神经编码、神经数据分析和神经网络的组合几何学
Combinatorial geometry of neural codes, neural data analysis, and neural networks
摘要：
本论文探讨离散几何在数学神经科学中的应用。我们从凸神经代码开始，它模拟了海马位置细胞和其他具有凸感受野的神经元的活动。在第四章中，我们介绍了一种约束代码凸实现的工具--强制序，并使用它来构造没有局部障碍的非凸代码的新例子。在第五章中，我们将定向拟阵与凸神经码联系起来，证明了一个码有凸多面体的实现当且仅当它是一个可表示的定向拟阵在神经码态射下的像。我们还表明，确定一个代码是否是凸的，至少是困难的，因为确定一个定向拟阵是否是可表示的，这意味着确定一个代码是否是凸的问题是NP-难的。接下来，我们转向矩阵的基本秩的问题。这个问题的动机是确定的（神经）数据已被破坏的未知单调变换的维数的问题。在第六章中，我们介绍了两种计算底层秩的工具：最小结点和Radon秩。我们应用这些来分析钙成像数据从斑马鱼幼虫。在第7章中，我们将更详细地探讨底层秩，建立与定向拟阵理论的联系，并证明计算底层秩也是NP难的。最后，我们研究了阈值线性网络（TLN）的动力学，TLN是神经回路活动的一个简单模型。在第九章中，我们描述了一个阈值线性网络的零倾排列，并证明了它的一个腔室子集是一个吸引集。在第10章中，我们将重点介绍组合阈值线性网络（CTLN），它是从有向图中定义的TLN。我们证明了如果CTLN的图是有向无圈图，则CTLN的所有轨迹都趋近于一个不动点。
英文摘要：
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.
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