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

## TL;DR

This paper introduces a mathematical model inspired by Purkinje cells that explains fractional diffusion as a superposition of delayed traveling waves, suggesting an evolutionary advantage in signal smoothing through high ramification.

## Contribution

The paper presents a simple toy-model linking hyperbolic wave superposition to fractional diffusion, offering a new perspective on neuronal signal transmission and its mathematical representation.

## Key findings

- Fractional diffusion can arise from superimposing delayed hyperbolic waves.
- High ramification in Purkinje cells may provide an evolutionary advantage by smoothing signals.
- Explicit computation for a traveling concave parabola demonstrates the model's applicability.

## Abstract

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.