I am interested primarily in the modeling and analysis of the dynamics of networks of active cells, with application to specific problems in neuroscience and mathematical biology. My work in this area has been focused on the question of how interacting neurons detect features in visual scenes. This interdisciplinary work involved creating biologically specific models where I could compare computational results with measured data and formulating and analyzing simple qualitative model networks for mathematical understanding.

I have developed a large-scale numerical model of primary visual cortex and have shown how, within the architectural constraints of V1, a single network circuit can provide a unified account of spatial summation and orientation selectivity. Furthermore, I showed that the generation of orientation selectivity in highly coupled networks can be related to bifurcations seen in much simpler networks of all-to-all coupled integrate-and-fire neurons (see Fig. 1). Through asymptotic reduction of the neuronal population dynamics, I show that a firing rate equation (a generalization of the Wilson-Cowan equations) can explain the bifurcations in the all-to-all network. Furthermore, within this reduced model, there is a subcritical bifurcation in the infinite size limit, which, in the finite-size case, allows the I & F network to be in a critical state. These results imply that network-based computations can take advantage of the structured recurrent coupling between neurons in the network without resorting to biologically complex properties of single neurons. It is conceivable that these features can be implemented in biological circuits in general.

Figure 1. Firing rate vs. G{Input} curves for four networks with different N = 25, 50, 100, and 200. These curves were obtained by first increasing and then decreasing the strength of the feedforward excitation G{Input}. The larger networks (N=100 and 200) show strong hysteresis in the feedforward excitation. This hystersis can be tuned to a near critical curve at N=50.