Pith. sign in

REVIEW 5 major objections 5 minor 1 cited by

A sub-Riemannian model of neural states in the primary motor cortex

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The eight neural states observed in motor cortex can be recovered from kinematics alone.

desk verdict A genuinely new geometric construction in service of an overstated empirical claim: the paper never compares its clustering to the data it claims to recover. read the letter →

arxiv 2501.03247 v4 pith:LLOS7YN4 submitted 2024-12-22 q-bio.NC

Caterina Mazzetti , Jawad Ali , Alessandro Sarti , Giovanna Citti This is my paper · ORCID
classification q-bio.NC MSC 92C2053C17
keywords sub-Riemanniangeometrymotorcortexneuralstatesmovementfragmentsspectralclusteringconnectivitykernelmean-fieldequationkinematicfeatures
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 aims to show that the eight neural states observed in the primary motor cortex during arm reaching can be produced from purely kinematic and geometric principles, with no access to neural recordings. It represents each movement fragment—a short hand trajectory with nearly constant direction and a bell-shaped speed profile—as a curve in a space of time, direction, speed, and acceleration, and equips that space with a sub-Riemannian distance that ignores hand position. From this distance it builds a connectivity kernel, inserts it into a mean-field equation for cortical activity, and groups the kernel's leading eigenvectors. The resulting eight groups reproduce the neural states that an earlier experimental study [11] found with a hidden Markov model on measured neural activity, a grouping that study could not recover from kinematics alone. The paper reads this match as evidence that the chosen kinematic variables and distance capture the mechanism behind neural-state formation.

What carries the argument

The load-bearing mechanism is a pseudo-metric on the space of fragments, $$d_F(\gamma_1,\gamma_2)=\$int_0^{1}$\|\gamma_1'(t)-\gamma_2'(t)\|_{M_1}\,dt+d_{M_1}(\gamma_1(1),\gamma_2(1)),$$ built from a sub-Riemannian distance on a four-dimensional manifold that drops the position coordinates and factors locally as the product of the Heisenberg group and a circle; the distance is estimated through canonical coordinates using a standard local bound. This metric defines the Gaussian-type connectivity kernel $K_F(\gamma,\bar\gamma)=e^{-d_F(\gamma,\bar\gamma)^2}$, which enters a mean-field equation for cortical activity. Linearizing that equation reduces stable states to the leading eigenvectors of the kernel's discretization, and the argument uses a normalized Markov matrix followed by k-means to obtain the clusters. The modularity of the construction—features first, then fragments, then states—is itself part of the proposed neural architecture.

What would settle it

Run the same spectral-clustering pipeline on the fragments of [11]—or on a fresh motor-cortex dataset—and compare the k-means labels on the leading eigenvectors of $K_F$ with the hidden-Markov-model states; if the match is at chance level, or if the eigenvalue threshold predicts a number of clusters different from eight, the central claim is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that neural states are not hidden in the population activity but are already present in the geometry of hand movement, once the right kinematic variables and distance are used. Each fragment is modeled as an admissible horizontal curve in the sub-Riemannian feature space; projecting out the position variables defines a pseudo-metric on fragment space in which fragments that differ only by hand location are equivalent. The heat-kernel estimate $K_F(\gamma,\bar\gamma)=e^{-d_F(\gamma,\bar\gamma)^2}$ is then treated as the cortical connectivity kernel, and a linearized mean-field equation shows that stable activation patterns correspond to the principal eigenvectors of $K_F$. Applying spectral clustering and k-means to those eigenvectors yields eight clusters whose (x,y) direction histograms and (t,v) speed profiles coincide with the neural states of [11]. The paper concludes that this kinematic construction is sufficient to explain neural-state formation, and that its two-stage grouping mirrors the brain's hierarchical processing of motor primitives.

Load-bearing premise

The argument collapses if the leading eigenvectors of the kinematic connectivity kernel do not actually give the same grouping as the hidden-Markov-model neural states, because that equivalence is imported from earlier work on visual cortex and is not re-derived or tested for motor cortex here.

