  Asynchronous irregular (AI) and critical states are two competing frameworks
proposed to explain spontaneous neuronal activity. Here, we propose a
mean-field model with simple stochastic neurons that generalizes the
integrate-and-fire network of Brunel (2000). We show that the point with
balanced inhibitory/excitatory synaptic weight ratio $g_c \approx 4$
corresponds to a second order absorbing phase transition usual in
self-organized critical (SOC) models. At the synaptic balance point $g_c$, the
network exhibits power-law neuronal avalanches with the usual exponents,
whereas for nonzero external field the system displays the four usual
synchronicity states of balanced networks. We add homeostatic inhibition and
firing rate adaption and obtain a self-organized quasi-critical balanced state
with avalanches and AI-like activity. Our model might explain why different
inhibition levels are obtained in different experimental conditions and for
different regions of the brain, since at least two dynamical mechanisms are
necessary to obtain a truly balanced state, without which the network may hover
in different regions of the presented theoretical phase diagram.
