# A theoretical connection between the noisy leaky integrate-and-fire and escape rate models: the non-autonomous case

## TL;DR

This paper establishes a theoretical link between the noisy leaky integrate-and-fire model and the escape rate model in neuroscience, especially under time-dependent inputs, providing a unified framework that explains their statistical similarities.

## Contribution

It introduces a new general stochastic framework that encompasses both models, revealing their connection in non-autonomous conditions.

## Key findings

- The framework unifies the two models under a common theoretical structure.
- It explains the observed statistical similarities between the models.
- The results have implications for understanding neuronal noise processes.

## Abstract

One of the most important challenges in mathematical neuroscience is to properly illustrate the stochastic nature of neurons. Among different approaches, the noisy leaky integrate-and-fire and the escape rate models are probably the most popular. These two models are usually chosen to express different noise action over the neural cell. In this paper we investigate the link between the two formalisms in the case of a neuron subject to a time dependent input. To this aim, we introduce a new general stochastic framework. As we shall prove, our general framework entails the two already existing ones. Our result has theoretical implications since it offers a general view upon the two stochastic processes mostly used in neuroscience, upon the way they can be linked, and explain their observed statistical similarity.