SIAM-AMS Proceedings Volume 13 1981
GAIL A. CARPENTER1
1. Introduction. A fundamental problem of psychophysics is an inverse problem: induce underlying cellular or network mechanisms from signal patterns. The patterns may be observed at various levels: from EEG, extracellular, or intracellular recordings or from psychological experiments. In each case the recorded pattern is an ensemble of events taking place at a finer, and usually unobservable, level.
At the center of the work presented in this article is the following simple question:
Which intracellular signal patterns are the result of basic single-cell membrane mechanisms and how are these patterns altered as membrane parameters change?
In other words, we will be considering the inverse problem at the level of the single cell.
At the heart of all models of the nerve impulse is the ionic hypothesis, which assumes that the main factor in transmembrane voltage shifts is the flow of selected ions into and out of the nerve cell. §2 contains an outline of the ionic hypothesis and its classical realization, the Hodgkin-Huxley equations ( Hodgkin and Huxley [ 1952]). The latter is a nonlinear diffusion equation coupled with three first-order ordinary differential equations. The model was constructed partly from classical electrical theory and partly from a careful series of experiments which allowed Hodgkin, Huxley, et al., to separate the various components of current and to model them empirically. These experiments were performed on the fortuitously large giant axon of the squid. Hodgkin-Huxley (HH) models of other experimental preparations have been obtained and a large
© 1981 American Mathematical Society____________________