  We consider an excitatory random network of leaky integrate-and-fire pulse
coupled neurons. The neurons are connected as in a directed Erd\"os-Renyi graph
with average connectivity $<k>$ scaling as a power law with the number of
neurons in the network. The scaling is controlled by a parameter $\gamma$,
which allows to pass from massively connected to sparse networks and therefore
to modify the topology of the system. At a macroscopic level we observe two
distinct dynamical phases: an Asynchronous State (AS) corresponding to a
desynchronized dynamics of the neurons and a Partial Synchronization (PS)
regime associated with a coherent periodic activity of the network. At low
connectivity the system is in an AS, while PS emerges above a certain critical
average connectivity $<k>_c$. For sufficiently large networks, $<k>_c$
saturates to a constant value suggesting that a minimal average connectivity is
sufficient to observe coherent activity in systems of any size irrespectively
of the kind of considered network: sparse or massively connected. However, this
value depends on the nature of the synapses: reliable or unreliable. For
unreliable synapses the critical value required to observe the onset of
macroscopic behaviors is noticeably smaller than for reliable synaptic
transmission. Due to the disorder present in the system, for finite number of
neurons we have inhomogeneities in the neuronal behaviors, inducing a weak form
of chaos, which vanishes in the thermodynamic limit. In such a limit the
disordered systems exhibit regular (non chaotic) dynamics and their properties
correspond to that of a homogeneous fully connected network for any
$\gamma$-value. Apart for the peculiar exception of sparse networks, which
remain intrinsically inhomogeneous at any system size.
