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Solving the voltage of a simple cable WITHOUT Fourier Theory

Posted: Mon Dec 07, 2015 11:49 am
by kburke

I hope this is the right place to ask this question. I am intending to use NEURON to teach basic electrophysiology to undergraduate students. However it is unlikely many of them will have "advanced" mathematics coursework by the time they enter my class. I believe I can derive the cable equation with them using only their previous mathematical knowledge in class. I feel it would really be beneficial to show them how a passive cable's voltage really does follow simply from solving this equation in space and time.

I read what I thought were the pertinent chapters of the NEURON book regarding numerical solutions to the equation. The temporal discretization with CN or backwards Euler seems straightforward enough to explain, but the spatial discretization is less obvious. The students (and myself, I confess) do not have enough of a background in Fourier theory to find that analytical solution intuitive. I was trying to figure out a way to solve each spatial dx individually (obviously only for pedagogical purposes) for a given dt.

Simply put, is there a different, perhaps more intuitive way to numerically solve the cable equation one dx at a time?

Re: Solving the voltage of a simple cable WITHOUT Fourier Th

Posted: Mon Dec 07, 2015 8:00 pm
by ted
If you're looking for a development of a formula that combines finite difference approximations to both the temporal and second order spatial derivatives of membrane potential, that's not in the NEURON Book. However, I seem to recall something like that in one of the chapters of
Koch, C. & Segev, I. (Eds.) Methods in Neuronal Modeling MIT Press, 1989 (a more recent edition might be available)
--maybe the chapter by Yamada et al..

The aim of the first sections of chapter 4 of the NEURON Book is to set the stage for a principled consideration of discretization error. It's actually a pretty gentle introduction to that topic. There is one sentence that begins with the phrase "From Fourier theory" and then goes on to assert that any spatial pattern can be represented as an infinite sum of cosine waveforms. I think that is the only place where Fourier theory is mentioned, and it is mentioned only in passing as a justification for an assertion, analogous to how one might mention quantum theory as a justification for introducing the notion of energy levels, e.g. in the context of a discussion about fluorescence, without expecting the reader to know anything about Schroedinger's equation etc..