A basic lattice model of an excitable medium: Kinetic Monte Carlo simulations

Abstract

The simplest stochastic lattice model of an excitable medium is considered. Each lattice cell can be in one of three states: excited, refractory, or quiescent. Transitions between different cell states occur with the prescribed probabilities. The model is designed for studying the transfer of excitation in the cardiac muscle and nerve fiber at the cellular and subcellular levels, and also for modeling the spreading of epidemics. Elementary events on the lattice are simulated by the kinetic Monte Carlo method, which consists in constructing a Markov chain of the lattice states corresponding to the solution of the master equation. An effective algorithm for implementing the Kinetic Monte Carlo simulations is suggested. The number of the arithmetic operations at each time step of the proposed algorithm is practically independent of the lattice size, which enables making calculations on two- and three-dimensional lattices of a very large size (more than 109 cells). It is shown that the model reproduces the basic spatiotemporal structures (solitary traveling pulses, pulse trains, concentric and spiral waves, and spiral turbulence) characteristic of an excitable medium. The basic properties of traveling pulses and spiral waves for the considered stochastic lattice model are studied and compared with the known properties of deterministic equations of the reaction-diffusion type, which are usually employed for modeling excitable media.