Distributions of Potential in Cylindrical Coordinates and Time Constants for a Membrane Cylinder

A mathematical problem relating to membrane cylinders is stated and solved; its implications are illustrated and discussed. The problem concerns the volume distribution, in cylindrical coordinates, of the electric potential inside and outside a membrane cylinder of finite length (with sealed ends), during passive decay of an initially nonuniform membrane potential. The time constants for equalization with respect to the angle, θ, are shown to be typically about ten thousand times smaller than the time constant, τm = RmCm, for uniform passive membrane potential decay. The time constants for equalization with respect to length are shown to agree with those from one-dimensional cable theory; typically, they are smaller than τm by a factor between 2 and 10. The relation of the membrane current density, Im(θ, x, t), to the values (at the outer membrane surface) of the extracellular potential φe(r, θ, x, t) and of ∂2φe/∂x2, is examined and it is shown that these quantities are not proportional to each other, in general; however, under certain specified conditions, all three of these quantities are proportional with each other and with φi(r, θ, x, t) and ∂2φi/∂x2 (at the inner membrane surface). The relation of these results to those of one-dimensional cable theory is discussed