Does the motor cortex draw on a wire plane?

TL;DR

This paper introduces the wire diffeology, a geometric framework where the equi-affine metric becomes a covariant tensor, aligning with motor cortex functions that trace curves rather than surfaces.

Contribution

It proposes a novel geometric setting using wire diffeology to make the equi-affine metric covariant under all diffeomorphisms, modeling motor primitives as curves.

Findings

The two-thirds power law emerges as a covariant invariant in the wire diffeology.

The framework aligns with motor control theories emphasizing elementary curve primitives.

The equi-affine metric becomes a true tensor under the full diffeomorphism group.

Abstract

The two-thirds power law of human motor control ($v \propto \kappa^{-1/3}$) is geometrically equivalent to constant equi-affine speed. In classical differential geometry, however, the equi-affine metric is not a tensor: it depends on acceleration, which does not transform covariantly under arbitrary coordinate changes. To recover tensorial behavior, one must either restrict the symmetry group to the affine group or introduce an affine connection -- sacrificing full diffeomorphism covariance. This article proposes a different geometric setting. We equip the Euclidean plane with the "wire diffeology', the smooth structure generated by all smooth curves. In this diffeological space, the equi-affine metric becomes a true covariant $3$-tensor under the **full** diffeomorphism group -- no restriction of symmetries, no additional structure required. The construction is motivated by a simple fact: the motor cortex traces curves, not two-dimensional patches. Accordingly, curves are taken as primitive, echoing the motor control literature in which movements are built from a repertoire of elementary building blocks -- motor primitives. The wire plane offers a geometric formalization of this idea in which the two-thirds power law emerges as a fully covariant invariant.