Coupled cell networks: Boolean perspective

Abstract

During the 1980s and early 1990s, Martin Golubitsky and Ian Stewart В formulated and developed a theory of "coupled cell networks" (CCNs). Their research was primarily focused onquadrupeds' gaits and they applied the framework of differential equations. Golubitsky and Stewart were particularly interested in change of synchrony between $4$ legs of an animal. For example what happens when the animal speeds up from walk to gallop.В The most important concept of their theory is a {\it cell}. The cell captures the dynamics of one unit and a dynamical system consists of many identical (governed by the same principles) cells influencing (coupling to) each other. Models based on identical cooperating units are fairly common in many areas, especially in biology, ecology and sociology.В The goal of investigation in Coupled Cell Networks theory В is understanding the dependencies and interplay between dynamics of an individual cell, graph of connections between cells, and the nature of couplings.В \vspace*{0.2em}In this paper, I redefine Coupled Cell Networks using framework of Boolean functions. This moves the entire theory to a new setting. Some phenomena proved to be very similar as for continuous networks and some are completely different. Also, for discrete networks we ask questions differently and study different phenomena. The paper presents two examples: networks that bring 2-cell bidirectional ring as a quotient and networks that bring 3-cell bidirectional ring as a quotient.