Statistics and Computing

, Volume 1, Issue 1, pp 23–39 | Cite as

Threshold-range scaling of excitable cellular automata

  • Robert Fisch
  • Janko Gravner
  • David Griffeath


Each cell of a two-dimensional lattice is painted one of κ colors, arranged in a ‘color wheel’. The colors advance (k tok+1 mod κ) either automatically or by contact with at least a threshold number of successor colors in a prescribed local neighborhood. Discrete-time parallel systems of this sort in which color 0 updates by contact and the rest update automatically are called Greenberg-Hastings (GH) rules. A system in which all colors update by contact is called a cyclic cellular automation (CCA). Started from appropriate initial conditions, these models generate periodic traveling waves. Started from random configurations the same rules exhibit complex self-organization, typically characterized by nucleation of locally periodic ‘ram's horns’ or spirals. Corresponding random processes give rise to a variety of ‘forest fire’ equilibria that display large-scale stochastic wave fronts. This paper describes a framework, theoretically based, but relying on extensive interactive computer graphics experimentation, for investigation of the complex dynamics shared by excitable media in a broad spectrum of scientific contexts. By focusing on simple mathematical prototypes we hope to obtain a better understanding of the basic organizational principles underlying spatially distributed oscillating systems.


Cellular automation excitable medium locally periodic spiral stable periodic object Greenberg-Hastings model cyclic CA self-organization phase transition, turbulence bug macaroni 


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Copyright information

© Chapman and Hall Ltd 1991

Authors and Affiliations

  • Robert Fisch
    • 1
  • Janko Gravner
    • 2
  • David Griffeath
    • 2
  1. 1.Department of MathematicsColby CollegeWatervilleUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA

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