Dynamics of Hodgkin–Huxley systems revisited

Abstract

We consider the singularly perturbed Hodgkin–Huxley system subject to Neumann boundary conditions. We construct a family of exponential attractors {ℳ_ε} which is continuous at ε = 0, ε being the parameter of perturbation. Moreover, this continuity result is obtained with respect to a metric independent of ε, compared with all previous results where the metric always depends on ε. In the latter case, one needs to consider more regular function spaces and more smoother absorbing sets. Our results show that we can construct and analyse the stability of exponential attractors in a natural phase-space as it is known for the global attractor. Also, a new proof of the upper semicontinuity of the global attractor 𝒜_ε at ε = 0 is given.

Keywords:

• Hodgkin–Huxley system
• singular perturbation
• exponential attractors

AMS Subject Classifications::

• 35B25
• 35B41
• 35L70
• 92C30

Acknowledgements

The author is grateful to the King Fahd University of Petroleum and Minerals for its support. The author is also grateful to the referees for their helpful and careful reading of previous versions of this manuscript. Their suggestions and remarks greatly improved the readability as well as the quality of this article.