Design of Hierarchical Excitable Networks

Abstract

We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbf{\Gamma}$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbf{\Gamma}$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbf{\Gamma}$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $\delta$-neighborhood of the first set contains an initial condition with $\omega$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.