2,594 research outputs found
Wheels: A New Criterion for Non-convexity of Neural Codes
We introduce new geometric and combinatorial criteria that preclude a neural
code from being convex, and use them to tackle the classification problem for
codes on six neurons. Along the way, we give the first example of a code that
is non-convex, has no local obstructions, and has simplicial complex of
dimension two. We also characterize convexity for neural codes for which the
simplicial complex is pure of low or high dimension.Comment: 25 pages, 3 figures, 2 table
Nondegenerate Neural Codes and Obstructions to Closed-Convexity
Previous work on convexity of neural codes has produced codes that are
open-convex but not closed-convex -- or vice-versa. However, why a code is one
but not the other, and how to detect such discrepancies are open questions. We
tackle these questions in two ways. First, we investigate the concept of
degeneracy introduced by Cruz et al., and extend their results to show that
nondegeneracy precisely captures the situation when taking closures or
interiors of open or closed realizations, respectively, yields another
realization of the code. Second, we give the first general criteria for
precluding a code from being closed-convex (without ruling out open-convexity),
unifying ad-hoc geometric arguments in prior works. One criterion is built on a
phenomenon we call a rigid structure, while the other can be stated
algebraically, in terms of the neural ideal of the code. These results
complement existing criteria having the opposite purpose: precluding
open-convexity but not closed-convexity. Finally, we show that a family of
codes shown by Jeffs to be not open-convex is in fact closed-convex and
realizable in dimension two.Comment: 32 pages, 12 figures. Corrected Examples 4.32 and 5.8, added two
figures to aid in understanding proofs, improved exposition throughout, and
corrected typo
Combinatorial geometry of neural codes, neural data analysis, and neural networks
This dissertation explores applications of discrete geometry in mathematical
neuroscience. We begin with convex neural codes, which model the activity of
hippocampal place cells and other neurons with convex receptive fields. In
Chapter 4, we introduce order-forcing, a tool for constraining convex
realizations of codes, and use it to construct new examples of non-convex codes
with no local obstructions. In Chapter 5, we relate oriented matroids to convex
neural codes, showing that a code has a realization with convex polytopes iff
it is the image of a representable oriented matroid under a neural code
morphism. We also show that determining whether a code is convex is at least as
difficult as determining whether an oriented matroid is representable, implying
that the problem of determining whether a code is convex is NP-hard. Next, we
turn to the problem of the underlying rank of a matrix. This problem is
motivated by the problem of determining the dimensionality of (neural) data
which has been corrupted by an unknown monotone transformation. In Chapter 6,
we introduce two tools for computing underlying rank, the minimal nodes and the
Radon rank. We apply these to analyze calcium imaging data from a larval
zebrafish. In Chapter 7, we explore the underlying rank in more detail,
establish connections to oriented matroid theory, and show that computing
underlying rank is also NP-hard. Finally, we study the dynamics of
threshold-linear networks (TLNs), a simple model of the activity of neural
circuits. In Chapter 9, we describe the nullcline arrangement of a threshold
linear network, and show that a subset of its chambers are an attracting set.
In Chapter 10, we focus on combinatorial threshold linear networks (CTLNs),
which are TLNs defined from a directed graph. We prove that if the graph of a
CTLN is a directed acyclic graph, then all trajectories of the CTLN approach a
fixed point.Comment: 193 pages, 69 figure
Oriented Matroids and Combinatorial Neural Codes
A combinatorial neural code is convex if it
arises as the intersection pattern of convex open subsets of . We
relate the emerging theory of convex neural codes to the established theory of
oriented matroids, both categorically and with respect to geometry and
computational complexity. On the categorical side, we show that the map taking
an acyclic oriented matroid to the code of positive parts of its topes is a
faithful functor. We adapt the oriented matroid ideal introduced by Novik,
Postnikov, and Sturmfels into a functor from the category of oriented matroids
to the category of rings; then, we show that the resulting ring maps naturally
to the neural ring of the matroid's neural code.
For geometry and computational complexity, we show that a code has a
realization with convex polytopes if and only if it lies below the code of a
representable oriented matroid in the partial order of codes introduced by
Jeffs. We show that previously published examples of non-convex codes do not
lie below any oriented matroids, and we construct examples of non-convex codes
lying below non-representable oriented matroids. By way of this construction,
we can apply Mn\"{e}v-Sturmfels universality to show that deciding whether a
combinatorial code is convex is NP-hard
Neural ring homomorphism preserves mandatory sets required for open convexity
It has been studied by Curto et al. (SIAM J. on App. Alg. and Geom., 1(1) :
222 \unicode{x2013} 238, 2017) that a neural code that has an open convex
realization does not have any local obstruction relative to the neural code.
Further, a neural code has no local obstructions if and only if
it contains the set of mandatory codewords,
which depends only on the simplicial complex . Thus
if , then
cannot be open convex. However, the problem of constructing for any given code is undecidable.
There is yet another way to capture the local obstructions via the homological
mandatory set, The significance of for a given code is that and so will have local obstructions if In this paper we study the
affect on the sets and
under the action of various surjective elementary code maps. Further, we study
the relationship between Stanley-Reisner rings of the simplicial complexes
associated with neural codes of the elementary code maps. Moreover, using this
relationship, we give an alternative proof to show that is preserved under the elementary code maps
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