Analytical solution for time-dependent potentials in a fiber stimulated by an external electrode
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This study provides an analytical solution for time-dependent potentials in a 3D cylindrical fiber stimulated by an extracellular point electrode. The membrane is passive and represented by surface resistance and surface capacitance. Separation of variables solution expresses intracellular and extracellular potentials as sums involving modified Bessel functions; the coefficients (\(A_n\) and \(B_n\)) depend on time. In contrast to previous analytical solutions, where \(A_n\) and \(B_n\) had to be determined numerically, here \(A_n\) and \(B_n\) are given by explicit formulas that resemble the formulas for potentials in a fiber stimulated by a transverse electric field. The comparison of the 3D analytical solution with the 1D cable model shows that the cable model approximates transmembrane potential with the error below 5 % when the distance between the electrode and the fiber is 0.2–4 mm and when the stimulus is longer than 3.3 ms. For stimuli between 0.43 and 3.3 ms, the range of fiber–electrode distances with error below 5 % shrinks, and it disappears completely for stimuli shorter than 0.43 ms. Thus, our study shows that the applicability of the 1D cable model may be more limited than previously considered.
KeywordsElectric potential Stimulation Passive fiber Cable model Separation of variables
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