Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1849–1870 | Cite as

A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion

  • Serena Dipierro
  • Enrico Valdinoci
Original Article


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). 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.


Mathematical models Deduction from basic principles Time fractional equations Wave equations Purkinje cells Dendrites Neuronal arbours 

Mathematics Subject Classification

35Q92 92B05 92C20 



It is a pleasure to thank Elena Saftenku and Fidel Santamaria for very interesting discussions on neural transmissions. We also thank the Referees for their very valuable comments. This work has been supported by the Australian Research Council Discovery Project “N.E.W. Nonlocal Equations at Work”.


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli studi di MilanoMilanItaly
  2. 2.School University of Western AustraliaCrawleyAustralia
  3. 3.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  4. 4.Istituto di Matematica Applicata e Tecnologie InformaticheConsiglio Nazionale delle RicerchePaviaItaly

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