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 activity. In general, however the interpretation of the observed correlations is diicult, since the correlation between a pair of neurons is innuenced not only by the direct interaction between them but also by the dynamic state of the entire network to which they belong. Thus, a comparison between the observed correlations and the predictions from speciic model networks is needed. In this paper we develop the theory of neuronal correlation functions in large networks comprising of several highly connected subpopulations, and obeying stochastic dynamic rules. When the networks are in asynchronous states, the cross-correlations are relatively weak, i.e., their amplitude relative to that of the auto-correlations is of order of 1=N, N being the size of the interacting populations. Using the weakness of the cross-correlations, general equations which express the matrix of cross-correlations in terms of the mean neuronal activities, and the eeective interaction matrix are presented. The eeective interactions are the synaptic eecacies multiplied by the the gain of the postsynaptic neurons. The time-delayed cross-correlation matrix can be expressed as a sum of exponentially decaying modes that correspond to the (non-orthogonal) eigenvectors of the eeective interaction matrix. The theory is extended to networks with random connectivity, such as randomly dilute networks. This allows for the comparison between the contribution from the internal common input and that from the direct interactions to the correlations of monosynaptically coupled pairs. A closely related quantity is the linear response of the neurons to external time-dependent perturbations. We derive the form of the dynamic linear response function of neurons in the above architecture, in terms of the eigenmodes of the eeective interaction matrix. The behavior of the correlations and the linear response when the system is near a bifurcation point is analyzed. Near a saddle-node bifurcation the correlation matrix is dominated by a single slowly decaying critical mode. Near a Hopf-bifurcation the correlations exhibit weakly damped sinusoidal oscillations. The general theory is applied to the case of randomly dilute network consisting of excitatory and inhibitory subpopulations, using parameters that mimic the local circuit of 1mm 3 of rat neocortex. Both the eeect of dilution as well as the innuence of a nearby bifurcation to an oscillatory states are demonstrated.

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