Analysis of a linear model for electrical stimulation of axons-critical remarks on the "activating function concept"

A comprehensive description of a linear model of an axon of infinite length exposed to an external voltage is presented. The steady-state transmembrane potential is derived as a function proportional to the convolution product of the second spatial difference s_n of the external potential (the "activating function") and the impulse response ψ_n of a spatial low-pass filter. The impulse response ψ_n represents the influence of the axon and is fully characterized by the axon\´s length constant λ. Q closed-form solution of the cable equation can be given in the spatial Fourier domain. Due to a "spectral acceleration effect", the overall transmembrane potential approximates the steady-state considerably faster than an exponential with the axon\´s membrane time constant τ. The effect is increasingly pronounced, the smaller the distance between the electrode and the axon. Regarding myelinated fibers and practically relevant electrode/axon distances and pulse widths, the transmembrane potential at the end of a stimulation pulse can be substantially better approximated by the steady-state condition than by the initial response as claimed by the "activating function concept." Quantitative limits for the range of validity of the activating function concept are derived.