Analytical solutions of the Frankenhaeuser-Huxley equations I: minimal model for backpropagation of action potentials in sparsely excitable dendrites.

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na(+) channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na(+) ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na(+) and K(+). In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Phi) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na(+) activation (micro) and a state-dependent inactivation (eta) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na(+) channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.