Liouville integrability in a four-dimensional model of the visual cortex

TL;DR

This paper extends a visual cortex model by incorporating curvature, analyzes the sub-Riemannian geodesics using control theory, and demonstrates Liouville integrability through explicit first integrals and numerical evidence.

Contribution

It introduces a four-dimensional model including curvature, proves controllability, derives Hamiltonian equations, and shows Liouville integrability of the system.

Findings

Complete controllability of the model.

Explicit Hamiltonian system for geodesics.

Numerical evidence of Liouville integrability.

Abstract

We consider a natural extension of the Petitot-Citti-Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taking into account such that occluded contours are completed using sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R^2 x SO(2) x R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system.