Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes -- open convex and closed convex codes -- are related. As a starting point, open convex codes have a desirable monotonicity property,  but here we show that  closed convex codes do not possess the same property. We additionally show that, even for 3-sparse codes, closed convexity is not equivalent to avoiding local obstructions to convexity .  Finally, we disprove a conjecture of Goldrup and Phillipson, and then present an example of a code that is neither open convex nor closed convex.
Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes -- open convex and closed convex codes -- are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserves open convexity; but here we show that this property fails to hold for closed convex codes . Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also present an example of a code that is neither open convex nor closed convex.
A
namely, adding non-maximal codewords preserves open convexity;
A
this property fails to hold for
D
do not possess the same property. We additionally show that, even for 3-sparse codes, closed convexity is not equivalent to avoiding local obstructions to convexity
A
Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount.
R
then
also