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Some Mathematical Problems Arising in Neurobiology

  • Chapter
Book cover Mathematics of Biology

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 80))

Abstract

Outline of Lectures

  1. 1.

    Physiological Background, work of Hodgkin and Huxley, Hodgkin-Huxley equations.

  2. 2.

    Mathematical problems for the Hodgkin-Huxley equations, simplified models.

  3. 3.

    Existence theorems for travelling waves.

  4. 4.

    Stability of travelling waves, multiple pulses.

  5. 5.

    Multi-cellular phenomena, relation to patterns seen in chemistry.

  6. 6.

    Discrete models of excitable media.

These lectures will largely be devoted to the study of mathematical problems originating in the work of A. L. Hodgkin and A. F. Huxley, who, in 1952, proposed a set of partial differential equations to describe the transmission of an electrical impulse by a nerve axon [2], [3]. This set of equations is probably the most important mathematical model in physiology, and yet for nearly twenty years few mathematicians were aware of its existence. Recently, however, a number of people have studied the Hodgkin-Huxley (HH) equations, and it has become clear that they lead to several interesting and fundamental mathematical problems. In some cases resolution of these problems may increase our understanding of message transmission in the nervous system. Further, it has developed that there is an intimate connection between nerve models and equations describing other phenomena, such as certain chemical reactions which can support time periodic two dimensional spatial patterns. It is suspected that similar processes may play a role in pattern formation in many other settings, but here I will concentrate almost entirely on developments related to neurobiology.

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References

References For Lecture I

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References For Lecture II

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  10. Feroe, J., Existence and Stability of Multiple Impulse Solutions of a Nerve Equation, to appear.

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References For Lecture VI

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  3. Greenberg, J. M., B. D. Hassard and S. P. Hastings, Pattern Formation and Periodic Structures in Systems Modeled by Reaction-Diffusion Equations. Bulletin Amer. Math. Soc. 84 (1978), 1296–1327.

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Authors and Affiliations

  1. State University of New York, New York, USA

    Stuart Hastings

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  1. Stuart Hastings
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Mimmo Iannelli

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Hastings, S. (2010). Some Mathematical Problems Arising in Neurobiology. In: Iannelli, M. (eds) Mathematics of Biology. C.I.M.E. Summer Schools, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11069-6_4

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