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Singular perturbation approach to traveling wave solutions of the Hodgkin-Huxley equations and its application to stability problems

  • Hideo Ikeda
  • Masayasu Mimura
  • Tohru Tsujikawa
Article

Abstract

We are concerned with traveling wave solutions for the Hodgkin-Huxley equations. Such solutions depend only on the single variablez=x+ct so that the equations are rewritten as a five-dimensional dynamical system with a velocity parameterc. By singular perturbation methods, we construct a homoclinic orbit of the resulting system for somec *, which tends to the resting state asz→±∞. It gives a traveling wave solution. We know the direction in which the unstable manifold passes through the stable manifold with respect to the resting state when the parameterc increases throughc *. This gives essential information on stability properties of the traveling wave solution.

Key words

the Hodgkin-Huxley equations singular perturbation stability of traveling wave solution 

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Copyright information

© JJAM Publishing Committee 1989

Authors and Affiliations

  • Hideo Ikeda
    • 1
  • Masayasu Mimura
    • 2
  • Tohru Tsujikawa
    • 3
  1. 1.Department of MathematicsToyama UniversityToyamaJapan
  2. 2.Department of MathematicsHiroshima UniversityHiroshimaJapan
  3. 3.Hiroshima Junior College of Automotive EngineeringHiroshimaJapan

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