  Many neurodegenerative diseases are connected to the spreading of misfolded
prionic proteins. In this paper, we analyse the process of misfolding and
spreading of both $\alpha$-synuclein and Amyloid-$\beta$, related to
Parkinson's and Alzheimer's diseases, respectively. We introduce and analyze a
positivity-preserving numerical method for the discretization of the
Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic
proteins. The proposed approximation method is based on the discontinuous
Galerkin method on polygonal and polyhedral grids for space discretization and
on $\vartheta-$method time integration scheme. We prove the existence of the
discrete solution and a convergence result where the Implicit Euler scheme is
employed for time integration. We show that the proposed approach is
structure-preserving, in the sense that it guaranteed that the discrete
solution is non-negative, a feature that is of paramount importance in
practical application. The numerical verification of our numerical model is
performed both using a manufactured solution and considering wavefront
propagation in two-dimensional polygonal grids. Next, we present a simulation
of $\alpha$-synuclein spreading in a two-dimensional brain slice in the
sagittal plane. The polygonal mesh for this simulation is agglomerated
maintaining the distinction of white and grey matter, taking advantage of the
flexibility of PolyDG methods in the mesh construction. Finally, we simulate
the spreading of Amyloid-$\beta$ in a patient-specific setting by using a
three-dimensional geometry reconstructed from magnetic resonance images and an
initial condition reconstructed from positron emission tomography. Our
numerical simulations confirm that the proposed method is able to capture the
evolution of Parkinson's and Alzheimer's diseases.
