Approximate analytical solution of a (V,m,h) reduced system for backpropagating action potentials in sparsely excitable dendrites

We derive an approximate analytical solution of a nonlinear cable equation describing the backpropagation of action potentials in sparsely excitable dendrites with clusters of transiently activating, TTX-sensitive Na+ channels of low density, discretely distributed as point sources of transmembrane current along a continuous (non-segmented) passive cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na+ ion channels, included known cumulative inactivation kinetics observed at the microscopic level. In such a reduced third-order system, the ‘recovery’ variable is an electrogenic sodium-pump and/or a Na+-Ca2+ exchanger imbedded in the passive membrane, and a high leakage conductance stabilizes the system. A nonlinear cable equation was used to investigate back-propagation and repetitive activity of action potentials, exhibiting characteristics of the modified Hodgkin-Huxley kinetics (in the presence of suprathreshold input). In particular, a time-dependent analytical solution was obtained through a perturbation expansion of the non-dimensional membrane potential (Φ) for all voltage dependent terms including the voltage dependent Na+ activation μ) and state-dependent inactivation (η) gating variables and then solving the resulting system of integral equations. It was shown that back-propagating action potentials attenuate in amplitude with the frequency following experimental findings and that the discrete and low-density distributions of transient Na+ channels along the cable structure contribute significantly to their discharge patterns. A major significance of integrative modelling is the provision of a continuous description of the non-dimensional membrane potential (Φ) as a function of position.