  Many developmental processes in biology utilize Notch-Delta signaling to
construct an ordered pattern of cellular differentiation. This signaling
modality is based on nearest-neighbor contact, as opposed to the more familiar
mechanism driven by the release of diffusible ligands. Here, exploiting this
"juxtacrine" property, we present an exact treatment of the pattern formation
problem via a system of nine coupled ordinary differential equations. The
possible patterns that are realized for realistic parameters can be analyzed by
considering a co-dimension 2 pitchfork bifurcation of this system. This
analysis explains the observed prevalence of hexagonal patterns with high Delta
at their center, as opposed to those with central high Notch levels. We show
that outside this range of parameters, in particular for low cis-coupling, a
novel kind of pattern is produced, where high Delta cells have high Notch as
well. It also suggests that the biological system is only weakly first order,
so that an additional mechanism is required to generate the observed
defect-free patterns. We construct a simple strategy for producing such
defect-free patterns.