Editorial extensions

If this is right

  • Neural-state classification no longer requires neural recordings; a kinematic kernel and spectral clustering reproduce the same eight groups.
  • Because the pseudo-metric ignores hand position, the model explains why neural states are invariant to where in the work space the hand moves.
  • The same sub-Riemannian framework that describes fragments can be extended upward to describe states, giving one geometry for multiple scales of motor coding.
  • The linearized mean-field equation ties stable cortical activity to the kernel's leading eigenvectors, so state formation is a pattern-formation phenomenon rather than a purely statistical clustering.
  • The eight-cluster structure emerges from the eigenvalue threshold on the connectivity kernel, matching the experimentally observed number of states.

Reading between the lines

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

  • The paper leaves implicit that the eigenvalue spectrum of $K_F$ could predict the number of neural states directly, rather than fixing eight from the experimental data.
  • Because the pseudo-metric is invariant to hand position, the same construction should transfer to other task geometries or effectors once their symmetry variables are quotiented out.
  • If the eigenvector-to-HMM correspondence holds, the model offers a behavioral readout: neural-state structure could be inferred from kinematic recordings alone, which would be useful for brain-machine interface decoders.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes a sub-Riemannian model of the primary motor cortex in which movement fragments are curves in a feature space, a pseudo-metric is defined on a reduced space independent of (x,y), and a connectivity kernel is used in a mean-field equation. The authors then apply a spectral clustering algorithm to the eigenvectors of the discretized kernel and claim that this procedure recovers the eight neural states found by Kadmon Harpaz et al. [11] from measured cortical activity. The only experimental evidence presented is clustering of synthetic fragments generated from the model's own family (16), with no comparison to the neural states or data of [11].

Significance. If the central claim were established, the model would provide a kinematic explanation for the formation of neural states without invoking neural data, which would be a valuable contribution to motor-control theory. The geometric construction is explicit, the exponential coordinates in Remark 1 are computed in closed form, and the synthetic experiments are reproducible in principle. However, the paper does not validate the claimed recovery of the biological states of [11]: the results in Section 4 cluster synthetic fragments only, and the link between eigenvectors of the connectivity kernel and neural states is imported from [45] without testing. The evidence is therefore circular relative to the main claim, and the manuscript currently does not deliver the promised biological validation.

major comments (5)
  1. [Section 4, Figures 6-11] The central claim of the paper, stated in the abstract and conclusion, is that the grouping algorithm recovers the neural states observed in [11]. Section 4 tests this only on synthetic fragments generated from the model's own family (16): uniform θ0 and j in Section 4.1 and fully random parameters in Section 4.2. There is no comparison, quantitative or visual, to the HMM-derived states of [11], no use of the neural data from [11], and no evaluation metric such as an adjusted Rand index. The text in Section 4.1 says the algorithm gives a 'correct clusterization' without defining the correctness criterion, and the number of clusters is set to eight, matching the target. The claimed recovery is therefore unsupported as stated.
  2. [Section 3.2-3.3, Eq. (16)] The fragment family in (16) already encodes the variables that separate the resulting clusters: the initial direction θ0 and the jerk coefficient j, which determines the sign of acceleration. Section 4.1 reports that the clusters are characterized by orientation quadrant and increasing versus decreasing velocity, so the algorithm partly redis covers labels that are built into the generative model. To break this circularity, the clustering should be applied to fragments whose direction and acceleration profile are not predetermined by the model, or to the real fragments of [11], and compared with the HMM states using a standard clustering agreement measure.
  3. [Section 3.1, Proposition 2 and Eq. (15)] Proposition 2 guarantees only a local estimate C0 dM1 ≤ |e| ≤ C1 dM1 with unspecified constants, but Eq. (15) is used as an exact distance formula in the kernel (17). The paper does not justify that the constants can be ignored, that the Heisenberg coordinate expression is valid globally, or that replacing e2 by 4 sin((θ0−θ1)/4) preserves the required estimate. Since the affinity matrix (21) depends exponentially on dF, uncontrolled constants in the distance estimate can materially change the spectral clustering output.
  4. [Sections 3.4-3.5, Eq. (20)] The identification of stable neural states with the principal eigenvectors of the connectivity kernel KF is assumed from [45] and is neither derived nor tested for motor cortex. The manuscript does not show that the HMM-derived states of [11] correspond to the groupings of eigenvectors of the linearized mean-field equation (18). If this spectral-to-biological correspondence fails, the synthetic clusters in Section 4 have no bearing on real neural states. This is a load-bearing assumption that needs either a derivation in the motor-cortex setting or a direct empirical test.
  5. [Definition 5, Eq. (17)] The pseudo-distance dF in Definition 5 contains the term ||γ′1(t)−γ′2(t)||M1, but the manuscript defines a horizontal norm only at a single point via Eq. (9). The difference of tangent vectors at different base points requires a connection or a global coordinate identification, which is not introduced. This makes the definition of dF incomplete and directly affects the kernel (17) used in all subsequent computations.
minor comments (5)
  1. [Section 1, first paragraph] The word 'esperiment' should be 'experiment'.
  2. [Section 2.1, paragraph on neural states] The word 'senslible' should be 'sensitive'.
  3. [Remark 1, proof] In the system displayed in the proof, '˙v=e1a+e5' uses an undefined variable e5; it should be e4, consistent with the expression for e4 immediately below.
  4. [Eqs. (6) and (16)] The coefficient α3 is written as j(t−T/2) in Eq. (6) but as j(s−T/2) in Eq. (16), with T undefined in (16). Clarify that the interval is rescaled and define T consistently.
  5. [Section 3.1, definition of M1] The space M1 is written as R^+_t × S^1 × R^2, but the original space M in Eq. (2) has t∈R. Justify the restriction to nonnegative time or state that the half-line is only a notational convenience for the interval of definition of fragments.

