arXiv:2511.10369v2 Announce Type: replace Abstract: The association of epileptic activity and Alzheimer's disease (AD) has been increasingly reported in both clinical and experimental studies, suggesting that amyloid-$\beta$ accumulation may directly affect neuronal excitability. Capturing these interactions requires a quantitative description that bridges the molecular alterations of AD with the fast electrophysiological dynamics of epilepsy. We introduce a novel mathematical model that extends the Barreto-Cressman ionic formulation by incorporating multiple mechanisms of calcium dysregulation induced by amyloid-$\beta$, including formation of $\mathrm{Ca}^{2+}$-permeable pores, overactivation of voltage-gated $\mathrm{Ca}^{2+}$ channels, and suppression of $\mathrm{Ca}^{2+}$-sensitive potassium currents. The resulting ionic model is coupled with the monodomain equation and discretized using a $p$-adaptive discontinuous Galerkin method on polytopal meshes, providing an effective balance between efficiency and accuracy in capturing the sharp spatiotemporal electrical wavefronts associated with epileptiform discharges. Numerical simulations performed on idealized and realistic brain geometries demonstrate that progressive amyloid-\textbeta{} accumulation leads to severe alterations in calcium homeostasis, increased neuronal hyperexcitability, and pathological seizure propagation. Specifically, high amyloid-$\beta$ concentrations produce secondary epileptogenic sources and spatially heterogeneous wavefronts, indicating that biochemical inhomogeneities play a critical role in shaping seizure dynamics. These results illustrate how multiscale modeling provides new mechanistic insights into the interplay between neurodegeneration and epilepsy in Alzheimer's disease.