Greenberg-Hastings Model

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The spatio-temporal patterns generated by means of the Greenberg–Hastings model, which is given by

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,

where represents the state of the cell at the time step, is the number of different states distinguished, the cardinality is determined within an Moore neighborhood, and is a threshold value representing the number of excited cells () that have to enclose an excitable cell in order for it to become excited itself. A cell is in an excitable, excited, or refractory state if , , or , respectively. This cyclic cellular automaton (CA) generates patterns that resemble the spiral waves generated by chemical clock reactions such as the Belousov–Zhabotinsky reaction. The evolution starts from random initial conditions, so the outcome will be different every time the model is run and might be homogeneously blank if there were not enough excited cells at the beginning.

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Contributed by: Jan Baetens (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The CA model used in this Demonstration was first published in:

J. M. Greenberg and S. P. Hastings, "Spatial Patterns for Discrete Models of Diffusion in Excitable Media," SIAM Journal on Applied Mathematics, 34, 1978 pp. 515–523.



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