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

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.

💡 Research Summary

This paper presents a neurogeometric framework for modeling the functional architecture of the primary motor cortex (M1) that underlies arm reaching movements. Building on the sub‑Riemannian geometry introduced in 2023, the authors describe motor cortical cells as points in a six‑dimensional feature space M = ℝ²_{(x,y)} × ℝ_t × S¹_θ × ℝ⁺_v × ℝ_a, where (x,y) is hand position, t is time, θ the movement direction, v the speed, and a the acceleration. Differential constraints among these variables are encoded by three one‑forms, whose kernels define three horizontal vector fields X₁, X₂, X₃. By endowing these fields with an orthonormal metric, a sub‑Riemannian manifold is obtained, and the associated Carnot‑Carathéodory distance d_M quantifies cortical connectivity.

Experimental observations (Georgopoulos, Hatsopoulos, Kadmon‑Harpaz et al.) show that neural activity in M1 is organized hierarchically: simple features (direction), short trajectory fragments, and higher‑order “neural states” that group fragments with similar direction and acceleration profiles. Crucially, neural states are invariant to the absolute hand position (x,y). To capture this invariance, the authors restrict M to the four‑dimensional sub‑manifold M₁ = {x = y = 0} = ℝ_t × S¹_θ × ℝ⁺_v × ℝ_a. On M₁ the original vector fields reduce to (\hat X_1 = a∂_v + ∂_t), (\hat X_2 = ∂_θ), (\hat X_3 = ∂a). Their Lie brackets satisfy Hörmander’s condition, guaranteeing that any two points in M₁ can be joined by a horizontal curve. The authors introduce a graded norm that assigns degree 1 to (\hat X{1,2,3}) and degree 2 to the commutator (\hat X_4 =