Designing Heteroclinic and Excitable Networks in Phase Space Using Two Populations of Coupled Cells

We give a constructive method for realising an arbitrary directed graph (with no one-cycles) as a heteroclinic or an excitable dynamic network in the phase space of a system of coupled cells of two types. In each case, the system is expressed as a system of first-order differential equations. One of the cell types (the p-cells) interacts by mutual inhibition and classifies which vertex (state) we are currently close to, while the other cell type (the y-cells) excites the p-cells selectively and becomes active only when there is a transition between vertices. We exhibit open sets of parameter values such that these dynamical networks exist and demonstrate via numerical simulation that they can be attractors for suitably chosen parameters.