Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems

TL;DR

This paper extends phase reduction theory to infinite-dimensional reaction-diffusion systems, enabling analysis of complex spatiotemporal rhythms and synchronization in chemical and biological patterns.

Contribution

It develops a novel phase reduction framework for reaction-diffusion systems, generalizing isochrons and phase sensitivity functions to functional spaces.

Findings

Revealed nontrivial phase response properties of FitzHugh-Nagumo models.

Analyzed synchronization dynamics of complex spatiotemporal patterns.

Provided a theoretical basis for controlling rhythms in reaction-diffusion systems.

Abstract

Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. These rhythmic dynamics can be considered limit cycles of reaction-diffusion systems. However, the conventional phase-reduction theory, which provides a simple unified framework for analyzing synchronization properties of limit-cycle oscillators subjected to weak forcing, has mostly been restricted to low-dimensional dynamical systems. Here, we develop a phase-reduction theory for stable limit-cycle solutions of infinite-dimensional reaction-diffusion systems. By generalizing the notion of isochrons to functional space, the phase sensitivity function - a fundamental quantity for phase reduction - is derived. For illustration, several rhythmic dynamics of the FitzHugh-Nagumo model of excitable media are considered. Nontrivial phase response properties and synchronization dynamics are revealed, reflecting their complex spatiotemporal organization. Our theory will provide a general basis for the analysis and control of spatiotemporal rhythms in various reaction-diffusion systems.