One-dimensional Deterministic Greenberg-Hastings Models

In this simple model for a one-dimensional array of excitable cells, each site x E Z is in one of I), states: 0 (rested state) , 1 (excited state), 2, . .. ,1), 1 (refract ory states) . The states update in discrete t ime according to a synchronous rule: changes 1 ~ 2, . .. , I), 1 ~ 0 happen auto matically, while t he 0 ~ 1 change is induced by at least a threshold numb er of I s in t he local neighborhoo d of x. If indestructible stable periodic objects exist, the model evolves into a locally periodic state. In parameter ranges when these st ructures are imposs ible, the syste m approaches the ground state 0: either the dynamics are dominated by annihilat ing waves, which cause power-law decay, or excitation is unable to propagate and the model experiences exponentially fast relaxat ion.