A low-dimensional dynamical basis for tongue shape control

I. INTRODUCTION

• The tongue’s many degrees of freedom motivate a low-dimensional account of how speakers reliably control speech movements.The paper identifies several possible control bases, including tract variables, privileged articulatory dimensions, and statistical modes.Source: §I · pp. 3–4

A. The dimensionality of vocal tract movements

• Prior work suggests that tongue behavior has low-dimensional structure, but shape modes have seldom served as coordinates for a control law.Empirical vocal-tract modes capture major components of articulatory variance, while remaining kinematic rather than neuromuscular.Source: A. The dimensionality of vocal tract… · pp. 5, 7

B. A dynamical model for tongue shape dynamics

• The proposed control basis uses holistic tongue-shape modes, treating tract and articulatory variables as projections from that space.The authors present this as an alternative coordinate system, not as a rejection of the task–articulator distinction.Source: B. A dynamical model for tongue shape… · p. 9
• Combining independent linear modes can make a one-dimensional projection appear nonlinear, even when each mode follows a linear law.For the illustrated mixture y = x1 + x2, the linear model fit has R2 = 0.683; the authors connect this to projection between coordinate systems [Mori, 1965; Zwanzig, 1961].Source: B. A dynamical model for tongue shape… · p. 10, Figure 2 · p. 11
• Mode mixing alone cannot generate amplitude-dependent kinematics, so such patterns would require genuine nonlinearity such as nonlinear coupling.The authors’ narrower claim is that apparent departures from second-order linear dynamics in one-dimensional variables can arise from projection.Source: B. A dynamical model for tongue shape… · pp. 10–12

C. This study

• The study tests whether a linear dynamical model of tongue-shape variables captures vowel lingual dynamics.The analysis focuses on vowel formation and release into a coronal consonant; more complex movements such as diphthongs are proposed as future extensions.Source: C. This study · p. 12

A. Data

• The dataset comprises short monophthongs from the TarDiS corpus, recorded with midsagittal ultrasound and repeated 4–6 times per speaker.The selected vowels are bid, bed, bad, bud, and bod, in two carrier phrases.Source: A. Data · p. 12
• Long vowels and diphthongs are outside this study’s short-vowel focus because they typically require two gestural activation intervals and a more complex piecewise model.The authors identify them as prospects for future analysis.Source: A. Data · pp. 12–13
• Analysis covers the full C1VC2 interval, with C1-to-V midpoint marking vowel formation and V midpoint-to-C2 marking release into /d/.Four speakers were excluded, leaving 36 speakers and 1,687 tokens for analysis.Source: A. Data · p. 13

B. Tongue shape representation

• The first three PCs of GPA-aligned contours form a shared tongue-shape state space, represent 75.1% of cumulative variance, and retain 93% of by-speaker PCA variance on average.The 93% figure ranges from 0.59 to 0.96 across speakers.Source: B. Tongue shape representation · p. 14

C. Dynamical model fitting

• Each of the three principal components is modeled as an independent damped oscillator, fitted directly through its analytic solution rather than numerical integration.The fitted parameters include a target, damping ratio, and natural frequency for each component.Source: C. Dynamical model fitting · p. 16

D. Fitting models to windowed segments

• Splitting each [C1VC2] window at the vowel midpoint gives low closure and release RMSE, and fits are relatively insensitive to split points from 40% to 60% of vowel duration.The midpoint is used as a proxy for the vowel target in short monophthongs.Source: D. Fitting models to windowed segments · p. 17

E. Parameter estimation and evaluation

• 0.00–0.14 mm model-versus-3-PC RMSE, compared with 0.73–1.52 mm model-versus-measured RMSE, shows the model nearly reaches the three-PC accuracy ceiling.The residual disagreement with raw contours reflects variance lost in PCA, not model-fit error.Source: E. Parameter estimation and evaluation · p. 18
• Model fit is scored on position only because PC-trajectory velocity estimates are noisy and smoothing them could bias the observed structure.Position trajectories alone are relatively insensitive to the order of the dynamics that generated them.Source: E. Parameter estimation and evaluation · p. 18

