  Many systems in Nature exhibit avalanche dynamics with scale-free features. A
general scaling theory has been proposed for critical avalanche profiles in
crackling noise, predicting the collapse onto a universal avalanche shape, as
well as the scaling behaviour of the activity power spectrum as Brown noise.
Recently, much attention has been given to the profile of neuronal avalanches,
measured in neuronal systems in vitro and in vivo. Although a universal profile
was evidenced, confirming the validity of the general scaling theory, the
parallel study of the power spectrum scaling under the same conditions was not
performed. The puzzling observation is that in the majority of healthy neuronal
systems the power spectrum exhibits a behaviour close to $1/f$, rather than
Brown, noise. Here we perform a numerical study of the scaling behaviour of
avalanche shape and power spectrum for a model of integrate and fire neurons
with a short-term plasticity parameter able to tune the system to criticality.
We confirm that, at criticality, the average avalanche size and the avalanche
profile fulfill the general avalanche scaling theory. However, the power
spectrum consistently exhibits Brown noise behaviour, for both fully excitatory
networks and systems with 30\% inhibitory networks. Conversely, a behaviour
closer to $1/f$ noise is observed in systems slightly off-criticality. Results
suggest that the power spectrum is a good indicator to determine how close
neuronal activity is to criticality.
