Rall W
Biophys J. 1969 Dec;9(12):1483-508. doi: 10.1016/S0006-3495(69)86467-2.
A theoretical basis is provided for the estimation of the electrotonic length of a membrane cylinder, or the effective electrotonic length of a whole neuron, from electrophysiological experiments. It depends upon the several time constants present in passive decay of membrane potential from an initially nonuniform distribution over the length. In addition to the well known passive membrane time constant, tau(m) = R(m)C(m), observed in the decay of a uniform membrane potential, there exist many smaller time constants that govern rapid equalization of membrane potential over the length. These time constants are present also in the transient response to a current step applied across the membrane at one location, such as the neuron soma. Similar time constants are derived when a lumped soma is coupled to one or more cylinders representing one or more dendritic trees. Different time constants are derived when a voltage clamp is applied at one location; the effects of both leaky and short-circuited termination are also derived. All of these time constants are demonstrated as consequences of mathematical boundary value problems. These results not only provide a basis for estimating electrotonic length, L = [unk]/lambda, but also provide a new basis for estimating the steady-state ratio, rho, of cylinder input conductance to soma membrane conductance.
为通过电生理实验估计膜圆柱体的电紧张长度或整个神经元的有效电紧张长度提供了理论基础。它取决于膜电位从沿长度方向最初的非均匀分布进行被动衰减时存在的几个时间常数。除了在均匀膜电位衰减中观察到的众所周知的被动膜时间常数τ(m)=R(m)C(m)之外，还存在许多较小的时间常数，它们控制着膜电位沿长度方向的快速均衡。这些时间常数也存在于在一个位置（如神经元胞体）跨膜施加电流阶跃时的瞬态响应中。当一个集中的胞体与代表一个或多个树突树的一个或多个圆柱体耦合时，会导出类似的时间常数。当在一个位置施加电压钳时，会导出不同的时间常数；还导出了漏电和短路终端的影响。所有这些时间常数都作为数学边值问题的结果得到了证明。这些结果不仅为估计电紧张长度L = [未知量]/λ提供了基础，还为估计圆柱体输入电导与胞体膜电导的稳态比率ρ提供了新的基础。
