- 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.
| # |
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 |
|
