A metric model for the functional architecture of the visual cortex

The purpose of this work is to construct a model for the functional architecture of the primary visual cortex (V1), based on a structure of metric measure space induced by the underlying organization of receptive profiles (RPs) of visual cells. In order to account for the horizontal connectivity of V1 in such a context, a diffusion process compatible with the geometry of the space is defined following the classical approach of K.-T. Sturm [Ann. Probab., 26 (1998), pp. 1-55]. The construction of our distance function neither requires any group parameterization of the family of RPs nor involves any differential structure. As such, it adapts to nonparameterized sets of RPs, possibly obtained through numerical procedures; it also allows us to model the lateral connectivity arising from nondifferential metrics such as the one induced on a pinwheel surface by a family of filters of vanishing scale. On the other hand, when applied to the classical framework of Gabor filters, this construction yields a distance approximating the sub-Riemannian structure proposed as a model for V1 by Citti and Sarti [J. Math. Imaging Vision Archive, 24 (2006), pp. 307-326], thus showing itself to be consistent with existing cortex models.