CAREER: Toward a second order description of neuronal networks

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

Understanding how the brain's neuronal networks perform computations remains a difficult challenge. The interaction of large populations of neurons leads to a complex repertoire of high-dimensional activity patterns that is difficult to analyze. The dimension of the space of connectivity patterns among neurons is likewise astronomical. To understand the behavior of neuronal networks, one would like to analyze the relationship between these connectivity patterns and neuronal activity. This project will develop an analytical approach to distill both network connectivity and activity down to first and second order statistics. This simplified framework will facilitate studying the relationships between key features of connectivity and activity. The project centers around a kinetic theory framework to represent populations of neurons by capturing the first and second order statistics among the states of neurons. The evolution of the kinetic theory equations will be determined by the first and second order statistics of the connectivity. An approach to estimating these first and second order connectivity statistics from experimental data will link the analysis with biological networks.

The central objective of this project is the development of mathematical tools to analyze the interrelationships between simplified descriptions of neuronal network connectivity and simplified descriptions of neuronal activity. When linked with experimental data, the analysis may provide insight into the behavior of the neuronal networks underlying the brain's computations. As this research may provide tools that help scientists better understand the effects of the degradation of neural connections, such as in neurodegenerative disease, it may eventually lead to strategies to mitigate or reverse these effects. Integrated into the project is interdisciplinary training in conjunction with the research as well as a plan to help young people discover exciting ways of applying mathematics to biology, with the goal of inspiring them to pursue further applications of mathematics to biology and other sciences.