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Greenberg-Hastings Models

The Greenberg-Hastings (GH) Model is perhaps the simplest CA prototype for an excitable medium. Real world excitable media include the Belousov - Zhabotinski reaction (see also ff.) and Slime Mold. GH dynamics have been discussed on numerous occasions in the Kitchen; our second-oldest recipe includes an explanation of its update rule. Here we present an interactive Java-based GH simulator with a dozen demo experiments to illustrate many of the model's basic features. Several of the 12 capsule descriptions below are linked to other pages from the Kitchen Shelf which provide additional detail. The Table at the bottom of the page prescribes each experiment's parameters. For convenience, since this page is fairly long, you can click on the number index of an experiment to go to its Table entry, and vice versa, or click on the upper left # symbol in the Table to jump to the applet. Try other parameter combinations to discover additional properties of this fascinating rule space. The brave hearted can download our research paper, Threshold-Range Scaling of Excitable Cellular Automata, for an in-depth empirical and theoretical analysis of GH and related dynamics.

1. Prototypical Spiral Pairs ("ram's horns")
Hit Random, then Start to see spirals emerge from randomness in the range 2 Box, threshold 3, 8 color GH rule. The palette evokes a fire: black is burnt (resting), orange is burning (excited), and shades of red represent the extinguishing (refractory) phase.

2. Spirals are robust
Hit Load Image to feed a filtered jill.gif to the same dynamics.

3. A Peter Max variant
Stop, then switch to the nostalgic CGA palette to see a more psychedelic transition from Jill to spirals.

4. Capture a weakly wrapping spiral
Next, enter the parameters for Experiment #4, Start, and hit Random repeatedly until a 7-color spiral emerges. These are the structures in the thumbnail graphic above and its corresponding Soup (suitable for postcard delivery to a friend by clicking on the envelope icon). You might need to try as many as 10 times before a spiral nucleates from the noise on such a small (60 by 60) grid.

5. Macaroni
As specified in the table below, a Kitchen favorite. See also the Spaghetti follow-up recipe. Even in such a small universe, these wave fragments, unable to form stable spiral cores, will eventually align almost perfectly in most cases. Our dedicated palette evokes a colander's monolayer of elbow pasta drenched in artifically-colored cheese product.

6. Complex 3-color feedback
Started from a sufficiently large solid seed of excitation on a latent background, GH dynamics typically evolve as a spreading ring. However, in 3-color models excitation can propagate across the back edge of a wave front, creating intricate interference patterns.

7. Wrap tests for varying thresholds
Load the Wave End initial condition, set the boundary condition to Free (so that waves move off the array without disturbing the opposite edge), and observe how the excitation bends less and less efficiently as the threshold increases from 1 to 5. The behavior of these wave ends for prescribed threshold and range plays a crucial role in the asymptotic self-organized behavior of the model started randomly. Click on the title of this experiment for a link to more details about spiral core design.

8. Puzzle I: What happens to this band?
Range 1, Threshold 1 GH models are simplest to analyze since the geometry of their wave propagation is particularly simple. Enter the parameters from the table, then try to predict the ultimate outcome starting from this simple band initial condition. Hit Start to see if your prediction was correct.

9. Puzzle II: What happens to a slightly longer one?
Title says it all: try to predict the outcome from a band just two cells wider. Watch the movie. Is there a morale here?

10. Bugs and Demons
Range 1 Box, Threshold 2 GH models tell a story all their own. Start the 4-color case from randomness to see the emergence of both bugs (= gliders or spaceships) and demons (indesctructable checkboard structures which eventually monopolize space).

11. Bugs alone
Raise the number of colors to 5 and the demons disappear. Apparently only bugs, which annihilate when they collide, can emerge from noise for this choice of parameters. Experiments on much larger arrays suggest the same conclusion. One is tempted to surmise that even an infinite version of this model would slowly relax to all 0's. But that would be wrong! Use paper and pencil to see if you can design a 5-color configuration which survives and emits periodic waves. (With enough patience, you can paint your pattern into the simulator by clicking on individual cells repeatedly until the desired colors are obtained.)

12. Pritikin's bowtie
Even for 6,7,8,9, and 10 colors, increasingly large and exotic stable periodic objects (SPOs) have been discovered by Dan Pritikin. Our final experiment exhibits his 6-color creation. See also Pritikin's Pagoda, an SPO for the 7-color case, and click on the title above for a link to Pritikin's crowning achievement, a monstrous SPO for the 10-color rule. It is not know whether any such object exists for 11 or more colors.

[This applet requires a Java-enabled browser...]

# Palette Range Threshold # Colors Initial Condition Boundary
1. Fire 2 3 8 Random Wrap
2. Fire 2 3 8 Jill Wrap
3. CGA 2 3 8 Jill Wrap
4. Bright 3 6 7 Random Wrap
5. Macaroni 2 4 5 Random Wrap
6. Bright 3 9 3 Seed Wrap
7. CGA 2 1 - 5 4 Wave End Free
8. CGA 1 1 16 Band1 Free
9. CGA 1 1 16 Band2 Free
10. CGA 1 2 4 Random Wrap
11. CGA 1 2 5 Random Wrap
12. CGA 1 2 5 Bowtie Free

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