III. VALIDATION EXPERIMENT

• Synthetic trajectories span underdamped, critically damped, and overdamped regimes, with frequencies and window durations chosen to cover the empirical range.The simulated durations span 80–300 ms; the empirical median is 207 ms, with an interquartile range of 171–256 ms.Source: A. Method · p. 20
• The simulations add position noise and pass trajectories through the empirical smoothing pipeline before fitting the analytical oscillator model.Position noise has a standard deviation of 2.5% of each component’s range.Source: A. Method · p. 20
• Parameter recovery is compared using true versus empirically estimated initial velocity, and measured as signed error in natural frequency, damping ratio, and target.Figure 5 displays recovery error by window duration and damping ratio.Source: A. Method · pp. 20–21, Figure 5 · p. 21

B. Results and summary

• Estimated initial velocity degrades synthetic parameter recovery, with up to ∼25% error for ω and ∼16% inaccuracy for ζ, while x∗ remains close to ground truth within empirical durations.The estimated-velocity condition systematically underestimates ω.Source: B. Results and summary · p. 22
• Because real data are noisier and more variable than the synthetic data, the authors treat empirical damping and frequency estimates as potentially moderately to significantly inaccurate.They address this by testing parameter constraints and treating estimates as unreliable in an under-determined system.Source: B. Results and summary · p. 22

B. Model fit and reconstruction accuracy

• The unconstrained model reaches RMSE of 0.177 mm for closure and 0.286 mm for release, but only 23/36 speakers have sepV > sepD, with median sepV = 0.57 and sepD = 0.51.Both separability medians are below the threshold of 1.Source: B. Model fit and reconstruction accuracy · p. 23
• Critical damping alone lowers the speakers with sepV > sepD from 23/36 to 19/36 for PC1 and 18/36 for PC1–3.It also increases RMSE for both PC1 and PC1–3 in closure and release.Source: B. Model fit and reconstruction accuracy · pp. 23–24
• Pinned endpoint targets yield sepV = 5.14 and sepD = 0.52, with sepV > sepD for all 36 speakers; RMSE rises by +0.053 mm for closure and +0.121 mm for release.The target is fixed at each segmented window’s endpoint rather than estimated by the model.Source: B. Model fit and reconstruction accuracy · p. 24
• Adding critical damping to pinned targets more than doubles RMSE—by 102% for closure and 126% for release—while leaving only one fitted ω per PC alongside fixed targets.The authors frame this as a trade-off between model fit and parameter interpretability, not a search for one true model.Source: B. Model fit and reconstruction accuracy · pp. 24–25

C. Target parameter

• Pinned endpoint targets show excellent vowel-category separation in PC1, smaller vowel-specific differences in PC2, and no systematic vowel pattern in PC3; release targets cluster together.The endpoint targets reflect the data and segmentation rather than model optimization.Source: C. Target parameter · p. 25, Figure 6 · p. 26
• All 36 speakers show significant closure-phase vowel separability (median sepV = 5.14), versus 31% in release (median sepD = 0.52); unconstrained fits exceed the null for 56% in closure and 22% in release.The permutation test reassigns vowel labels within each speaker and phase 1,000 times.Source: C. Target parameter · p. 26
• For speaker MCR14, tokens form a near-canonical vowel quadrilateral with little overlap, while release targets cluster toward the top left, suggesting a front-vowel-like /d/ target.The release target is often higher and more anterior.Source: C. Target parameter · pp. 26–27, Figure 7 · p. 28

D. Frequency and damping ratio

• Pinned-model closure dynamics are near-critically damped (ζ = 0.68–0.81), whereas release is heavily underdamped (ζ = 0.27–0.57).Release may be more underdamped because acoustic segmentation truncates the true /d/ target, with the fit compensating through a more extreme virtual target.Source: D. Frequency and damping ratio · p. 27
• Damping estimates are sensitive to smoothing-window length, although the qualitative pattern is fairly robust; the authors therefore treat frequency and damping as properties of model fits, not true physical quantities.Closure RMSE decreases from 0.255 to 0.136 mm across the tested 5- to 9-sample windows, while release damping remains heavily underdamped.Source: D. Frequency and damping ratio · pp. 28–29

