Poisson Hypothesis and large-population limit for networks of spiking neurons

TL;DR

This paper proves that in large populations of spiking neurons, the Poisson Hypothesis holds, allowing for simplified models where neurons act independently with Poisson-driven interactions, extending previous results to more complex neuron dynamics.

Contribution

It establishes the validity of the Poisson Hypothesis for large networks with quadratic neuron dynamics, enabling the derivation of a well-posed neural field model with resets.

Findings

Poisson Hypothesis holds for large-population limits of these networks.

Neurons become independent with Poisson interaction times in the limit.

The large-population limit results in a well-posed neural field model.

Abstract

We study mean-field descriptions for spatially-extended networks of linear (leaky) and quadratic integrate-and-fire neurons with stochastic spiking times. We consider large-population limits of continuous-time Galves-L\"ocherbach (GL) networks with linear and quadratic intrinsic dynamics. We prove that that the Poisson Hypothesis holds for the replica-mean-field limit of these networks, that is, in a suitably-defined limit, neurons are independent with interaction times replaced by independent time-inhomogeneous Poisson processes with intensities depending on the mean firing rates, extending known results to networks with quadratic intrinsic dynamics and resets. Proving that the Poisson Hypothesis holds opens up the possibility of studying the large-population limit in these networks. We prove this limit to be a well-posed neural field model, subject to stochastic resets.