An uncertainty principle underlying the pinwheel structure in the primary visual cortex

The visual information in V1 is processed by an array of modules called orientation preference columns. In some species including humans, orientation columns are radially arranged around singular points like the spokes of a wheel, that are called pinwheels. The pinwheel structure has been observed first with optical imaging techniques and more recently by in vivo two-photon imaging proving their organization with single cell precision. In this research we provide evidence that pinwheels are de facto optimal distributions for coding at the best angular position and momentum. In the last years many authors have recognized that the functional architecture of V1 is locally invariant with respect to the symmetry group of rotations and translations SE(2). In the present study we show that the orientation cortical maps used to construct pinwheels can be modeled as coherent states, i.e. the configurations best localized both in angular position and angular momentum. The theory we adopt is based on the well known uncertainty principle, introduced by Heisenberg in quantum mechanics and later extended to many other groups of invariance. Here we state a corresponding principle in the cortical geometry with SE(2) symmetry, and by computing its minimizers we obtain a model of orientation activity maps in the cortex. As it is well known the pinwheels configuration is directly constructed from these activity maps, and we will be able to formally reproduce their structure starting from the group symmetries of the functional architecture of the visual cortex. The primary visual cortex is then modeled as an integrated system in which the set of simple cells implements the SE(2) group, the horizontal connectivity implements its Lie algebra and the pinwheels implement its minimal uncertainty states.