Circularity Check

2 steps flagged · score 8.0 of 10

Central 'recovery' is in-sample clustering of the model's own fragment family, with the spectral-to-biological mapping imported from the authors' prior work [45].

  1. fitted input called prediction [Section 4.1, Eq. (16), Figs. 6–8, vs. Section 2.1]
    "We start by testing our model on samples of curves generated by the expression of fragments(see Figure 6) introduced in (16). ... θ0 uniformly distributed in [−pi,pi] α1=1, α2=0 and j uniformly distributed. ... We apply the clustering algorithm and in this case, we obtain a correct clusterization of the curves, in eight clusters, each one characterized by the orientation belonging to a specific quadrant and increasing or decreasing velocity (see Figures 7 and 8)."

    The target states of [11] are defined in Section 2.1 as 'a group of fragments all with compared direction in the (x,y) plane and with a specific acceleration and deceleration phase in the (t,v) plane.' The synthetic fragments in Section 4 are generated from the model's own fragment family (16), whose free parameters are the direction θ0 and the acceleration-shape parameter j. Clustering with an affinity built from these same kinematic variables therefore reproduces a partition by direction quadrant and sign of j. The paper calls this 'recover[ing] the same neural states obtained in [11]', but no [11] data or comparison metric are used; the eight clusters are a re-clustering of the model's own generative parameters, so the 'prediction' is forced by the construction of the input family.

  2. self citation load bearing [Section 3.4, Eq. (18)–(20), citation [45]; Section 3.5]
    "For this reason, stable neural states can be studied in terms of a spectral analysis of the connectivity kernel. This argument has been developed in the paper [45] with the scope of finding a strict link between emergence of patterns in the brain, and spectral clustering algorithms."

