Homoclinic organization in the Hindmarsh-Rose model: A three parameter study

Barrio, Roberto (Universidad de Zaragoza) ; Ibáñez, Santiago ; Pérez, Lucía

Resumen: 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.

Idioma: Inglés

DOI: 10.1063/1.5138919

Año: 2020

Publicado en: CHAOS 30, 5 (2020), 053132 [20 pp.]
ISSN: 1054-1500

Factor impacto JCR: 3.642 (2020)

Categ. JCR: PHYSICS, MATHEMATICAL rank: 4 / 55 = 0.073 (2020) - Q1 - T1
Categ. JCR: MATHEMATICS, APPLIED rank: 13 / 265 = 0.049 (2020) - Q1 - T1

Factor impacto SCIMAGO: 0.97 - Applied Mathematics (Q1) - Mathematical Physics (Q1) - Statistical and Nonlinear Physics (Q1) - Physics and Astronomy (miscellaneous) (Q1) - Medicine (miscellaneous) (Q1)

Financiación: info:eu-repo/grantAgreement/ES/DGA-FEDER/E24-17R
Financiación: info:eu-repo/grantAgreement/ES/DGA/LMP124-18
Financiación: info:eu-repo/grantAgreement/ES/MICINN/PGC2018-096026-B-I00

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)