Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T11:45:25.553Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical and numerical studies of the Bonhoeffer van der Pol system

Published online by Cambridge University Press:  17 February 2009

Belinda Barnes  and
Roger Grimshaw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Bonhoeffer Van der Pol system is a planar autonomous nonlinear system of differential equations which has been invoked as a qualitative model of physiological states in a nerve membrane. It contains three independent parameters and previous work has only studied a small portion of the parameter space, that part which is thought to be of physiological relevance. Here we give a complete study of the full parameter space, using both theoretical results and numerical solutions.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 4 , April 1997 , pp. 427 - 453
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Fitzhugh, R., “Threshholds and plateaus in the Hodgkin-Huxley nerve equations”, J. Gen. Physiol. 45 (1960) 867–896.CrossRefGoogle Scholar
[2]Fitzhugh, R., “Impulses and physiological states in models of nerve membrane”, Biophys. J. 1 (1961) 445–466.CrossRefGoogle ScholarPubMed
[3]Grimshaw, R., Nonlinear ordinary differential equations (Blackwell, Oxford, 1990).Google Scholar
[4]Guttman, R., Lewis, S. and Rinzel, J. R., “Control of repetitive firing in squid axon membrane as a model for a neuronoscillator”, J. Physiol. 305 (1980) 377–395.CrossRefGoogle Scholar
[5]Hodgkin, A. L. and Huxley, A. F. A., “A quantitative description of membrane current and its application to conduction and excitation in nerve”, J. Physiol. 117 (1952) 500–544.CrossRefGoogle ScholarPubMed
[6]Jordan, D. W. and Smith, P., Nonlinear ordinary differential equations, second ed. (Oxford University Press, Oxford, 1987).Google Scholar
[7]Kloeden, P. E., Platen, E. and Shurz, H., Numerical solution of stochastic differential equations through computer experiments (Springer, Berlin, 1994).Google Scholar
[8]Rinzel, J., “Excitation dynamics: insights from simplified membrane models”, Theoretical Trends on Neuroscience 44 (1985) 2945–2946.Google ScholarPubMed
[9]Rinzel, J. and Miller, R. N., “Numerical calculation of stable and unstable periodic solutions in the Hodgkin-Huxley equations”, Math. Biosciences 49 (1980) 27–59.CrossRefGoogle Scholar
[10]Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (Springer, New York, 1990).CrossRefGoogle Scholar