Traveling wave-like phenomena characterize various biological phenomena, such as electrical impulses in the nervous system and cardiac tissue. Accurately simulating these dynamics is numerically challenging because it requires high spatial and temporal resolution, resulting in high computational costs. At the tissue scale, neuronal activity can be described by the monodomain equation coupled with appropriate ionic models that account for underlying membrane dynamics and ion-channel mechanisms. The resulting system exhibits steep transmembrane potential variations that propagate along preferential directions in anisotropic media, giving rise to traveling wavefronts with multiscale temporal and spatial features. Among the different models suitable for neuronal representation, we consider a conductance-based ionic model that captures both the transmembrane potential dynamics and the evolution of multiple ionic concentrations. The resulting system exhibits steep transmembrane potential variations that propagate along preferential directions in anisotropic media, giving rise to traveling wavefronts with multiscale temporal and spatial features. Adaptive high-order discontinuous Galerkin methods on general polygonal and polyhedral meshes provide the accuracy and flexibility needed to capture these phenomena. Exploiting a suitable p-adaptive strategy based on an a posteriori error indicator enables locally adjusting the polynomial degree, thereby focusing computational resources where they are most needed. Numerical results show that this approach can accurately capture neuronal wave propagation while significantly reducing the total number of degrees of freedom and the overall computational effort.