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

摘要

Recently, several experiments have demonstrated the existence of fractionaldiffusion in the neuronal transmission occurring in 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) parabolicequations possess smoothing effects, in contrast with the case of hyperbolicequations, 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 fractionaldiffusion via the superposition of travelling waves that solve a hyperbolicequation. This could suggest that the high ramification of the Purkinje cells mighthave provided an evolutionary advantage of "smoothing" the transmission ofsignals and avoiding shock propagations (at the price of slowing a bit suchtransmission). Though an experimental confirmation of the possibility of suchevolutionary advantage goes well beyond the goals of this paper, we think thatis intriguing, as a mathematical counterpart, to consider the time fractionaldiffusion as arising from the superposition of delayed travelling waves inhighly ramified transmission media. The new link that we propose between time fractional diffusion and hyperbolicequation also provides a novelty with respect to the usual paradigm relatingtime fractional diffusion with parabolic equations in the limit. The paper is written in such a way to be usable by both the communities ofbiologists and mathematicians: to this aim, full explanations of the objectconsidered and detailed lists of references are provided.