  We present a detailed analysis of the dynamical regimes observed in a
balanced network of identical Quadratic Integrate-and-Fire (QIF) neurons with a
sparse connectivity for homogeneous and heterogeneous in-degree distribution.
Depending on the parameter values, either an asynchronous regime or periodic
oscillations spontaneously emerge. Numerical simulations are compared with a
mean field model based on a self-consistent Fokker-Planck equation (FPE). The
FPE reproduces quite well the asynchronous dynamics in the homogeneous case by
either assuming a Poissonian or renewal distribution for the incoming spike
trains. An exact self consistent solution for the mean firing rate obtained in
the limit of infinite in-degree allows identifying balanced regimes that can be
either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in
terms of circular cumulants is also considered. Two cumulants suffice to
reproduce the transition scenario observed in the network. The emergence of
periodic collective oscillations is well captured both in the homogeneous and
heterogeneous setups by the mean field models upon tuning either the
connectivity, or the input DC current. In the heterogeneous situation we
analyze also the role of structural heterogeneity.
