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23.—Oscillation Phenomena in the Hodgkin-Huxley Equations

Published online by Cambridge University Press:  14 February 2012

William C. Troy  and
J. B. McLeod
William C. Troy
Affiliation:
Department of Mathematics, University of Pittsburgh.
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Synopsis

A widely accepted model of nerve conduction in the squid axon is the systemof four non-linear partial differential equations developed by Hodgkin and Huxley. Under space clamp and current clamp conditions these equations are reduced to a system of ordinary differential equations.

We find that under appropriate assumptions on the functions and parameters in the resulting fourth order Hodgkin-Huxley equations there occurs a bifurcation of periodic solutions from the steady state. This bifurcation takes place as the current parameter, I, passes through a critical value.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 299 - 310
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

1Cole, K. S., Antosiewicz, H. A. and Rabinowitz, P.. Automatic computation of nerve excitation. J. Soc. Indust. Appl. Math. 3 (1955), 153–172.CrossRefGoogle Scholar
2FitzHugh, R.. Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. Gen. Phys. 43 (1960), 867–896.CrossRefGoogle ScholarPubMed
3FitzHugh, R.. Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1 (1961), 445–66.CrossRefGoogle ScholarPubMed
4Hale, J. K.. Ordinary differential equations (New York: Wiley-Interscience, 1969).Google Scholar
5Hastings, S. P.. Travelling wave solutions of the Hodgkin-Huxley equations. Arch. Rational Mech. Anal., to appear.Google Scholar
6Hodgkin, A. L. and Huxley, A. F.. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952), 500–544.CrossRefGoogle ScholarPubMed
7Hopf, E.. Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Verh. Sachs. Akad. Wiss. Leipzig. Math.-Nat. 94 (1942), 3–22.Google Scholar
8Noble, D., McAllister, D. E. and Tsien, R. W.. Reconstruction of the electrical activity of cardiac Purkinje fibers. J. Physiol, 251 (1975), 1–59.Google Scholar
9Hauswirth, O., Noble, D. and Tsien, R. W.. The mechanism of oscillatory activity at low membrane potentials in cardiac Purkinje fibers. J. Physiol. 200 (1969), 255–265.CrossRefGoogle Scholar
10Troy, W.. Bifurcation phenomena in FitzHugh's nerve conduction equations. J. Math. Anal. Appl., to appear.Google Scholar
11Uspensky, J. V.. Theory of equations (New York: McGraw-Hill, 1948).Google Scholar
12Yorke, J. A. and Alexander, J. C.. On Global Bifurcation of Periodic Solutions. Notices Amer. Math. Soc (Jan. 1974), 711–34–7.Google Scholar