Abstract
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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Aizenman, M. and Lebowitz, J. (1988) Metastability effects in bootstrap percolation.J. Phys. A: Math. Gen.,21, 3801–3813.
Bennett, C., Grinstein, G., He, Y., Jayaprakash, C. and Mukamel, D. (1990) Stability of temporally-periodic states of classical many-body systems.Phys. Rev. A,41, 1932–1935.
Bramson, M. and Griffeath, D. (1980) Flux and fixation in cyclic particle systems.Ann. Probability,17, 26–45.
Dewdney, A. K. (1988) Computer recreations: the hodgepodge machine makes waves.Scientific American, August, 104–107.
Dewdney, A. K. (1989) Computer recreations: a cellular universe of debris, droplets, defects and demons.Scientific American, August, 102–105.
Durrett, R. (1991) Multicolor particle systems with large threshold and range. Submitted.
Durrett, R. and Neuhauser, C. (1991) Epidemics with regrowth ind=2.Ann. Appl. Probability, to appear.
Durrett, R. and Steif, J. (1991) Some rigorous results for the Greenberg-Hastings model.J. Theoretical Prob., to appear.
Fisch, R. (1990) The one-dimensional cyclic cellular automaton: a system with deterministic dynamics which emulates an interacting particle system with stochastic dynamics.J. Theoretical Probl.,3, 311–338.
Fisch, R. (1991) Clustering in the one-dimensional 3-color cyclic cellular automaton.Ann. Probability, to appear.
Fisch, R. and Griffeath, D. (1991)excite: a periodic wave modeling environment. (1 disk for IBM 286/386 and compatibles). Freeware. Write: D. Griffeath, Mathematics Dept, University of Wisconsin, Madison, WI 53706, USA.
Fisch, R., Gravner, J. and Griffeath, D. (1991 a) Cyclic cellular automata in two dimensions, inSpatial Stochastic Processes. A festschrift in honor of the seventieth birthday of T. E. Harris, Birkhäuser, Boston.
Fisch, R., Gravner, J. and Griffeath, D. (1991 b) Metastability in the Greenberg-Hastings model. In preparation.
Freedman, W. and Madore, B. (1983) Time evolution of disk galaxies undergoing stochastic self-propagating star formation.Astrophysical J. 265, 140–147.
Gerhardt, M. and Schuster, H. (1989) A cellular automaton describing the formation of spatially ordered structures in chemical systems.Physica D,36, 209–221.
Gerhardt, M., Schuster, H. and Tyson, J. (1990) A cellular automaton model of excitable media including curvature and dispersion.Science,247, 1563–1566.
Gravner, J. (1991) Mathematical aspects of excitable media. Ph.D thesis, University of Wisconsin.
Greenberg, J., Hassard, B. and Hastings, S. (1978) Pattern formation and periodic structures in systems modeled by reaction-diffusion equations.Bull. AMS. 84, 1296–1327.
Greenberg, J. and Hastings, S. (1978) Spatial patterns for discrete models of diffusion in excitable media.SIAM J. Appl. Math.,34, 515–523.
Griffeath, D. (1988) Cyclic random competition: a case history in experimental mathematics, in ‘Computers and Mathematics’AMS Notices, 1472–1480.
Hodgkin, A. and Huxley, A. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve.J. Physiology,117, 500–544.
Kapral, R. (1991) Discrete models for chemically reacting systems.J. Math. Chem.,6, 113–163.
Markus, M., Krafczyk, M. and Hess, B. (1991) Randomized automata for isotropic modelling of two- and three-dimensional waves and spatiotemporal chaos in excitable media, inNonlinear Wave Processes in Excitable Media, A. Holden, M. Markus and H. Othmer (eds), Plenum Press New York. Fortheoming.
Mollison, D. and Kuulasmaa, K. (1985) Spatial epidemic models: theory and simulations, inPopulation Dynamics of Rabies in Wildlife, P. J. Bacon (ed), Academic Press, London, pp. 291–309.
Murray, J., Stanley, E. and Brown, D. (1986) On the spatial spread of rabies among foxes.Proc. Roy. Soc. London B,229, 111–150.
Toffoli, T. and Margolus, N. (1987)Cellular Automata Machines. MIT Press, Cambridge, Mass.
Tomchik, K. and Devreotes, P. (1981) Adenosine 3′,5′-monophosphate waves in Dictyostelium discoideum: a demonstration by isotope dilution-fluorography.Science,212, 443–446.
Waldrop, M. (1990), Spontaneous order evolution, and life.Science,247, 1543–1545.
Weiner N. and Rosenbluth, A. (1946) The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle.Arch. Inst. Cardiol. Mexico,16, 205–265.
Winfree, A. (1974) Rotating chemical reactions.Scientific American, June, 82–95.
Winfree, A. (1987)When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrythmias. Princeton University Press, Princeton, N.J.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fisch, R., Gravner, J. & Griffeath, D. Threshold-range scaling of excitable cellular automata. Stat Comput 1, 23–39 (1991). https://doi.org/10.1007/BF01890834
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01890834