Impact of local timescales in a cellular automata model of excitable media

Abstract

A cellular automata system known as the Greenberg-Hastings cellular automata (GHCA), with cyclic and excitable individual dynamics, provides a useful paradigm for modeling emergent phenomena in an excitable medium. Most past studies of GHCA have investigated the asymptotic dynamics of the system, assuming fixed extents for the active and passive dynamical phases. This study is a systematic investigation of the relationship between the dynamical properties of individual cells of a small system implementing excitable dynamics, and the emergent system wide asymptotic state. The impact of the temporal extents of individual dynamical phases (τa, τp), on the probability of persistence P(τa,τp), defined as the fraction of initial configurations which sustain spatiotemporal oscillations asymptotically, is examined for each unique configuration of the system. The main result obtained is that the probability of persistence P(τa, τp) assumes a characteristic sigmoidal form, as the dynamical behaviour of the system transitions from absolute quiescence to persistent spatiotemporal oscillations, with respect to the parameters (τa, τp). A plot of the numerical findings in the (τa, τp) space provides a consolidated view of their impact on the emergent behaviour of the system. In conclusion, some implications of the methods and results presented in this work, to the study of emergent phenomena in excitable systems, are discussed.