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 $(r,\theta,\phi )$ . 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.

中文翻译：

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神经组织的解析模型：I.球形双域。

这里介绍的是外部注入电流刺激的导电介质中神经组织的模型。组织被描述为导电的各向同性双畴，即由占据相同空间的细胞内和细胞外区域以及将它们隔开的膜组成，注入电流被描述为一对源点和下沉点。该问题在三个空间维度上得到解决，并定义为球坐标$（r，\ theta，\ phi）$。耦合偏微分方程组的系统是通过将问题改写为膜和单畴来解决的，该问题被解释为胞内和胞外域的加权平均值。膜和单畴分别由标量Helmholtz和Laplace方程定义，它们在球坐标系中都是可分离的。因此，产品解决方案是通过某些先验功能来假定和给出的。从这些电势中，得出电流密度的解析表达式，并从这些场中计算出磁通密度。考虑了数值示例，其中间隙电导率是变化的，以及由于方位角独立性而使问题简化到二维的极限情况。最后，讨论了将来的建模工作。以及由于方位角独立性而使问题简化到二维的极限情况。最后，讨论了将来的建模工作。以及由于方位角独立性而使问题简化到二维的极限情况。最后，讨论了将来的建模工作。