Stochastic Models of Neuronal Growth

TL;DR

The paper addresses how neuronal growth, particularly growth cone trajectories, can be predicted in noisy microenvironments by linking intracellular regulation to macroscopic motion. It uses drift–diffusion formalisms via Langevin dynamics and Fokker–Planck equations, coupled to a minimal mechanochemical actin–myosin–clutch model, to connect cytoskeletal dynamics and adhesions to ensemble trajectory statistics. Key contributions include explicit FP descriptions for position, speed, and angle; a mechanical beam model to infer bending modulus; and a linear stability analysis showing transitions among steady extension, damped oscillations, and limit-cycle behavior as feedback gains vary. The results offer practical design rules for engineered substrates and neuroprosthetic scaffolds to enhance nerve repair, with measurable parameters that enable hypothesis testing and parameter inference from trajectory data.

Abstract

Neuronal circuits arise as axons and dendrites extend, navigate, and connect to target cells. Axonal growth, in particular, integrates deterministic guidance from substrate mechanics and geometry with stochastic fluctuations generated by signaling, molecular detection, cytoskeletal assembly, and growth cone dynamics. A comprehensive quantitative description of this process remains incomplete. We review stochastic models in which Langevin dynamics and the associates Fokker-Planck equation capture axonal motion and turning under combined biases and noise. Paired with experiments, these models yield key parameters, including effective diffusion (motility) coefficients, speed and angle distributions, mean-square displacement, and mechanical measures of cell-substrate coupling, thereby linking single-cell biophysics and intercellular interactions to collective growth statistics and network formation. We further couple the Fokker-Planck description to a mechanochemical actin-myosin-clutch model and perform a linear stability analysis of the resulting dynamics. Routh--Hurwitz criteria identify regimes of steady extension, damped oscillations, and Hopf bifurcations that generate sustained limit cycles. Together, these results clarify the mechanisms that govern axonal guidance and connectivity and inform the design of engineered substrates and neuroprosthetic scaffolds aimed at enhancing nerve repair and regeneration.

Figures (4)

• Figure 1: : (a) Fluorescence image showing examples of axonal growth for cortical neurons cultured on a poly-D-lysine (PDL) coated surface with periodic micropatterns. Cortical neurons typically grow a long process (axon) and several minor processes (dendrites). Axonal growth is directed by the growth cone. The axons is identified by its morphology and the growth cone is identified as the tip of the axon. The angular coordinate $\theta$ used in this paper is also defined in this figure. The figure inset shows the parallel and perpendicular components of the acceleration in the reference frame used in the paper (see main text). (b) Schematic representation of the growth cone structure, illustrating the major cytoskeletal components: actin filaments and microtubules. The actin filaments are mechanically linked to the growth substrate through point contacts formed by transmembrane cell adhesion molecules, such as integrins and cadherins. The interaction among integrins, adhesion proteins, actin filaments and microtubules generates traction forces that promote the advancement of the growth cone.
• Figure 2: (a) Example of normalized experimental angular distributions for axonal growth for neurons cultured on micropatterned surfaces. The vertical axis (labeled Normalized Frequency) represents the ratio between the number of axonal segments growing in a given direction and the total number N of axon segments. Each axonal segment is of 10 $\mu m$ in length. The plot shows the angular distribution for $N$ = 510 different axon segments (total of 67 axons). The data shows that the axons display strong directional alignment along the surface patterns (peaks at $\theta = \pi/2$ and $\theta = 3\pi/2$. The continuous red curve represents a fit to the data with the Fokker-Planck model discussed in the text (b) Normalized speed distributions measured for the growth cones of the axons shown in (a). Data points were collected at $t = 32$ hr after neuron plating.
• Figure 3: Phase portraits of the reduced mechanochemical actin--myosin--clutch model illustrating the transition from stable to oscillatory to unstable regimes as the feedback parameter $\mu$ is varied. Here $x$ denotes the dimensionless actin-driven protrusion (tip) velocity and $y$ the normalized density of point-contact adhesions (integrins bound to the substrate), while $\mu$ represents the effective strength of feedback coupling polymerization, contractility, and adhesion reinforcement (defined in the main text). Trajectories evolve in the $(x,y)$ phase plane from initial conditions on a circle of radius 1.2. (a) For $\mu=-1$ the system exhibits stable growth (focus), with trajectories spiraling inward toward a fixed point, representing steady extension with stabilized adhesions. (b) At $\mu=0$ a Hopf bifurcation occurs and trajectories approach a closed orbit, corresponding to sustained oscillations of growth speed and clutch engagement. (c) For $\mu=+1$ the origin becomes unstable and trajectories spiral outward, reflecting runaway protrusion due to dominant polymerization or weakened clutch coupling. These regimes correspond to experimentally observed transitions between steady extension, oscillatory behavior, and instability in axonal tip dynamics Betz2006Vensi2019Descoteaux2022Kumarasinghe2022
• Figure 4: Time evolution of the dimensionless growth cone velocity $v_g(t)$ in the reduced mechanochemical model for different regimes of the control parameter $\mu = \operatorname{Tr}(\mathcal{J})$. The growth cone velocity $v_g(t)$ represents actin-driven protrusion modulated by nonlinear feedback between myosin contraction, adhesion-mediated resistance, and polymerization dynamics. Each panel corresponds to a different value of $\mu$, illustrating characteristic behaviors: (a) monotonic decay to a steady state for $\mu = -1.5$ (stable node); (b) damped oscillations for $\mu = -0.3$ (stable spiral); (c) sustained periodic oscillations (limit cycle) for $\mu = 0.05$; (d) exponential divergence for $\mu = 1.2$ (unstable node).