López-Sánchez Erick J, Romero Juan M
Posgrado en Ciencias Naturales e Ingeniería, Universidad Autónoma Metropolitana, Cuajimalpa and Vasco de Quiroga 4871, Santa Fe Cuajimalpa, Ciudad de México 05300, México.
Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana-Cuajimalpa, Vasco de Quiroga 4871, Santa Fe Cuajimalpa, Ciudad de México 05300, México.
Phys Rev E. 2017 Feb;95(2-1):022403. doi: 10.1103/PhysRevE.95.022403. Epub 2017 Feb 13.
The cable equation describes the voltage in a straight cylindrical cable, and 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 nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus 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 by 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-sectional 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.
电缆方程描述了直圆柱形电缆中的电压，该模型已被用于对树突和轴突中的电势进行建模。然而，有时这个方程对于某些实际几何形状可能会给出错误的预测，特别是当电缆半径变化显著时。具有非恒定半径的电缆对于某些现象很重要，例如，在阿尔茨海默病、帕金森病、人类免疫缺陷病毒相关痴呆症和多发性硬化症等神经退行性疾病中，轴突上会出现离散的肿胀。在本文中，我们使用弗伦内 - 塞雷标架，为一般电缆几何形状提出了一个广义电缆方程。这个广义方程取决于电缆的曲率和挠率等几何量。我们表明，当电缆具有恒定的圆形横截面时，电缆的第一基本形式可以简化，广义电缆方程既不取决于电缆的曲率也不取决于挠率。此外，我们找到了具有特定可变圆形横截面和零曲率的理想电缆的精确解。对于这种情况，我们表明当电缆的横截面积增加时，电压会降低。受此理想情况的启发，我们将广义电缆方程重写为一个带有由电缆几何形状产生的源项的扩散方程。这个源项取决于电缆横截面积及其导数。此外，我们研究了不同的肿胀电缆并提供了它们的数值解。数值解表明，当电缆的横截面有突变时，其电压低于圆柱形电缆中的电压。此外，这些数值解表明电压会受到电缆上几何不均匀性的影响。
