Finite propagation enhances Turing patterns in reaction-diffusion networked systems

TL;DR

This paper extends the theory of Turing instability to reaction-diffusion systems on complex networks with finite propagation speed, broadening the conditions under which patterns can emerge and aligning theory with more realistic physical assumptions.

Contribution

It introduces a finite propagation framework for networked reaction-diffusion systems, expanding the classical Turing instability conditions and providing analytical and simulation validation.

Findings

Turing patterns emerge under broader conditions, including faster activator diffusion.

Finite propagation speed allows for wave-like and stationary Turing patterns.

Analytical results are confirmed by simulations on the FitzHugh-Nagumo model.

Abstract

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on complex networks assuming finite propagation. Extending to networked systems the framework introduced by Cattaneo in the 40's, we remove the unphysical assumption of infinite propagation velocity holding for reaction-diffusion systems, thus allowing to propose a novel view on the fine tuning issue and on existing experiments. We analytically prove that Turing instability, stationary or wave-like, emerges for a much broader set of conditions, e.g., once the activator diffuses faster than the inhibitor or even in the case of inhibitor-inhibitor systems, overcoming thus the classical Turing framework. Analytical results are compared to direct simulations made on the FitzHugh-Nagumo model, extended to the relativistic reaction-diffusion framework with a complex network as substrate for the dynamics.