Cluster formations in rule dynamical systems: Emergence of non-local effects

A rule dynamical system is constructed using one-dimensional elementary cellular automata, and the mechanism of pattern clustering is studied in detail. We are able to explain cluster format on in terms of the spatio-temporal action of at least two local rules and their interplay. Among the various self-organizing patterns which are induced by the relationships between the local rules, 'switching phenomena' are especially interesting. The pattern of one cluster can be exchanged with the pattern of another cluster due to boundary interactions between nearest neighboring clusters. That is, a non-local effect can be generated instantaneously beyond the direct interaction range assumed in the original system. The significance of the switching phenomena is briefly discussed in relation to the information transfer in biological systems.