Analytical Dynamics of Neuron Pulse Propagation

The four-dimensional Hodgkin–Huxley equations describe the propagation in space and time of the action potential v(z) along a neural axon with z = x+ ct and c being the pulse speed. The potential v(z), which is parameterized by the temperature, is driven by three gating functions, m(z), n(z) and h(z), each of which obeys formal first order kinetics with rate constants that are represented as nonlinear functions of the potential v. It is shown that this system can be analytically simplified (i) in the number of gating functions and (ii) in the form of associated rate functions while retaining to close approximation quantitative fidelity to computer solutions of the exact equations over the complete temperature range for which stable pulses exist. At a given temperature we record two solutions (T < Tmax) corresponding to a high-speed and a low-speed branch in speed-temperature plots, c(T ), or no solution (T > Tmax). The pulse is considered as composed of two contiguous parts: (i) a pulse front extending from v(0) = 0 to a pulse maximum v = Vmax, and (ii) a pulse back extending from Vmax through a pulse minimum Vmin to a final regression back to v(z → ∞) = 0. An approximate analytic solution is derived for the pulse front, which is predicted to propagate at a speed c(T ) = 1203Θ 3 8 (T ◦C) cm/sec, Θ = 3 T−6.3 10 in close agreement with computer solution of the exact Hodgkin–Huxley equations for the entire pulse. These results provide the basis for a derivation of two-dimensional differential equation systems for the pulse front and pulse back, which predict the pulse maximum and minimum over the operational temperature range 0 ≤ T ≤ 25◦C, in close agreement with the exact equations. Most neuron dynamics studies have been based on voltage clamp experiments featuring external current injection in place of self-generating pulse propagation. Since the behaviors of the gating functions are similar, it is suggested that the present approximations might be applicable to such situations as well as to the dynamics of myelinated fibers.