一般几何的电缆方程
Cable equation for general geometry
摘要：
电缆方程描述了圆柱形电缆中的电压，该模型已被用来模拟树突和轴突中的电势。然而，有时该方程可能会对某些实际几何形状给出不正确的预测，特别是当电缆的半径发生显著变化时。具有非恒定半径的缆线对于某些现象是重要的，例如沿着轴突的离散轴突出现在神经退行性疾病中，例如阿尔茨海默病、帕金森病、HIV相关的痴呆症和多发性硬化症。本文利用Frenet-Serret标架，提出了一种适用于一般索几何的广义索方程。这个广义方程取决于几何量，如电缆的曲率和扭矩。当索的截面为等圆截面时，第一基本形式可以简化，广义索方程既不依赖于曲率，也不依赖于挠率.此外，我们找到了一个精确的解决方案，具有特定的可变圆截面和零曲率的理想电缆。对于这种情况下，我们表明，当电缆的横截面增加的电压降低。在这种理想情况下的启发，我们重写的广义电缆方程作为一个扩散方程与电缆的几何形状产生的源项。该源项取决于电缆横截面积及其导数。此外，我们研究了不同的电缆膨胀，并提供了它们的数值解。数值解表明，当电缆截面发生突变时，其电压小于圆柱形电缆中的电压。此外，这些数值解表明，电缆上的几何不均匀性会影响电压。
英文摘要：
The cable equation describes the voltage in a straight cylindrical cable, this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a non constant radius are important for some phenomena, for example discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimer, Parkinson, HIV-associated dementia and Multiple Sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross-section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross-section and zero curvature. For this case we show that when the cross-section of the cable increases the voltage decreases. Inspired in this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-section area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross-section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.
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