Spatio-temporal dynamics of relaxation oscillations in a reaction-diffusion-ODE system

This paper investigates the emergence of time-periodic solutions and spatiotemporal patterns in a reaction-diffusion-ODE system, inspired by biological models such as FitzHugh-Nagumo for neuronal activity. We focus on a variant where the activator lacks diffusion, a feature known to support discontinuous patterns. By employing multiple time scale analysis, we rigorously establish the existence of relaxation oscillations, which are fundamental to rhythmic phenomena in biology, such as neuronal spiking. For distinct parameter regimes, we also derive regular periodic solutions via Hopf bifurcation theory. Extending the analysis to the spatial domain, we characterize the Hopf bifurcation that gives rise to both spatially homogeneous and inhomogeneous periodic solutions, elucidating the mechanism behind spatio-temporal pattern formation. Our theoretical findings are corroborated by numerical simulations, which reveal a rich variety of dynamics, including discontinuous stationary states and quasi-periodic patterns. This work provides a mathematical framework for understanding how coupling between non-diffusive and diffusive components can generate complex rhythms and patterns relevant to electrophysiology and ecology.