Proving the Poisson Hypothesis for intensity-based models through the replica-mean-field approach
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 an 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. After a short primer on intensity-based point process models, we will present the replica-mean-field approach and the Poisson Hypothesis, contrasting it with classical mean-field models. We will then discuss the two main ideas of the proof which may be of independent interest.