Neural dynamics is determined by the transmission of discrete synaptic pulses (synaptic shot noise) among neurons. However, the neural responses are usually obtained within the diffusion approximation modeling synaptic inputs as continuous Gaussian noise. Here we present a rigorous mean-field theory that encompasses synaptic shot noise for sparse balanced inhibitory neural networks driven by an excitatory drive. Our theory predicts alternative dynamical regimes, in agreement with numerical simulations, which are not captured by the classical diffusion approximation. Notably, these regimes feature self-sustained global oscillations emerging at low connectivity (in-degree) via either continuous or hysteretic transitions and characterized by irregular neural activity, as expected for balanced dynamics. For sufficiently weak (strong) excitatory drive (inhibitory feedback) the transition line displays a peculiar reentrant shape revealing the existence of global oscillations at low and high in-degrees, separated by an asynchronous regime at intermediate levels of connectivity. The mechanisms leading to the emergence of these global oscillations are distinct: drift-driven at high connectivity and cluster activation at low connectivity. The frequency of these two kinds of global oscillations can be varied from slow (10. 16em{0ex}Hz) to fast (1000. 16em{0ex}Hz) without altering their microscopic and macroscopic features by adjusting the excitatory drive and the synaptic inhibition strength in a prescribed way. Furthermore, the cluster-activated oscillations at low in-degrees could correspond to the rhythms reported in mammalian cortex and hippocampus and attributed to ensembles of inhibitory neurons sharing few synaptic connections Buzs\'aki and Wang, Annu. Rev. Neurosci. 35, 203 (2012).
Goldobin et al. (Tue,) studied this question.