Greenberg-Hastings Automata

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A cellular automaton is defined by

Classical cellular automata are iterated synchronously, the local function is applied to all cells at the same moment. With asynchronous dynamics a cell is chosen and iterated, teh the next cell and so on. A cellular automaton is a dynamical systen discrete in time, space and state. A crucial property is the locality, i.e. the new state of a cell does not depend on the state of the whole grid but only on the states of the cells in the neighborhood.

A classical model of excitable media was introduced 1978 by Greenberg and Hastings. Thery consider a two-dimensional square grid G=Z2. The cells are in one of a excited, one of g refractory or in the resting state. Neighbors are the five nearest cells, including the cell itself. A cell in the resting state with at least s excited neighbors becomes excited itself, runs through all excited and resting states and returns finally to the resting state. A resting cell with less than s excited neighbors stays in th eresting state. Parameters are a,g, and s. In an epidemc model one looks at the cells as infectious, immune and susceptible.

Here we show seven small examples of these automata, the neighborhood are the nine nearest cells. One observes a wavelike spread, the shape of the waves depends on the threshold s (Example1, Example 2). In other cases the patterns spreads but travels only once across the grid and dies out on finite grids (Example 3). At some parameter sets on finds only this spread without persistence, at other parameters the behaviour depends on the initial configuration (Example 2 and 3). One observes also persistence without spread like in Example 4. Example 5 shows a travelling patterns which finally ends in spirals -- an effect of the finite grid with susceptible occupied border cells. Examples 6 and 7 show that on finite grids with some paraters one can reach different periods -- only depending on the initial configuration.

This applet was desinged by Dieter Moroff, it works fine with Netscape 3.01 (32 bit),there may be problems with other browsers. If you are using a X-Window emulation, unfortunately the performance is low.


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Last modified 8/1/2000
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