Interaction of Pulses in Dissipative Systems: FitzHugh-Nagumo Equations

Oscillatory pulses in FitzHugh-Nagumo type systems with cross-diffusion.

We study FitzHugh-Nagumo type reaction-diffusion systems with linear cross-diffusion terms. Based on an analytical description using piecewise linear approximations of the reaction functions, we completely describe the occurrence and properties of wavy pulses, patterns of relevance in several biological contexts, in two prototypical systems. The pulse wave profiles arising in this treatment con...

Spatially Discrete FitzHugh--Nagumo Equations

We consider pulse and front solutions to a spatially discrete FitzHugh–Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discre...

Qualitative Theory of the FitzHugh-Nagumo Equations

FitzHugh-Nagumo equations with generalized diffusive coupling.

The aim of this work is to investigate the dynamics of a neural network, in which neurons, individually described by the FitzHugh-Nagumo model, are coupled by a generalized diffusive term. The formulation we are going to exploit is based on the general framework of graph theory. With the aim of defining the connection structure among the excitable elements, the discrete Laplacian matrix plays a...

Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo System

Numerical studies indicate that the FitzHugh–Nagumo system exhibits stable traveling pulses with oscillatory tails. In this paper, the existence of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh–Nagumo system exhibits folds. In addition, the stability of these pulses is investigated numerically, and a mechanism is proposed that exp...