    This sentence is the only argument connecting eigenvectors of the model's connectivity kernel KF to the biological 'neural states' of [11]. It rests entirely on reference [45] (Sarti & Citti, 2015), prior work by two of the present authors, and is not re-derived or validated for motor cortex in this paper. The subsequent k-means-on-eigenvectors algorithm in Section 3.5 therefore inherits its biological interpretation from an unverified self-citation chain; if that imported equivalence fails, the synthetic clusters in Section 4 say nothing about the states measured by [11].

full rationale

The geometric construction itself (sub-Riemannian distance estimates, kernel definition, diffusion-map equivalence) is mostly self-contained and cites external mathematical results ([33], [51], [52]). However, the paper's central advertised claim—'we successfully recover the neural states observed in Kadmon-Harpaz et al'—is not evaluated against [11]'s data. Section 4 clusters only synthetic curves drawn from the model's own family (16); the cluster structure (direction quadrant × acceleration sign) is exactly the structure used to define the [11] states in Section 2.1, and the number of clusters is taken to be eight. Thus the empirical 'recovery' is a self-consistency check, not an independent prediction. The additional step identifying spectral eigenvectors of KF with stable neural states is imported from the authors' own [45] and functions as a load-bearing self-citation. Together these make the central claim substantially circular: the output clusters are built into the generative family and the spectral-biological equivalence is self-referential. Score 8.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model's central claim depends on a small set of modeling choices: the parametric fragment family, the reduced submanifold, the explicit distance formula, and the spectral-clustering equivalence. Several of these choices are made to match the target [11] classification, which raises circularity burden; none are fitted to experimental data in the paper.

free parameters (4)
  • number of clusters k = 8
    The algorithm is run to produce eight clusters in Sections 4.1 and 4.2, matching the eight neural states of [11]; no data-driven criterion is given.
  • spectral threshold epsilon = not specified
    Step 3 of the spectral clustering algorithm selects eigenvectors with eigenvalues above 1 - epsilon; epsilon is not quantified, yet it determines the effective dimension and cluster count.
  • weights in the pseudo-metric d_F = all equal to 1
    Definition 5 sums the integral of the M1-norm of the velocity difference and the endpoint M1-distance with implicit unit weights; different weights would change the clustering.
  • weights in the distance estimate (15) = all coefficients 1
    The components e1, e2_hat, e3, e4 are combined with unit weights, although Proposition 2 only guarantees equivalence up to unknown constants C0, C1; these constants are effectively absorbed into the explicit formula.
assumptions (5)
  • standard math Chow's theorem and the Hormander condition guarantee horizontal connectivity and a finite distance d_M1 on M1.
    Used in Section 3.1 to define the sub-Riemannian distance on the reduced manifold M1.
  • standard math The Nagel-Stein-Wainger local distance estimate (14) is valid and can be replaced by the explicit homogeneous norm (15).
    Proposition 2 is the basis for the connectivity kernel; the paper substitutes an explicit formula despite the estimate being two-sided with unspecified constants.
  • domain assumption Fragments are exactly the curves in (16), with constant direction and a linear derivative of acceleration.
    This parametric family is taken from [21] and used as the space F for distances and clustering; real fragments may not fit this family.
  • domain assumption The neural-state classification of [11] is invariant to (x,y) and therefore reducible to the submanifold M1.
    Section 3 states this invariance and uses it to drop position variables; if wrong, the pseudo-metric omits a relevant variable.
  • domain assumption Stable neural states correspond to principal eigenvectors of the connectivity kernel via linearization of the mean-field equation.
    Sections 3.4-3.5 import this equivalence from Sarti-Citti [45] and use it to justify spectral clustering; the present paper does not derive or test it for motor cortex.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A sub-Riemannian model of neural states in the primary motor cortex." pith.science (2026). https://pith.science/paper/LLOS7YN4

@misc{pith2026250103247,
  author       = {Pith},
  title        = {Pith review of: A sub-Riemannian model of neural states in the primary motor cortex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLOS7YN4}},
  note         = {Machine review of arXiv:2501.03247}
}
abstract

We develop a neurogeometric model for the arm area of motor cortex, which encodes complex motor primitives, ranging from simple movement features like movement direction, to short hand trajectories, termed fragments, and ultimately to more complex patterns known as neural states (Georgopoulos, Hatsopoulos, Kadmon-Harpaz et al). Based on the sub-riemannian framework introduced in 2023, we model the space of fragments as a set of short curves defined by kinematic parameters. We then introduce a geometric kernel that serves as a model for cortical connectivity and use it in a differential equation to describe cortical activity. By applying a grouping algorithm to this cortical activity model, we successfully recover the neural states observed in Kadmon-Harpaz et al, which were based on measured cortical activity. This confirms that the choice of kinematic variables and the distance metric used here are sufficient to explain the phenomena of neural state formation. The modularity of our model reflects the brain's hierarchical structure, where initial groupings in the kinematic space $\mathcal{M}$ lead to more abstract representations. This approach mimics how the brain processes stimuli at different scales, extracting both local and global properties.

