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We quantify the finite size effects in a stochastic network made up of rateneurons, for several kinds of recurrent connectivity matrices. This analysis isperformed by means of a perturbative expansion of the neural equations, wherethe perturbative parameters are the intensities of the sources of randomness inthe system. In detail, these parameters are the variances of the background orinput noise, of the initial conditions and of the distribution of the synapticweights. The technique developed in this article can be used to study systemswhich are invariant under the exchange of the neural indices and it allows usto quantify the correlation structure of the network, in terms of pairwise andhigher order correlations between the neurons. We also determine the relationbetween the correlation and the external input of the network, showing thatstrong signals coming from the environment reduce significantly the amount ofcorrelation between the neurons. Moreover we prove that in general thephenomenon of propagation of chaos does not occur, even in the thermodynamiclimit, due to the correlation structure of the 3 sources of randomnessconsidered in the model. Furthermore, we show that the propagation of chaosdoes not depend only on the number of neurons in the network, but also andmainly on the number of incoming connections per neuron. To conclude, we provethat for special values of the parameters of the system the neurons becomeperfectly correlated, a phenomenon that we have called stochasticsynchronization. These discoveries clearly prevent the use of the mean-fieldtheory in the description of the neural network.

Finite size effects in the correlation structure of stochastic neural  networks: analysis of different connectivity matrices and failure of the  mean-field theory