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We consider complex dynamical systems composed of many individual subsystems linked by a network of interactions. As a specific example of the general theory, a model of a neural network, previously introduced by other authors to describe the electrical activity of the cerebral cortex, is analyzed in detail. On the basis of this analysis, we propose a novel definition of a neuronal cascade, a physiological phenomenon of primary importance. We derive extreme value laws for the statistics of these cascades, both from the point of view of exceedances (that satisfy critical scaling theory in a certain regime) and of block maxima.<\/jats:p>","DOI":"10.1063\/1.5120570","type":"journal-article","created":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T16:36:53Z","timestamp":1586882213000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Extreme value theory of evolving phenomena in complex dynamical systems: Firing cascades in a model of a neural network"],"prefix":"10.1063","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8960-7279","authenticated-orcid":false,"given":"Theophile","family":"Caby","sequence":"first","affiliation":[{"name":"Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Universit\u00e0 degli Studi dell\u2019 Insubria , 22100 Como, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9604-4283","authenticated-orcid":false,"given":"Giorgio","family":"Mantica","sequence":"additional","affiliation":[{"name":"Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Universit\u00e0 degli Studi dell\u2019 Insubria , 22100 Como, Italy"}]}],"member":"317","published-online":{"date-parts":[[2020,4,14]]},"reference":[{"key":"2023062700420982900_c1","doi-asserted-by":"publisher","first-page":"423","DOI":"10.2307\/1968974","article-title":"Sur la distribution limite du terme maximum d\u2019une serie aleatoire","volume":"44","year":"1943","journal-title":"Ann. 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