Figures

Figures reproduced from arXiv: 2501.03247 by the authors.

Figure 1
Figure 1. Here, a RTP task is represented: the starting point of the motion is the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. clusterization of fragments in neural states obtained in [ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. decomposed in fragments of the (x,y) components: fragments are not organized in neural states. Source: [21]. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: decomposed in fragments of the (t,v) components. Source: [21] [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: movement is decomposed in fragments, according to the decomposition [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: A family of curves starting from the origin with constant direction [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: A visualization of the grouping of fragments in neural states: the first [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: A visualization of the grouping of fragments in neural states: the last 4 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: A family of curves with all parameters randomly choosen [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Visualization of the grouping of fragments in neural states: the first 4 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Visualization of the grouping of fragments in neural states: the last 4 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    q-bio.NC 2026-03 conditional novelty 5.0 of 10

    Sub-Riemannian horizontal curves on (x,y,t,θ,v,a) automatically satisfy the speed-curvature relation, and Wasserstein distance clusters real movement fragments into the neural states of Kadmon et al.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [21]

    Functional architecture of m1 cells encoding movement direction,

    C. Mazzetti, A. Sarti, and G. Citti, “Functional architecture of m1 cells encoding movement direction,” Journal of Computational Neuroscience, pp. 1–29, 2023

    work page 2023

  2. [11]

    Movement decomposition in the primary motor cortex,

    N. Kadmon Harpaz, D. Ungarish, N. G. Hatsopoulos, and T. Flash, “Movement decomposition in the primary motor cortex,” Cerebral cortex, vol. 29, no. 4, pp. 1619– 1633, 2019

    work page 2019

  3. [45]

    The constitution of visual perceptual units in the functional architecture of v1,

    A. Sarti and G. Citti, “The constitution of visual perceptual units in the functional architecture of v1,” Journal of computational neuroscience, vol. 38, no. 2, pp. 285– 300, 2015

    work page 2015

  4. [1]

    Static spatial effects in motor cortex and area 5: quantitative relations in a two-dimensional space,

    A. Georgopoulos, R. Caminiti, and J. Kalaska, “Static spatial effects in motor cortex and area 5: quantitative relations in a two-dimensional space,” Experimental Brain Research, vol. 54, no. 3, pp. 446–454, 1984

    work page 1984

  5. [2]

    R. E. Kettner, A. B. Schwartz, and A. P. Georgopoulos, “Primate motor cortex and free arm movements to visual targets in three-dimensional space. iii. positional gradients and population coding of movement direction from various movement origins,”Journal of Neuroscience, vol. 8, no. 8, pp. 2938–2947, 1988

    work page 1988

  6. [3]

    On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex,

    A. P. Georgopoulos, J. F. Kalaska, R. Caminiti, and J. T. Massey, “On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex,” Journal of Neuroscience, vol. 2, no. 11, pp. 1527–1537, 1982

    work page 1982

  7. [4]

    Primate motor cortex and free arm movements to visual targets in three-dimensional space. i. relations between single cell discharge and direction of movement,

    A. B. Schwartz, R. E. Kettner, and A. P. Georgopoulos, “Primate motor cortex and free arm movements to visual targets in three-dimensional space. i. relations between single cell discharge and direction of movement,” Journal of Neuroscience, vol. 8, no. 8, pp. 2913–2927, 1988

    work page 1988

  8. [5]

    Movement parameters and neural activity in motor cortex and area 5,

    J. Ashe and A. P. Georgopoulos, “Movement parameters and neural activity in motor cortex and area 5,” Cerebral cortex, vol. 4, no. 6, pp. 590–600, 1994

    work page 1994

Show all 53 references
  1. [6]

