Non-Trivial Dynamics in the FizHugh–Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions Systems

Abstract

This article focuses on the qualitative analysis of complex dynamics arising in a few mathematical models in neuroscience context. We first discuss the dynamics arising in the three-dimensional FitzHugh–Rinzel (FHR) model and then illustrate those arising in a class of non-homogeneous FitzHugh–Nagumo (Nh-FHN) reaction-diffusion systems. FHR and Nh-FHN models can be used to generate relevant complex dynamics and wave-propagation phenomena in neuroscience context. Such complex dynamics include canards, mixed-mode oscillations (MMOs), Hopf-bifurcations and their spatially extended counterpart. Our article highlights original methods to characterize these complex dynamics and how they emerge in ordinary differential equations and spatially extended models.

Synchronization in a Three Level Network of All-to-All Quasi-Periodically Forced Hodgkin-Huxley Reaction-Diffusion Equations

“…For other values, such as a ∈ (0.04465, 0.3) some spike arise. For some values, one can observe the appearance of the so called Mixed Mode Oscillations (MMOs), see for example [9][10][11][12][13][14][15] and references therein cited.…”

“…Of note, this picture illustrates a geometry appearing in some slow-fast systems in which the switching between small and large oscillations occur as canard solutions and in a tiny space region. We refer to [11,13] for such systems derivated from the FitzHugh-Nagumo system and with three time scale system. Although there is no small parameter in the HH equation, its hidden slow-fast nature has been studied for a long time, see [18][19][20].…”