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 derive coupling equations for the populations based on first and second order statistics of 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. We implement a kinetic theory representation of a simple feed-forward network and demonstrate that the 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.
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Acknowledgements This work was supported in part by NSF grant DMS 0719724 (DQN) and NIH training grant R90 DK71500 (CYL). We thank Dan Tranchina, Brent Doiron, Michael Buice, Carson Chow, and Hide Câteau for helpful discussions.
- Connectivity patterns
- Maximum entropy
- Neuronal network
- Population density
- Syn-fire chain