The electrotonic length constant: A theoretical estimate for neuroprosthetic electrical stimulation

The electrotonic length constant of a neurite is theoretically estimated taking into account the extremely constricted extracellular space and the high spectral frequencies of electrical pulses used during neuroprosthetic stimulation of neurons. The analysis proceeds by rigorously deriving a one dimensional cable equation from the full problem with three-dimensional geometry and a known current passed into a constricted extracellular space. Conditions for the validity of the derivation are given. An analysis of the cable equation gives estimates for the electronic length constant of between 2 μm and 20 μm, which are between one and three orders of magnitude smaller than estimates in the literature for neuroprosthetic electrical stimulation. The implication of this is that passive changes in the membrane potential due to electrical stimulation are determined by the local extracellular current density, on the scale of a few to a few tens of micrometers.