THE POTASSIUM A-CURRENT, LOW FIRING RATES AND REBOUND EXCITATION IN HODGKIN-HUXLEY MODELS

It is widely believed, following the work of Connor and Stevens (1971, J. Physiol. Lond. 214, 31-53) that the ability to fire action potenti als over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I-A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects of I-A on st eady firing rate, especially in the near-threshold regime for the onse t of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based g eneric I-A parameters, the model does not fire at arbitrarily low rate s, although it can for the more atypical I-A parameters given by Conno r and Stevens for the crab axon. When the I-A inactivation rate is slo w, we find that the transient potassium current can mediate more compl ex firing patterns, such as periodic bursting in some parameter regime s. The number of spikes per burst increases as g(A) decreases and as i nactivation rate decreases. We also study how I-A affects properties o f transient voltage responses, such as threshold and firing latency fo r anodal break excitation. We provide mathematical explanations for se veral of these dynamic behaviors using bifurcation theory and averagin g methods.