Finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed Hodgkin-Huxley systems (Q1923669)

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.