  The existence of power-law distributions is only a first requirement in the
validation of the critical behavior of a system. Long-range spatio-temporal
correlations are fundamental for the spontaneous neuronal activity to be the
expression of a system acting close to a critical point. This chapter focuses
on temporal correlations and avalanche dynamics in the spontaneous activity of
cortex slice cultures and in the resting fMRI BOLD signal. Long-range
correlations are investigated by means of the scaling of power spectra and of
Detrended Fluctuations Analysis. The existence of 1/f decay in the power
spectrum, as well as of power-law scaling in the root mean square fluctuations
function for the appropriate balance of excitation and inhibition suggests that
long-range temporal correlations are distinctive of "healthy brains". The
corresponding temporal organization of neuronal avalanches can be dissected by
analyzing the distribution of inter-event times between successive events. In
rat cortex slice cultures this distribution exhibits a non-monotonic behavior,
not usually found in other natural processes. Numerical simulations provide
evidences that this behavior is a consequence of the alternation between states
of high and low activity, leading to a dynamic balance between excitation and
inhibition that tunes the system at criticality. In this scenario, inter-times
show a peculiar relation with avalanche sizes, resulting in a hierarchical
structure of avalanche sequences. Large avalanches correspond to low-frequency
oscillations, and trigger cascades of smaller avalanches that are part of
higher frequency rhythms. The self-regulated balance of excitation and
inhibition observed in cultures is confirmed at larger scales, i.e. on fMRI
data from resting brain activity, and appears to be closely related to critical
features of avalanche activity.
