A stochastic system with infinite interacting components to model the time evolution of the membrane potentials of a population of neurons

TL;DR

This paper introduces a new stochastic model for the membrane potential dynamics of an infinite neuron population, providing proofs of existence, uniqueness, and perfect simulation algorithms.

Contribution

It develops a novel class of interacting particle systems for infinite neurons, with methods for perfect simulation and error bounds for finite approximations.

Findings

Proved existence and uniqueness of the process.

Developed a successful perfect simulation algorithm.

Established bounds for finite neuron sampling.

Abstract

We consider a new class of interacting particle systems with a countable number of interacting components. The system represents the time evolution of the membrane potentials of an infinite set of interacting neurons. We prove the existence and uniqueness of the process, using a perfect simulation procedure. We show that this algorithm is successful, that is, we show that the number of steps of the algorithm is almost surely finite. We also construct a perfect simulation procedure for the coupling of a process with a finite number of neurons and the process with a infinite number of neurons. As a consequence, we obtain an upper bound for the error that we make when sampling from a finite set of neurons instead of the infinite set of neurons.