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 reaction-diffusion systems with infinite-dimensional state space. 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.
- Received 26 September 2013
DOI:https://doi.org/10.1103/PhysRevX.4.021032
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Published by the American Physical Society
Popular Summary
A wide range of rhythmic spatiotemporal patterns exist in nature, including the target and spiral waves observed extensively in excitable or oscillatory media such as cardiac tissues and the brain. The mathematical equations of reaction-diffusion systems can describe these patterns, and synchronization of such spatiotemporal rhythms has been experimentally observed. However, the standard theory for analyzing synchronization dynamics—phase-reduction theory—has thus far been restricted to finite-dimensional dynamical systems described by ordinary differential equations. We develop a theory to describe rhythmic spatiotemporal patterns of infinite-dimensional reaction-diffusion systems, which allows us to use a simple one-dimensional phase equation to describe the dynamics of rhythmic spatiotemporal patterns.
Our formulation is free of conventional assumptions that the spatiotemporal pattern is rigid and simply translating or rotating in its medium. As a result, our proposed theory is generally applicable to various spatiotemporal rhythms in reaction-diffusion systems. As a simple example, we analyze synchronization dynamics between a pair of weakly coupled FitzHugh-Nagumo systems exhibiting circulating pulses, oscillating spots, target waves, and rotating spirals. In particular, the oscillatory spots and target waves of these systems are not rigid and lack translational/rotational symmetry, so conventional analyses that presume these characteristics cannot be applied. We demonstrate that our formulation correctly predicts the nontrivial synchronization properties of coupled FitzHugh-Nagumo systems and the times when their phase synchronizations occur.
This general basis for developing methods to control and design spatiotemporal rhythms in various reaction-diffusion systems has applications in a variety of fields, including chemical and biomedical engineering, where controlling the dynamics of these rhythms is desirable.