RUI: Computational algebraic geometry and combinatorial algorithms for neuroscience and biological networks

这个项目的重点是计算数学在神经科学和系统生物学中的应用。除了以创造性的方式利用目前可用的方法外，主要目标是开发新的计算方法，以满足特定生物学应用的需要，特别是在系统生物学和神经科学方面。在系统生物学中，目标是通过研究网络环境中的相互作用而不是孤立地来理解复杂的生物过程。而在神经科学中，目标是更好地了解大脑中的神经连接，以及大脑如何与外部环境相互作用。在系统生物学方面，拟议项目中的工作将有助于理解两种类型的网络：生化反应网络，分子相互作用的确定性模型，以及蛋白质-蛋白质相互作用网络，过去十年收集、确认和改进的关系数据。在神经科学方面，重点将放在神经元网络、记录神经元之间联系的图解以及组合神经代码(一种离散化细胞放电数据的形式)上。所使用的技术将来自组合学和计算代数几何，这两个子领域在统计、物理、工程和生物等一系列领域中有应用。该提案有三个主要的研究部分。首先，将使用计算代数几何的技术和理论，如环面理想和Groebner基，来理解和可视化神经科学数据。其次，将开发具有固定属性的随机图抽样的新算法，以检验关于蛋白质-蛋白质相互作用和神经元网络的统计假设。第三，将开发计算消除理想的新算法，目标是将这些新方法应用于生化反应网络理论中的模型选择。

This project is focused on applications of computational mathematics to problems arising in neuroscience and systems biology. Beyond utilizing currently available methods in creative ways, the main aim is to develop new computational methods to meet the needs of specific biological applications, particularly in systems biology and neuroscience. In systems biology, the goal is to understand complicated biological processes by studying interactions in a network setting rather than in isolation. While in neuroscience, the goal is to better understand neural connections in the brain, and how the brain interacts with it's external environment. On the systems biology side, the work in the proposed project will help make sense of two types of networks: biochemical reaction networks, deterministic models for molecular interactions, and protein-protein interaction networks, relational data collected, confirmed, and refined over the past decade. On the neuroscience side, the focus will be on neuronal networks, schematics that record connections between neurons, and combinatorial neural codes, a form of discretized cell firing data.The techniques employed will come from combinatorics and computational algebraic geometry, two subfields with applications in an array of fields including statistics, physics, engineering, and biology. The proposal has three main research components. First, techniques and theory from computational algebraic geometry, such as toric ideals and Groebner bases, will be used to understand and visualize neuroscience data. Second, new algorithms for sampling random graphs with fixed properties will be developed for testing statistical hypotheses about protein-protein interaction and neuronal networks. Third, new algorithms for computing elimination ideals will be developed with the goal of applying these new methods to model selection in biochemical reaction network theory.