  Determining how the brain stores information is one of the most pressing
problems in neuroscience. In many instances, the collection of stimuli for a
given neuron can be modeled by a convex set in $\mathbb{R}^d$. Combinatorial
objects known as \emph{neural codes} can then be used to extract features of
the space covered by these convex regions. We apply results from convex
geometry to determine which neural codes can be realized by arrangements of
open convex sets. We restrict our attention primarily to sparse codes in low
dimensions. We find that intersection-completeness characterizes realizable
$2$-sparse codes, and show that any realizable $2$-sparse code has embedding
dimension at most $3$. Furthermore, we prove that in $\mathbb{R}^2$ and
$\mathbb{R}^3$, realizations of $2$-sparse codes using closed sets are
equivalent to those with open sets, and this allows us to provide some
preliminary results on distinguishing which $2$-sparse codes have embedding
dimension at most $2$.
