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

TL;DR

Problem: understanding how motor cortical activity organizes into neural states from simple movement features and short fragments. Approach: a sub-Riemannian neurogeometric model on a six-dimensional feature space $\mathcal{M}$, reduced to a sub-manifold $\mathcal{M}_1$, with a fragment distance $d_{\mathcal{F}}$, a connectivity kernel $K_{\mathcal{F}}$, and a mean-field dynamic enabling spectral clustering. Contributions: explicit construction of the feature-space geometry, a pseudo-metric ignoring spatial coordinates, a kernel-based fragmentation model, and a clustering pipeline that reproduces Kadmon-Harpaz 2019 neural states from kinematics. Significance: provides a neurally plausible, modular framework linking local fragments to global neural states, with potential to explain hierarchical motor encoding.

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 (11)

• Figure 1: Here, the random target pursuit task is represented: the starting point of the motion is the red dot, and subsequent targets are black circles. The hand trajectory is represented by two images: (A) represents position in a 2D plane, and (B) represents the speed profile. Movement is segmented into fragments, which are characterized by almost constant orientation (see (A)) and acceleration or deceleration phase (B). Finally, each color represents a single neural state. Image taken from kadmon2019movement.
• Figure 2: Clusterization of fragments in neural states obtained in kadmon2019movement. The two images visualized in each column repreasent the fragment: above is represented it $(x,y)-$ section and below a normalized profile of the $(t, v)-$section). Radial histograms show the mean directions of all the trajectories within each neural state.
• Figure 3: Movement is decomposed in fragments of the $(x,y)$ components: fragments are not organized in neural states. Source: mazzetti2023functional.
• Figure 4: Movement is decomposed in fragments of the $(t, v)$ components. Source: mazzetti2023functional.
• Figure 5: Movement is decomposed in fragments, according to the decomposition from mazzetti2023functional.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• proof
• Proposition 2
• Definition 4
• Definition 5
• Proposition 3