  The Wilson-Cowan model constitutes a paradigmatic approach to understanding
the collective dynamics of networks of excitatory and inhibitory units. It has
been profusely used in the literature to analyze the possible phases of neural
networks at a mean-field level, e.g., assuming large fully-connected networks.
Moreover, its stochastic counterpart allows one to study fluctuation-induced
phenomena, such as avalanches. Here, we revisit the stochastic Wilson-Cowan
model paying special attention to the possible phase transitions between
quiescent and active phases. We unveil eight possible types of phase
transitions, including continuous ones with scaling behavior belonging to known
universality classes -- such as directed percolation and tricritical directed
percolation -- as well as novel ones. In particular, we show that under some
special circumstances, at a so-called Hopf tricritical directed percolation
transition, rather unconventional behavior including an anomalous breakdown of
scaling emerges. These results broaden our knowledge of the possible types of
critical behavior in networks of excitatory and inhibitory units and are of
relevance to understanding avalanche dynamics in actual neuronal recordings.
From a more general perspective, these results help extend the theory of
non-equilibrium phase transitions into quiescent or absorbing states.
