Rall W, Rinzel J
Biophys J. 1973 Jul;13(7):648-87. doi: 10.1016/S0006-3495(73)86014-X.
Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.
当仅向一个树突分支注入稳定电流时，给出了广泛分支神经元模型中膜电位稳态分布的数学解和数值示例。得到了分支输入位点处的输入电阻以及从输入位点到胞体的电压衰减的显式表达式；还给出了交流稳态输入阻抗和衰减的表达式。理论模型假设膜具有被动特性以及对分支直径的等效圆柱约束。数值示例说明了分支输入电阻和稳态衰减如何取决于以下因素：树突树的数量、树突分支的阶数、树突树的电紧张长度、树突输入位点的位置以及胞体处的输入电阻。讨论了该模型在猫脊髓运动神经元以及其他神经元类型中的应用。说明了并讨论了大的树突输入电阻对突触位点处局部膜去极化量以及到达胞体的去极化量的影响；一般来说，与输入电阻的简单比例关系并不成立。此外，分支输入电阻超过胞体处输入电阻的量总是小于从输入位点到胞体路径上的核心电阻之和。
