{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:15:06Z","timestamp":1760148906930,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Lokenath Debnath Endowed Professorship"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This work aims to study the interplay between the Wilson\u2013Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson\u2013Cowan equations provide a dynamical description of neural interaction. We formulate Wilson\u2013Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson\u2013Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson\u2013Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson\u2013Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson\u2013Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.<\/jats:p>","DOI":"10.3390\/e25060949","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T10:19:22Z","timestamp":1686910762000},"page":"949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Hierarchical Wilson\u2013Cowan Models and Connection Matrices"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6332-2520","authenticated-orcid":false,"given":"W. A.","family":"Z\u00fa\u00f1iga-Galindo","sequence":"first","affiliation":[{"name":"School of Mathematical & Statistical Sciences, University of Texas Rio Grande Valley, One West University Blvd., Brownsville, TX 78520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5949-416X","authenticated-orcid":false,"given":"B. A.","family":"Zambrano-Luna","sequence":"additional","affiliation":[{"name":"School of Mathematical & Statistical Sciences, University of Texas Rio Grande Valley, One West University Blvd., Brownsville, TX 78520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0006-3495(72)86068-5","article-title":"Excitatory and inhibitory interactions in localized populations of model neurons","volume":"12","author":"Wilson","year":"1972","journal-title":"Biophys. 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