  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.
