The electrophysiological behavior of the brain is a highly complex, multiscale phenomenon. It is the result of interactions between electrical and chemical processes that link the activity of neuron's membrane to the functioning of the whole organ. This complexity is further amplified when examining pathological conditions, such as epilepsy. In this context, numerical simulation represents a valuable tool for enhancing the understanding of pathophysiological mechanisms, providing access to high-resolution reconstructions of spatiotemporal physical quantities. However, to ensure the accuracy required to capture all the relevant scales, an extremely large number of degrees of freedom are necessary for characterizing the numerical solutions on the intricate brain geometry, resulting in a significant increase in the computational burden of the modeling and simulation approach. To improve the accuracy/efficiency balance of standard numerical methods, we numerically approximate multiscale differential models of brain electrophysiology using a high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG). During the talk, we will demonstrate how PolyDG schemes are tailored to investigate the role of functional and structural heterogeneities in the onset and prolongation of seizure episodes. Specifically, we will show how spatially distributed heterogeneities of ion channels' properties, anisotropy and heterogeneity in conduction and geometry complexity can contribute to the abnormal transmembrane potential behavior that is observed during seizures.