After a long break (we went to Hamburg for New Year's Day: it was cold and rainy but the Kunsthalle and the Hafenrundfahrt have been fantastic) I'm back on math.
Delio Mugnolo and me just uploaded on arXiv our last work on parabolic systems.
If you are analysing a diffusion equation on a graph, you usually define a measure space representing the network, and then define a scalar-valued diffusion on this network. Of course, you could take the symmetric way and choose to use a single interval, and to represent your diffusion equation as a 'diagonal', vector-valued diffusion. The graph structure is in both cases encoded in the non-diagonal coupling of the boundary conditions.
If you use this point of view, you maybe come up with the idea of use exactly the same approach on domains in order to obtain vector-valued diffusion equations with non-diagonal couplings in the boundary conditions.
This is exactly what Delio and me do in the paper: we study such systems, also because we are interested in understanding the connections of the these systems to gauge symmetries - if there are some.
PS: of course, you can also introduce coupling in the coefficients of the diffusion, and then it is no longer a 'diagonal' diffusion, but I only wanted to write a post, not a full article...
PPS: probably, in two or three week I also will upload on arXiv my first neuro-paper!