimension estimates for EEGs(electroencephalograms) (see Layne et al. 1986: Rapp et al. 1987) and time series of economics (see Scheinkman & Le Baron 1986) are not at all suggestive of pure randomness. Let me talk of EEG data that I have seen (some from Rapp and some from Lehmann, analysed in Geneva in collaboration with J.-P. Eckmann and S. Kamphorst). They suggest that there are many degrees of freedom, or 'modes', with decreasing amplitudes, and that computations of information dimension yield variable results depending on which modes have amplitudes sufficiently large to be captured by a given calculation. This type of 'explanation' is, however, basically unsatisfactory because 'modes' cannot in general be separated in a truly nonlinear theory.
What then ? I suggest that some interesting time evolutions occurring in Nature, those with adiabatically fluctuating parameters (AFPS), although not represented by an autonomous dynamical system, are accessible to analysis. I have in mind evolutions of the type
dx/dt = F(x, λ(t)) (continuous time)
or xn+1 = f(x, λ(n)) (discrete time),
where the time dependence of λ is assumed to be adiabatic (slow compared with the characteristic times of the autonomous systems obtained by fixing λ), and not too large. Then, instead of a fixed attractor A, we have a family (Aλ) depending on λ = λ(t) or λ(n). The evolution of λ might itself be determined by a dynamical system, but we consider it as arbitrarily given a priori. Note that a time evolution of the above type is expected both for EEGS and in economics. Note also that noise can be accommodated in our λ-dependence provided that it satisfies the requirement of adiabaticity.
A first remark is that in a system with AFPS the information dimension will be considerably messed up, because instead of looking at an attractor A we are looking at a union [union] Aλ. The observed information dimension will thus be the dimension of the attractors Aλ (supposed to be independent ofλ) plus the dimension of the set of λs in parameter space. If we observe, for instance, an attracting periodic orbit with slowly decreasing amplitude (mechanical oscillations with friction) we shall obtain a dimension equal to 1 + 1 = 2. Long-term evolution in economics would similarly increase the dimension by 1. All we can say in general is that the observed dimension is an upper bound to the dimensions of the Aλ (more precisely one should speak of the information dimension of invariant probability measures carried by Aλ. If one suspects (as in economics) that there is a long time 'secular' evolution of the system, this can be checked by taking an early point X(t0) on the reconstruced attractor and looking at the statistics of times at which the point X(t) comes back close to X(t0). For a long time series these times will be predominantly at the beginning of the series if there is a secular evolution of the system
Contrary to the dimension, the higher characteristic exponents may not be much perturbed by the fluctuation of the parameters λ. In other words, it will often be the case that the higher characteristic exponents are stable practically under small changes of λ; note however that this is not a mathematical statement of continuity. (If the time evolution of λ(t) is given by a dynamical system, adiabaticity will correspond to small exponents for the λ-evolution, and those will not interfere with the higher characteristic exponents of the global system.) Therefore the determination of the higher characteristic exponent (or exponents) is very desirable for systems with AFPS , because it (or they) can provide more unambiguous information than the information dimension.
We turn now to the problem of short-term predictions for the time evolution of dynamical systems. Consider for simplicity a time series (ul) corresponding to a system with discrete time, and chose an embedding dimension n such that the points (xi, ..., xl+n-1) [member of] Rn give a faithful representation of the dynamical system on its attractor. Then one can determine a continuous function Φ such that
ul+n = Φ (ul, ul+1, ..., ul+n-1)
(see Ruelle 1987, §3). The introduction of AFPS will not make this representation useless. Therefore short-term predictions of the evolution of a dynamical system with fluctuating parameters remain possible.
CHAPTER 2
Nonlinear dynamics, chaos and complex cardiac arrhythmias
By L. Glass, A.L. Goldberger, M. Courtemanche and A. Shrier
Periodic stimulation of a nonlinear cardiac oscillator in vitro gives rise to complex dynamics that is well described by one-dimensional finite difference equations. As stimulation parameters are varied, a large number of different phase locked and chaotic rhythms is observed. Similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites. Simplified models are analysed, which show some correspondence to clinical observations.
