A dynamical system for action potentials in the giant axon of the squid

We introduce a novel quantitative approach to describe ionic gating and use the Kramers equation for electrodiffusion of ions through membrane channels to construct a simple dynamical system for transient action potentials and resting potentials in giant axons of the squid on a better physicochemical basis than the Hodgkin-Huxley (HH) model and the Goldman-Hodgkin-Katz (GHK) model. Like the HH dynamical system, our present model describes many features of excitable membranes such as sharp firing thresholds, latency, refractory periods, repetitive firings with a sustained stimulating current, excitation blocking, and propagating action potentials. It differs from the HH dynamical system in having three fixed points, the first of which corresponds to the electrodiffusive resting potential. The second fixed (or saddle) point corresponds to the threshold for generation of local action potentials. It predicts monotonic rather than oscillatory decay of the membrane potential following subthreshold stimulation by microelectrodes. The ratio of sodium to potassium currents in the resting state of the membrane is set at 3:2. In the electrodiffusive "resting" state, all potassium and sodium activation gates are postulated to be closed, whereas, according to the HH model, about 32% of the potassium gates and 5.3% of the sodium activation gates are open. As our electrodiffusive "resting" state, described by a generalization of the GHK model, emerges as the stable fixed point of our dynamical system, our new model provides a unified treatment of both transient action potentials and electrodiffusive "resting" potentials in perfused axons.