Neurodegenerative diseases have a significant global impact affecting millions of individuals worldwide. Some of them, known as proteinopathies, are characterized by the accumulation and propagation of toxic proteins, known as prions. Alzheimer's and Parkinson's diseases are relevant of protheinopathies. Mathematical models of prion dynamics play a crucial role in understanding disease progression and could be of help to potential interventions. This article focuses on the heterodimer model: a system of two partial differential equations that describe the evolution of healthy and misfolded proteins. In particular, we propose a space discretization based on a Discontinuous Galerkin method on polygonal/polyhedral grids, which provides flexibility in handling meshes of complex brain geometries. Concerning the semi-discrete formulation we prove stability and a-priori error estimates. Next, we adopt a $\vartheta$-method scheme for time discretization. Some convergence tests are performed to confirm the theoretical bounds and the ability of the method to approximate travelling wave solutions. The proposed scheme is also tested to simulate the spread of $\alpha$-synuclein in a realistic test case of Parkinson's disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

翻译：神经退行性疾病对全球数百万人产生显著影响。其中被称为蛋白质病的疾病以毒性蛋白（即朊病毒）的积累和传播为特征。阿尔茨海默病和帕金森病属于蛋白质病的范畴。朊病毒动力学的数学模型在理解疾病进展中起着关键作用，并可能有助于潜在干预措施的开发。本文聚焦于异二聚体模型：一个由两个偏微分方程组成的系统，描述健康蛋白和错误折叠蛋白的演化过程。具体而言，我们提出了一种基于多边形/多面体网格的不连续伽辽金方法的空间离散化方案，该方法在处理复杂脑部几何结构的网格时具有灵活性。针对半离散格式，我们证明了其稳定性和先验误差估计。随后采用θ-方法进行时间离散化。通过相关收敛性测试验证了理论界的可靠性及该方法逼近行波解的能力。所提出的方案还应用于模拟真实帕金森病案例中α-突触核蛋白的扩散——该案例基于医学影像重建的二维大脑矢状切面几何结构。