Turing patterns in a network-reduced FitzHugh-Nagumo model

TL;DR

This paper investigates how Turing patterns can form in a simplified network model derived from the FitzHugh-Nagumo system, highlighting the role of long-range connections and stability conditions.

Contribution

It introduces a novel single-component network model with long-range interactions derived from the FitzHugh-Nagumo system and analyzes pattern formation conditions.

Findings

Turing patterns can emerge in both original and reduced models.

The reduced model incorporates long-range connections naturally from the adiabatic elimination.

Conditions for instability of homogeneous states are established.

Abstract

Reduction of a two-component FitzHugh-Nagumo model to a single-component model with long-range connection is considered on general networks. The reduced model describes a single chemical species reacting on the nodes and diffusing across the links of a multigraph with weighted long-range connections that naturally emerge from the adiabatic elimination, which defines a new class of networked {dynamical} systems with local and nonlocal Laplace matrices. We study the conditions for the instability of homogeneous states in the original and reduced models and show that Turing patterns can emerge in both models.