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Resumen: Understanding common dynamical principles underlying an abundance of widespread brain behaviors is a pivotal challenge in the new century. The bottom-up approach to the challenge should be based on solid foundations relying on detailed and systematic understanding of dynamical functions of its basic components—neurons—modeled as plausibly within the Hodgkin-Huxley framework as phenomenologically using mathematical abstractions. Such one is the Hindmarsh-Rose (HR) model, reproducing fairly the basic oscillatory activities routinely observed in isolated biological cells and in neural networks. This explains a wide popularity of the HR-model in modern research in computational neuroscience. A challenge for the mathematics community is to provide detailed explanations of fine aspects of the dynamics, which the model is capable of, including its responses to perturbations due to network interactions. This is the main focus of the bifurcation theory exploring quantitative variations and qualitative transformations of a system in its parameter space. We will show how generic homoclinic bifurcations of equilibria and periodic orbits can imply transformations and transitions between oscillatory activity types in this and other bursting models of neurons of the Hodgkin-Huxley type. The article is focused specifically on bifurcation scenarios in neuronal models giving rise to irregular or chaotic spiking and bursting. The article demonstrates how the combined use of several state-of-the-art numerical techniques helps us confine “onion”-like regions in the parameter space, with macro-chaotic complexes as well as micro-chaotic structures occurring near spike-adding bifurcations.
Idioma: Inglés
DOI: 10.1063/1.4882171
Año: 2014
Publicado en: CHAOS 24 (2014), 023128 [11 pp]
ISSN: 1054-1500
Factor impacto JCR: 1.954 (2014)
Categ. JCR: PHYSICS, MATHEMATICAL rank: 9 / 54 = 0.167 (2014) - Q1 - T1
Categ. JCR: MATHEMATICS, APPLIED rank: 17 / 255 = 0.067 (2014) - Q1 - T1
Financiación: info:eu-repo/grantAgreement/ES/DGA/E48
Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2012-31883
Tipo y forma: Artículo (Versión definitiva)
Área (Departamento): Matemática Aplicada (Departamento de Matemática Aplicada)

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