A stochastic walker model is proposed to describe the chemotactic guidance of growth cones, which are the tips of developing neurites. The model accounts for the influence of both attractive and repulsive chemical cues, which are emitted by the growth cones and the somas. The system couples stochastic differential equations governing the motion of the growth cones with reaction-diffusion equations that describe the dynamics of the chemical concentrations. The existence of a unique solution to this coupled system is proved. Numerical experiments are performed to investigate the sensitivity of the model to key biological parameters. The impact of the nonlocal regularization of point sources in the reaction-diffusion equations is analyzed in a simplified deterministic setting.
| [1] |
Evans LPartial Differential Equations, 1998, Providence: American Mathematical Society
|
| [2] |
Flandoli F, Gubinelli M, Priola EFlow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bull. Sci. Math., 2010, 134: 405-422.
|
| [3] |
Fujii M, Okada I, Yanagida EIsolated singularities in the heat equation behaving like fractional Brownian motions. J. Math. Anal. Appl., 2021, 504125322
|
| [4] |
Gafarov FSelf-wiring in neural nets of point-like cortical neurons fails to reproduce cytoarchitectural differences. J. Integr. Neurosci., 2006, 5: 159-169.
|
| [5] |
Grubb GRegularity in $L_p$ Sobolev spaces of solutions to fractional heat equations. J. Funct. Anal., 2018, 274: 2634-2660.
|
| [6] |
Hentschel H, van Ooyen AModels of axon guidance and bundling during development. Proc. R. Soc. Lond. B, 1999, 266: 2231-2238.
|
| [7] |
Mortimer D, Fothergill T, Pujic Z, Richards L, Goodhill GGrowth cone chemotaxis. Trends Neurosci., 2008, 31: 90-98.
|
| [8] |
Nasser, T., Spencer, G.: Neurite outgrowth. In: Reference Module in Biomedical Sciences. Elsevier, Amsterdam (2017). https://doi.org/10.1016/B978-0-12-801238-3.99507-2
|
| [9] |
Oliveri H, Goriely AMathematical models of neuronal growth. Biomech. Model. Mechan., 2022, 21: 89-118.
|
| [10] |
Pearson Y, Castronovo E, Lindsley T, Drew DMathematical modeling of axonal formation. Part I: geometry. Bull. Math. Biol., 2011, 73: 2837-2864.
|
| [11] |
Protter PStochastic Integration and Differential Equations, 2005, Berlin: Springer
|
| [12] |
Segev R, Ben-Jacob EGeneric modeling of chemotactic based self-wiring of neural networks. Neural Netw., 2000, 13: 185-199.
|
| [13] |
Takahashi, J., Yanagida, E.: Removability of time-dependent singularities in the heat equation. arXiv:1304.5147 (2013)
|
| [14] |
Xu, J.-X., Deng, X.: Biological neural network based chemotaxis behaviors modeling of C. elegans. In: The International Joint Conference on Neural Networks, pp. 1–8. Barcelona, Spain (2010)
|
| [15] |
Yurchenko I, Farwell M, Brady D, Staii CNeuronal growth and formation of neuron networks on directional surfaces. Biomimetics, 2021, 641.
|
Funding
FWF(F65)
H2020 European Research Council(101018153)
Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie(MultHeFlo)
Rights & permissions
The Author(s)
PDF
