Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which provides a general mathematical modelling framework of spatio-temporal voltage dynamics. Typically such models need to be solved numerically unless the cell membrane is modelled either by passive or quasi-active dynamics, in which cases analytical solutions can be reduced to calculation of the Green’s function describing the fundamental input-output relationship in a given morphology. Such analytically tractable models often assume individual dendritic segments to be cylinders. However, it is known that dendritic segments in many types of neurons taper, i.e. their radii decline from proximal to distal ends. Here we consider a generalised form of cable theory which takes into account both branching and tapering structures of dendritic trees. We demonstrate that analytical solutions can be found in compact algebraic forms in an arbitrary branching neuron with a class of tapering dendrites studied earlier in the context of single neuronal cables by Poznanski (Bull. Math. Biol. 53(3):457–467, 1991). We apply this extended framework to a number of simplified neuronal models and contrast their output dynamics in the presence of tapering versus cylindrical segments.

中文翻译：

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具有锥形树突的分支神经元中电缆方程的精确解。


神经元是具有其他细胞类型中不存在的独特复杂树突形态的生物细胞。具有分支树突的神经元中的电信号可以通过电缆理论来研究，该理论提供了时空电压动力学的通用数学建模框架。通常，此类模型需要进行数值求解，除非细胞膜是通过被动或准主动动力学建模的，在这种情况下，解析解可以简化为描述给定形态中基本输入输出关系的格林函数的计算。这种分析上易于处理的模型通常假设各个树突片段是圆柱体。然而，众所周知，许多类型的神经元中的树突段逐渐变细，即它们的半径从近端到远端逐渐减小。在这里，我们考虑电缆理论的广义形式，该理论考虑了树突树的分支结构和锥形结构。我们证明，可以在具有一类锥形树突的任意分支神经元中找到紧凑代数形式的解析解，Poznanski 之前在单神经元电缆的背景下研究过它（Bull. Math. Biol. 53(3):457–467， 1991）。我们将这个扩展框架应用于许多简化的神经元模型，并在存在锥形段和圆柱形段的情况下对比它们的输出动态。