Traveling wave-like phenomena characterize various biological phenomena, such as electrical impulses in the nervous system and cardiac tissue. Numerical simulation of these events presents multiple challenges. For example, to capture rapid and sharp wavefronts, propagating numerical schemes must have a sufficiently high resolution in space and time, resulting in high computational costs. From this perspective, the electrophysiology of the brain at the tissue level is one pivotal example: the dynamics of the transmembrane potential are characterized by steep and fast wavefronts propagating through different brain regions along preferential axonal directions. Modeling brain electrophysiology requires integrating multiple scales and dynamics in complex domains, as rapid ion concentration fluctuations give rise to electrical signals in anisotropic tissues. From a numerical perspective, the computational costs of exploiting a sufficient degree of accuracy are extremely high. Employing the high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) is still computationally intensive. To further improve its performance, we develop a p-adaptive algorithm that takes advantage of the traveling wave-like structure of the solution: the transmembrane potential undergoes rapid variations in a small tissue region while the remaining domain is stationary. Specifically, we investigate how to construct efficient a-posteriori error indicators to identify the wavefront accurately. We devise an algorithm to automatically and locally adjust the polynomial degree, fully leveraging the discretization provided by the discontinuous Galerkin method. This talk will discuss numerical results to verify our approach numerically and showcase the capability of simulating epileptic events in heterogeneous domains, such as grey and white matter, in which polynomial adaptivity plays a crucial role in reducing the total number of degrees of freedom and the computational costs of the simulation. Our numerical results demonstrate that our approach is capable of maintaining high-order accuracy while significantly reducing computational efforts.