Finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed Hodgkin-Huxley systems
type: Item
instance of: scholarly article
title: Finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed Hodgkin-Huxley systems
zbMATH Open document ID: 0859.34040
published in: Journal of Differential Equations
publication date: 1996-10-10T00:00:00Z
review text: The authors consider the Hodgkin-Huxley system \[ {{\partial\nu}\over{\partial t}}-{{\partial^2\nu} \over {\partial x^2}}= g_{N_a}m^3h (\nu_{N_a}-\nu)+ g_Kn^4 (\nu_K-\nu)+ g_L(\nu_L-\nu) \] with some given conditions. Due to the fact that the inductance in this system is small and can be ignored the authors perturb this system to the system \[ \varepsilon {{\partial^2\nu}\over{\partial t^2}}+ (\varepsilon f(m,h)+1) {{\partial\nu}\over{\partial t}}-{{\partial^2\nu}\over {\partial x^2}}= g_{N_a}m^3h (\nu_K-\nu)+ g_Kn^4 (\nu_K-\nu)+ g_L(\nu_L-\nu). \] The solution of the perturbed system is used to approximate the solution of the original system. This article is of many articles on this subject published by the authors. Using normal singular perturbation techniques it is shown that the solution of the perturbed system converges to the solution of the unperturbed system.
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 933274
zbMATH Keywords: Hodgkin-Huxley system
zbMATH Keywords: perturbed system
zbMATH Keywords: singular perturbation techniques
author: W. E. Fitzgibbon
author: Mary E. Parrott
author: Yuncheng You
reviewed by: Hussain S. Nur
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0116
OpenAlex ID: W2064044540
DOI: 10.1006/JDEQ.1996.0116
description: scientific article; zbMATH DE number 933274