  Neurodegenerative diseases are closely associated with the amplification and
invasion of toxic proteins. In particular Alzheimer's disease is characterized
by the systematic progression of amyloid-$\beta$ and $\tau$-proteins in the
brain. These two protein families are coupled and it is believed that their
joint presence greatly enhances the resulting damage. Here, we examine a class
of coupled chemical kinetics models of healthy and toxic proteins in two
spatial dimensions. The anisotropic diffusion expected to take place within the
brain along axonal pathways is factored in the models and produces a
filamentary, predominantly one-dimensional transmission. Nevertheless, the
potential of the anisotropic models towards generating interactions taking
advantage of the two-dimensional landscape is showcased. Finally, a reduction
of the models into a simpler family of generalized
Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type systems is examined. It is
seen that the latter captures well the qualitative propagation features,
although it may somewhat underestimate the concentrations of the toxic
proteins.
