GH model) is a three state two dimensional cellular automaton (abbrev CA) named after James M. Greenberg and Stuart Hastings,[1] designed to model excitable media,[2] One advantage of a CA model is ease of computation.
The model can be understood quite well using simple "hand" calculations, not involving a computer.
[2] Another advantage is that, at least in this case, one can prove a theorem characterizing those initial conditions which lead to repetitive behavior.
[3] As in a typical two dimensional cellular automaton,[1] consider a rectangular grid, or checkerboard pattern, of "cells".
See the first animation in Belousov–Zhabotinsky reaction for a striking example of behavior that can be exhibited by this model.
To describe the GH model more mathematically, consider the simplest case of a grid of square cells
The type of neighborhood is not important, so long as each cell has some neighbors.
In a square grid (as opposed to hexagonal), either four or eight cell neighborhoods work fine.
[1][2] While the GH model has often been compared with the ground breaking model of Wiener and Rosenblueth [4] developed earlier for the same purpose, the analogy is incorrect because the latter is not a CA.
See for example,[5] in which it is stated that "The organization of muscle cells, the contraction of muscle, the dependence of the activity of the medium on the activity of its component elements, problems of memory, reliability, and mobility were formulated by Wiener in the form of definitions and theorems for a three-phase threshold-invariant continuous excitable medium."
By reading the original paper [4] carefully, it can be seen that neither the time, the medium, nor the state are discrete.
Wiener and Rosenblueth do say, though, that "There are three conditions in which any given region of the fiber can exist."
And in the next paragraph we find "..behind every wave front moving freely there will be a band of fixed width within which the recovery process is taking place."
A wave therefore consists of a "front", a smooth curve of points with epoch number 0 which moves in the plane with constant speed, followed by a refractory region of points with epoch number in (0,1), of constant width (depending on the velocity), and leaving behind a rest region of points with epoch number 1.
This is far from a cellular automata, and is more correctly called a "geometric" model.
Further on in [5] it is stated that "Automaton models of excitable media were investigated in [9] and [10].
The earliest paper found by Google Scholar with this designation clearly stated is.
[6] However, as mentioned above, the continuity of the Wiener-Rosenblueth medium has not so far allowed as precise a theorem about persistence of patterns as the one for GH which is described below.
See also [7] for an often cited computer study based on a model which is similar to that of Wiener and Rosenblueth.
[2] This spiral can easily be seen by following the pattern forward for a few iterations "by hand".
As stated in,[2] and proved in [3] (which also considered n-state models), if the set of initial 1's is finite, then every individual cell oscillates forever through the cycle 0,1,2,0 if and only if at the start there is at least one square of four neighboring cells with one of the following patterns: or some reflection or rotation of one of these.
[2] The key tool in the proof in [3] is a "winding number" which is shown to be invariant for this model.
One easy consequence of the theorem stated above is that if there are no "refractory" cells initially then the pattern will die out in any bounded region (whether the total grid is finite or infinite in extent).
This property of an excitable medium was found earlier, in the paper of Wiener and Rosenblueth,[4] and does not hold if there are "holes" in the region.