- 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.