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A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion
Serena Dipierro1,2, Enrico Valdinoci3,4,5,6
1Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133, Milan, Italy. serena.dipierro@unimi.it.
Bulletin of Mathematical Biology
|April 27, 2018
Summary
Fractional diffusion in neuronal signals may arise from ramified structures like Purkinje cells. This mathematical model suggests a biological advantage in smoothing signal transmission and preventing disruptions.
Area of Science:
- Neuroscience
- Mathematical Biology
- Physics
Background:
- Fractional diffusion observed in Purkinje cell neuronal transmission.
- Purkinje cell dysfunction linked to coordination issues and tremors.
- Parabolic equations exhibit smoothing, unlike hyperbolic equations with discontinuities.
Purpose of the Study:
- Investigate fractional diffusion in a model inspired by Purkinje cells.
- Explore the link between hyperbolic equations and fractional diffusion.
- Propose a novel mechanism for signal smoothing in biological systems.
Main Methods:
- Developed a toy model of a highly ramified structure.
- Utilized the superposition of traveling waves solving a hyperbolic equation.
- Computed the specific case of a traveling concave parabola.
Main Results:
- Demonstrated that superposition of traveling waves can produce fractional diffusion.
- Showed a connection between hyperbolic equations and fractional diffusion.
- Identified a potential evolutionary advantage for Purkinje cell ramification in signal smoothing.
Conclusions:
- Fractional diffusion may emerge from hyperbolic wave superposition in ramified media.
- High ramification in Purkinje cells could enhance signal transmission smoothing.
- This work offers a new perspective linking fractional diffusion to hyperbolic phenomena.
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