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

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in 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). Though an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that 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 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. The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.