  In the last decade, several models with network adaptive mechanisms (link
deletion-creation, dynamic synapses, dynamic gains) have been proposed as
examples of self-organized criticality (SOC) to explain neuronal avalanches.
However, all these systems present stochastic oscillations hovering around the
critical region that are incompatible with standard SOC. This phenomenology has
been called self-organized quasi-criticality (SOqC). Here we make a linear
stability analysis of the mean field fixed points of two SOqC systems: a fully
connected network of discrete time stochastic spiking neurons with firing rate
adaptation produced by dynamic neuronal gains and an excitable cellular
automata with depressing synapses. We find that the fixed point corresponds to
a stable focus that loses stability at criticality. We argue that when this
focus is close to become indifferent, demographic noise can elicit stochastic
oscillations that frequently fall into the absorbing state. This mechanism
interrupts the oscillations, producing both power law avalanches and dragon
king events, which appear as bands of synchronized firings in raster plots. Our
approach differs from standard SOC models in that it predicts the coexistence
of these different types of neuronal activity.
