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Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,7,28]]},"abstract":"<jats:p>Stability is a fundamental requirement for both biological and engineered neural circuits, yet it is surprisingly difficult to guarantee in the presence of recurrent interactions. Standard linear dynamical models of recurrent networks are sensitive to the precise values of the synaptic weights, since stability requires all eigenvalues of the recurrent matrix to lie within the unit circle. Here we demonstrate, both theoretically and numerically, that an arbitrary recurrent neural network can remain stable even when its spectral radius exceeds 1, provided it incorporates divisive normalization, a dynamical neural operation that suppresses the responses of individual neurons. Sufficiently strong recurrent weights lead to instability, but the approach to the unstable phase is preceded by a regime of critical slowing down, a well-known early warning signal for loss of stability. Remarkably, the onset of critical slowing down coincides with the breakdown of normalization, which we predict analytically as a function of the synaptic strength and the magnitude of the external input. Our findings suggest that the widespread implementation of normalization across neural systems may be linked to its essential role in ensuring dynamical stability in the presence of the strong recurrent interactions required for complex cortical computations.<\/jats:p>","DOI":"10.1073\/pnas.2601841123","type":"journal-article","created":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T17:34:43Z","timestamp":1784828083000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Stabilization of recurrent neural networks through divisive normalization"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8830-1179","authenticated-orcid":false,"given":"Flaviano","family":"Morone","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Neural Science, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Center for Soft Matter Research, New York University","place":["New York"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-7282-6884","authenticated-orcid":false,"given":"Shivang","family":"Rawat","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Center for Soft Matter Research, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Courant Institute of Mathematical Sciences, New York University","place":["New York"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3282-9898","authenticated-orcid":false,"given":"David J.","family":"Heeger","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Neural Science, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Psychology, New York University","place":["New York"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2028-2175","authenticated-orcid":false,"given":"Stefano","family":"Martiniani","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Neural Science, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Center for Soft Matter Research, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Courant Institute of Mathematical Sciences, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Chemistry, Simons Center for Computational Physical Chemistry, New York University","place":["New York"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,7,23]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0952523800009640"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.8191289"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2010.04.009"},{"key":"e_1_3_4_4_2","first-page":"157","article-title":"Natural sound statistics and divisive normalization in the auditory system","volume":"13","author":"Schwartz O.","year":"2000","unstructured":"O. 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Rawat F. Morone D. J. Heeger S. Martiniani perturbed_organics: Source code to stabilize recurrent neural networks through divisive normalization. GitHub. https:\/\/github.com\/shivangrawat\/perturbed_organics. 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