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Math 636 - Mathematical Modeling |
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San Diego State University -- This page last updated 29-Oct-07 |
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Nerve Impulses and Excitation
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Math 636 - Mathematical Modeling |
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San Diego State University -- This page last updated 29-Oct-07 |
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Nerve Impulses and Excitation
This webpage is primarily to provide links to important references for this section. The primary reference is a few sections from the book by Professor Leah Edelstein-Keshet on Hodgkin-Huxley Theory of Nerve Membranes. Central to this theory is the original work by Hodgkin and Huxley (1952). Their work is a beautiful mix of experiment and theory, resulting in an excellent understanding of the giant squid axon for conductance of a signal. Remarkably, their model is still part of research efforts, over 50 years after the publication of their results.
The Hodgkin-Huxley model for nerve impulses is a complicated system of differential and algebraic equations. The complete mathematical analysis has never been done (as far as I know). It still is used as a souce for a number of interesting studies. At the core of their model is a much simplified model known as the FitzHugh-Nagumo Model. (See Scholarpedia for more information on this model from one of its creators.) This model can be analyzed to a large extent.
MatLab M-files were created for the Hodgkin-Huxley model and the FitzHugh-Nagumo model. The Hodgkin-Huxley model was obtained from the book by Leah Edelstein-Keshet, and the numerical simulation at this time gives a reverse electrical spike from the one usually seen for nerve impulses. In addition, you can download the pplane 7 file for the FitzHugh-Nagumo model. More development showing these programs and important graphs will be developed in the future. In addition, this section will include information on Hopf bifurcations.