Rigorous Green's function formulation for transmembrane potential induced along a 3-D infinite cylindrical cell

The quasi-static electromagnetic field interaction with three-dimensional infinite-cylindrical cell is investigated for both intracellular (IPS) and extracellular (EPS) current point-source excitation. The induced transmembrane potential (TMP), expressed conventionally via Green's function, may alternatively be expanded into a faster-converging representation using a complex contour integration, consisting of an infinite-discrete set of exponentially decaying oscillating modes (corresponding to complex eigenvalues) and a continuous source-mode convolution integral. The dominant contributions for both the IPS and EPS problems are obtained in simple closed-form expressions, including well documented special mathematical functions. In the IPS case, the dominant modal contribution (of order zero)-an exact solution of the well-known cable equation-is explicitly and analytically corrected by the imaginary part of its eigenvalue and the source-mode convolution contribution. However, the TMP along a fiber was shown to decay at infinity algebraically and not exponentially, as predicted by the classic cable equation solution. In the EPS case, the dominant contribution is expressed as a source-mode convolution integral. However, for a long EPS distance (e.g., >10 cable length constant) the order-one-modes involved in the convolution is not a solution of the cable equation. Only for shorter EPS distance should the cable equation solution (i.e., the order zero dominant mode) be included in addition to the modes of order one. For on-membrane EPS location, additional modes should be included as well. In view of our EPS result, we suggest that the cable equation modeling existing in the literature and related to functional electrical stimulation for EPS problems, should be critically reviewed and corrected.