  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.
