Biophysical Journal 8: 358-379 (1968)
© 1968 the Biophysical Society

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Passive Membrane Potentials

A Generalization of the Theory of Electrotonus

ABSTRACT

The theory of electrotonus, which has been well developed for small cylinders, is extended: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.