Bruce J. West

Applications of Nonlinear Dynamics to Clinical Cardiologya

(1) Nonlinear mechanisms may apply both to the understanding of SA-AV node interactions and to bifurcations leading to certain types of AV block. (2) The fractal His-Purkinje system serves as the structural substrate for the generation of the broadband, inverse power-law spectrum of the stable ventricular depolarization (QRS) waveform. (3) Fractal anatomy is also seen in multiple other systems: pulmonary, hepatobiliary, renal, etc. Fractal morphogenesis may reflect a type of critical phenomenon that results in the generation of these irregular, but self-similar structures. (4) Self-similar (fractal) scaling may underlie the 1/f-like spectra seen in multiple systems (e.g., interbeat interval variability, daily neutrophil fluctuations). This fractal scaling may provide a mechanism for the constrained randomness that appears to underlie physiological variability and adaptability. (5) Behavior consistent with subharmonic bifurcations is seen in cardiac electrophysiology (e.g., sick sinus syndrome) and hemodynamic perturbations (e.g., swinging heart phenomenon in pericardial tamponade). (6) Ventricular tachyarrhythmias associated with sudden cardiac death (e.g., torsades de pointes, ventricular fibrillation) appear to reflect relatively periodic, not chaotic (turbulent) processes resulting from disruption of the physiologic fractal depolarization sequence. (7) Spectral analysis of Holter monitor data may help in the detection of patients at high risk for sudden death.

Some observations on the questions: is ventricular fibrillation chaos?

Abstract Ventricular fibrillation has been modeled as cardiac chaos occuring after a series of subharmonic bifurcations. However, previous experimental studies have suggested that fibrillatory oscillations have a relatively narrow-band frequency spectrum inconsistent with a turbulent process. Similarly, during the first minute of canine fibrillation we observed only a few localized frequency peaks from the epicardial and body surface electrocardiogram rather than a broadband type of spectrum as would be predicted for chaotic dynamics. Further narrowing of the frequency spectrum occured during the second minute of fibrillation. The frequency spectrum of ventricular fibrillation contrasts with scaled, broadband spectra observed in normal cardiac function. We suggest that ventricular fibrillation may serve as a general model for transitions from broadband stability to certain types of pathological periodicities in other physiological perturbations.

Chaos in Physiology: Health or Disease?

The universality of ‘chaotic’ dynamics in mathematical and physical systems [1–4] has prompted renewed interest in the application of nonlinear analysis to biological processes [4,5]. Attention has also focused on the physiological and medical implications of these concepts [4,6–11]. The prevailing viewpoint is that the dynamics of health are ordered arra regular and that a variety of pathologies represent a bifurcation to chaos [6,9,12]. For example, Smith and Cohen [9] advanced the hypothesis that ventricular fibrillation, the arrhythmia most commonly associated with sudden cardiac death, is a turbulent process (cardiac chaos) that may result from a subharmonic bifurcation (period-doubling) mechanism.

Nonlinear dynamics of the heartbeat: I. The AV junction: Passive conduit or active oscillator?

Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V0) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3:2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1:1 (SA:AV) phase-locking of the oscillators occurs over a critical range of V0. Externally driving the SA oscillator at increasing rates results in 3:2 AV Wenckebach periodicity and a 2:1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1:1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.

The First, The Biggest, and Other Such Considerations

We investigate the relation between the underlying dynamics of randomly evolv ing systems and the extrema statistics for such systems. Independent processes, Fokker-Planck processes and Lévy processes are considered.

Stochastic processes with non-additive fluctuations

We present a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations. Our main conclusion is that the Stratonovich prescription is the one that should always be used to describe physical systems. This conclusion is shown to be consistent with results obtained from path integral and Master equation approaches.

Nonlinear dynamics of the heartbeat: II. Subharmonic bifurcations of the cardiac interbeat interval in sinus node disease

Abstract Changing the coupling of electronic relaxation oscillators may be associated with the emergence of complex periodic behavior. The electrocardiographic record of a patient with the “sick sinus syndrome” demonstrated periodic behavior including subharmonic bifurcations in an attractor of his interbeat interval. Such nonlinear dynamics which may emerge from alterations in the coupling of oscillating pacemakers are not predicted by traditional models in cardiac electrophysiology. An understanding of the nonlinear behavior of physical and mathematical systems may generalize to pathophysiological processes.

Nonlinear dynamics, electrical alternans, and pericardial tamponade

Total electrical alternans is a highly specific ECG marker of pericardial tamponade.’ This phenomenon is characterized by periodic changes in the amplitude or morphology of the P, QBS, and ST-T waves in any single lead from one beat to the next. A feature of particular interest is that this cycle typically repeats itself after every second heartbeat (2:l alternans). Thus the second ECG complex differs from the first, while the third is similar to the first, the fourth similar to the second, and so on. Angiographic2 and echocardiographic3-5 studies have convincingly demonstrated that such total electrical alternation reflects mechanical oscillation of the heart in the pericardial fluid-the “swinging heart” phenomenon. However, a major unanswered question remains: What is the underlying mechanism responsible for the characteristic two-beat periodicity associated with electrical alternans in pericardial tamponade? Nonlinear dynamics. A potentially useful approach to explaining these frequency changes in pericardial disease may come from a new branch of physics called nonlinear dynamics.6-8 A major focus of nonlinear dynamics is modeling transitions in systems which show abrupt changes in their patterns of behavior. Such models have proved to be essential in understanding a variety of previously refractory problems in such fields as fluid turbulence and meteorology, as well as in providing new insights into anomalous oscillations in mathematical, chemical, and biological systems. Oscillations and subharmonic bifurcations. For a few simple processes, the output of the system is linearly

Polaron formation in the acoustic chain

Exact calculations are presented which detail the process of polaron formation in the one‐dimensional acoustic chain. Exact solution is possible because the limit considered is the transportless limit in which the Hamiltonian matrix elements responsible for the motion of an excitation among site states have been set to zero. The polaron formation process is found to be decomposable into two subprocesses having distinct time scales. A disparity of time scales is possible in the case of long wavelength excitations. The consistency of our conclusions is demonstrated through the consensus of results obtained for the discrete acoustic chain, the Debye model, and the elastic continuum.

Stochastic processes with non-additive fluctuations: II. Some applications of Itô and Stratonovich calculus

Abstract We present by way of examples a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations. The particular examples we consider include Kubo line broadening and chemically reacting systems. We conclude that the Stratonovich prescription should always be used to analyze physical systems.