Leon Glass

Synchronization and rhythmic processes in physiology

Complex bodily rhythms are ubiquitous in living organisms. These rhythms arise from stochastic, nonlinear biological mechanisms interacting with a fluctuating environment. Disease often leads to alterations from normal to pathological rhythm. Fundamental questions concerning the dynamics of these rhythmic processes abound. For example, what is the origin of physiological rhythms? How do the rhythms interact with each other and the external environment? Can we decode the fluctuations in physiological rhythms to better diagnose human disease? And can we develop better methods to control pathological rhythms? Mathematical and physical techniques combined with physiological and medical studies are addressing these questions and are transforming our understanding of the rhythms of life.

The logical analysis of continuous, non-linear biochemical control networks.

Abstract We propose a mapping to study the qualitative properties of continuous biochemical control networks which are invariant to the parameters used to describe the networks but depend only on the logical structure of the networks. For the networks, we are able to place a lower limit on the number of steady states and strong restrictions on the phase relations between components on cycles and transients. The logical structure and the dynamical behavior for a number of simple systems of biological interest, the feedback (predator-prey) oscillator, the bistable switch, the phase dependent switch, are discussed. We discuss the possibility that these techniques may be extended to study the dynamics of large many component systems.

PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS

A large number of human diseases are characterized by changes in the qualitative dynamics of physiological control systems: Systems that normally oscillate. stop oscillating, or begin to oscillate in a new and unexpected fashion, and systems that normally d o not oscillate, begin oscillating. These changes in qualitative dynamics often have a sudden onset, and in many instances it has not been possible to identify the factors that lead to the disease. By dynarnical disease we mean a disease that occurs in an intact physiological control system operating in a range of control parameters that leads to abnormal dynamics and human p a t h ~ l o g y . ~ ’ In this paper, the changes in qualitative dynamics associated with the onset of the disease are identified with bifurcations in the dynamics of mathematical models of the physiological control systems. We shall consider in some detail dynamical diseases in the respiratory and haematopoietic systems. Our starting point is the ordinary differential equation

Classification of biological networks by their qualitative dynamics

Techniques are given to classify biological networks into classes having similar qualitative dynamics. The following steps are used: (1) A discretization is proposed which identifies a sequence of Boolean N -vectors, each of which differs from the preceding one in one locus, with transients and cycles in continuous N variable biological systems. (2) The sequences are represented as directed edges on Boolean N -cubes. (3) Any two dynamical systems which have identical steady states and cycles on the Boolean N -cube, under some symmetry operation of the N -cube, are defined to be in the same dynamical equivalence class and to have the same deep structure . We consider systems in which there is no self-input (each edge of the N -cube representation is directed in only one orientation) and enumerate the three deep structures for N = 2 and the 13 deep structures with at least a single steady state for N = 3. We illustrate these techniques by considering dynamic data from a number of biological systems and showing how the deep structure of each system can be determined.

Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias.

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 2∶2 and 4∶2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.

Combinatorial and topological methods in nonlinear chemical kinetics

Combinatorial and topological techniques are developed to classify nonlinear chemical reaction networks in terms of their qualitative dynamics. A class of N coupled equations, based on a hypothesis concerning biological control by Monod and Jacob is derived. Transitions between volumes in concentration space for these equations are represented as directed edges on N cubes (hypercubes in N dimensions). A classification of the resulting state transition diagrams for N=2,3 is given. A version of a topological theorem by Poincare and Hopf is derived which is appropriate for application to chemical systems. This theorem is used to predict the existence of critical points in continuous nonlinear equations with oscillation and bistability on the basis of their state transition diagrams. A large number of nonlinear kinetic equations proposed in previous studies by other authors are classified in terms of their state transition diagrams.

Automatic detection of atrial fibrillation using the coefficient of variation and density histograms of RR and ΔRR intervals

The paper describes a method for the automatic detection of atrial fibrillation, an abnormal heart rhythm, based on the sequence of intervals between heartbeats. The RR interval is the interbeat interval, and ΔRR is the difference between two successive RR intervals. Standard density histograms of the RR and ΔRR intervals were prepared as templates for atrial fibrillation detection. As the coefficients of variation of the RR and ΔRR intervals were approximately constant during atrial fibrillation, the coefficients of variation in the test data could be compared with the standard coefficients of variation (CV test). Further, the similarities between the density histograms of the test data and the standard density histograms were estimated using the Kolmogorov-Smirnov test. The CV test based on the RR intervals showed a sensitivity of 86.6% and a specificity of 84.3%. The CV test based on the ΔRR intervals showed that the sensitivity and the specificity are both approximately 84%. The Kolmogorov-Smirnov test based on the RR intervals did not improve on the result of the CV test. In contrast, the Kolmogorov-Smirnov test based on the ΔRR intervals showed a sensitivity of 94.4% and a specificity of 97.2%.

Theory of heart

In recent years there has been a growth in interest in studying the heart from the perspective of the physical sciences: mechanics, fluid flow, electromechanics. This volume is the result of a workshop held in July 1989 at the Institute for Nonlinear Sciences at the University of California at San Diego that brought together scientists and clinicians with graduate students and postdoctoral fellows who share an interest in the heart. The chapters were prepared by the invited speakers as didactic reviews of their subjects but also include up-to-date results in their fields. Topics covered include the structure, mechanical properties, and function of the heart and the myocardium, electrical activity of the heart and myocardium, and mathematical models of heart function. Individual chapters are abstracted separately.

Pattern Recognition in Humans: Correlations Which Cannot be Perceived

Random-dot Moiré patterns are manipulated to destroy our ability to perceive the spatial correlations which remain present in the patterns.

Stable oscillations in mathematical models of biological control systems

SummaryOscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.