    Motor cortical representation of speed and direction during reaching,

    D. W. Moran and A. B. Schwartz, “Motor cortical representation of speed and direction during reaching,” Journal of neurophysiology, vol. 82, no. 5, pp. 2676–2692, 1999. 19

    1999

  2. [7]

    Temporal complexity and heterogeneity of single-neuron activity in premotor and motor cortex,

    M. M. Churchland and K. V . Shenoy, “Temporal complexity and heterogeneity of single-neuron activity in premotor and motor cortex,” Journal of neurophysiology, vol. 97, no. 6, pp. 4235–4257, 2007

    2007

  3. [8]

    Spatiotem- poral tuning of motor cortical neurons for hand position and velocity,

    L. Paninski, M. R. Fellows, N. G. Hatsopoulos, and J. P. Donoghue, “Spatiotem- poral tuning of motor cortical neurons for hand position and velocity,” Journal of neurophysiology, vol. 91, no. 1, pp. 515–532, 2004

    2004

  4. [9]

    Encoding of movement fragments in the motor cortex,

    N. G. Hatsopoulos, Q. Xu, and Y . Amit, “Encoding of movement fragments in the motor cortex,” Journal of Neuroscience, vol. 27, no. 19, pp. 5105–5114, 2007

    2007

  5. [10]

    The problem of parametric neural coding in the motor system,

    J. Reimer and N. G. Hatsopoulos, “The problem of parametric neural coding in the motor system,” in Progress in motor control, pp. 243–259, Springer, 2009

    2009

  6. [12]

    Optimal control theory,

    E. Todorov, “Optimal control theory,” Bayesian brain: probabilistic approaches to neural coding, pp. 268–298, 2006

    2006

  7. [13]

    An organizing principle for a class of voluntary movements,

    N. Hogan, “An organizing principle for a class of voluntary movements,”Journal of neuroscience, vol. 4, no. 11, pp. 2745–2754, 1984

    1984

  8. [14]

    The coordination of arm movements: an experimentally confirmed mathematical model,

    T. Flash and N. Hogan, “The coordination of arm movements: an experimentally confirmed mathematical model,”Journal of neuroscience, vol. 5, no. 7, pp. 1688–1703, 1985

    1985

  9. [15]

    Formation and control of optimal trajectory in human multijoint arm movement,

    Y . Uno, M. Kawato, and R. Suzuki, “Formation and control of optimal trajectory in human multijoint arm movement,” Biological cybernetics, vol. 61, no. 2, pp. 89–101, 1989

    1989

  10. [16]

    Affine differential geometry analysis of human arm movements,

    T. Flash and A. A. Handzel, “Affine differential geometry analysis of human arm movements,” Biological cybernetics, vol. 96, no. 6, pp. 577–601, 2007

    2007

  11. [17]

    A computational model for redundant human three-dimensional pointing movements: integration of independent spatial and temporal motor plans simplifies movement dynamics,

    A. Biess, D. G. Liebermann, and T. Flash, “A computational model for redundant human three-dimensional pointing movements: integration of independent spatial and temporal motor plans simplifies movement dynamics,” Journal of Neuroscience, vol. 27, no. 48, pp. 13045–13064, 2007

    2007

  12. [18]

    Computational approaches to motor control,

    T. Flash and T. J. Sejnowski, “Computational approaches to motor control,”Current opinion in neurobiology, vol. 11, no. 6, pp. 655–662, 2001

    2001

  13. [19]

    Optimal control models of the goal-oriented human locomotion,

    F. Jean, “Optimal control models of the goal-oriented human locomotion,” in Talk given at the “Workshop on Nonlinear Control and Singularities”, Porquerolles, France, pp. 24–28, 2010

    2010

  14. [20]

    Jean,Control of nonholonomic systems: from sub-Riemannian geometry to motion planning

    F. Jean,Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. Springer, 2014. 20

    2014

  15. [22]

    Higher visual perception as prolongation of the basic lie transforma- tion group,

