THEORY OF CORRELATIONS IN STOCHASTIC NEURAL NETWORKS

One of the main experimental tools in probing the interactions between neurons has been the measurement of the correlations in their activit y. In general, however the interpretation of the observed correlations is difficult since the correlation between a pair of neurons is influ enced not only by the direct interaction between them but also by the dynamic state of the entire network to which they belong. Thus a compa rison between the observed correlations and the predictions from speci fic model networks is needed. In this paper we develop a theory of neu ronal correlation functions in large networks comprising several highl y connected subpopulations and obeying stochastic dynamic rules. When the networks are in asynchronous states, the cross correlations are re latively weak, i.e., their amplitude relative to that of the autocorre lations is of order of 1/N, N being the size of the interacting popula tions. Using the weakness of the cross correlations, general equations that express the matrix of cross correlations in terms of the mean ne uronal activities and the effective interaction matrix are presented. The effective interactions are the synaptic efficacies multiplied by t he gain of the postsynaptic neurons. The time-delayed cross-correlatio n matrix can be expressed as a sum of exponentially decaying modes tha t correspond to the (nonorthogonal) eigenvectors of the effective inte raction matrix. The theory is extended to networks with random connect ivity, such as randomly dilute networks. This allows for a comparison between the contribution from the internal common input and that from the direct interactions to the correlations of monosynaptically couple d pairs. A closely related quantity is the linear response of the neur ons to external time-dependent perturbations. We derive the form of th e dynamic linear response function of neurons in the above architectur e in terms of the eigenmodes of the effective interaction matrix. The behavior of the correlations and the linear response when the system i s near a bifurcation point is analyzed. Near a saddle-node bifurcation , the correlation matrix is dominated by a single slowly decaying crit ical mode. Near a Hopf bifurcation the correlations exhibit weakly dam ped sinusoidal oscillations. The general theory is applied to the case of a randomly dilute network consisting of excitatory and inhibitory subpopulations, using parameters that mimic the local circuit of 1 mm( 3) of the rat neocortex. Both the effect of dilution as well as the in fluence of a nearby bifurcation to an oscillatory state are demonstrat ed.