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[Iris Ginzburg]

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42 printed pages


         THEORY OF CORRELATIONS IN STOCHASTIC NEURAL NETWORKS 


                           Iris Ginzburg

                 School of Physics and Astronomy
           Tel-Aviv University, Tel-Aviv 69978, Israel
                               and

                         Haim Sompolinsky

     Racah Institute of Physics and Center for Neural Computation
            Hebrew University, Jerusalem 91904, Israel
       and AT&T Bell Laboratories, Murray Hill, NJ 07974, USA

Submitted to Physical Review E, March 1994

ABSTRAT:
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 
difficult,
since the correlation between a pair of neurons is influenced 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 specific 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 
obey 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 effective interaction 
matrix are presented.  The effective interactions are the synaptic 
efficacies multiplied by the the gain of the postsynaptic neurons.
The time-delayed cross-correlations can be expressed as a sum of
exponentially decaying modes that correspond to the 
eigenvectors of the effective 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 effective 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 1 cube mm of rat neocortex.  Both the effect of 
dilution as well as the influence of a nearby bifurcation to an oscillatory 
states are demonstrated.


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