Complex Systems

One-dimensional Deterministic Greenberg--Hastings Models Download PDF

Robert Fisch
Department of Computer Science,
University of Wisconsin,
Madison, WI 53706, USA

Janko Gravner
Mathematics Department,
University of California,
Davis, CA 95616, USA


In this simple model for a one-dimensional array of excitable cells, each site is in one of states: 0 (rested state), 1 (excited state), (refractory states). The states update in discrete time according to a synchronous rule: changes happen automatically, while the change is induced by at least a threshold number of 1s in the local neighborhood of . If indestructible stable periodic objects exist, the model evolves into a locally periodic state. In parameter ranges when these structures are impossible, the system approaches the ground state 0: either the dynamics are dominated by annihilating waves, which cause power-law decay, or excitation is unable to propagate and the model experiences exponentially fast relaxation.