1. Introduction
The normal adult human heart at rest usually beats at a rate of between 50 and 100 times per minute. In many circumstances, some of which are life-threatening, but most of which are not, the normal rhythmicity is altered, resulting in abnormal rhythms called cardiac arrhythmias. The point of this paper is to show that a branch of mathematics called nonlinear dynamics may be useful in the analysis of physiological processes believed to underlie normal heart rate regulation and some cardiac arrhythmias.
The idea that mathematical analysis can play a role in understanding cardiac arrhythmias is not novel. Indeed, in the 1920s it was demonstrated that as parameters in mathematical models for the heart were varied, several different rhythms that resembled clinically observed arrhythmias could be generated (Mobitz 1924; van de Pol & van der Mark 1928). In nonlinear mathematics, these changes in the qualitative features of the rhythms that are observed as parameters vary are called bifurcations. Thus the problem of understanding cardiac arrhythmias in the human heart is identified with understanding the bifurcations and complex dynamics in mathematical models of the human heart.
One type of dynamic behaviour that is the object of intensive analysis in mathematics is chaos. Loosely, chaos is defined as a periodic dynamics in deterministic systems in which there is sensitive dependence to the initial conditions. This means that although in principle one could determine precisely the future evolution of the system starting from some initial condition, for chaotic dynamics any difference in the initial condition, no matter how small, will eventually lead to marked differences in the future evolution of the system. Although the existence of chaos was known to Poincaré and others since the end of the last century, in the past decade there has been a recognition of the potential significance of chaos in understanding the genesis of aperiodic dynamics experimentally observed in the natural sciences (Cvitanovic 1984). Unfortunately, there is in our view not yet an adequate operational definition for chaos in experimental or naturally occurring systems, but see Mayer-Kress (1986) for recent advances. The concept of chaos excludes non-deterministic stochastic processes, such as the Poisson process or random walk. It is not yet known how to measure the relative contribution of chaos as opposed to non-deterministic stochastic processes in experimental data.
Normal individuals show marked fluctuations in heart rate (Kitney & Rompelman 1980: Kobayashi & Musha 1982; Pomeranz et al. 1985; De Boer et al. 1985). In addition, cardiac arrhythmias are often extremely irregular and unstable (Pick & Langendorf 1979: Schamroth 1980) The adjective chaotic' is sometimes used to characterize cardiac arrhythmias that are believed to arise when there are several pacemaker sites competing for control of the myocardium (Katz 1946; Phillips et al. 1969; Chung 1977). It has been proposed that chaotic dynamics, in the mathematical sense, may underlie normal heart-rate variability (Goldberger et al. 1984; Goldberger & West 1987) as well as certain cardiac arrhythmias in humans (Guevara & Glass 1982; Smith & Cohen 1984: Glass et al. 1986b). The absence of a clear definition for chaos in experimental data has led to controversy. For example, ventricular fibrillation, an arrhythmia that leads to rapid death, is frequently called chaotic by clinicians, and it has been proposed that it may be associated with chaos in deterministic systems (Smith & Cohen 1984). However, there are marked periodicities during ventricular fibrillation, and the presence of deterministic chaos in this arrhythmia has been questioned (Goldberger et al. 1985, 1986).
In humans it is frequently difficult to analyse the mechanism underlying an arrhythmia, and systematic experimental studies are usually not feasible. One means of analysis is from the electrocardiogram (ECG). a record of electrical potential differences on the surface of the body that reflects the electrical activity associate!· with the heartbeat. Because the ECG can be obtained with lightweight monitor, it can be readily recorded over long time intervals. The ambulatory (Holter) ECG is an important means for evaluating patients. Holter recordings for as long as 24 h can be readily obtained, but conventional analysis of such records is limited. The great wealth of data about the dynamics of the heart that is contained in such records is generally distilled to characterize the mean heart rate and range. The presence and frequency of abnormal electrocardiographic complexes, which reflect abnormalities in cardiac impulse formation and propagation, are also determined. However, the analysis of long-term fluctuations in the Holter ECG is largely ignored.
