Asymptotic behavior of excitable cellular automata. (English) Zbl 0796.60100

Durrett, Richard; Griffeath, David

Exp. Math. 2, No. 3, 183-208 (1993).

Summary: We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range \(\rho\) of interaction, \(l^ p\) shape of its neighbor set, threshold \(\theta\) for contact updating, and number \(\kappa\) of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science. R. Fisch, J. Gravner and D. Griffeath [Stat. Comput. 1, No. 1, 23-39 (1991)] studied experimentally the ergodic behavior of these models on \({\mathbb{Z}}^ 2\), started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on \({\mathbb{R}}^ 2\)) in the threshold-range scaling limit as \(\rho,\theta\to\infty\) with \(\theta/\rho^ 2\) constant.
Here we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend\((p)\), the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo\((p)\), the cutoff for existence of stable periodic objects that arise as spiral cores. Some subtle consequences of anisotropic neighbor sets \((p\neq 2)\) are also discussed; the case of box neighborhoods \((p=\infty)\) is examined in detail.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
68Q80	Cellular automata (computational aspects)

Keywords:

excitable cellular automata; Greenberg-Hastings model; asymptotic phase diagrams; stable periodic objects

References:

[1]	Dewdney A. K., Scientific American pp 104– (1988) · doi:10.1038/scientificamerican0888-104
[2]	Dewdney A. K., Scientific American pp 102– (1989) · doi:10.1038/scientificamerican0889-102
[3]	Durrett R., Probability: Theory and Examples (1991) · Zbl 0709.60002
[4]	Durrett R., J. Theoretical Prob. 4 pp 127– (1992) · Zbl 0751.60095 · doi:10.1007/BF01046781
[5]	Durrett R., 1993 Saint Flour Probability Summer School
[6]	Durrett R., Ann. Appl. Probability 1 pp 189– (1991) · Zbl 0733.92022 · doi:10.1214/aoap/1177005933
[7]	Durrett R., J. Theoretical Prob. bf 3 pp 669– (1991) · Zbl 0741.92001 · doi:10.1007/BF01259549
[8]	Durrett R., Ann. Probability 21 pp 232– (1993) · Zbl 0769.60092 · doi:10.1214/aop/1176989403
[9]	Fisch R., Excite!: a periodic wave modeling environment.
[10]	Fisch R., Statistics and Computing 1 pp 23– (1991) · doi:10.1007/BF01890834
[11]	Fisch R., Spatial Stochastic Processes (1992)
[12]	Fisch R., Ann. Appl. Probability 3 (1993)
[13]	Gerhardt M., Physica 46 pp 392– (1990)
[14]	Gravner J., ”Ring dynamics in the Greenberg–Hastings model” · Zbl 0945.60092
[15]	Gravner J., Trans. Amer. Math. Soc. (1994)
[16]	Greenberg J., Bull. Amer. Math. Soc. 84 pp 1296– (1978) · Zbl 0424.35011 · doi:10.1090/S0002-9904-1978-14560-1
[17]	Greenberg J., SIAM J. Appl. Math. 34 pp 515– (1978) · Zbl 0398.92004 · doi:10.1137/0134040
[18]	Griffeath D., Notices Amer. Math. Soc. pp 1472– (1988)
[19]	Kapral R., J. Math. Chem. 6 pp 113– (1991) · doi:10.1007/BF01192578
[20]	Markus M., Nonlinear Wave Processes in Excitable Media (1991)
[21]	Mikhailov A., Quantum pp 12– (1991)
[22]	Muller S., Physica 24 pp 71– (1986)
[23]	Newell P. C., Fungal Differentiation: A Contemporary Synthesis pp 43– (1983)
[24]	Toffoli T., Cellular Automata Machines (1987) · Zbl 0655.68055
[25]	Weiner N., Arch. Inst. Cardiol. Mexico 16 pp 205– (1946)
[26]	Winfree A., Scientific American pp 82– (1974) · doi:10.1038/scientificamerican0674-82
[27]	Winfree A., When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. (1987)