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Neurodegenerative diseases are closely associated with the amplification andinvasion of toxic proteins. In particular Alzheimer's disease is characterizedby the systematic progression of amyloid-$\beta$ and $\tau$-proteins in thebrain. These two protein families are coupled and it is believed that theirjoint presence greatly enhances the resulting damage. Here, we examine a classof coupled chemical kinetics models of healthy and toxic proteins in twospatial dimensions. The anisotropic diffusion expected to take place within thebrain along axonal pathways is factored in the models and produces afilamentary, predominantly one-dimensional transmission. Nevertheless, thepotential of the anisotropic models towards generating interactions takingadvantage of the two-dimensional landscape is showcased. Finally, a reductionof the models into a simpler family of generalizedFisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type systems is examined. It isseen that the latter captures well the qualitative propagation features,although it may somewhat underestimate the concentrations of the toxicproteins.

Anisotropic Diffusion and Traveling Waves of Toxic Proteins in Neurodegenerative Diseases