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An initial-boundary value problem of physiological significance for equations of nerve conduction. (English) Zbl 0664.92007
A model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid Loligo of A. L. Hodgkin and A. F. Huxley [J. Physiol. 117, 500-544 (1952)]. This nonlinear parabolic partial differential equation (PDE), when considered as part of a particular initial-boundary value problem (IBVP), models the electrical activity of an axon under conditions found both in nature and in the laboratory. This IBVP for the Hodgkin-Huxley equations is proved to be well posed in the sense of Hadamard, and a priori bounds on the solution are derived.

MSC:
92Cxx Physiological, cellular and medical topics
35K55 Nonlinear parabolic equations
78A70 Biological applications of optics and electromagnetic theory
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[1] The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Applications, vol. 23, Addison-Wesley Publ. Co., Reading (Mass.), London, Amsterdam, Don Mills (Ont.), Sydney, Tokyo, 1984.
[2] and , Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
[3] and , Methods of Mathematical Physics, vols. 1 and 2, Wiley-Interscience, New York-London-Sydney, 1962.
[4] Evans, Biophys. J. 6 pp 583–
[5] Hodgkin, J. Physiol. 117 pp 500– (1952) · doi:10.1113/jphysiol.1952.sp004764
[6] , and , Electric Current Flow in Excitable Cells, Clarendon Press, Oxford, 1975.
[7] Partial Differential Equations, 4th ed., Springer-Verlag, New York-Heidelberg-Berlin, 1982. · doi:10.1007/978-1-4684-9333-7
[8] Negative feedback in neural networks, Doctoral Dissertation, Courant Institute of Mathematical Sciences, New York University, 1987.
[9] The backward Euler method for numerical solution of the Hogkin-Huxley equations of nerve conduction, submitted to SIAM J. Numer. Anal.
[10] Partial Differential Equations in Biology, Courant Institute of Mathematical Sciences Lecture Notes, New York, 1976.
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