Stability and Bifurcation Analysis of Coupled Fitzhugh-Nagumo Oscillators

TL;DR

This paper analyzes the stability and bifurcations of small networks of coupled Fitzhugh-Nagumo neurons using algebraic geometry methods, providing insights into their dynamical behavior within biologically relevant parameters.

Contribution

It applies the Discriminant Variety method to study bifurcations in coupled FHN neuron networks, a novel approach in this context.

Findings

Identification of stability regions for coupled FHN neurons.

Characterization of bifurcation points in small neural networks.

Insights into how coupling affects neuronal dynamics.

Abstract

Neurons are the central biological objects in understanding how the brain works. The famous Hodgkin-Huxley model, which describes how action potentials of a neuron are initiated and propagated, consists of four coupled nonlinear differential equations. Because these equations are difficult to deal with, there also exist several simplified models, of which many exhibit polynomial-like non-linearity. Examples of such models are the Fitzhugh-Nagumo (FHN) model, the Hindmarsh-Rose (HR) model, the Morris-Lecar (ML) model and the Izhikevich model. In this work, we first prescribe the biologically relevant parameter ranges for the FHN model and subsequently study the dynamical behaviour of coupled neurons on small networks of two or three nodes. To do this, we use a computational real algebraic geometry method called the Discriminant Variety (DV) method to perform the stability and bifurcation analysis of these small networks. A time series analysis of the FHN model can be found elsewhere in related work[15].