{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T13:02:07Z","timestamp":1777813327335,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T00:00:00Z","timestamp":1761350400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e27111104","type":"journal-article","created":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T00:51:42Z","timestamp":1761526302000},"page":"1104","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Entropy Production and Irreversibility in the Linearized Stochastic Amari Neural Model"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9070-2326","authenticated-orcid":false,"given":"Dario","family":"Lucente","sequence":"first","affiliation":[{"name":"Department of Mathematics & Physics, University of Campania \u201cLuigi Vanvitelli\u201d, Viale Lincoln 5, 81100 Caserta, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5860-7214","authenticated-orcid":false,"given":"Giacomo","family":"Gradenigo","sequence":"additional","affiliation":[{"name":"Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, 67100 L\u2019Aquila, Italy"},{"name":"INFN-LNGS, Via Giovanni Acitelli, 22, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0817-4753","authenticated-orcid":false,"given":"Luca","family":"Salasnich","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia \u201cGalileo Galilei\u201d, Universit\u00e0 di Padova, Via Marzolo 8, 35131 Padova, Italy"},{"name":"INFN, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"051919","DOI":"10.1103\/PhysRevE.75.051919","article-title":"Field-theoretic approach to fluctuation effects in neural networks","volume":"75","author":"Buice","year":"2007","journal-title":"Phys. Rev. E Stat. 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