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Recently, several experiments have demonstrated the existence of fractionaldiffusion in the neuronal transmission occurringin the Purkinje cells, whosemalfunctioning is known to be related to the lack of voluntary coordination andthe appearance of tremors. Also, a classical mathematical feature is that(fractional) parabolic equations possess smoothing effects, in contrast withthe case of hyperbolic equations, which typically exhibit shocks anddiscontinuities. In this paper, we show how a simple toy-model of a highlyramified structure, somehow inspired by that of the Purkinje cells, may producea fractional diffusion via the superposition of travelling waves that solve ahyperbolic equation. This could suggest that the high ramification of thePurkinje cells might have provided an evolutionary advantage of "smoothing" thetransmission of signals and avoiding shock propagations (at the price ofslowing a bit such transmission). Although an experimental confirmation of thepossibility of such evolutionary advantage goes well beyond the goals of thispaper, we think that it is intriguing, as a mathematical counterpart, toconsider the time fractional diffusion as arising from the superposition ofdelayed travelling waves in highly ramified transmission media. The case of atravelling concave parabola with sufficiently small curvature is explicitlycomputed. The new link that we propose between time fractional diffusion andhyperbolic equation also provides a novelty with respect to the usual paradigmrelating time fractional diffusion with parabolic equations in the limit. Thispaper is written in such a way as to be of interest to both biologists andmathematician alike. In order to accomplish this aim, both completeexplanations of the objects considered and detailed lists of references areprovided.

A simple mathematical model inspired by the Purkinje cells: from delayed travelling waves to fractional diffusion