# Entropy Production and Irreversibility in the Linearized Stochastic Amari Neural Model

## TL;DR

This paper explores how entropy production reveals whether the brain's neural network model is in equilibrium or not, using a stochastic version of the Amari model.

## Contribution

The study provides explicit entropy production formulas for a stochastic Amari model and links it to Shannon entropy dynamics.

## Key findings

- Entropy production distinguishes equilibrium and non-equilibrium states in the stochastic Amari model.
- Noise properties and model features determine whether the stationary state is in thermal equilibrium.
- The entropy production rate is connected to the time variation of the system's Shannon entropy.

## Abstract

One among the most intriguing results coming from the application of statistical mechanics to the study of the brain is the understanding that it, as a dynamical system, is inherently out of equilibrium. In the realm of non-equilibrium statistical mechanics and stochastic processes, the standard observable computed to determine whether a system is at equilibrium or not is the entropy produced along the dynamics. For this reason, we present here a detailed calculation of the entropy production in the Amari model, a coarse-grained model of the brain neural network, consisting of an integro-differential equation for the neural activity field, when stochasticity is added to the original dynamics. Since the way to add stochasticity is always to some extent arbitrary, particularly for coarse-grained models, there is no general prescription to do so. We precisely investigate the interplay between noise properties and the original model features, discussing in which cases the stationary state is in thermal equilibrium and which cases it is out of equilibrium, providing explicit and simple formulae. Following the derivation for the particular case considered, we also show how the entropy production rate is related to the variation in time of the Shannon entropy of the system.