Propagation of chaos and Poisson Hypothesis for replica mean-field models of intensity-based neural networks

TL;DR

This paper proves that in large neural network models, the Poisson Hypothesis accurately describes neuron interactions by using a novel mathematical approach, enabling more tractable analysis of complex neural dynamics.

Contribution

It establishes the validity of the Poisson Hypothesis in replica-mean-field neural models through a novel application of the Chen-Stein method and fixed point techniques.

Findings

Poisson Hypothesis holds at the limit of infinite replicas.

Introduces a new application of Chen-Stein method for neural models.

Provides a law of large numbers for exchangeable random variables.

Abstract

Neural computations arising from myriads of interactions between spiking neurons can be modeled as network dynamics with punctuate interactions. However, most relevant dynamics do not allow for computational tractability. To circumvent this difficulty, the Poisson Hypothesis regime replaces interaction times between neurons by Poisson processes. We prove that the Poisson Hypothesis holds at the limit of an infinite number of replicas in the replica-mean-field model, which consists of randomly interacting copies of the network of interest. The proof is obtained through a novel application of the Chen-Stein method to the case of a random sum of Bernoulli random variables and a fixed point approach to prove a law of large numbers for exchangeable random variables.