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

Abstract

We simulate the effect of periodic stimulation on a strand of ventricular muscle by numerically integrating the one-dimensional cable equation using the Beeler-Reuter model to represent the transmembrane currents. As stimulation frequency is increased, the rhythms of synchronization { 1 : 1 -* 2 : 2 -* 2 : 1 -* 4: 2 -* irregular-* 3 : 1 -* 6: 2 -* irregular-* 4: 1 -* 8 : 2 -*... --~ 1 : 0} are successively encountered. We show that this sequence of rhythms can be accounted for by considering the response of the strand to premature stimulation. This involves deriving a one-dimensional finiteditierence equation or "'map" from the response to premature stimulation, and then iterating this map to predict the response to periodic stimulation. There is good quantitative agreement between the results of iteration of the map and the results of the numerical integration of the cable equation. Calculation of the Lyapunov exponent of the map yields a positive value, indicating sensitive dependence on initial conditions ("chaos"), at stimulation frequencies where irregular rhythms are seen in the corresponding numerical cable simulations. The chaotic dynamics occurs via a previously undescribed route, following two period-doubling bifurcations. Bistability (the presence of two different synchronization rhythms at a fixed stimulation frequency) is present both in the simulations and the map. Thus, we have been able to directly reduce consideration of the dynamics of a partial differential equation (which is of infinite dimension) to that of a one-dimensional map, incidentally demonstrating that concepts from the field of non-linear dynamics--such as perioddoubling bifurcations, bistability, and chaotic dynamics--can account for the phenomena seen in numerical simulations of the cable equation. Finally, we sketch out how the one-dimensional description can be extended, and point out some implications of our work for the generation of malignant ventricular arrhythmias.

Subharmonic resonance and chaos in forced excitable systems

“…From this bifurcation diagram it appears that the phase-locking regions are separated by intervals of irregular dynamics. Such a sequence has also observed in numerical simulations of the Beeler-Reuter model (Lewis and Guevara 1990). Fig.…”

“…6. This type of bistability was "rst found in maps by Guevara et al (1984) and later found by Lewis and Guevara (1990). Bistability between two stable solutions with di!erent rotation numbers occurs in many other dynamical systems as well (Aronson et al 1986, Boyland 1986, Rinzel and Troy 1983.…”

“…Excitable or oscillatory dynamics occur in many models of biological systems, including cardiac tissue (Glass et al 1987, Chialvo 1990a, Glass and Zeng 1990, Lewis and Guevara 1990, neuronal systems, and secretory systems (Wetsel et al 1992, Sturis et al 1991. Such systems are often subject to a periodic stimulus and the question arises as to when they can be entrained.…”

A normal form for excitable media

“…To go beyond the case of a single pulse circulating in a ring, periodically stimulated excitable media have been studied in the context of alternans [41,42,35,34,36,37]. In [41,36] one-dimensional maps were developed to study the Hopf bifurcation and the transition to conduction blocks.…”

Dynamics of sustained reentry in a loop model with discrete gap junction resistances

“…[1,2,3,4] For the 1-D loop, most work has been done assuming the membrane to be a continuous and uniform cable with constant intracellular axial resistivity. [5,6,7,8,9,10,11,12,13,14,15] For different models representing the ionic properties of the membrane, propagation was found to change from stable period-1 propagation to quasi-periodic reentry when the length of the loop was reduced below a critical length. The quasi-periodic reentry was characterized by a spatial oscillation of the action potential duration as propagation proceeded around the loop.…”