Dynamics of the FitzHugh-Nagumo equation with numerical methods

Abstract

In this paper, our research focuses on investigating the behavior of the FitzHugh-Nagumo (FHN) equation dynamics, a mathematical model that describes the spiking behavior of neurons. We begin by presenting the mathematical formulation of the FHN equation as a system of two coupled ordinary differential equations. We then apply methods from the theory of dynamical systems to analyze the behavior of the system, including stability analysis and phase-plane analysis. We also discuss the bifurcations that occur in the system as its parameters are varied. In addition, we present numerical methods for solving the FHN equation, including explicit and implicit methods. We compare the accuracy and efficiency of these methods and discuss their suitability for different types of problems.Our results show that the FHN equation exhibits a rich range of dynamical behaviors, including periodic and chaotic solutions, bistability, and hysteresis.Additionally, FHN equation can be used to simulate the behavior of networks of excitable cells, such as neural circuits or cardiac tissue,and FHN models can also be used to study the effects of drugs or other interventions on the behavior of excitable cells. The numerical methods we present provide a powerful tool for studying the FHN equation and its variants, allowing us to explore the parameter space and predict the behavior of the system under different conditions.