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A sub-Riemannian model of the motor cortex with Wasserstein distance

T0 review · 2 major / 4 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Horizontal curves in a sub-Riemannian model of motor cortex automatically satisfy the observed speed-curvature law, and Wasserstein clustering of fragments recovers experimental neural states.

desk verdict Solid incremental step: Wasserstein lets their prior sub-Riemannian fragment model finally work on real variable-duration M1 data and recover the Kadmon states, while the speed-curvature relation falls out of the horizontal distribution by construction. read the letter →

arxiv 2603.20756 v2 pith:6OO3BXMF submitted 2026-03-21 q-bio.NC

Jawad Ali , Giovanna Citti , Alessandro Sarti This is my paper · ORCID
classification q-bio.NC MSC 53C1792C2049Q22
keywords primarymotorcortexmovementfragmentssub-RiemanniangeometryWassersteindistanceneuralstatesspeed-curvaturelawneurogeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper models the primary motor cortex as a six-dimensional feature space of hand position, time, orientation, velocity and acceleration equipped with a sub-Riemannian structure. Its horizontal curves are forced by the differential constraints to obey the experimentally observed relation between trajectory curvature and movement speed. Short hand trajectories called fragments are turned into vector-valued probability measures; clustering them with the Wasserstein distance recovers the same grouping into neural states previously extracted from multi-electrode recordings, and does so without the artificial time reparametrization required by earlier Sobolev distances. A reader who accepts the model therefore obtains a single geometric account of both the kinematics of reaching and the hierarchical organization of motor primitives in cortex.

What carries the argument

The sub-Riemannian structure generated by the vector fields X1 = v cos θ ∂x + v sin θ ∂y + a ∂v + ∂t, X2 = ∂θ, X3 = ∂a, whose horizontal curves automatically obey R = v/α₂, together with the Wasserstein distance on the vector-valued densities μ = (sin θ, cos θ, a)/∫√(1+a²) that supplies the affinity matrix for spectral clustering into neural states.

What would settle it

Apply the same Wasserstein affinity matrix (same density definition and metric weights) to a new multi-electrode reaching dataset recorded under comparable conditions; if the resulting spectral clusters fail to match the experimentally reported neural states, the central claim is falsified.

Watch

Extended reading notes

Core claim

A sub-Riemannian geometry on the feature space of position, time, orientation, velocity and acceleration makes its horizontal curves satisfy the speed-curvature relation observed in reaching movements; the Wasserstein distance between the vector-valued measures associated with those curves then clusters real movement fragments into the same neural states reported from cortical recordings, outperforming the Sobolev distance that required all trajectories to share a common time interval.

Load-bearing premise

The model assumes that the particular vector-valued density built from orientation and acceleration, together with metric weights that nearly discard absolute time, correctly represents how motor-cortex cells compare movement fragments.

Editorial extensions

If this is right

  • The speed-curvature law need not be imposed by hand; it is a differential consequence of the chosen feature geometry.
  • Cortical connectivity among fragments can be modelled by a Wasserstein kernel that tolerates variable durations without reparametrization.
  • Spectral clustering of that kernel recovers the experimentally observed neural states from real reaching data.
  • The same geometric setting that organizes elementary features into fragments also organizes fragments into higher-order neural states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Wasserstein-on-horizontal-curves construction may apply to other cortical areas whose cells are selective to short spatiotemporal trajectories.
  • Metric weights that effectively discard absolute time imply a short-term shape memory that is shift-invariant but not scale-invariant in time.
  • Replacing hand-tuned coefficients with anatomically or data-derived weights would yield a fully parameter-free prediction of neural-state boundaries.
  • Population codes in motor cortex should be more sensitive to the joint distribution of orientation and acceleration than to absolute position or absolute time.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper models primary motor cortex (M1) features as the 6-dimensional manifold M = R^{2}_{(x,y)} imes R^{+}_t imes S^{1}_ heta imes R_v imes R_a equipped with the sub-Riemannian structure generated by the horizontal fields X1, X2, X3 of Eq. (3). Horizontal (admissible) curves are proposed as models of movement fragments; the differential constraints immediately yield the curvature–speed relation k = heta'/v (Eq. 8) and hence R = v/ heta' (Eq. 9), recovering the experimentally observed speed–curvature law (and the 2/3-power law by the special choice heta' = v^{-2}). Fragments are then represented by real- or vector-valued probability measures (Eqs. 13–14) and clustered by spectral analysis of a Gaussian kernel of the Wasserstein distance. On synthetic data the method recovers coherent groups (silhouette 0.65); on the real reaching trajectories of Kadmon-Harpaz et al. it recovers a partition visually comparable to the experimentally reported neural states (silhouette 0.35), outperforming the earlier Sobolev distance that required identical time domains.

