Chemotaxis guidance of random walkers modeling self-wiring of neural networks

TL;DR

The work addresses chemotaxis-guided neural wiring by coupling stochastic growth-cone motion with reaction–diffusion dynamics for chemical cues, using mollified point sources to ensure well-posedness. It proves a unique, strong-in-SDE and classical-in-PDE solution for the coupled system and develops a Galerkin–Euler–Maruyama numerical scheme in 2D. An analysis of the mollifier limit shows that nonlocal regularization is essential to avoid trivial dynamics as $\varepsilon\to 0$, informing the modeling choice. The study combines rigorous analysis with simulations to explore how diffusion, emission, and interaction strengths shape neurite trajectories and network wiring.

Abstract

A stochastic walker model is proposed to describe the chemotactic guidance of growth cones, i.e. 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.

Figures (6)

• Figure 1: Coefficient functions $a_{1j}^{(1)}$, $a_{3j}^{(1)}$ (left) and $b_{1j}^{(1)}$, $b_{2j}^{(1)}$ (right).
• Figure 2: Simulation of 9 somas and 9 neurites (left) and 27 neurites (right). The somas are distributed on a grid with random variation in their positions.
• Figure 3: Simulation of six neurites and six somas with diffusion coefficient $\sigma=0.05$ (left) and $\sigma=0.2$ (right).
• Figure 4: Simulation of two neurites and two somas using $\sigma=0.1$ and $\varepsilon=0.005$ (left), $\varepsilon=0.02$ (middle), and $\varepsilon=0.04$ (right).
• Figure 5: Four neurites and four somas using $\sigma=0.2$, $\varepsilon=0.01$, $T=5$, and $(\alpha,\beta)=(5,10)$ (top left), $(\alpha,\beta)=(20,10)$ (top right), $(\alpha,\beta)=(15,5)$ (bottom left), $(\alpha,\beta)=(15,20)$ (bottom right).

Theorems & Definitions (15)

• Theorem 1: Existence and uniqueness
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof