Periodic Solutions and Global Bifurcations for Nerve Impulse Equations

  • M. Rosário
  • L. M. S. Álvares-Ribeiro
Part of the NATO ASI Series book series (NSSA, volume 138)


Software for the Apple Macintosh microcomputer has been developed using the harmonic balance method for the detection of periodic solutions of feedback systems. This software, based on graphical criteria, also provides a good notion of the system’s dynamics and the way bifurcations occur, as well as the stability characteristics of the limit cycles. Here we report the results of its application to the FitzHugh equations for the nerve impulse. We numerically detect periodic solutions and global bifurcation points that, although theoretically predicted, had never been located. We describe amplitude, frequency and stability characteristics for these solutions, as well as the type and location of the bifurcation points.


Periodic Solution Hopf Bifurcation Bifurcation Diagram Bifurcation Point Stability Characteristic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1: 445–466 (1961).PubMedCrossRefGoogle Scholar
  2. [2]
    A. Gelb and W. van der Velde, Multiple-input Describing Functions and Nonlinear System Design, McGraw-Hill (1968).Google Scholar
  3. [3]
    Isabel S. Labouriau and Nelma R.A. Moreira, Solucões Periódicas das equacões de FitzHugh para o impulso nervoso, VII Congresso dos Matematicos de Expressão Latina (1985).Google Scholar
  4. [4]
    Alistair Mees, Dynamics of Feedback Systems. John Wiley (1981).Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • M. Rosário
    • 1
  • L. M. S. Álvares-Ribeiro
    • 1
  1. 1.Grupo de Matemática AplicadaUniversidade do PortoPortoPortugal

Personalised recommendations