Liouville Integrability in a Four-Dimensional Model of the Visual Cortex

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 taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R2×SO(2)×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 the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the 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.

Geometric Methods for Efficient Planar Swimming of Copepod Nauplii

Copepod nauplii are larval crustaceans with important ecological functions. Due to their small size, they experience an environment of low Reynolds number within their aquatic habitat. Here we provide a mathematical model of a swimming copepod nauplius with two legs moving in a plane. This model allows for both rotation and two-dimensional displacement by the periodic deformation of the swimmer’s body. The system is studied from the framework of optimal control theory, with a simple cost function designed to approximate the mechanical energy expended by the copepod. We find that this model is sufficiently realistic to recreate behavior similar to those of observed copepod nauplii, yet much of the mathematical analysis is tractable. In particular, we show that the system is controllable, but there exist singular configurations where the degree of non-holonomy is non-generic. We also partially characterize the abnormal extremals and provide explicit examples of families of abnormal curves. Finally, we numerically simulate normal extremals and observe some interesting and surprising phenomena.