Hodgkin-Huxley Equations

The Hodgkin-Huxley equations are a nonlinear system of ordinary differential equations for the voltage and three gating variables , , and of an excitable membrane (Hodgkin and Huxley 1952). In a voltage convention with currents proportional to , the spatially uniform system is

			(1)
			(2)

where the second equation applies to each in . Here is the capacitance, is the applied current, and the and are maximal conductances and reversal potentials for the sodium, potassium, and leak terms. The voltage-dependent rates and are nonnegative. They make the interval invariant for each gating variable since at and at .

If , a fixed voltage makes relax exponentially to

(3)

For a constant applied current, substituting these values in the voltage equation reduces the search for an equilibrium point to a scalar equation for . Varying the applied current in the original parameter regime gives Hopf bifurcations and branches of periodic orbits (Hassard 1978).

The Morris-Lecar model uses a related conductance-based voltage equation with a single dynamic gate and an instantaneous calcium gate.