The torus \(T\) of projective space also acts on the Hilbert scheme of subschemes of projective space. The \(T\)-graph of the Hilbert scheme has vertices the fixed points of this action, and edges connecting pairs of fixed points in the closure of a one-dimensional orbit. In general this graph depends on the underlying field. We construct a subgraph, which we call the spine, of the \(T\)-graph of \(\operatorname{Hilb}^m(\mathbb A^2)\) that is independent of the choice of infinite field. For certain edges in the spine we also give a description of the tropical ideal, in the sense of tropical scheme theory, of a general ideal in the edge. This gives a more refined understanding of these edges, and of the tropical stratification of the Hilbert scheme.

Mathematics Subject Classifications: 14C05, 14T10, 14L30

Keywords: Hilbert scheme, \(T\)-graph, tropical ideal

Specialized sensorimotor systems allow us to perform dexterous, graceful movements in complicated, dynamic environments. One remarkable feature of the sensorimotor system is its ability to learn new skills. The neural underpinnings of sensorimotor learning have traditionally been studied at two levels: low-level changes between individual neurons, and high-level changes in behavior. Directly connecting these distant levels of study is challenging. In this thesis, I try to bridge this gap by focusing on plasticity at an intermediate level that considers how neuron populations change with respect to each other, i.e., meso-scale plasticity.

In Chapter 2, I study changes in meso-scale connectivity in response to neurostimulation. A large-scale optogenetic interface enabled us to simultaneously stimulate and record population activity across primary somatosensory (S1) and primary motor (M1) cortex. We tracked two measures of network connectivity—one based on responses to focal stimulation and the other based on spontaneous activity patterns. Within minutes of stimulation, the inter-area functional connectivity strengthened. At a finer scale, stimulation led to heterogeneous changes across the network, which reflected the correlations introduced by stimulation-evoked activity, consistent with Hebbian models of synaptic plasticity. This work extends Hebbian plasticity models to meso-scale circuits.

In Chapter 3, I connect low-level changes in M1 spiking structure to two meso-scale oscillations—spindles and slow oscillations (SOs), which are thought to support sensorimotor learning and memory consolidation. During spindles, individual neurons fired at a preferred phase of spindle cycles and neuron pairs synchronized activity during spindle peaks, signifying an increase in pairwise correlations and local functional connectivity. We found a direct relationship between the temporal proximity of SO and spindles (which are thought to interact), and changes to the distribution of spike correlations; closer oscillations were associated with narrowing of the distribution of correlations, with a reduction in low- and high-correlation pairs. Such narrowing is consistent with exploration of novel neural states and may be a key mechanism through which the interaction of meso-scale oscillations can support sensorimotor consolidation.

Throughout this thesis I advocate for studying and modeling neuron populations. I provide insight into meso-scale plasticity through a direct study of meso-scale connectivity changes and by relating the meso-scale to coordinated spiking activity.

The protein YbiV from Escherichia coli K12 MG1655 is a hypothetical protein with sequence homology to the haloacid dehalogenase (HAD) superfamily of proteins. Although numerous members of this family have been identified, the functions of few are known. Using the crystal structure, sequence analysis, and biochemical assays, we have characterized ybiV as a HAD phosphatase. The crystal structure of YbiV reveals a two domain protein, one with the characteristic HAD hydrolase fold, the other an inserted a/b fold. In an effort to understand the mechanism we also solved and report the structures of YbiV in complex with beryllofluoride (BeF3-) and aluminum trifluoride (AlF3) which have been shown to mimic the phosphorylated intermediate and transition state for hydrolysis, respectively, in analogy to other HAD phosphatases. Analysis of the structures reveals the substrate binding cavity, which is hydrophilic in nature. Both structure and sequence homology indicate ybiV may be a sugar phosphatase, which is supported by biochemical assays which measured the release of free phosphate on a number of sugar-like substrates. We also investigated available genomic and functional data in an effort to determine the physiological substrate.