The geometry of the Hodgkin-Huxley Model

The Hodgkin-Huxley model for the "space-clamped" (i.e., having variables independent of position) squid giant axon is described by a system of four coupled nonlinear ordinary differential equations. In this study, the behavior under various conditions of the solution vector of these equations is predicted qualitatively. This is done by studying the geometry of the phase space of an approximation to the four dimensional system obtained by assuming that the two "rapid" components of the system, V and m, are described by algebraic rather than differential equations. Using this method, known properties of the solution of the Hodgkin-Huxley equation, such as threshold, repeated oscillation under constant current stimulus, etc., are explained qualitatively. The direct relation between the Hodgkin-Huxley equations and simpler systems such as the Fitz Hugh equations, which have been used as an aid to the understanding of the Hodgkin-Huxley model, is examined. The roles of the various activation and inactivation components of the Hodgkin-Huxley model are clarified and the effect of modification of those components is studied.