  Neurocognitive disorders, such as Alzheimer's and Parkinson's, have a wide
social impact. These proteinopathies involve misfolded proteins accumulating
into neurotoxic aggregates. Mathematical and computational models describing
the prion-like dynamics offer an analytical basis to study the diseases'
evolution and a computational framework for exploring potential therapies. This
work focuses on the heterodimer model in a three-dimensional setting, a
reactive-diffusive system of nonlinear partial differential equations
describing the evolution of both healthy and misfolded proteins. We investigate
traveling wave solutions and diffusion-driven instabilities as a mechanism of
neurotoxic pattern formation. For the considered mathematical model, we propose
a space discretization, relying on the Discontinuous Galerkin method on
polytopal/polyhedral grids, allowing high-order accuracy and flexible handling
of the complicated brain's geometry. Further, we present a priori error
estimates for the semi-discrete formulation and we perform convergence tests to
verify the theoretical results. Finally, we conduct simulations using realistic
data on a three-dimensional brain mesh reconstructed from medical images.
