Epilepsy is a clinical neurological disorder characterized by recurrent and spontaneous seizures consisting of abnormal high-frequency electrical activity in the brain. In this condition, the transmembrane potential dynamics are characterized by rapid and sharp wavefronts traveling along the heterogeneous and anisotropic conduction pathways of the brain. This work employs the monodomain model, coupled with specific neuronal ionic models characterizing ion concentration dynamics, to mathematically describe brain tissue electrophysiology in grey and white matter at the organ scale. This multiscale model is discretized in space with the high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and advanced in time with a Crank-Nicolson scheme. This ensures, on the one hand, efficient and accurate simulations of the high-frequency electrical activity that is responsible for epileptic seizure and, on the other hand, keeps reasonably low the computational costs by a suitable combination of high-order approximations and agglomerated polytopal meshes. We numerically investigate synthetic test cases on a two-dimensional heterogeneous squared domain discretized with a polygonal grid, and on a two-dimensional brainstem in a sagittal plane with an agglomerated polygonal grid that takes full advantage of the flexibility of the PolyDG approximation of the semidiscrete formulation. Finally, we provide a theoretical analysis of stability and an a-priori convergence analysis for a simplified mathematical problem.

翻译：癫痫是一种临床神经系统疾病，其特征为由脑内异常高频电活动引起的反复自发性发作。在该疾病中，跨膜电位动力学表现为沿脑组织异质各向异性传导通路传播的快速锐利波前。本研究采用单域模型，并结合表征离子浓度动力学的特定神经元离子模型，在器官尺度上数学描述灰质和白质的脑组织电生理特性。该多尺度模型在空间上采用高阶间断伽辽金方法（PolyDG）于多边形及多面体网格上进行离散，在时间上采用Crank-Nicolson格式推进。这一方法一方面能对引发癫痫发作的高频电活动进行高效精确模拟，另一方面通过高阶近似与聚合多面体网格的合理组合，有效控制计算成本。我们在二维异质方形域（使用多边形网格离散）及二维脑干矢状面（使用充分发挥半离散格式PolyDG近似灵活性的聚合多边形网格）上对合成测试案例进行了数值研究。最后，针对简化数学问题，我们给出了稳定性理论分析及先验收敛性分析。