A Systematic Review of Differential Equation Models of Neurodegenerative Disease Dynamics: Methods, Architectures, and Future Research Directions

Abstract

Neurodegenerative diseases such as Alzheimer’s disease, Parkinson’s disease, and related disorders are characterized by progressive neuronal loss and complex spatiotemporal pathological processes. Differential equation-based mathematical modelling has emerged as a powerful framework for understanding disease progression by capturing interactions among biochemical processes, neural connectivity, and tissue-level dynamics. This review examines advances in ordinary differential equation, partial differential equation, and hybrid dynamical system models for neurodegenerative disease analysis. ODE-based models are widely used to represent intracellular mechanisms such as amyloid-beta aggregation, tau protein dynamics, and neuronal degradation, while PDE-based models effectively describe the spatial propagation of pathology across brain regions. Network diffusion models have further enhanced understanding by simulating disease spread along structural brain connectivity using graph-based approaches. Multiscale frameworks integrating molecular, cellular, and network-level processes have improved predictive accuracy and biological realism. Emerging techniques include fractional differential equations for anomalous diffusion, stochastic models for biological variability, and coupled systems for interacting disease mechanisms. Despite progress, challenges such as parameter estimation, model validation, and integration of heterogeneous biological data persist, highlighting the need for robust, data-driven, and personalized modelling approaches.