Bifurcation Portrait of a Hybrid Spiking Neuron Model

Abstract

A bifurcation portrait classifies the behavior of a dynamical system and how it transitions between different behaviors. A hybrid dynamical system displays both continuous and discrete dynamics and may display nonsmooth bifurcations. Herein, we analyze a novel hybrid model of a spiking neuron proposed by E.M. Izhikevich [9] that is based on a previous hybrid model with a convex spike-activation function f(x), but modified with a conductance reversal potential term. We analyze the model proposed by Izhikevich and obtain a bifurcation portrait for the continuous dynamics for an arbitrary convex spike activation function f(x). Both subcritical and supercritical Andronov-Hopf bifurcations are possible, and we numerically confirm the presence of a Bautin bifurcation for a particular choice of spike activation function. The model is capable of simulating common cortical neuron types and presents several possibilities for generalizations that may be capable of more complicated behavior.