Toric Ideals, Polytopes, and Convex Neural Codes

How does the brain encode the spatial structure of the external world? A partial answer comes through place cells, hippocampal neurons which become associated to approximately convex regions of the world known as their place fields. When an organism is in the place field of some place cell, that cell will fire at an increased rate. A neural code describes the set of firing patterns observed in a set of neurons in terms of which subsets fire together and which do not. If the neurons the code describes are place cells, then the neural code gives some information about the relationships between the place fields–for instance, two place fields intersect if and only if their associated place cells fire together. Since place fields are convex, we are interested in determining which neural codes can be realized with convex sets and in finding convex sets which generate a given neural code when taken as place fields. To this end, we study algebraic invariants associated to neural codes, such as neural ideals and toric ideals. We work with a special class of convex codes, known as inductively pierced codes, and seek to identify these codes through the Gröbner bases of their toric ideals

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 $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. 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

The 2014 M_w 6.1 South Napa Earthquake: A Unilateral Rupture with Shallow Asperity and Rapid Afterslip

The Mw 6.1 South Napa earthquake occurred near Napa, California, on 24 August 2014 at 10:20:44.03 (UTC) and was the largest inland earthquake in northern California since the 1989 Mw 6.9 Loma Prieta earthquake. The first report of the earthquake from the Northern California Earthquake Data Center (NCEDC) indicates a hypocentral depth of 11.0 km with longitude and latitude of (122.3105° W, 38.217° N). Surface rupture was documented by field observations and Light Detection and Ranging (LiDAR) imaging (Brooks et al., 2014; Hudnut et al., 2014; Brocher et al., 2015), with about 12 km of continuous rupture starting near the epicenter and extending to the northwest. The southern part of the rupture is relatively straight, but the strike changes by about 15° at the northern end over a 6 km segment. The peak dextral offset was observed near the Buhman residence with right‐lateral motion of 46 cm, near the location where the strike of fault begins to rotate clockwise (Hudnut et al., 2014). The earthquake was well recorded by the strong‐motion network operated by the NCEDC, the California Geological Survey and the U.S. Geological Survey (USGS). There are about 12 sites within an epicentral distance of 15 km that had relatively good azimuthal coverage (Fig. 1). The largest peak ground velocity (PGV) of nearly 100  cm/s was observed on station 1765, which is the closest station to the rupture and lies about 3 km east of the northern segment (Fig. 1). The ground deformation associated with the earthquake was also well recorded by the high resolution COSMO–SkyMed (CSK) satellite and Sentinel-1A satellite, providing independent static observations

Geodetic Constraints on San Francisco Bay Area Fault Slip Rates and Potential Seismogenic Asperities on the Partially Creeping Hayward Fault

The Hayward fault in the San Francisco Bay Area (SFBA) is sometimes considered unusual among continental faults for exhibiting significant aseismic creep during the interseismic phase of the seismic cycle while also generating sufficient elastic strain to produce major earthquakes. Imaging the spatial variation in interseismic fault creep on the Hayward fault is complicated because of the interseismic strain accumulation associated with nearby faults in the SFBA, where the relative motion between the Pacific plate and the Sierra block is partitioned across closely spaced subparallel faults. To estimate spatially variable creep on the Hayward fault, we interpret geodetic observations with a three-dimensional kinematically consistent block model of the SFBA fault system. Resolution tests reveal that creep rate variations with a length scale of \u3c15 km are poorly resolved below 7 km depth. In addition, creep at depth may be sensitive to assumptions about the kinematic consistency of fault slip rate models. Differential microplate motions result in a slip rate of 6.7 ± 0.8 mm/yr on the Hayward fault, and we image along-strike variations in slip deficit rate at ∼15 km length scales shallower than 7 km depth. Similar to previous studies, we identify a strongly coupled asperity with a slip deficit rate of up to 4 mm/yr on the central Hayward fault that is spatially correlated with the mapped surface trace of the 1868 MW = 6.9–7.0 Hayward earthquake and adjacent to gabbroic fault surfaces

Wood dynamics in headwater streams of the Colorado Rocky Mountains

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