Planar Lattice Subsets with Minimal Vertex Boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

Olfactory Cortex

Although the piriform cortex anatomical structure is simple and well defined compared to neocortical structures, it subserves higher brain function like other associative cortices. While the piriform cortex is engaged in relatively basic functions like identifying familiar odors and discriminating between similar odors, it also has a central role in complex tasks, such as integration of information about the identity and the reward value of odors. Such a variegated capability is enabled by its intrinsic hardwire, which resembles that seen in high neocortical areas, and by efficient sets of connections with ascending and descending brain areas with which the cortex generates synchronized activity.