    W. C. Hoffman, “Higher visual perception as prolongation of the basic lie transforma- tion group,” Mathematical Biosciences, vol. 6, pp. 437–471, 1970

    1970

  16. [23]

    Vers une neurog´eom´etrie. fibrations corticales, structures de contact et contours subjectifs modaux,

    J. Petitot and Y . Tondut, “Vers une neurog´eom´etrie. fibrations corticales, structures de contact et contours subjectifs modaux,”Math´ematiques et sciences humaines, vol. 145, pp. 5–101, 1999

    1999

  17. [24]

    A cortical based model of perceptual completion in the roto-translation space,

    G. Citti and A. Sarti, “A cortical based model of perceptual completion in the roto-translation space,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 307–326, 2006

    2006

  18. [25]

    Cortical spatiotemporal dimensionality reduction for visual grouping,

    G. Cocci, D. Barbieri, G. Citti, and A. Sarti, “Cortical spatiotemporal dimensionality reduction for visual grouping,” Neural computation, vol. 27, no. 6, pp. 1252–1293, 2015

    2015

  19. [26]

    Agrachev, D

    A. Agrachev, D. Barilari, and U. Boscain, A comprehensive introduction to sub-Riemannian geometry, vol. 181. Cambridge University Press, 2019

    2019

  20. [27]

    Lecture notes on sub-riemannian geometry,

    E. Le Donne, “Lecture notes on sub-riemannian geometry,” preprint, 2010

    2010

  21. [28]

    A sub-riemannian model of the functional ar- chitecture of m1 for arm movement direction,

    C. Mazzetti, A. Sarti, and G. Citti, “A sub-riemannian model of the functional ar- chitecture of m1 for arm movement direction,” In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, pp. 483–492, 2023

    2023

  22. [29]

    Rectifiability and perimeter in the heisen- berg group,

    B. Franchi, R. Serapioni, and F. Cassano, “Rectifiability and perimeter in the heisen- berg group,” Math Ann, vol. 321, pp. 479–531, 2001

    2001

  23. [30]

    Regular submanifolds, graphs and area formula in heisenberg groups,

    B. Franchi, R. Serapioni, and F. Cassano, “Regular submanifolds, graphs and area formula in heisenberg groups,” Advances in Mathematics, vol. 211, pp. 152–203, 2007

    2007

  24. [31]

    Intrinsic regular hypersurfaces in heisenberg groups,

    L. Ambrosio, F. Serra Cassano, and D. Vittone, “Intrinsic regular hypersurfaces in heisenberg groups,” J. Geom. Anal., vol. 16, pp. 187–232, 2006

    2006

  25. [32]

    Implicit function theorem in carnot carath´eodory spaces,

    G. Citti and M. Manfredini, “Implicit function theorem in carnot carath´eodory spaces,” Communications in Contemporary Mathematics, vol. 08, pp. 657–680, 2006

    2006

  26. [33]

    Balls and metrics defined by vector fields i: Basic properties,

    A. Nagel, E. M. Stein, and S. Wainger, “Balls and metrics defined by vector fields i: Basic properties,” Acta Mathematica, vol. 155, pp. 103–147, 1985

    1985

  27. [34]

    From intention to action: motor cortex and the control of reaching movements,

    J. F. Kalaska, “From intention to action: motor cortex and the control of reaching movements,” Progress in Motor Control, pp. 139–178, 2009

    2009

  28. [35]

    The cortical control of movement revisited,

    M. S. Graziano, C. S. Taylor, T. Moore, and D. F. Cooke, “The cortical control of movement revisited,”Neuron, vol. 36, no. 3, pp. 349–362, 2002. 21

    2002

  29. [36]

    Detecting neural-state transitions using hidden markov models for motor cortical prostheses,

    C. Kemere, G. Santhanam, B. M. Yu, A. Afshar, S. I. Ryu, T. H. Meng, and K. V . Shenoy, “Detecting neural-state transitions using hidden markov models for motor cortical prostheses,” Journal of neurophysiology, vol. 100, no. 4, pp. 2441–2452, 2008

