Finite volume and asymptotic methods for stochastic neuron models with correlated inputs

TL;DR

This paper introduces a fast finite volume method for modeling correlated stochastic neuron pairs, providing accurate probability distributions and insights into how input variance affects joint neuron output.

Contribution

It develops a novel finite volume approach for the Fokker-Planck equation in neuron models, improving computational efficiency over Monte Carlo methods.

Findings

The finite volume method is significantly faster than Monte Carlo simulations.

The joint output of neuron pairs is most sensitive to input variance.

The method ensures accurate, nonnegative probability density approximations.

Abstract

We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker-Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein--Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.