BIFURCATIONS IN A DISCONTINUOUS CIRCLE MAP: A THEORY FOR A CHAOTIC CARDIAC ARRHYTHMIA

Abstract

The dynamics of discontinuous circle maps are investigated in the context of modulated parasystole, a cardiac arrhythmia in which there is an interaction between normal (sinus) and abnormal (ectopic) pacemaking sites in the heart. A class of noninvertible discontinuous circle maps with slope greater than 1 displays banded chaos under certain conditions. Banded chaos in these maps is characterized by a zero rotation interval width in the presence of a positive Lyapunov exponent. The bifurcations of a simple piecewise linear circle map are investigated. Parameters that result in banded chaos are organized into discrete, nonoverlapping zones in the parameter space. We apply these results to analyze a circle map that models modulated parasystole. Analysis of the model is complicated by the fact that the map has slope less than 1 for part of its domain. However, numerical simulations indicate that the modulated parasystole map displays banded chaos over a wide range of parameters. Banded chaos in this map produces rhythms with a relatively constant sinus-ectopic coupling interval, long trains of uninterrupted sinus beats, and patterns of successive sinus beats between ectopic beats characteristic of those found clinically.