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

Jawad Ali, Giovanna Citti, Alessandro Sarti

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.

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

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.
Paper Structure (20 sections, 1 theorem, 37 equations, 11 figures)

This paper contains 20 sections, 1 theorem, 37 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The state of the art
  3. A sub-Riemannian geometry of the cortex and fragments
  4. A Sobolev distance in the space of fragments
  5. The Wasserstein distance
  6. Distance for signed and vector valued probability measures
  7. The one-dimensional case
  8. The geometry of the motor cortex
  9. Geometry of spaces of features and fragments
  10. Kinematic and geometric variables
  11. Pseudo-distances induced by restriction of subspaces
  12. Wasserstein distances in the space of fragments
  13. Clustering Algorithm and generation of fragments
  14. Brain activity
  15. Implementation of the algorithm
  16. ...and 5 more sections

Key Result

Proposition 1

where $F_\mu^{-1}$ and $F_\nu^{-1}$ are the inverses of the CDFs for the probability measures $\mu$ and $\nu$.

Figures (11)

  • Figure 1: The results of clusterization made by kadmon2019movement 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).
  • Figure 2: Visualization of the grouping of fragments in neural states obtained in mazzetti2024sub.
  • Figure 3: A family of curves with different curvature (left) and its clustering with respect to curvature (right).
  • Figure 4: Trajectories with random initial positions, orientations, and velocity profiles (accelerating or decelerating).
  • Figure 5: Wasserstein affinity matrix.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • Definition 7