Coupled Spatio-Temporal Dynamics and Nonlocality in Advanced Mathematical Models for the Analysis of Complex Neurodegenerative Disease Pathologies

Abstract

One in six of the world's population has to deal with neurodegenerative disorders, and while medical devices exist to detect, prevent, and treat such disorders, some fundamentals of the progression of associated diseases remain ambiguous. In this contribution, we focus on Alzheimer's disease (AD), where amyloid-beta (Aβ) and tau proteins are among the main contributors to the development or propagation of AD. The Aβ proteins clump together to form plaques and disrupt cell functions. Moreover, the abnormal chemical change in the brain helps to build sticky tau tangles that block the neuron's transport system. Astrocytes generally maintain a healthy balance in the brain by clearing the Aβ toxic plaques. Even so, over-activated astrocytes release chemokines and cytokines and also react to pro-inflammatory cytokines, further increasing the production of Aβ. We have provided details of a novel coupled mathematical model that can capture astrocytes' dual behaviour, emphasizing the importance of spatio-temporal coupling and nonlocality. We have demonstrated that the disease propagation depends on memory effects, that is the disease's earlier status, which involves non-Markovian processes. We have explained how to integrate brain connectome data in the network model and to study this effect, as well as the dual role of astrocytes as a coupled phenomenon. Depending on toxic loads in the brain, we have also discussed details of the analysis of the neuronal damage in the brain. We have explained how the memory effect can slow down the propagation of toxic proteins in the brain, decreasing the rate of neuronal damage. Representative numerical examples have been given, and special attention has been paid to nonequilibrium considerations and stochastic modelling frameworks in the study of neurodegenerative diseases.