The symplectic structure of the primary visual cortex

Abstract

We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation-dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

“…A notable class of intrinsic C 1 and Lipschitz surfaces has been identified by means of their invariance with respect to the sub-Riemannian structure. They have been defined in the framework of rectifiable sets (see, for instance, [2,3,15,23,24,26,29,32,35,40]), with several applications to geometry of Banach spaces, theoretical computer science, mathematical models in neurosciences (see, for instance, [12,17,19,30,42]). We refer the reader to the monograph [11] and the references therein for a more detailed introduction to the Heisenberg group and the afore-mentioned arguments.…”

Dihedral Fourier analysis in phase-space

“…Bivariate D 4  D 4 orbit generated by q ¼ (1,0) and p ¼ (1,1). þ þ þ À À þ À À q x q y p x p y q x q y p x p y q x q y p x p y q x q y p x p y of the receptive profiles in response to a stimulus signal of intensity I is clearly an example of data indexed by Z m  D n when the dilation-rotation group of [39] is replaced by the (finite) dilation-rotation-reversal group Z m  D n in the operator (with q ¼ 1, say)…”

“…The decompositions associated with Z m  D n may be of interest to study the (symplectic) structure of the visual cortex, as described in [39], where the intensity output …”

Spatiotemporal Dynamics of Neural Fields on Product Spaces

“…In particular, the effects of patternforming instabilities of a stationary, homogeneous (x and θ independent) solution have been investigated using weakly nonlinear analysis and bifurcation theory. Neural field equations of the form (1.6) also provide a framework for developing geometric-based approaches to vision [43].…”