Coupled Cellular Automata: Symbiotic Patterns and Synchronization

Modeling of cooperative behavior has great relevance to a wide range of scientific fields. Often trivial systems with primitive standalone behavior produce rich dynamics working in collaboration. This Demonstration allows you to couple up to four different one-dimensional elementary cellular automata (1D-eCA) and produce behaviors beyond those of single 1D-eCA. Patterns of these 1D-eCA in the standalone case are shown in four rectangles on the left. They can be compared to the two patterns of coupled evolution on the right (the rectangles labeled -output and -output). Often, simple evolution of standalone 1D-eCAs results in rich coupled dynamics of and . Such behavior is "symbiotic", since the coupled behavior of and cannot be obtained from standalone 1D-eCA. Another feature of coupled behavior is "synchronization". Namely, the pattern of is generally different from the pattern of , yet often similar "twin" sub-patterns can be noticed in and . In all images, the time of evolution goes from top to bottom. The four images on the left are just samples of standard 1D-eCA Wolfram rules. These four images are not used directly to create and patterns and are given only for comparison with and . Browse the snapshots for interesting patterns. See the "Details" section for a more technical discussion of the implemented coupling.

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DETAILS

Controls:
zoom—increase or decrease the size of the initial condition and the length of evolution
seed—randomize the initial condition
, , —change the rules of the coupled cellular automata
and buttons—choose whether the or output is fed back in the specific cellular automaton (values of , , , and in the formula below)
and —choose functions (multiplication "×" or addition "+" modulo 2) that couple evolution of and outputs ( and in the formula below)
Coupling formula description:
Two random binary vectors of the same length (labeled and ) are chosen as initial conditions (IC). They are fed into four cellular automata (, , , ) for one evolution step and the result is evaluated further, according to
,
.
Each parameter , , , can be or . and are coupling functions; each can be set to element-by-element multiplication "×" or element-by-element addition "+" modulo 2 of the vectors and . The result is interpreted again as an IC and repeatedly fed back into the same formulas. The evolution of and is shown in vertical rectangles on the right, labeled "-output" and "-output".
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