  Cortical networks, in-vitro as well as in-vivo, can spontaneously generate a
variety of collective dynamical events such as network spikes, UP and DOWN
states, global oscillations, and avalanches. Though each of them have been
variously recognized in previous works as expressions of the excitability of
the cortical tissue and the associated nonlinear dynamics, a unified picture of
their determinant factors (dynamical and architectural) is desirable and not
yet available. Progress has also been partially hindered by the use of a
variety of statistical measures to define the network events of interest. We
propose here a common probabilistic definition of network events that, applied
to the firing activity of cultured neural networks, highlights the
co-occurrence of network spikes, power-law distributed avalanches, and
exponentially distributed `quasi-orbits', which offer a third type of
collective behavior. A rate model, including synaptic excitation and inhibition
with no imposed topology, synaptic short-term depression, and finite-size
noise, accounts for all these different, coexisting phenomena. We find that
their emergence is largely regulated by the proximity to an oscillatory
instability of the dynamics, where the non-linear excitable behavior leads to a
self-amplification of activity fluctuations over a wide range of scales in
space and time. In this sense, the cultured network dynamics is compatible with
an excitation-inhibition balance corresponding to a slightly sub-critical
regime. Finally, we propose and test a method to infer the characteristic time
of the fatigue process, from the observed time course of the network's firing
rate. Unlike the model, possessing a single fatigue mechanism, the cultured
network appears to show multiple time scales, signalling the possible
coexistence of different fatigue mechanisms.
