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They are associated with various brain disorders such as migraine and ischemia. We consider a reduced version of a biophysical model of a neuron\u2013astrocyte network for the initiation and propagation of CSD waves [Huguet et al., Biophys. J. 111(2), 452\u2013462, 2016], consisting of reaction-diffusion equations. The reduced model considers only the dynamics of the neuronal and astrocytic membrane potentials and the extracellular potassium concentration, capturing the instigation process implicated in such waves. We present a computational and mathematical framework based on the parameterization method and singular perturbation theory to provide semi-analytical results on the existence of a wave solution and to compute it jointly with its velocity of propagation. The traveling wave solution can be seen as a heteroclinic connection of an associated system of ordinary differential equations with a slow\u2013fast dynamics. The presence of distinct time scales within the system introduces numerical instabilities, which we successfully address through the identification of significant invariant manifolds and the implementation of the parameterization method. Our results provide a methodology that allows to identify efficiently and accurately the mechanisms responsible for the initiation of these waves and the wave propagation velocity.<\/jats:p>","DOI":"10.1063\/5.0160509","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T13:15:24Z","timestamp":1693314924000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":["Traveling waves in a model for cortical spreading depolarization with slow\u2013fast dynamics"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6185-311X","authenticated-orcid":false,"given":"David","family":"Reyner-Parra","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya 1 , Avda. Diagonal 647, 08028 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4413-7952","authenticated-orcid":false,"given":"Carles","family":"Bonet","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya 1 , Avda. Diagonal 647, 08028 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8421-8717","authenticated-orcid":false,"given":"Teresa M.","family":"Seara","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya 1 , Avda. Diagonal 647, 08028 Barcelona, Spain"},{"name":"Institut de Matem\u00e0tiques de la UPC - Barcelona Tech (IMTech) 2 , Pau Gargallo 14, 08028 Barcelona, Spain"},{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra 3 , 08193 Bellaterra, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3932-948X","authenticated-orcid":false,"given":"Gemma","family":"Huguet","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya 1 , Avda. Diagonal 647, 08028 Barcelona, Spain"},{"name":"Institut de Matem\u00e0tiques de la UPC - Barcelona Tech (IMTech) 2 , Pau Gargallo 14, 08028 Barcelona, Spain"},{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra 3 , 08193 Bellaterra, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2023,8,29]]},"reference":[{"key":"2024012215444259500_c1","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1038\/nm.2333","article-title":"The role of spreading depression, spreading depolarization and spreading ischemia in neurological disease","volume":"17","year":"2011","journal-title":"Nat. Med."},{"key":"2024012215444259500_c2","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jocn.2015.08.004","article-title":"Cortical spreading depolarization: Pathophysiology, implications, and future directions","volume":"24","year":"2016","journal-title":"J. Clin. 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