    2008

  30. [37]

    Horizontal mean curvature flow as a scaling limit of a mean field equation in the heisenberg group,

    G. Citti, N. Dirr, F. Dragoni, and R. Grande, “Horizontal mean curvature flow as a scaling limit of a mean field equation in the heisenberg group,” arXiv preprint arXiv:2411.15814, 2024

    2024 arXiv

  31. [38]

    ¨Uber systeme von liearren partiellen differentialgleichungen erster ordnung,

    W.-L. Chow, “ ¨Uber systeme von liearren partiellen differentialgleichungen erster ordnung,” Mathematische Annalen, vol. 117, no. 1, pp. 98–105, 1940

    1940

  32. [39]

    A model of reaching via subriemannian geodesics in engel-type group,

    C. Mazzetti, A. Sarti, and G. Citti, “A model of reaching via subriemannian geodesics in engel-type group,” arXiv preprint arXiv:2301.05765, 2023

    2023 arXiv

  33. [40]

    Characteristics of random nets of analog neuron-like elements,

    S. I. Amari, “Characteristics of random nets of analog neuron-like elements,”IEEE Transactions on Systems, Man, and Cybernetics, vol. 2, no. 5, pp. 643–657, 1972

    1972

  34. [41]

    Excitatory and inhibitory interactions in localized populations of model neurons,

    H. R. Wilson and J. D. Cowan, “Excitatory and inhibitory interactions in localized populations of model neurons,” Biophysical Journal, vol. 12, no. 1, pp. 1–24, 1972

    1972

  35. [42]

    Large scale spatially organized activity in neural nets,

    G. B. Ermentrout and J. D. Cowan, “Large scale spatially organized activity in neural nets,” SIAM Journal on Applied Mathematics, vol. 38, no. 1, pp. 1–21, 1980

    1980

  36. [43]

    The functional geometry of local and horizontal connections in a model of v1,

    P. C. Bressloff and J. D. Cowan, “The functional geometry of local and horizontal connections in a model of v1,” Journal of Physiology-Paris, vol. 97, no. 2-3, pp. 221– 236, 2003

    2003

  37. [44]

    Some theoretical and numerical results for delayed neural field equations,

    G. Faye and O. Faugeras, “Some theoretical and numerical results for delayed neural field equations,” Physica D: Nonlinear Phenomena, vol. 239, no. 9, pp. 561–578, 2010

    2010

  38. [46]

    A factorization approach to grouping,

    P. Perona and W. Freeman, “A factorization approach to grouping,” European Conference on Computer Vision, pp. 655–670, 1998

    1998

  39. [47]

    e. a. Steve Butler, Fan Chung, Spectral graph theory. Handbook of linear algebra. Springer, 2006

    2006

  40. [48]

    On spectral clustering: Analysis and an algorithm,

    M. J. Ng and Y . Weiss, “On spectral clustering: Analysis and an algorithm,”Advances in neural information processing systems, vol. 14, 2001

    2001

  41. [49]

    Normalized cuts and image segmentation,

    J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE Transactions on pattern analysis and machine intelligence, vol. 22(8), pp. 888–905, 2000. 22

    2000

  42. [50]

    A random walks view of spectral segmentation,

    M. Meil˘a and J. Shi, “A random walks view of spectral segmentation,” inInternational Workshop on Artificial Intelligence and Statistics, pp. 203–208, 2001

    2001

  43. [51]

    Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization,

    S. Lafon and A. Lee, “Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization,” IEEE transactions on pattern analysis and machine intelligence, vol. 28(9), pp. 1393–1403, 2006

    2006

  44. [52]

    Diffusion maps,

    R. Coifman and S. Lafon, “Diffusion maps,” Applied and computational harmonic analysis, vol. 21(1), pp. 5–30, 2006

    2006

  45. [53]

    On clusterings: Good, bad and spectral,

    R. Kannan, S. Vempala, and A. Vetta, “On clusterings: Good, bad and spectral,” Journal of the ACM (JACM), vol. 51, no. 3, pp. 497–515, 2004. 23

    2004

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.