Greenberg-Hastings model

Citing a number of earlier - and, in their opinion, unnecessarily complicated -research efforts to duplicate the characteristic spatial patterns of BZ-like reactions, Greenberg and Hastings wanted to construct a model that reproduced such patterns in the simplest possible manner, doing away with the need for extensive computation as the only means by which the desired results can be obtained. Previous studies. 

The Greenberg-Hastings model [green78] uses the two-dimensional von-Neumann neighborhood and assumes that each site is populated by variables a that take on one of three values 0,1,2. It is important to have at least three values 

Heuristically, the dynamics proceed as follows the reaction term makes all active ((7 = 1) and refractory a = 2) sites cycle to their respective successor states. The diffusion term defines the manner in which activity (defined by sites with value (7 = 1) diffuses through the lattice. 

Specifically, a given site becomes active on the next time step if at least one of its neighbors is already active otherwise it becomes quiescent (cr = 0). Once a given site has become active, it becomes refractory a = 2) on the next time step. Refractory sites are both insensitive to the values of neighboring sites and incapable of activating them. Upon becoming refractory, sites attain the quiescent state on the next time step. 

The spirals are composed of active wavefronts propagating over regions of quiescent sites. Simple singularities around which long-wavelength spirals form include 

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Fig. 8.14-b Greenberg-Hastings model of the BZ-reaction 320 x 200 lattice after 500 iteration steps, starting from a random initial state consisting of 5% active and 5% refractory sites. 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

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