Homoclinic and Periodic Solutions of Nerve Impulse Equations

  • Isabel Salgado Labouriau
Part of the NATO ASI Series book series (NSSA, volume 138)


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).


Periodic Solution Hopf Bifurcation Homoclinic Orbit Centre Manifold Homoclinic Solution 


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Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Isabel Salgado Labouriau
    • 1
  1. 1.Grupo de Matemática Aplicada Faculdade de CienciasUniversidade do PortoPortoPortugal

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