A Cellular Automaton Model of Excitable Media Including Curvature and Dispersion

Abstract

Excitable media are spatially distributed systems characterized by their ability to propagate signals undamped over long distances. Wave propagation in excitable media has been modeled extensively both by continuous partial differential equations and by discrete cellular automata. Cellular automata are desirable because of their intuitive appeal and efficient digital implementation, but until now they have not served as reliable models because they have lacked two essential properties of excitable media. First, traveling waves show dispersion, that is, the speed of wave propagation into a recovering region depends on the time elapsed since the preceding wave passed through that region. Second, wave speed depends on wave front curvature: curved waves travel with normal velocities noticeably different from the plane-wave velocity. These deficiencies of cellular automation models are remedied by revising the classical rules of the excitation and recovery processes. The revised model shows curvature and dispersion effects comparable to those of continuous models, it predicts rotating spiral wave solutions in quantitative accord with the theory of continuous excitable media, and it is parameterized so that the spatial step size of the automation can be adjusted for finer resolution of traveling waves.