The Cook Book

Recipe for the week of November 7 - 13

Excitable Sponge

Perhaps the simplest cellular automaton rule for an excitable medium is known as the Greenberg-Hastings Model. A prescribed number of colors N are arranged cyclically in a "color wheel." Each color can only advance to the next, the last cycling to 0. Every update, cells change from color 0 (resting) to 1 (excited) if they have at least threshold 1's in their neighbor set. All other colors (refractory) advance automatically. Starting from a uniform random soup of the available colors, the excitation dies out if the threshold is too large compared to the size of the neighbor set, while a disordered soup virtually indistinguishable from noise results if the threshold is too low. For intermediate thresholds, however, waves of excitation self-organize into large-scale spiral pairs that stabilize in a locally periodic state.

This week's soup deals with the question of how many colors can support self-organization if the initial mix is not uniform, but instead is optimally chosen to induce spiral formation. It turns out that in the nearest-neighbor case (N,S,E,W), a mix of about 61% 0's and 39% 2's together with a very small smattering of excited states can nucleate on a grid containing 1 million cells and nearly 200 colors. The current graphic shows the tenuous fronts of excitation (in red) after 100 updates. It was featured on the poster for the Seventh Annual Complex Systems Summer School of the Santa Fe Institute.

For more about the mathematics behind these dynamics, which is intimately connected with percolation theory, see R. Fisch, J. Gravner and D. Griffeath, "Metastability in the Greenberg-Hastings model." Special Invited Paper. Ann. Appl. Probability 3 (1993), 935-967.

Take me higher...
Introduction to the PSK PSK Search Recent Additions CA Archive CA Links Feedback Appreciated !