One class of arrhythmias that has recently been the subject of much attention results from the presence of two pacemakers· the normal (sinus) pacemaker and a pacemaker at an ectopic (non-sinus) location. Such rhythms, whose existence has been recognized since the start of this century (Fleming 1912; Kaufmann & Rothberger 1917) are now called parasystolic rhythms. The possibility for interactions between the sinus rhythm and the ectopic rhythm often complicates interpretation of such rhythms. However, recent workers have made great progress in developing both experimental (Jalife & Moe 1976; Jalife & Michaels 1985 and theoretical (Moe et al. 1977; Swenne et al. 1981; Ikeda et al. 1983) models for parasystole. Interpretation of ECG records has led to the recognition of the importance of parasystolic mechanisms (Jalife et al. 1982; Nau et al. 1982; Castellanos et al. 1984).
Here we consider the interaction between a fixed periodic stimulus and a cardiac oscillator. Such a problem is of interest because it is amenable to experimental and theoretical analysis and because of its relevance to the interpretation of parasystolic rhythms. In §2 we consider the effects of periodic stimulation of spontaneously beating aggregates of cells from embryonic chick heart (Guevara et al. 1981; Glass et al. 1983, 1984, 1986b). Theoretical analysis of this system shows that periodic dynamics are expected at some stimulation frequencies and amplitudes, whereas chaotic dynamics are expected for other stimulation parameters. Experiments are in close agreement with the theory. In §3 we develop a theoretical model for parasystole. The model extends previous theoretical models of parasystole (Moe et al. 1977; Swenne et al. 1981; Ikeda et al. 1983; Glass et al. 1986a). We describe the bifurcations in the theoretical model and show that chaotic dynamics is expected over some regions of parameter space. In §4 we discuss Holter ECG records from ambulatory patients who display frequent ectopic beats. These records may show extremely irregular dynamics which we discuss in the context of chaotic dynamics and modulated parasystole. Finally, the significance of this approach to the analysis of cardiac dynamics is discussed.
2. Periodic stimulation of a cardiac oscillator
In this section we describe the effects of single and periodic stimulation of an aggregate of spontaneously beating cells from embryonic chick heart. As this work has been described in several recent publications, we briefly summarize the main results and refer the reader elsewhere for more details (Guevara et al. 1981; Glass et al. 1983; Glass et al. 1984; Glass et al. 1986b; Guevara et al. 1986).
Spontaneously beating aggregates of ventricular heart cells are formed by dissociating the ventricles of seven-day embryonic chicks and allowing the cells to reaggregate in tissue culture medium. The resulting aggregates are approximately 100–200 µm in diameter and each beats with its own intrinsic frequency, which lies in a range of about 60–120 times per minute (DeHaan & Fozzard 1975). A glass microelectrode is inserted intracellularly and can be used to inject single and periodic current pulses into the aggregate. In the present context, the electrical stimulator is analogous to the sinus rhythm, and the aggregate is analogous to an ectopic focus. Clearly, this represents a gross oversimplification of the anatomically complex heart, as it in no sense takes into account the spatial heterogeneity of cardiac tissue nor the various feedback mechanisms that act to modulate cardiac activity in vivo. Nevertheless, as stimulation parameters are varied, this model system generates a great variety of rhythms that resemble clinically observed arrhythmias. Some of these rhythms are periodic with N cycles of the periodic stimulation for each M cycles of the cardiac oscillation (N:M phase locking). Other rhythms are aperiodic (figure 1). The dynamics of this system can only be understood by using techniques in nonlinear dynamics. Thus, this model system is useful to fix ideas and to form a foundation for the analysis of more complex situations.
(Continues...)
Excerpted from Dynamical Chaos by Michael V. Berry, Ian C. Percival, Nigel Oscar Weiss. Copyright © 1987 The Royal Society of London and authors of the individual papers. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
