Bifurcation phenomena in FitzHugh's nerve conduction equations

We consider the second order nonlinear ordinary differential equations developed by FitzHugh to simplify the fourth order current clamped Hodgkin-Huxley nerve conduction equations. We demonstrate the bifurcation, direction, and stability of a family of small periodic solutions as the current parameter I passes through a critical value. Arguments are given which suggest that this family grows to become a large periodic solution, then shrinks, collapsing onto the steady state as I passes through a second critical value. The usefulness of these results in studying the Hodgkin-Huxley equations is discussed.