Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations

We have calculated periodic solutions to the space-clamped Hodgkin-Huxley equations for constant applied depolarizing current I. For each I, I_v<I<I₂, there is a stable periodic solution which corresponds to repetitive firing. For I<I₁ and I>I₂, where I_v<I₁, the unique singular point is stable; for other values of I it is unstable. These results have been found by other investigators as well. In the overlap range I_v<I<I₁, where a stable periodic solution and a stable singular point coexist, we have verified computationally the conjecture that there is an additional periodic solution which is unstable. In an analysis of linear stability we demonstrate instability by direct calculation of growth rates of unstable modes. The two sets of periodic solutions (stable and unstable) form one continuous family with two limbs which coalesce at I=I_v. The periodic solutions emerge with zero amplitude through Hopf bifurcation for I=I₁,I₂. The temperature dependence of the solutions is examined. For 6.3°C the limb of unstable solutions has a curious switchback region so that for some I values this limb exhibits three unstable limit cycles. Moreover, additional branches of unstable periodic solutions emerge from this region through successive period-doubling bifurcations. The numerical method we use to calculate a periodic solution is insensitive to the stability of the solution; a full description is provided.