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 potentials 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 steady firing rate, especially in the near-threshold regime for the onset 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 generic I_A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypical I_A parameters given by Connor and Stevens for the crab axon. When the I_A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases as g_A decreases and as inactivation rate decreases. We also study how I_A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.