In this paper, we investigate the dynamics of a neuron–glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro–scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single–cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters.

中文翻译：

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神经胶质细胞系统的动力学：癫痫发作的发生和电惊厥刺激的影响：数学和数值研究。

在本文中，我们研究了神经元-神经胶质细胞系统的动力学以及癫痫发作的潜在机制。为了进行细胞模型的数学和数值研究，我们将使用分叉分析和一些计算方法。事实证明，贮库中钾浓度的升高是癫痫发作的一个诱因，并且与环面分叉有关。此外，我们将通过考虑简化模型来研究该模型的钾动力学，并且我们将展示两种机理如何相互联系。此外，钾泄漏电流的降低也将引起癫痫发作。我们的研究将表明，细胞外钾浓度的增加会影响钾电流的能斯特势，这可能导致癫痫发作。此外，我们将显示外部强迫项（例如，电击为单向矩形脉冲，也称为电抽搐治疗）会导致癫痫发作，类似于非强迫系统，其细胞外钾浓度增加。为此，我们将单向矩形脉冲描述为常微分方程的自治系统。这些方法将解释癫痫发作在细胞模型中的出现。此外，癫痫发作通过脑电图（EEG）进行测量，并在宏观范围内扩散（我们将单向矩形脉冲描述为常微分方程的自治系统。这些方法将解释癫痫发作在细胞模型中的出现。此外，癫痫发作通过脑电图（EEG）进行测量，并在宏观范围内扩散（我们将单向矩形脉冲描述为常微分方程的自治系统。这些方法将解释癫痫发作在细胞模型中的出现。此外，癫痫发作通过脑电图（EEG）进行测量，并在宏观范围内扩散（厘米）。因此，我们用合适的均质化单域模型扩展了细胞模型，提出了一组（数值）实验来补充对单细胞模型进行的分叉分析。基于这些实验，我们引入了一个双域模型来对大脑的白和灰质进行更真实的建模。对单域模型执行类似（数字）实验会导致对两种模型进行适当的比较。单个细胞模型及其癫痫发作以环分叉来解释，直接扩展到单域模型和双域模型中的相应结果，其中对于生理学相关的参数，神经放电几乎像快速传播的波一样在整个域中同步传播。