What this project is about

Your brain constructs edges and surfaces that are not physically present in the light hitting your retina. Jean Petitot's 2003 neurogeometry model shows this modal completion is mathematically equivalent to finding shortest paths on the Lie group SE(2) — the group of rigid motions of the plane — under a sub-Riemannian metric. The geodesics of that metric are Euler's elastica, parametrised by Jacobi elliptic functions. This series develops the full theory from first principles, culminating in an open problem on the exact cut time.

The Series

Part	Title	Key objects	Status
1	The Visual Cortex as a Contact Manifold	SE(2), contact structure, Kanizsa	DRAFT
2	Euler's Elastica and Jacobi Elliptic Functions	sn, cn, dn, K(k²), elastica	DRAFT
3	Maxwell Strata: When Optimal Paths Fork	discrete symmetries, first Maxwell time	SOON
4	The Open Problem: Exact Cut Time on SE(2)	cut locus, conjugate time, conjecture	SOON

Mathematical Setting

The model rests on three ingredients.

The state space is SE(2). A neuron in V1 is sensitive not just to position $(x,y)$ in the visual field but also to the local orientation $\theta$ of the edge it detects. The lifted state $(x, y, \theta) \in \mathbb{R}^2 \times S^1 \cong \mathrm{SE}(2)$ carries both position and direction. The group law — translation composed with rotation — is the group of rigid motions of the plane.

The metric is sub-Riemannian. Not all directions in $\mathrm{SE}(2)$ are allowed at unit cost. A neuron at orientation $\theta$ can move cheaply along its preferred direction $(\cos\theta, \sin\theta)$ and rotate cheaply by $d\theta$, but moving transversally is penalised. This defines a rank-2 distribution (a contact structure) with the Pontryagin Hamiltonian

\[H = \frac{1}{2}(p_x \cos\theta + p_y \sin\theta)^2 + \frac{1}{2} p_\theta^2.\]

Geodesics are Euler’s elastica. The projection of SE(2) geodesics onto the $(x,y)$-plane satisfies the elastica ODE — the same curves Euler studied in 1744 when minimising the integral of squared curvature. The curvature along these curves is $\kappa(s) = 2k\,\mathrm{sn}(s \mid k^2)$, directly expressed through Jacobi’s elliptic sine, with spatial period $T = 4K(k^2)$.

Key Results Covered

• Petitot’s contact model (Part 1): why V1 lifts the image to SE(2) and why modal completion is a geodesic problem.
• Complete parametrisation (Part 2): all three inflectional families, the Euler separatrix, and the non-inflectional family, written in closed form using $\mathrm{sn}, \mathrm{cn}, \mathrm{dn}$.
• Maxwell strata (Part 3): discrete symmetry group \(\mathbb{Z}_2 \times \mathbb{Z}_2\), the first Maxwell time \(t_{\max}^1 = 2\pi/\sqrt{H}\), loss of optimality.
• The open problem (Part 4): the cut time is bounded above by \(t_{\max}^1\) but the exact value for $k \in (0,1)$ remains unproved.

Code and Data

The interactive figures use the elliptic library — Jacobi elliptic functions and complete/incomplete integrals implemented without Maple calls, accepting tensors as input.

• GitHub: moiseevigor/elliptic
• arXiv: 0807.4731 — Moiseev & Sachkov (2010)
• arXiv: 0903.0727 — Sachkov (2011)

Core References

1. J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." J. Physiology–Paris 97(2–3): 265–309.
2. I. Moiseev & Yu. L. Sachkov (2010). "Maxwell strata in sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16(2): 380–399. arXiv:0807.4731
3. Yu. L. Sachkov (2011). "Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 17(4): 293–321. arXiv:0903.0727
4. G. Citti & A. Sarti (2006). "A cortical based model of perceptual completion in the roto-translation space." J. Math. Imaging Vision 24(3): 307–326.