Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133, Milan, Italy.
School University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia.
Bull Math Biol. 2018 Jul;80(7):1849-1870. doi: 10.1007/s11538-018-0437-z. Epub 2018 Apr 25.
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.
最近，几项实验证明了在浦肯野细胞的神经元传递中存在分数扩散现象，已知其功能障碍与缺乏自愿协调和出现震颤有关。此外，一个经典的数学特征是（分数）抛物线方程具有平滑效应，与双曲线方程形成对比，双曲线方程通常表现出冲击波和不连续性。在本文中，我们展示了一个高度分支结构的简单玩具模型（在某种程度上受到浦肯野细胞的启发）如何通过解决双曲线方程的传播波的叠加产生分数扩散。这可能表明浦肯野细胞的高分支结构提供了一种进化优势，可以“平滑”信号的传输并避免冲击波的传播（代价是稍微减缓这种传输）。尽管实验证实这种进化优势的可能性远远超出了本文的目标，但我们认为，从高度分支的传输介质中延迟传播波的叠加中产生时间分数扩散的数学对应物是很有趣的。明确计算了具有足够小曲率的移动凹抛物线的情况。我们在分数扩散与双曲线方程之间提出的新联系也为通常在极限中与抛物线方程相关的时间分数扩散范式提供了新颖性。本文的编写方式既对生物学家也对数学家都有兴趣。为了实现这一目标，提供了所考虑对象的完整解释和详细的参考文献列表。
