Mathematics and brain diseases: a (partial) overview and a recent model

In the first part of this talk I will try to give a (very partial) overview on some of the phenomena of interest in the area of brain diseases, on the kind of contributions mathematical modelling could give in this respect and on which mathematical instruments can be used when it comes to models. In the second part I will present a mathematical model for the onset and progression of Alzheimer’s disease. The synergistic interplay of proteins Amyloid-beta and tau is a subject of considerable interest when it comes to the study of Alzheimer’s disease. The model I will describe is based on transport and reaction-diffusion equations for the two proteins. In the model neurons are treated as nodes of a graph (the connectome) and structured by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid-beta, ii) effects of misfolded tau protein and iii) neuron-to- neuron prion-like transmission of the disease. These processes are modelled by a system of Smoluchowski equations for the Amyloid-beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. I will explain the structure of the model, give a hint of the main analytical results obtained and I will show the output of some numerical simulations