  Recently, several experiments have demonstrated the existence of fractional
diffusion in the neuronal transmission occurringin the Purkinje cells, whose
malfunctioning is known to be related to the lack of voluntary coordination and
the appearance of tremors. Also, a classical mathematical feature is that
(fractional) parabolic equations possess smoothing effects, in contrast with
the case of hyperbolic equations, which typically exhibit shocks and
discontinuities. In this paper, we show how a simple toy-model of a highly
ramified structure, somehow inspired by that of the Purkinje cells, may produce
a fractional diffusion via the superposition of travelling waves that solve a
hyperbolic equation. This could suggest that the high ramification of the
Purkinje cells might have provided an evolutionary advantage of "smoothing" the
transmission of signals and avoiding shock propagations (at the price of
slowing a bit such transmission). Although an experimental confirmation of the
possibility of such evolutionary advantage goes well beyond the goals of this
paper, we think that it is intriguing, as a mathematical counterpart, to
consider the time fractional diffusion as arising from the superposition of
delayed travelling waves in highly ramified transmission media. The case of a
travelling concave parabola with sufficiently small curvature is explicitly
computed. The new link that we propose between time fractional diffusion and
hyperbolic equation also provides a novelty with respect to the usual paradigm
relating time fractional diffusion with parabolic equations in the limit. This
paper is written in such a way as to be of interest to both biologists and
mathematician alike. In order to accomplish this aim, both complete
explanations of the objects considered and detailed lists of references are
provided.
