This award supports fundamental research into mathematics that will enable a better understanding of the connection between neural rhythms and cognitive function. The research represents a necessary first step in the development of new treatment options for brain disorders caused by abnormal brain rhythms including schizophrenia, autism spectrum disorder, attention deficit hyperactivity disorder, and Parkinson's disease, promoting science and advancing national prosperity. Results of the project will contribute to the understanding of the biophysical mechanisms that support short-term memory, enabling a better understanding of how patients with disorders such as Alzheimer’s disease lose their ability to remember where they are. The award will support graduate students and researchers at the interface of neuroscience, mathematics, and engineering. The research plan includes two outreach efforts to encourage high school students to pursue careers in engineering and mathematics: 1) Week-long, hands-on research experiences with activities coordinated by the University of Tennessee, and 2) National mathematical modeling contests for regional high school and undergraduate students at the University of Florida. This project will advance the mathematical understanding of emergent behaviors in populations of coupled neural oscillators. Due to the size and dynamical complexity of most neural systems, model order reduction is often an imperative first step for mathematical analy