E. Summary of results

• A linear damped point attractor provides an excellent fit to tongue-shape dynamics, but phonetically interpretable targets require positions specified from segmented-window endpoints.The fitted estimates indicate near-critical damping for vowel formation and heavy underdamping for release.Source: E. Summary of results · p. 30

A. Shape modes as a potential basis for lingual control

• Holistic tongue shape modes are proposed as control variables, with modal variables combining linearly to generate observed tongue-shape dynamics.The model is entirely linear in control space.Source: A. Shape modes as a potential basis… · p. 30
• The proposed framing offers a cross-speaker coordinate space and explains how projecting tongue shape into one-dimensional articulatory measures can yield nonlinear artifacts.These are described as theoretical advantages of the shape-mode approach.Source: A. Shape modes as a potential basis… · p. 30
• The evaluation fits tongue positions only; velocity and acceleration predictions were not scored because time-varying principal components are significantly noisier than EMA or flesh-point trajectories.The authors note that higher-order derivatives provide stronger signatures of (non)linearity.Source: A. Shape modes as a potential basis… · p. 31

B. From tongue shape modes to muscle synergies?

• Empirical tongue shape modes are proposed as a possible kinematic expression of muscle synergies, but the relationship remains speculative.Muscle synergies are described as co-activating muscle modules that combine to generate functional behavior.Source: B. From tongue shape modes to muscle… · p. 31
• Principal components were chosen pragmatically and are not claimed to represent the empirical correlates of muscle synergies.The authors note that synergy dimensionality from muscle models can exceed that derived from data, so the appropriate modal basis requires further work.Source: B. From tongue shape modes to muscle… · p. 32
• The shape modes indirectly encode jaw information, but do not show whether the jaw is independently controlled.Source: B. From tongue shape modes to muscle… · p. 32

C. Limitations and future work

• A linear model's sufficiency for shape dynamics does not establish that low-dimensional control dynamics are truly linear.A single tongue point in these vowels may not reveal significant nonlinearity, while the model cannot capture nonlinear dependence between movement amplitude and time-to-peak velocity.Source: C. Limitations and future work · p. 33
• Allowing modes to couple could place additional temporal complexity in the dynamical law, but the added parameters create severe identifiability problems with the present data.Off-diagonal damping terms depend on velocity, which the authors could not reliably estimate from time-varying principal components.Source: C. Limitations and future work · pp. 33–34
• The analysis models only lax vowels, so highly dynamic movements are not captured, and the damped linear point attractor is not claimed to be optimal.Short piecewise movements also leave insufficient signal to disambiguate different model forms.Source: C. Limitations and future work · p. 34
• Generalized Procrustes Analysis standardizes rotation and displacement to create a cross-speaker coordinate system, but the resulting modes describe shape-space variation rather than specific constriction degrees.Source: C. Limitations and future work · p. 34
• Future work will examine relationships between tongue shape modes and re-derived movement and constriction variables.Source: C. Limitations and future work · p. 34

VI. CONCLUSION

• Low-dimensional empirical variables serve as the control basis for a dynamical model that sufficiently captures tongue-shape dynamics in English monophthongal vowels.Source: §VI · p. 35
• Tract and articulatory variables are recast as projections from tongue-shape space to other coordinate spaces.Source: §VI · p. 35
• Apparent nonlinearities in one-dimensional articulatory measures can arise as projection artifacts rather than properties of the underlying control system.Source: §VI · p. 35

• The authors declare no conflicts of interest.Source: AUTHOR DECLARATIONS · p. 35
• Ethical approval came from the University of Lancaster Ethics Committee (ref. no. FL18188) and the University of Manchester’s Proportionate University Research Ethics Committee.The supplied passage cuts off before giving the Manchester reference number.Source: AUTHOR DECLARATIONS · p. 35

AUTHOR DECLARATIONS

• The supplied passages are reference-list entries, with no author-declaration content.They include citations to works by Strycharczuk and colleagues, Takano and Honda, and others.Source: AUTHOR DECLARATIONS · pp. 41–43