CAREER: Mathematical Questions Arising from Neural Systems - Research and Education

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The neural circuits of the brain are central to human actions, human thoughts, and human feelings. Because these circuits are so complicated and the details of the circuitry vary from person to person, our understanding of the brain remains limited. In this project the Principal Investigator will study fundamental questions relating circuit structure and dynamics in the brain. One thing we do know about the brain is that neural circuits change, from infancy through young adulthood and into our later years. Some of these changes are part of healthy developmental transitions, such as an infant becoming able to sleep through the night or an adolescent entering puberty; other changes are pathological, such as the loss of a particular group of neurons in the sleep disorder narcolepsy. In close collaboration with laboratory scientists, the Principal Investigator seeks both to characterize these neural changes and to determine the underlying neural mechanisms. She will focus on three areas: (1) puberty, (2) sleep-wake regulation, and (3) systems of neurotransmitters such as dopamine and serotonin. Changes in the cells of neural circuits can alter the activity of these neurotransmitters, with sometimes profound effects on important human behaviors such as feeding, mood, and social interactions. The Principal Investigator will carry out these investigations by studying mathematical models of different brain circuits, using mathematics that she is developing for this purpose.Mathematical investigations will include developing the theory of mixed mode oscillations, studying how stochastic dynamics on graphs change as the graph changes, using spectral analysis to understand how graph structure determines the general character of the distribution of absorption times of Markov processes, and developing the theory of doubly stochastic processes. These investigations will give understanding of the normal functioning of particular neural circuits and also help to explain what goes wrong in neuropsychiatric illnesses and neurodegenerative diseases. The Investigator will use her extensive experience in teaching, recruiting, and mentoring students to design a new pipeline program in Ohio that will encourage students to choose careers in science.