Designing heteroclinic and excitable networks in phase space using two populations of coupled cells

TL;DR

This paper presents a constructive method to realize arbitrary directed graphs as heteroclinic or excitable networks in coupled cell systems, enabling complex dynamic behaviors with specific phase space structures.

Contribution

It introduces a novel approach to design heteroclinic and excitable networks using two populations of coupled cells, with explicit construction and parameter conditions.

Findings

The method can realize any directed graph without one-cycles as a heteroclinic or excitable network.

Numerical simulations confirm these networks can serve as attractors under certain parameters.

Open sets of parameters are identified where the networks exist and are stable.

Abstract

We give a constructive method for realizing 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.