  How does the brain encode spatial structure? One way is through hippocampal
neurons called place cells, which become associated to convex regions of space
known as their receptive fields: each place cell fires at a high rate precisely
when the animal is in the receptive field. The firing patterns of multiple
place cells form what is known as a convex neural code. How can we tell when a
neural code is convex? To address this question, Giusti and Itskov identified a
local obstruction, defined via the topology of a code's simplicial complex, and
proved that convex neural codes have no local obstructions. Curto et al. proved
the converse for all neural codes on at most four neurons. Via a counterexample
on five neurons, we show that this converse is false in general. Additionally,
we classify all codes on five neurons with no local obstructions. This
classification is enabled by our enumeration of connected simplicial complexes
on 5 vertices up to isomorphism. Finally, we examine how local obstructions are
related to maximal codewords (maximal sets of neurons that co-fire). Curto et
al. proved that a code has no local obstructions if and only if it contains
certain "mandatory" intersections of maximal codewords. We give a new criterion
for an intersection of maximal codewords to be non-mandatory, and prove that it
classifies all such non-mandatory codewords for codes on up to 5 neurons.
