Epilepsy is a neurological disorder marked by recurring and spontaneous seizures, which involve abnormal high-frequency electrical activity in the brain. In this state, the dynamics of transmembrane potential are characterized by fast and sharp wavefronts that travel along different brain regions according to preferential axonal directions. For their mathematical description, we employ the monodomain model coupled with specific ionic models to consider the ion concentration dynamics. This multiscale coupled problem is extremely challenging as it involves small spatial (and temporal) scales to capture and approximate seizure accurately. To face these challenges and the geometric complexity of the brain, we propose to discretize the mathematical model by employing a high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG). The flexibility of PolyDG methods allows a high-order accuracy and at reasonable computational costs. In this talk, we will present numerical simulations of an epileptic event on a two-dimensional heterogeneous squared domain discretized with a polygonal grid, considering isotropic grey matter and anisotropic white matter. We then present a simulation on a two-dimensional brainstem in a sagittal plane with an agglomerated polygonal grid that fully exploits the geometric flexibility of the PolyDG approximation of the semidiscrete formulation. Finally, under simplifying assumption, we provide a theoretical stability analysis and a priori error estimates.