Null space in the Hodgkin-Huxley Equations. A critical test

Voltage perturbation methods based upon topological concepts are used to elicit responses from the Hodgkin-Huxley (HH) nonlinear differential equations. These responses present a critical check upon the validity of the HH model for electrical activity across squid axon membrane. It is shown that when a constant current is applied such that a stable equilibrium and rhythmic firing are present, the following predictions are inherent in the HH system of equations: (a) Small instantaneous voltage perturbations to the axon given at points along its firing spike result in phase resetting curves (when new phase versus old phase is plotted) with an average slope of 1. (b) A larger voltage perturbation (from certain points along the firing spike) results in the permanent cessation of periodic firing, with membrane voltage rapidly approaching the equilibrium value. (c) A still larger perturbation yields phase resetting curves with an average slope equal to 0. These predictions, coupled with Tasaki's experimental demonstration that squid axons in excellent condition do give repetitive firing under constant current, provide a critical test of the validity of the HH model.