Posgrado en Ciencias Naturales e Ingeniería, Universidad Autónoma Metropolitana, Cuajimalpa and Vasco de Quiroga 4871, Santa Fe Cuajimalpa, Ciudad de México 05300, Mexico.
Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana-Cuajimalpa, Vasco de Quiroga 4871, Santa Fe Cuajimalpa, Ciudad de México 05300, Mexico.
Phys Rev E. 2017 Sep;96(3-1):032411. doi: 10.1103/PhysRevE.96.032411. Epub 2017 Sep 20.
Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.
不同的实验研究报告称，脑组织中存在异常扩散，特别是这种异常扩散通过分数导数来表示。轴突对于理解多发性硬化症、阿尔茨海默病和帕金森病等神经退行性疾病很重要。事实上，轴突中蛋白质和细胞器的异常积累是这些疾病的标志。由于这种异常，轴突中的扩散可能变得异常。在这种情况下，轴突中的电压传播会受到影响。不同神经退行性疾病的另一个标志是轴突上离散的肿胀。为了对具有异常扩散和肿胀的轴突中的电压传播进行建模，本文提出了一种用于一般几何形状的分数电缆方程。这个广义方程依赖于分数参数和几何量，如电缆的曲率和扭转。对于具有恒定半径的电缆，我们表明当分数效应增加时，电压会降低。在具有肿胀的电缆中，我们发现当分数效应或肿胀半径增加时，电压会降低。当肿胀的数量和分数效应增加时，会得到类似的行为。此外，我们发现当肿胀半径（或肿胀数量）和分数效应同时增加时，电压会急剧下降。
