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.
In this project, we developed a new mathematical description of what it means to say that animals swarm. More generally, whether animals are swarming, herding, schooling, mobbing, or otherwise cooperating, the idea is that there is some coordination and coincidence of their behaviors. In terms of their actions and movements, this coincidence will be realized as the presence of low-dimensional manifolds in the underlying phase space of their configurations. In other words, if we know the motion of one animal, then the nearby animal will be functionally related to the first, rather than randomly and independently different. That is, like a dance, If one follows the other, then simply knowing the functional relationships is enough to predict one from the other. See Fig. 1. Formally this idea is stated that there is a manifold and all motions evolve on that manifold. With this simple but powerful definition of swarming, there follows computational methods to detect this prospect. See Fig. 2. Specifically, within the language of machine learning, and from wide ranging data sets and forms, from tracking to simply using video, we have demonstrated that the simplicity of this concept leads to powerful methods of detection. The suggestion moving forward would be then how might one design or control the manifold and the transverse stability of the manifold, and in so doing perhaps control the group behaviors.