Cortical Functional Architectures as Contact and Sub-riemannian Geometry

In the paper (Sarti et al. in Biol Cyb 98:33–48, 2008), Jean Petitot together with the authors of the present paper describe the functional geometry of the visual cortex as the symplectization of a contact form to describe the family of cells sensitive to position, orientation and scale. In the present paper, as a “homage” to the enormous contribution of Jean Petitot to neurogeometry, we will extend this approach to more complex functional architectures built as a sequence of contactization or a symplectization process, able to extend the dimension of the space. We will also outline a few examples where a sub-Riemannian lifting is needed.