  Neural dynamics is triggered by discrete synaptic inputs of finite amplitude.
However, the neural response is usually obtained within the diffusion
approximation (DA) representing the synaptic inputs as Gaussian noise. We
derive a mean-field formalism encompassing synaptic shot-noise for sparse
balanced networks of spiking neurons. For low (high) external drives (synaptic
strengths) irregular global oscillations emerge via continuous and hysteretic
transitions, correctly predicted by our approach, but not from the DA. These
oscillations display frequencies in biologically relevant bands.
