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

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Abstract

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.

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Notes

  1. As a matter of fact, from the biology perspective, it also makes sense of looking at signal travelling from the right to the left in a medium as the one described in Fig. 1. In this case, the left end of Fig. 1 would act as a “soma” (the cell body of a neuron), which is the site of final integration of the signals and ultimately is responsible to generate action potentials to send to downstream neurons. In our mathematical deduction, this situation can be also taken into account (what counts is just the average delayed obtained by a signal travelling from one end of the transmission medium to the other).

  2. Other models with different structures may also be taken into account to address anomalous diffusion in related contexts. For instance, it is interesting to investigate whether some sort of hyperbolic superposition with delay can also be applied to cases that do not need the presence of multiple branches. In particular, in Santamaria et al. (2011) the pyramidal cells in the hippocampus are studied, showing that the presence of dendritic spines causes anomalous diffusion: indeed, these small protrusions force diffusing molecules to undergo a continuous random walk with random waiting times that result in anomalous diffusion.

  3. For the sake of simplicity, here we take into account equations in the simplest possible form: in particular, we are not explicitly considering forcing terms in the equation (which can be anyway added in a more general analysis). Also, in order to develop formal expansions, we assume \(f_o\) to be smooth, with a smoothness independent of the structural parameters N, L and c. The advantage of the scaling in the variable of \(f_o\) is that its dependence on the spatial variables is weighted by the natural scale of the problem, thus \(f_o\) is a function of “adimensional” coordinates.

  4. We observe that, with respect to the parameters c and L in (1), it is possible to choose \(\delta \) small, without making the equation in (1) degenerate. For instance, if one takes \(c\sim L^{\frac{s-2}{s}}\), then the coefficients in (1) do not degenerate and \(\delta \) is small if L is large.

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Acknowledgements

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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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Serena Dipierro & Enrico Valdinoci

  2. School University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Serena Dipierro & Enrico Valdinoci

  3. School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, VIC, 3010, Australia

    Enrico Valdinoci

  4. Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100, Pavia, Italy

    Enrico Valdinoci

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  1. Serena Dipierro
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Correspondence to Serena Dipierro.

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Dipierro, S., Valdinoci, E. A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion. Bull Math Biol 80, 1849–1870 (2018). https://doi.org/10.1007/s11538-018-0437-z

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