  Power laws in nature are considered to be signatures of complexity. The
theory of self-organized criticality (SOC) was proposed to explain their
origins. A longstanding principle of SOC is the \emph{separation of timescales}
axiom. It dictates that external input is delivered to the system at a much
slower rate compared to the timescale of internal dynamics. The statistics of
neural avalanches in the brain was demonstrated to follow a power law,
indicating closeness to a critical state. Moreover, criticality was shown to be
a beneficial state for various computations leading to the hypothesis, that the
brain is a SOC system. However, for neuronal systems that are constantly
bombarded by incoming signals, separation of timescales assumption is
unnatural. Recently it was experimentally demonstrated that a proper correction
of the avalanche detection algorithm to account for the increased drive during
task performance leads to a change of the power-law exponent from $1.5$ to
approximately $1.3$, but there is so far no theoretical explanation for this
change. Here we investigate the importance of timescales separation, by partly
abandoning it in various models. We achieve it by allowing for external input
during the avalanche, without compromising the separation of avalanches. We
develop an analytic treatment and provide numerical simulations of a simple
neuronal model.
