Finite Dimensionality and Upper Semicontinuity of the Global Attractor of Singularly Perturbed Hodgkin–Huxley Systems

Abstract

In an effort to show that the standard Hodgkin–Huxley system approximates the hyperbolic Hodgkin–Huxley system we view the hyperbolic system as a singular perturbation of the standard system. We establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and Hausdorff dimensions. We then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero.