  Self-organization of neural circuitry is an appealing framework for
understanding cortical development, yet its applicability remains unconfirmed.
Models for the self-organization of neural circuits have been proposed, but
experimentally testable predictions of these models have been less clear. The
visual cortex contains a large number of topological point defects, called
pinwheels, which are detectable in experiments and therefore in principle well
suited for testing predictions of self-organization empirically. Here, we
analytically calculate the density of pinwheels predicted by a pattern
formation model of visual cortical development. An important factor controlling
the density of pinwheels in this model appears to be the presence of non-local
long-range interactions, a property which distinguishes cortical circuits from
many nonliving systems in which self-organization has been studied. We show
that in the limit where the range of these interactions is infinite, the
average pinwheel density converges to $\pi$. Moreover, an average pinwheel
density close to this value is robustly selected even for intermediate
interaction ranges, a regime arguably covering interaction-ranges in a wide
range of different species. In conclusion, our paper provides the first direct
theoretical demonstration and analysis of pinwheel density selection in models
of cortical self-organization and suggests to quantitatively probe this type of
prediction in future high-precision experiments.
