Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities

TL;DR

This paper investigates how Cauchy noise and different types of heterogeneity influence the collective behavior of large networks of quadratic integrate-and-fire neurons, providing exact mean-field models and bifurcation analysis.

Contribution

It introduces exact mean-field equations for networks with Cauchy noise and non-Cauchy heterogeneities, analyzing their impact on neural dynamics.

Findings

Noise and heterogeneity affect collective neural dynamics differently

Derived exact mean-field equations for complex heterogeneity distributions

Bifurcation analysis reveals qualitative changes in network behavior

Abstract

We analyze the dynamics of large networks of pulse-coupled quadratic integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous inputs. Two types of heterogeneities defined by families of $q$-Gaussian and flat distributions are considered. Both families are parametrized by an integer $n$, so that as $n$ increases, the first family tends to a normal distribution, and the second tends to a uniform distribution. For both families, exact systems of mean-field equations are derived and their bifurcation analysis is carried out. We show that noise and heterogeneity can have qualitatively different effects on the collective dynamics of neurons.