Horizontal Mean Curvature Flow in the Heisenberg Group as Scaling Limit of an Interacting Particle SystemMI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing.