Rall W
Biophys J. 1969 Dec;9(12):1509-41. doi: 10.1016/S0006-3495(69)86468-4.
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, theta, are shown to be typically about ten thousand times smaller than the time constant, tau(m) = R(m)C(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 tau(m) by a factor between 2 and 10. The relation of the membrane current density, I(m)(theta, x, t), to the values (at the outer membrane surface) of the extracellular potential phi(e)(r, theta, x, t) and of partial differential(2)phi(e)/ partial differentialx(2), 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 phi(i)(r, theta, x, t) and partial differential(2)phi(i)/ partial differentialx(2) (at the inner membrane surface). The relation of these results to those of one-dimensional cable theory is discussed.
阐述并解决了一个与膜圆柱体相关的数学问题；说明了并讨论了其影响。该问题涉及在初始非均匀膜电位的被动衰减过程中，有限长度（两端密封）的膜圆柱体内外的电势在柱坐标下的体积分布。相对于角度θ的均衡时间常数通常比均匀被动膜电位衰减的时间常数τ(m)=R(m)C(m)小约一万倍。相对于长度的均衡时间常数与一维电缆理论中的时间常数一致；通常，它们比τ(m)小2到10倍。研究了膜电流密度I(m)(θ,x,t)与细胞外电势φ(e)(r,θ,x,t)及其关于x的二阶偏导数在（外膜表面）的值之间的关系，结果表明这些量一般不成正比；然而，在某些特定条件下，这三个量相互之间以及与内膜表面的φ(i)(r,θ,x,t)及其关于x的二阶偏导数成正比。讨论了这些结果与一维电缆理论结果的关系。
