Determining the correlational structure in large neuronal networks is critical for understanding everything from information storage to network dynamics to computing to learning. Historically, the main approach has been experimental, but the amount of data required to accurately assess correlations scales very badly with the number of neurons. An alternative approach is to directly compute the correlations as a function of network connectivity and single neuron properties. So far this computational approach has been applied solely to weakly connected networks -- networks in which the connection strength is inversely proportional to the number of connections per neuron [1-3]. Here we consider the much more realistic regime in which there is strong, but random, background connectivity, and weak, structured connectivity. Using a simple model, we find an analytic expression that relates correlations to the connectivity matrix. Our main result is that the background connectivity, which is random and therefore does not play a major computational role, can have a large effect on the correlational structure. This makes it hard to relate measured correlations to the part of the connectivity matrix we care about -- the structured part.
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