A kinetic theory approach to capturing interneuronal correlation in feed-forward networks.

University of Minnesota Ph.D. dissertation. July 2009. Major: Mathematics. Advisor: Duane Q. Nykamp. 1 computer file (PDF); x, 104 pages, appendices A-B. Ill. (some col.)We present an approach for using kinetic theory to capture first and second order statistics of neuronal activity. We coarse grain neuronal networks into populations of neurons and calculate the population average firing rate and output cross-correlation in response to time varying correlated input. We initially derive coupling equations for the populations based only on first and second order statistics of neuronal activity and the network connectivity. This coupling scheme is based on the hypothesis that second order statistics of the network connectivity are sufficient to determine second order statistics of neuronal activity. Using this coupling scheme, we implement a kinetic theory representation of a simple feed-forward network and demonstrate that this kinetic theory model captures key aspects of the emergence and propagation of correlations in the network, as long as the correlations do not become too strong. By analyzing the correlated activity of feed-forward networks with a variety of connectivity patterns, we provide evidence supporting our hypothesis of the sufficiency of second order connectivity statistics. To improve the kinetic theory performance under high correlation in feed-forward networks, we propose an inference method to estimate the rate of synchronous firing by more than two neurons. Then we include the effect of such events in the evolution of the postsynaptic populations by deriving improved coupling equations for populations. With these improved coupling equations, we obtain an improved kinetic theory representation of the simple feed-forward network. To implement it, we make truncation approximations at different levels in the input and demonstrate that our improved kinetic theory model can capture the behavior of first and second order firing activity under higher correlation