Obstructions to convexity in neural codes

Abstract

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.Comment: 21 pages, 1 table; published versio

What can topology tell us about the neural code?

“…In [26], the following counterexample to the converse of Lemma 6.2 was discovered. That this code has no local obstructions can be easily seen using Theorem 6.3.…”

“…The fact that there is no convex open cover, however, relies on geometric arguments that are not obviously topological. Moreover, this code does have a good cover [26], suggesting the existence of a new class of obstructions to convexity which may or may not be topological in nature.…”

“…The proof of the following lemma is essentially the argument outlined above. The idea of using local obstructions to determine whether or not a neural code has a convex realization has been recently followed up in a series of papers [8][9][10]26]. In particular, local obstructions have been characterized in terms of links,…”

Algebraic signatures of convex and non-convex codes

“…The previous example showed that some codes satisfying signature C-1 are convex; however, this signature does not guarantee convexity. Specifically, code C 2 from Example 1.11 also satisfies this signature, and in fact has no local obstructions, yet that code is not convex [10].…”

“…However, Example 2.6 showed that even in the absence of local obstructions corresponding to elements of CF(J C ), a code C may still have local obstructions and thus be provably non-convex. Additionally, even when a code has no local obstructions of any type, it may still be nonconvex (see C 2 in Example 1.11(b), first observed to be non-convex in [10]). Despite these complicating factors, it may still be useful to identify when a code cannot have any local obstructions "arising" from canonical form elements; more precisely, it has no CF-detectable local obstructions, as defined below.…”