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

Abstract

Abstract.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 (1977Winfree ( , 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 Poincar6 or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincar6 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 Poincar6 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.

A qualitative model of a motor circadian rhythm

“…Also, this simplified model is closely related to one due to Guevara and Glass (1982) that has been extensively analyzed. This can be used to obtain theoretical conclusions that give reasonable explanations of some of the characteristics of the complete model and hopefully some clues towards a rigorous proof of them.…”

Chaotic dynamics in an ionic model of the propagated cardiac action potential

“…In many of these reports, the alternans is often seen just prior to the phase of induction of irregular rhythms, including ventricular fibrillation. Since it is wellknown that a cascade of period-doubling bifurcations can lead to chaotic dynamics (May, 1976;Thompson & Stewart, 1986), the hypothesis has been put forth that irregular rhythms, such as fibrillation, might be manifestations of chaotic dynamics occurring as the result of a cascade of period-doubling bifurcations (Guevara et al, 1981;Guevara & Glass, 1982;Adam et al, 1982;Chialvo & Jalife, 1987;Smith et al, 1988;Savino et al, 1989). We therefore decided to carry out a systematic study of the effect of pacing at different rates in an ionic model of a strand of ventricular muscle.…”

Global dynamics of a stochastic neuronal oscillator

“…The Poincaré oscillator and its variant are a member of a set of systems that are widely used in the analysis of neuronal or biological oscillators [4,9,[17][18][19][20][21][22][23][24][29][30][31]. Based on [5,20], we introduce the Poincaré oscillator here and summarize its properties.…”

“…Since the discretized SPTO is a positive matrix, the product of discretized SPTOs is weakly ergodic [44,45]. (15) and (16)], and the SPTO P K, ,A i ,I i expresses the relationship between the density just before the ith impulse to that just before the (i + 1)th impulse: The weak ergodicity leads to the following property for any densities h and h : (29) where H n,n 0 = P K, ,A n ,I n P K, ,A n−1 ,I n−1 . .…”