Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway.
Centre for Integrative Genetics, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway.
PLoS Comput Biol. 2013;9(12):e1003386. doi: 10.1371/journal.pcbi.1003386. Epub 2013 Dec 19.
The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K(+)-concentration to increase by several millimolars. The clearance of this excess K(+) depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K(+)-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K(+), (ii) suppresses extracellular transport of K(+), (iii) increases axial transport of K(+) within astrocytes, and (iv) facilitates astrocytic relase of K(+) in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K(+).
电缆方程是建模电神经信号的适当框架，适用于离子浓度变化不大的时间尺度。然而，在神经组织中，也存在关键的动态过程，这些过程发生在更长的时间尺度上。例如，持续的强烈神经信号可能导致局部细胞外 K(+)浓度增加几个毫摩尔。清除这种多余的 K(+)部分取决于细胞外空间的扩散，部分取决于星形胶质细胞的局部摄取，部分取决于星形胶质细胞内的细胞内转运（空间缓冲）。这些过程发生在秒的时间尺度上，需要一种数学描述，能够解释离子浓度的时空变化，以及这些变化对膜电位的后续影响。在这里，我们提出了一种用于一维几何模型中离子浓度动力学的一般电扩散形式，包括细胞内和细胞外区域。基于能斯特-普朗克方程，这种形式确保了膜电位和离子浓度的一致性，确保了全局粒子/电荷守恒，并考虑了扩散和浓度相关的电阻率变化。我们将该形式应用于与细胞外空间交换离子的星形胶质细胞模型。模拟结果表明，高浓度区域的 K(+)清除是由星形胶质细胞膜的局部去极化驱动的，这共同（i）增加了星形胶质细胞对 K(+)的局部摄取，（ii）抑制了 K(+)的细胞外转运，（iii）增加了星形胶质细胞内 K(+)的轴向转运，以及（iv）促进了细胞外浓度低的区域中 K(+)的星形胶质细胞释放。这些机制似乎为屏蔽细胞外空间免受多余 K(+)的影响提供了一种稳健的调节方案。
