Biophysical Journal 9: 1509-1541 (1969)
© 1969 the Biophysical Society

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

ABSTRACT

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, , are shown to be typically about ten thousand times smaller than the time constant, _m = R_mC_m, 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 _m by a factor between 2 and 10. The relation of the membrane current density, I_m(, x, t), to the values (at the outer membrane surface) of the extracellular potential _e(r, , x, t) and of ²_e/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 _i(r, , x, t) and ²_i/x² (at the inner membrane surface). The relation of these results to those of one-dimensional cable theory is discussed.