School of Biological and Health Systems Engineering, Arizona State University, 501 E Tyler Mall, Tempe, AZ, 85287-9709, USA.
J Math Neurosci. 2016 Dec;6(1):9. doi: 10.1186/s13408-016-0041-1. Epub 2016 Sep 9.
Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.
这里呈现的是一个在导电介质中受外部注入电流刺激的神经组织模型。组织被描述为一个具有各向同性导电性的双域，即由占据相同空间的细胞内和细胞外区域以及分隔它们的膜组成，并且注入电流被描述为一对源和汇点。该问题在三个空间维度上进行求解，并在球坐标系中定义 [公式：见正文]。通过将问题转换为膜和单域问题来求解耦合的偏微分方程组，该单域被解释为细胞内和细胞外区域的加权平均值。膜和单域分别由标量亥姆霍兹和拉普拉斯方程定义，它们在球坐标系中都是可分离的。因此，假设并给出了乘积解，通过某些超越函数。从这些电势中，推导出电流密度的解析表达式，并且从这些场中计算出磁通密度。考虑了一些示例，其中间质电导率发生变化，以及由于方位角独立性，问题简化为二维的极限情况。最后，讨论了未来的建模工作。
