Collaborative Research: Geometry of Group Behaviors with Application to Fish Schooling

本奖项的研究目的是检验集体行为是群体运动背后固有的低维结构的表现这一假设。该项目将通过对相应流形的拓扑分析和对其定义的可测量属性的研究来对集体行为进行分类。本研究将建立计算方案，通过尊重底层测地线结构的无向图来模拟这种低维流形，并通过计算拓扑学和信息论的现代方法分析这些图。该项目主要研究斑马鱼普遍动物模型中的鱼群现象。实验和模拟将协同集成，以产生一个全面的数据集，用于验证所提出的数据驱动的动态系统框架，同时有助于生物群体的数学建模。如果成功，这个项目的结果将提供原则性和健壮的计算套件，用于从原始数据中描述生物群体的集体行为，这些数据可以被广泛的动物行为社区使用。在这个项目中开发的数学工具将通过提供基于大规模数据的模型验证工具和数据挖掘的图像处理工具，通过为复杂系统开发新的几何分类器，帮助建立网络动态系统的模型。此外，本项目开发的方法将产生新的计算工具，用于时间序列的误差分析和具有快速和缓慢时间尺度的动态系统的数据驱动研究。研究结果将整合到机械工程和数学的课程、教材和项目中，以培养本科生和研究生的创新思维和多学科思维。K-12学生和教师将通过课堂内外的正式和非正式学习经验，结合鱼类、流形和复杂性，介绍动态系统。

The research objective of this award is to test the hypothesis that collective behavior is a manifestation of an inherently low-dimensional structure underlying the group motion. This project will classify collective behaviors through the topological analysis of the corresponding manifolds and the study of measurable properties defined thereon. This research will establish computational schemes to model such low-dimensional manifolds through undirected graphs which respect underlying geodesic structures and analyze these graphs through modern methods from computational topology and information theory. The project focuses on the fish schooling phenomenon in the universal animal model of zebrafish. Experiments and simulations will be synergistically integrated to produce a comprehensive dataset for validation of the proposed data-driven dynamical systems framework while contributing to mathematical modeling of biological groups.If successful, the results of this project will provide principled and robust computational suites for the characterization of collective behavior in biological groups from raw data that can be used by a broad animal behavior community. Mathematical tools developed in this project will aid modeling of networked dynamical systems, by providing tools for model validation based on large-scale data, and image processing for data mining, by developing novel geometric classifiers for complex systems. In addition, methods developed in this project will result into new computational tools for error analysis of time-series and data-driven studies of dynamical systems with fast and slow time scales. The results will be integrated in curricula, course materials, and projects for courses in Mechanical Engineering and Mathematics to fostering outside-the-box and multidisciplinary thinking in undergraduate and graduate students. K-12 students and teachers will be introduced to dynamical systems through formal and informal learning experiences in and out of the classroom combining fish, manifolds, and complexity.