Recurrent ring dynamics in two-dimensional excitable cellular automata

Abstract

The Greenberg–Hastings model (GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external stimulation can make ring dynamics in the GHM recurrent. The first scheme involves the initial placement of excitation centres which gradually lose strength, i.e. each time they become inactive (and then stay inactive forever) with probability 1 − p f. In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition as p f increases from 0 to 1. The second proposed rule utilizes continuous nucleation in that new rings are started at every rested site with probability p s . We show that, for small p s , these dynamics make a site excited about every p s −1/3 time units. This result yields some information about the asymptotic shape of a closely related random growth model.