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Recurrent ring dynamics in two-dimensional excitable cellular automata

Part of:Special processes

Published online by Cambridge University Press:  14 July 2016

Janko Gravner
Janko Gravner*
Affiliation:
University of California
*
∗ Postal address: Mathematics Department, University of California, Davis, CA 95616, USA. Email address: gravner@math.ucdavis.edu.
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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.

Keywords

Asymptotic shapeexcitable mediaGreenberg–Hastings modelinterface dynamicspercolation

MSC classification

Primary:60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999, pp. 492 - 511
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work was partly supported by the research grant J1-6157-0101-94 from Slovenia's Ministry of Science and Technology.

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