Significance. If the geometric derivation and the clustering results hold, the work supplies a single sub-Riemannian framework that simultaneously (i) explains the classical speed–curvature relation from first principles and (ii) organizes experimentally recorded fragments into the same neural states previously obtained from multi-electrode recordings. The introduction of Wasserstein distance into cortical connectivity modelling is novel and removes the artificial re-parametrization required by Sobolev-type metrics. The paper therefore offers a concrete, falsifiable link between differential geometry, optimal transport and motor-cortex physiology that can be tested on further data sets and with alternative measures.

major comments (2)
  1. Section 4.3 and Definition 7 / Eq. (11): the recovery of the Kadmon-Harpaz neural states rests on the explicit choice c5 = 0.008 for the temporal weight, which the authors state “completely discard[s] the time variable,” followed by re-parametrization of every fragment to a common initial time. Once absolute timing is removed, the remaining features (orientation and acceleration profile) are precisely those already used by the experimental clustering. No sensitivity analysis or ablation with non-vanishing c5 (or with alternative normalizations of the measure μ) is provided; without it the claimed agreement with experiment cannot be regarded as an independent validation of the sub-Riemannian + Wasserstein model.
  2. Section 3.1.3, Eq. (14) and Section 4.3: the vector-valued density μ = (sin heta, cos heta, a)/∫√(1+a^{2}) is introduced without physiological or information-theoretic justification, and the kernel widths σ1 = 0.7, σ2 = 0.09 are hand-tuned. The silhouette scores (0.65 synthetic, 0.35 real) are reported, yet no quantitative comparison against alternative distances (e.g., dynamic time warping, plain L^{2} on the same features) or against shuffled baselines is given. Consequently the claim that Wasserstein distance is “much more efficient” remains only qualitative.
minor comments (4)
  1. Eq. (14) writes the components of μ as (sin heta, cos heta, a) while the surrounding text lists (cos heta, sin heta, a); the order should be made consistent.
  2. Figure 7 caption and the accompanying text refer to a “center-out task,” yet the data provenance (Hatsopoulos laboratory) is only acknowledged in the acknowledgments; a brief methods paragraph describing the recording and pre-processing pipeline would improve reproducibility.
  3. The parameters c1…c4 of the ball-box norm (Eq. 11) are said to be “tuned o experimentally observed clusterization,” but their concrete values are never listed; they should be reported for the real-data experiment.
  4. Typographical inconsistencies appear throughout (e.g., “Hatsoupolos,” “real-valued probability: speed and curvature,” missing spaces after commas in several equations).

Circularity Check

3 steps flagged · score 5.0 of 10

Neural-state recovery is obtained by tuning metric weights (esp. c5=0.008 discarding absolute time) and by defining the vector measure on precisely the orientation+acceleration features that already define the target clusters of Kadmon-Harpaz et al.

  1. fitted input called prediction [§3.1.2 (after Def. 7 / Eq. 11) and §4.3]
    "The parameters c1, ...,c4 in (11) can be arbitrary. In the sequel, they will be tuned in such a way to obtain the experimentally observed clusterization. ... we use a small coefficient c5=0.008 for the temporal increment e5 ... Setting the coefficient c5=0.008 means that we completely discard the time variable, which is consistent with the classification proposed in [19]"

    The free metric weights (especially the near-zero temporal weight that forces time-shift invariance and common reparametrization) are adjusted until the spectral clusters match the experimental neural states; the subsequent claim that the model “recovers” those states is therefore a fit, not an independent prediction.

  2. self definitional [§3.1.1 Eqs. (8)–(9)]
    "k= α2 / v ... R=1/α2 v ... Here, we claim that this relation is already encoded in our choice of variables. We can also recover the relation R=Cv3 by simply choosing α2=v−2."

    The experimentally observed speed-curvature relation is an immediate algebraic consequence of the differential constraints built into the generators X1,X2,X3; the stronger power-law form is obtained only by a free functional choice of the control α2. The geometry therefore does not derive the law but encodes it by definition.

1 more flagged steps
  1. fitted input called prediction [§3.1.3 Def. of μ(γ) and §4.3 (affinity parameters)]
    "we consider only three components (cos(θ(s)),sin(θ(s)),a(s)). Then we define μ(γ(s))=(sin(θ(s)),cos(θ(s)),a(s))/∫√(1+a2(s))ds ... we set the parameters (σ1=0.7) and (σ2=0.09) to enhance the sensitivity of the similarity measure to both orientation and acceleration features."

