Theoretical stability properties of a space-clamped axon

Abstract

A mathematical method for determining the stability properties of a uniform nerve membrane is developed. Two basically similar tests of stability are considered: examination of the real characteristic roots of the linearized equations and application of a modified Nyquist criterion to the linearized alternating current admittance. The method is applied to the Hodgkin-Huxley equations for the squid axon membrane at 6.3 degrees C to decide theoretically whether stable membrane behavior might be expected in a space clamp experiment. The equations are solved for step depolarizations similar to those used in voltage clamp experiments. Each solution can be represented by a trajectory in the phase space of the variables V, m, h, and n. The stability of motion of a phase point on a given trajectory, and hence the adequacy of the control of the membrane potential, is shown to be a function of the effective conductance in series with the membrane. (For a patch of membrane away from the point controlled by feedback, the effective conductance is the combined conductance of the axial current electrode, axoplasm, and an external layer of sea water, all in series.) In particular, there is a (uniquely determined) critical conductance, defined as the minimum effective series conductance consistent with stability, associated with each point on the trajectory. During a step depolarization the critical conductance goes through a maximum. The values of such maxima as a function of voltage are closely similar to the negative slopes of the peak inward current versus voltage curve. This empirical correlation may be helpfup in the prediction of stability in experimental situations.