  We study Boolean networks which are simple spatial models of the highly
conserved Delta-Notch system. The models assume the inhibition of Delta in each
cell by Notch in the same cell, and the activation of Notch in presence of
Delta in surrounding cells. We consider fully asynchronous dynamics over
undirected graphs representing the neighbour relation between cells. In this
framework, one can show that all attractors are fixed points for the system,
independently of the neighbour relation, for instance by using known properties
of simplified versions of the models, where only one species per cell is
defined. The fixed points correspond to the so-called fine-grained "patterns"
that emerge in discrete and continuous modelling of lateral inhibition. We
study the reachability of fixed points, giving a characterisation of the trap
spaces and the basins of attraction for both the full and the simplified
models. In addition, we use a characterisation of the trap spaces to
investigate the robustness of patterns to perturbations. The results of this
qualitative analysis can complement and guide simulation-based approaches, and
serve as a basis for the investigation of more complex mechanisms.
