Wan F Y, Tuckwell H C
Biol Cybern. 1979 Jun 29;33(1):39-55. doi: 10.1007/BF00337416.
The depolarization of passive nerve cylinder or dendritic tree in the equivalent cylinder representation is assumed to satisfy the cable equation. We consider in detail the effects of white noise current injection at a given location for the case of sealed end boundary conditions and for an initial resting state. The depolarization at a point is a Gaussian random process but is not Markovian. Expression (infinite series) are obtained for the expectation, variance, spatial and temporal covariances of the depolarization. We examine the steady state expectation and variance and investigate how these are approached in time over the whole neuronal surface. We consider the relative contributions of various terms in the series for the expectation and variance of the depolarization at x = 0 (soma, trigger zone, recording electrode) for various positions of the input process. It is found that different numbers of terms must be taken to obtain a reasonable approximation depending on whether the stimulus is at proximal, central or distal parts of the dendritic tree. We consider briefly the interspike time problem and see in an approximate way how spatial effects are important in determining the mean time between impulses.
在等效圆柱体表示中，假定被动神经纤维或树突状树突的去极化满足电缆方程。对于密封端边界条件和初始静息状态的情况，我们详细考虑了在给定位置注入白噪声电流的影响。某一点的去极化是一个高斯随机过程，但不是马尔可夫过程。我们得到了去极化的期望、方差、空间和时间协方差的表达式（无穷级数）。我们研究了稳态期望和方差，并研究了在整个神经元表面上这些量是如何随时间趋近的。对于输入过程的不同位置，我们考虑了在x = 0（胞体、触发区、记录电极）处去极化的期望和方差的级数中各项的相对贡献。结果发现，根据刺激是在树突状树突的近端、中部还是远端，必须采用不同数量的项才能获得合理的近似值。我们简要考虑了峰间期问题，并以近似方式了解空间效应在确定脉冲之间的平均时间方面是如何重要的。
