Non-smooth Bifurcations of Mean Field Systems of Two-Dimensional Integrate and Fire Neurons

TL;DR

This paper investigates the complex bifurcation structures of non-smooth mean-field models of large neural networks, revealing how co-dimension two and three bifurcations organize neural dynamics and their relation to non-smooth bifurcations.

Contribution

It provides a detailed analysis of non-smooth bifurcations in mean-field systems of integrate-and-fire neurons, highlighting the role of co-dimension two and three bifurcations.

Findings

Identification of co-dimension two bifurcations involving Hopf and saddle-node points with switching manifolds.

Analysis of how these bifurcations organize neural network dynamics.

Discussion of regularizations and their relation to generic non-smooth bifurcations.

Abstract

Mean-field systems have been recently derived that adequately predict the behaviors of large networks of coupled integrate-and-fire neurons [14]. The mean-field system for a network of neurons with spike frequency adaptation is typically a pair of differential equations for the mean adaptation and mean synaptic gating variable of the network. These differential equations are non-smooth, and in particular are piecewise smooth continuous (PWSC). Here, we analyze the smooth and non-smooth bifurcation structure of these equations and show that the system is organized around a pair of co-dimension two bifurcations that involve, respectively, the collision between a Hopf equilibrium point and a switching manifold, and a saddle-node equilibrium point and a switching manifold. These two co-dimension 2 bifurcations can coalesce into a co-dimension 3 non-smooth bifurcation. As the mean-field system we study is a non-generic piecewise smooth continuous system, we discuss possible regularizations of this system and how the bifurcations which occur are related to non-smooth bifurcations displayed by generic PWSC systems.