Recurrent Ring Dynamics in Two–dimensional Excitable Cellular Automata

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 initial placement of excitation centers which gradually lose strength, i.e. each time they become inactive (and then stay so forever) with probability 1 − pf . In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition as pf increases from 0 to 1. The second proposed rule utilizes continuous nucleation: new rings are started at every rested site with probability ps. We show that, for small ps, this dynamics makes a site excited about every p −1/3 s time units. This result yields some information about the asymptotic shape of a closely related random growth model. 1991 Mathematics Subject Classification. Primary 60K35.