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Mathematical aspects of Hodgkin-Huxley neural theory. (English) Zbl 0627.92005

Cambridge Studies in Mathematical Biology, 7. Cambridge etc.: Cambridge University Press. XI, 261 p.; £35.00; $ 49.50 (1987).
In 1951, Hodgkin and Huxley made their remarkable contributions to the quantitative study of electrically active cells by using voltage-clamp methods and proposing a system of differential equations as a model. They received the Nobel prize in physiology for their research in 1961. Their pioneering effort has been continued by many researchers, the experimental and mathematical results are scattered through the voluminous literature. The purpose of this monograph is:
(i) To provide an introduction to the work of Hodgkin and Huxley and the later work based on the techniques which they introduced. Emphasis is on the theoretical aspects. The account is accessible to mathematicians with little or no background in physiology.
(ii) To summarize some of the mathematics that is used to study these differential equations. This brings the mathematicians to the research level in certain aspects of O.D.E., and also indicates to the interested biologists the kinds of mathematics that may be useful in such studies.
The contents: 1. Introduction (5 pages); 2. Nerve conduction: The work of Hodgkin and Huxley (61 pages); 3. Nerve conduction: Other mathematical models (7 pages) 4. Models of other electrically excitable cells (27 pages); 5. Mathematical theory (79 pages); 6. Mathematical analysis of physiological models (69 pages); Appendix (some background in electricity) (4 pages).
There are 121 references cited (up to 1985).
Reviewer: Li Bingxi

MSC:

92Cxx Physiological, cellular and medical topics
92-02 Research exposition (monographs, survey articles) pertaining to biology
34C25 Periodic solutions to ordinary differential equations
92C50 Medical applications (general)
34E15 Singular perturbations for ordinary differential equations

Software:

AUTO
Full Text: DOI

References:

[1] Hille, B.: Ionic channels of excitable membranes. (1984)
[2] Doedel, E.: AUTO: A program for the automatic bifurcation analysis of autonomous systems. Congr. numer. 30, 265-284 (1981) · Zbl 0511.65064
[3] Rinzel, J.: On repetitive activity in nerve. Fed. proc. 37, No. No. 14 (Dec. 1978) · Zbl 0384.92003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.