Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model

TL;DR

This paper establishes a novel mathematical connection between the noisy leaky integrate-and-fire neuron model and the age-structured escape-rate model, providing an explicit integral transform linking their solutions.

Contribution

It derives an explicit integral transform that connects the solutions of the Fokker-Planck equation and the age-structured model, bridging two key formalisms in neural variability modeling.

Findings

Derived an integral transform linking the two models

Established an explicit mathematical correspondence between solutions

Enhanced understanding of stochastic neuron dynamics

Abstract

Providing an analytical treatment to the stochastic feature of neurons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker-Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition famous in the field of population dynamics, where the {\it age} of a neuron is the amount of time passed by since its previous spike. In this theoretical paper, we investigate the mathematical connection between the two formalisms. We shall derive an integral transform of the solution to the age-structured model into the solution of the Fokker-Planck equation. This integral transform highlights the link between the two stochastic processes. As far as we know, an explicit mathematical correspondence between the two solutions has not been introduced until now.