Understanding the mechanisms of neuronal development is vital for elucidating the intricate processes that govern the nervous system. In a neuron’s final form, it is composed of a long axon and multiple dendrites. Going out from a state with multiple undifferentiated neurites, the growth to this final shape is not entirely understood yet. While many models have been developed in an attempt to explain this process on a molecular level, exploring the mathematics of free boundary problems may offer a promising approach to capture some parts of the dynamics of neurite growth.

A new paper in the European Journal of Applied Mathematics, “A Free Boundary Model for Transport-Induced Neurite Growth” by Greta Marino, Jan-Frederik Pietschmann, and Max Winkler, proposes a novel mathematical framework to describe how vesicle transport contributes to neurite elongation and retraction. The authors couple a system of drift-diffusion equations for vesicle movement within neurites with systems of ordinary differential equations to capture interactions at the soma and growth cones.

Central to the model is the concept that neurite length dynamically adjusts based on vesicle concentration at the growth cones. Vesicles, which are transported anterogradely (toward the growth cone) and retrogradely (toward the soma), are modelled using nonlinear transport terms that account for the finite size of the vesicles, an extension to earlier models. Coupling the lengths of neurites to the respective vesicle concentrations renders the proposed system a free boundary problem.

A major theoretical contribution of the study is the proof of existence and uniqueness of weak solutions to the proposed model. The authors further discuss the existence of constant stationary solutions.

Numerically, a finite volume numerical scheme is developed, preserving key constraints such as vesicle density limits. Simulations reveal biologically relevant behaviours, including cycles of neurite extension and retraction. This is consistent with experimental observations of neurite dynamics during neuronal development. The model also highlights the interplay between vesicle production at the soma and consumption at growth cones, demonstrating how these processes balance to regulate neurite growth.

This work lays a robust foundation for future studies to explore additional factors, such as microtubule dynamics, which were deliberately excluded here to maintain model simplicity. By doing so, it provides a stepping stone for understanding the complex mechanisms underlying neuronal polarization and maturation.

‘A free boundary model for transport-induced neurite growth‘, by Greta Marino, Jan-Frederik Pietschmann and Max Winkler. This open access paper is part of Parabolic Equations and Systems in European Journal of Applied Mathematics. Thanks to open access, this paper is free to read, download, and share.

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