102097 20220426091134.0 doi 10.1063/1.5138919 sideral 118682 ART-2020-118682 eng Barrio, Roberto Universidad de Zaragoza (orcid)0000-0002-8089-343X Homoclinic organization in the Hindmarsh-Rose model: A three parameter study 2020 Access copy available to the general public Unrestricted Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book, " gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme. info:eu-repo/grantAgreement/ES/DGA-FEDER/E24-17R info:eu-repo/grantAgreement/ES/DGA/LMP124-18 info:eu-repo/grantAgreement/ES/MICINN/PGC2018-096026-B-I00 info:eu-repo/semantics/openAccess by http://creativecommons.org/licenses/by/3.0/es/ 3.642 2020 PHYSICS, MATHEMATICAL 4 / 55 = 0.073 2020 Q1 T1 MATHEMATICS, APPLIED 13 / 265 = 0.049 2020 Q1 T1 0.97 2020 Applied Mathematics 2020 Q1 Mathematical Physics 2020 Q1 Statistical and Nonlinear Physics 2020 Q1 Physics and Astronomy (miscellaneous) 2020 Q1 Medicine (miscellaneous) 2020 Q1 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Ibáñez, Santiago Pérez, Lucía 2005 595 Universidad de Zaragoza Dpto. Matemática Aplicada Área Matemática Aplicada 30, 5 (2020), 053132 [20 pp.] Chaos CHAOS 1054-1500 5978825 http://zaguan.unizar.es/record/102097/files/texto_completo.pdf Versión publicada 2651693 http://zaguan.unizar.es/record/102097/files/texto_completo.jpg?subformat=icon icon Versión publicada oai:zaguan.unizar.es:102097 articulos driver 2022-04-26-08:52:44 ARTICLE