An overview of synchrony in coupled cell networks

Abstract

One of the key aspects in the theory of coupled cell networks concerns the existence of synchrony subspaces. That is, subspaces defined in terms of equalities between cell coordinates which are flow-invariant for all coupled cell systems that respect a given coupled cell network structure. We review some recent concepts and results concerning synchrony subspaces on coupled cell networks. The existence of such subspaces naturally restricts the dynamics that can occur at the coupled cell systems, as in general it is the case for any dynamical system admitting flow-invariant spaces. We focus at some of the aspects that make important and special the existence of synchrony subspaces for coupled cell systems. Namely, their existence depend on the network structure and not on the specific form of the differential equations that are chosen to govern the dynamics; the solutions of the restricted coupled cell systems represent dynamics where groups of cells are dynamically behaving exactly in the same way; the restricted coupled cell systems are again coupled cell systems that are consistent with a network structure with a fewer cells. We review some results on how synchrony changes, or it is combined, in evolving networks. More precisely, in networks where their topology changes with time, either to a rewiring of a link, appearance or removal of a link or a node, or by merging smaller networks into larger ones. Finally, we consider the complement network of a network remarking that both networks have the same set of synchrony subspaces.