  We study the statistical properties of the stationary firing-rate states of a
neural network model with quenched disorder. The model has arbitrary size,
discrete-time evolution equations and binary firing rates, while the topology
and the strength of the synaptic connections are randomly generated from known,
generally arbitrary, probability distributions. We derived semi-analytical
expressions of the occurrence probability of the stationary states and the mean
multistability diagram of the model, in terms of the distribution of the
synaptic connections and of the external stimuli to the network. Our
calculations rely on the probability distribution of the bifurcation points of
the stationary states with respect to the external stimuli, which can be
calculated in terms of the permanent of special matrices, according to extreme
value theory. While our semi-analytical expressions are exact for any size of
the network and for any distribution of the synaptic connections, we also
specialized our calculations to the case of statistically-homogeneous
multi-population networks. In the specific case of this network topology, we
calculated analytically the permanent, obtaining a compact formula that
outperforms of several orders of magnitude the
Balasubramanian-Bax-Franklin-Glynn algorithm. To conclude, by applying the
Fisher-Tippett-Gnedenko theorem, we derived asymptotic expressions of the
stationary-state statistics of multi-population networks in the
large-network-size limit, in terms of the Gumbel (double exponential)
distribution. We also provide a Python implementation of our formulas and some
examples of the results generated by the code.
