Generic balanced synchrony patterns in network dynamics

Abstract

A coupled cell network is a type of ordinary differential equation $\dot x(t)=f(x(t))$, with structural constraints on the vector field $f$, encoded in a directed graph, whose cells and arrows are labeled by type. The generated dynamics can model, for example, those of neural networks or ecological systems. These systems and the synchrony patterns observed in their solutions have been intensely studied, particularly by Golubitsky, Stewart, and their coauthors. In the present article, we show that, for a generic vector field $f$, the synchrony patterns of the solutions of $\dot x(t)=f(x(t))$ are always balanced. This roughly means that for almost all $f$, the observed synchrony patterns, such as synchronization in two different cells, are inherited from the structural symmetries imposed by the graph and the cell types. Any other synchronization, not directly imposed by the geometry of the graph and the cell types, cannot occur. By doing so, we are completing the proof of several conjectures, including the rigid synchrony conjecture, the full oscillation conjecture and the observation of constant states. This article is the published version of the results stated by the second author in his PhD thesis.