  The observation of apparent power-laws in neuronal systems has led to the
suggestion that the brain is at, or close to, a critical state and may be a
self-organised critical system. Within the framework of self-organised
criticality a separation of timescales is thought to be crucial for the
observation of power-law dynamics and computational models are often
constructed with this property. However, this is not necessarily a
characteristic of physiological neural networks - external input does not only
occur when the network is at rest/a steady state. In this paper we study a
simple neuronal network model driven by a continuous external input (i.e.\ the
model does not have a separation of timescales) and analytically tuned to
operate in the region of a critical state (it reaches the critical regime
exactly in the absence of input - the case studied in the companion paper to
this article). The system displays avalanche dynamics in the form of cascades
of neuronal firing separated by periods of silence. We observe partial
scale-free behaviour in the distribution of avalanche size for low levels of
external input. We analytically derive the distributions of waiting times and
investigate their temporal behaviour in relation to different levels of
external input, showing that the system's dynamics can exhibit partial
long-range temporal correlations. We further show that as the system approaches
the critical state by two alternative `routes', different markers of
criticality (partial scale-free behaviour and long-range temporal correlations)
are displayed. This suggests that signatures of criticality exhibited by a
particular system in close proximity to a critical state are dependent on the
region in parameter space at which the system (currently) resides.
