Homoclinic and Periodic Solutions of Nerve Impulse Equations
We study the clamped Hodgkin and Huxley equations for the nerve impulse of the squid giant axon (HH).
Results on generalized Hopf bifurcation are used to describe the periodic solutions of HH in a region of the parameter space where there is a single equilibrium solution. The equations are shown to have a two dimensional attracting centre manifold, and a parameter value is found for which HH are equivalent to a Hopf-Takens bifurcation of codimension 2. In this way we obtain a description of the periodic solution branch and of its stability when a special parameter, the stimulus intensity, is varied. Other codimension 2 singularities present in HH are the cusp catastrophe and the Bogdanov-Takens cusp. The study of these singularities provides a description of the way a nerve cell may switch from repetitive activity (periodic solutions) to action potentials (homoclinic solutions).
KeywordsPeriodic Solution Hopf Bifurcation Homoclinic Orbit Centre Manifold Homoclinic Solution
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