Scaling laws for nonlinear dynamical models of articulatory control

TL;DR

This paper addresses how to parameterize nonlinear restoring forces in nonlinear task-dynamic models of speech articulation, where a cubic term $-d(x - T)^3$ enriching the linear model $m \ddot{x} + b \dot{x} + k(x - T) = 0$ introduces nontrivial distance-dependent effects on velocity profiles. It proposes two scaling laws—local inverse-square normalization $d' = \frac{d k}{|x_0 - T|^2}$ and global scaling with $D$, $\lambda$, and $\alpha' = \frac{\lambda \alpha k}{|x_0 - T|^{n-1}}$—to render nonlinearity interpretable and numerically stable across trajectories. The main contributions are (i) formalizing these scaling laws, (ii) demonstrating how they preserve or reintroduce distance-velocity relationships in simulations of the cubic model, and (iii) linking the scaling to anatomic-motoric and cognitive constraints. The work advances dynamical phonological theory by providing simple, physically grounded constraints that improve cross-tract simulations and robustness of parameter estimation for articulatory control models.

Abstract

Dynamical theories of speech use computational models of articulatory control to generate quantitative predictions and advance understanding of speech dynamics. The addition of a nonlinear restoring force to task dynamic models is a significant improvement over linear models, but nonlinearity introduces challenges with parameterization and interpretability. We illustrate these problems through numerical simulations and introduce solutions in the form of scaling laws. We apply the scaling laws to a cubic model and show how they facilitate interpretable simulations of articulatory dynamics, and can be theoretically interpreted as imposing physical and cognitive constraints on models of speech movement dynamics.

Figures (4)

• Figure 1: TOP LEFT: Stiffness functions of the linear, cubic and summed restoring forces, where $k = 1$ and $F$ refers to the forces specified in the legend as a function of $x$. TOP RIGHT: A comparison of position and velocity trajectories generated by the linear ($d = 0$) and nonlinear ($d = 0.95$) models, where $x_{0} = 1, \dot{x}_{0} = 1, T = 0, k = 2000$. BOTTOM LEFT: Power function of $k$ against time-to-peak velocity (top) and peak velocity (bottom). BOTTOM RIGHT: Power function of the natural logarithms of $k$ against time-to-peak velocity (top) and peak velocity (bottom).
• Figure 2: TOP LEFT: Simulated position and velocity trajectories, with $x_{0} = 1$, $\dot{x}_{0} = 0$, $k = 2000$, $T = 0.0$ with varying values of $d$; and TOP RIGHT: The same simulations but across varying values of $T$, where $d = 0.95k, k = 2000$. BOTTOM LEFT: Nonlinear restoring force $-kx + dx^3$ ($k = 1$) for values of $d$ corresponding to top left plot, where $F$ refers to the forces specified in the legend as a function of $x$. BOTTOM RIGHT: The restoring forces for $d = 0.95k$ over the range $[-10, 10]$ without scaling.
• Figure 3: TOP LEFT: The relationship between distance-to-target $|x_{0}-T|$ and $d$ follows an inverse square law. TOP RIGHT: The inverse square law allows for appropriate scaling of larger movement distances, with $x_{0} = 10$, $T = 0$, $k = 2000$ across varying values of $d$. BOTTOM LEFT: The restoring forces for $d = 0.95k$ over the range $[-10, 10]$ without scaling. BOTTOM RIGHT: The restoring forces for $d = 0.95k$ over the range $[-10, 10]$ scaled by an inverse square law.
• Figure 4: TOP LEFT: Cubic model with scaling across different targets in the range $[0, 0.8]$ using an inverse square law. TOP RIGHT: Forces corresponding to the scaled cubic model in top left. BOTTOM LEFT: Cubic model with parameter-range scaling across different targets in the range $[0, 8]$. BOTTOM RIGHT: Cubic model with restricted parameter-range scaling to allow nonlinearity to only operate when $|x_{0}-T| < 8$.