Revealing pairwise coupling in linear-nonlinear networks

Through an asymptotic analysis of a simple network, we derive an estimate of the coupling between a pair of units when all other units are unobservable. The analysis is based on a model where the response of each unit is a linear-nonlinear function of a white noise stimulus. The results accurately determine the coupling when all unmeasured units respond to the stimulus differently than the measured pair. To account for the possibility of unmeasured units similar to the measured pair, we cast our results in the framework of "subpopulations," which are defined as a group of units who respond to the stimulus similarly. We demonstrate that we can determine when correlations between two units are caused by a connection between their subpopulations, although the precise identity of the units involved in the connection may remain ambiguous. The result is rigorously valid only when the coupling is sufficiently weak to justify a second-order approximation in the coupling strength. We demonstrate through simulations that the results are still valid even with stronger coupling and in the presence of some deviations from the linear-nonlinear model. The analysis is presented in terms of neuronal networks, although the general framework is more widely applicable.