# Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

## TL;DR

This paper develops a theoretical framework using phase reduction to optimize the stability of synchronized rhythmic patterns in coupled reaction-diffusion systems, with practical illustrations on FitzHugh-Nagumo models.

## Contribution

It derives the optimal linear filter and nonlinear interaction function to maximize synchronization stability in coupled reaction-diffusion systems.

## Key findings

- Derived the optimal linear filter for synchronization stability.
- Identified the nonlinear optimal interaction function.
- Validated theory with FitzHugh-Nagumo system examples.

## Abstract

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are linearly coupled. The nonlinear optimal interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.