    The vector measure is deliberately restricted to the two kinematic features (orientation and acceleration phase) that already define the experimental neural states; the kernel widths are then hand-chosen to emphasize exactly those features. Clustering on this pre-selected representation necessarily groups by the same criteria used in the target partition.

full rationale

The paper has two main claims. The speed-curvature relation follows immediately from the chosen generators of the horizontal distribution (k=α2/v), so it is encoded by construction rather than independently predicted; the 2/3-power law is recovered only by the free choice α2=v^{-2}. The more load-bearing empirical claim—that Wasserstein clustering recovers the experimental neural states—is obtained after the free coefficients of the ball-box metric are explicitly tuned “to obtain the experimentally observed clusterization,” with c5 set to 0.008 so that absolute time is discarded and every fragment is reparametrized to a common start (exactly the normalization already used by the experimental paper). The vector-valued density is likewise defined on (sin θ, cos θ, a), the two features that already label the experimental states. Once these choices are made, spectral clustering on the resulting affinity matrix reproduces the target partition; the match is therefore partly engineered by free parameters rather than a parameter-free prediction of the sub-Riemannian+Wasserstein model. The geometry itself and the superiority of Wasserstein over Sobolev on variable-length trajectories retain independent content, so the circularity is only partial (score 5).

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The central claims rest on the sub-Riemannian feature space and admissible-curve model imported from the authors’ prior work, on a hand-chosen vector-valued density for Wasserstein, and on several free metric and kernel parameters tuned to match the target experimental partition. No new physical entity is postulated, but the connectivity kernel is an ad-hoc modeling choice.

free parameters (3)
  • metric weights c_i (esp. c5=0.008) = c5=0.008, others=1
    Coefficients in the ball-box norm on M1; c5 is set by hand so that absolute time is effectively discarded, matching the visualization convention of Kadmon et al.
  • kernel widths σ1,σ2 = σ1=0.7, σ2=0.09
    Gaussian affinity parameters in the spectral clustering step; chosen to emphasize orientation and acceleration features.
  • α2 control function
    Free function that sets curvature; special choice α2=v^{-2} recovers the classical R∝v^3 law.
assumptions (3)
  • domain assumption Motor-cortex cells are selective to the six features (x,y,t,θ,v,a) and the admissible curves are integral curves of the given horizontal distribution with the specific α3 form that fits measured acceleration profiles.
    Imported from the authors’ earlier papers [25,26] and from Hatsopoulos/Kadmon experimental literature; not re-derived here.
  • standard math The Hörmander condition holds for the chosen generators, so a Carnot-Carathéodory distance exists and can be estimated by the ball-box metric.
    Standard sub-Riemannian fact used to define d_M1.
  • ad hoc to paper Cortical connectivity in the space of fragments is well-modeled by a Gaussian kernel of the Wasserstein distance on the chosen vector-valued density.
    Postulated in Section 3.1.3; no independent physiological measurement of the kernel is supplied.
invented entities (1)
  • vector-valued density μ=(sinθ,cosθ,a)/∫√(1+a²) on fragments
    purpose: Turns each trajectory into a signed measure so that Wasserstein distance can be applied without time reparametrization.
    Defined ad hoc in Eq. (14); no external evidence that cortex computes this particular density.

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Pith. "Pith review of A sub-Riemannian model of the motor cortex with Wasserstein distance." pith.science (2026). https://pith.science/paper/6OO3BXMF

@misc{pith2026260320756,
  author       = {Pith},
  title        = {Pith review of: A sub-Riemannian model of the motor cortex with Wasserstein distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OO3BXMF}},
  note         = {Machine review of arXiv:2603.20756}
}
read the original abstract

This study aims to better understand the functional geometry of the motor cortex, starting from different sources of experimental evidence. Recent studies have proved that cells of the primary motor cortex (M1) are sensitive to short hand trajectories called fragments. Here, we propose a sub-Riemannian higher-dimensional geometry accounting for geometric and kinematic properties. Due to the constraints of the geometry, horizontal curves naturally satisfy a relation between geometric and kinematic properties experimentally observed. In the space of trajectories, we also apply a clustering algorithm based on the Wasserstein distance: we obtain a grouping which nicely fits the observed experimental data much more efficiently than the Sobolev distance.

Figures

Figures reproduced from arXiv: 2603.20756 by the authors.

Figure 1
Figure 1. The results of clusterization made by [19] group fragments in neural states: each one is a set of fragments with homogeneous orientation and increasing or decreasing acceleration phase. Within each column, we see the (x,y) section of a set of fragments (above) and the corresponding profile in the (t,v) plane (below). Mathematical models of this brain area were proposed by various authors, with the specific scope to … view at source ↗
Figure 2
Figure 2. Visualization of the grouping of fragments in neural states obtained in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A family of curves with different curvature (left) and its clustering with [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Trajectories with random initial positions, orientations, and velocity [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Wasserstein affinity matrix. Eight clusters were identified, exhibiting high intra-cluster similarity. Silhouette score: 0.65, indicating well-separated clusters. In [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Visualization of the grouping of fragments in neural states with Wasser [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Reaching path and speed profile of a center-out task. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Space of features decomposition into fragments. Each curve shows a [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Wasserstein affinity matrix. Indices have been re-ordered for visualization [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Grouping of fragments in neural states with Wasserstein distance. [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Visualization of the grouping of fragments in neural states with Wasser [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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