Holmes W R
Biol Cybern. 1986;55(2-3):115-24. doi: 10.1007/BF00341927.
Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment delta t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.
使用电缆方程的模型在神经生理学分析中越来越多地被采用，但对于许多目的和许多实验室来说，其实现所需的计算机时间和内存量过大。本文提出了一种数学方法，用于确定已知几何形状的被动树突状树中对注入电流或突触输入的瞬态电压响应，该方法易于实现，因为它基于一个方程。当将结果与满足等效圆柱约束的树突状树的解析结果进行比较时，该方法被证明具有很高的准确性。在该方法中，被动电缆方程用于表示树突状树每个分支间段的电位。在分支点和末端应用边界条件后，通过检查树突状树可以很容易地得到段末端电位的拉普拉斯变换方程组。除了起始方程外，所有方程都具有简单的形式，仅随在分支点相遇的分支数量而变化。该方程组在拉普拉斯域中求解，并将结果数值反演回每个指定时间点的时域（该方法与任何时间增量Δt无关）。使用该方法可以获得树突状树中任何选定位置的电位。由于每个分支间段只需要一个方程，因此该过程使用的方程比类似的房室方法少得多。通过使用比其他方法少得多的计算机内存和时间来达到相似的精度，该方法允许在小型计算机上快速进行广泛的分析。人们希望这将使更多的研究人员参与建模研究，并促进他们建立实际复杂树突模型的积极性。
