Stochastic Integrate and Fire Models: a review on mathematical methods and their applications

TL;DR

This review comprehensively summarizes mathematical methods and analytical results for stochastic Integrate and Fire neuron models, aiming to unify notation and facilitate their application to network studies.

Contribution

It updates and consolidates existing methodologies and analytical results for one-dimensional stochastic Integrate and Fire models, filling a gap left by previous reviews.

Findings

Compilation of analytical solutions and closed-form expressions.

Overview of mathematical and statistical methods used.

Unification of mathematical notation for these models.

Abstract

Mathematical models are an important tool for neuroscientists. During the last thirty years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Reviews appeared recently collect the main features of these models but do not focus on the methodologies employed to obtain them. Aim of this paper is to fill this gap by upgrading old reviews on this topic. The idea is to collect the existing methods and the available analytical results for the most common one dimensional stochastic Integrate and Fire models to make them available for studies on networks. An effort to unify the mathematical notations is also made. This review is divided in two parts: Derivation of the models with the list of the available closed forms expressions for their characterization; Presentation of the existing mathematical and statistical methods for the study of these models.