WAVE-FRONT PROPAGATION IN A DISCRETE MODEL OF EXCITABLE MEDIA

We generalize our recent discrete cellular automata (CA) model of exci table media [Y. B. Chernyak, A. B. Feldman, and R. J. Cohen, Phys. Rev . E 55, 3215 (1997)] to incorporate the effects of inhibitory processe s on the propagation of the excitation wave front. In the common two v ariable reaction-diffusion (RD) models of excitable media, the inhibit ory process is described by the upsilon ''controller'' variable respon sible for the restoration of the equilibrium state following excitatio n. In myocardial tissue, the inhibitory effects are mainly due to the inactivation of the fast sodium current. We represent inhibition using a physical- model in which the ''source'' contribution of excited ele ments to the excitation of their neighbors decreases with time as a si mple function with a single adjustable parameter (a rate constant). We sought specific solutions of the CA state transition equations and ob tained (both analytically and numerically) the dependence of the wave- front speed c on the four model parameters and the wave-front curvatur e kappa. By requiring that the major characteristics of c(kappa) in ou r CA model coincide with those obtained from solutions of a specific R D model, we find a unique set of CA parameter values for a given excit able medium. The basic structure of our CA solutions is remarkably sim ilar to that found in typical RD systems (similar behavior is observed when the analogous model parameters are varied). Most notably, the '' turn-on'' of the inhibitory process is accompanied by the appear ance of a solution branch of slow speed, unstable waves. Additionally, when kappa is small, we obtain a family of ''eikonal'' relations c(kappa) that are suitable for the kinematic analysis of traveling waves in the CA medium. We compared the solutions of the CA equations to CA simula tions for the case of plane waves and circular (target) waves and foun d excellent agreement. We then studied a spiral wave using the CA mode l adjusted to a specific RD system and found good correspondence betwe en the shapes of the RD and CA spiral arms in the region away from the tip where kinematic theory applies. Our analysis suggests that only f our physical parameters control the behavior of wave fronts in excitab le media.