This paper studies the dynamics of a network of sparsely connected inhibitory integrate-and-fire neurons in a regime where individual neurons fire irregularly and at low rates. In the limit as the number of neurons $ N \rightarrow \infty $, the network exhibits a sharp transition between a stationary and an oscillatory global activity regime, where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is mainly controlled by synaptic times, but also depends on the characteristics of the external input. In large but finite networks, global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of system parameters in these two different regimes. The results are in good agreement with numerical simulations. The paper analyzes a sparsely connected network of identical inhibitory integrate-and-fire neurons to understand the coexistence of individual neurons with low firing rates and fast collective oscillations. The analysis shows that the global oscillation only appears above a well-defined parameter threshold. A linear stability analysis shows that the time-independent solution becomes unstable when the strength of recurrent inhibition exceeds a critical level. When this critical level is reached, the stationary solution becomes unstable and an oscillatory solution develops via a Hopf bifurcation. The time scale of the period of the corresponding global oscillations is set by a synaptic time, independently of the firing rate of individual neurons, but the period's precise value also depends on the characteristics of the external input. The analysis is then pushed to higher orders, leading to a reduced evolution equation describing the network's collective dynamics. The effects of the finite size of the network are also discussed. It is shown that having a large but finite number of neurons gives a small stochastic component to the collective evolution equation. As a result, cross-correlations in a finite network present damped oscillations both above and below the critical inhibition level. Below the critical level, the noise controls the oscillation amplitude, which decreases as the number of neurons increases. Above the critical level, the main effect of the noise is to produce a phase diffusion of the global oscillation. An increase in the number of neurons results in an increase of the global oscillation coherence time and a reduced damping in average cross-correlations. The effect of some simplifying assumptions is studied. The effect of allowing variability in synaptic times and number of synaptic connections from neuron to neuron is discussed. The effect of introducing a more detailed description of postsynaptic currents into the model is also considered. The technical aspects of the computations are detailed in several appendices. The paper also discusses the effect of inhomogeneous synaptic times and connectivity on the instability line in the plane $ (\mu_{ext}, \sigma_{ext}) $. The results show that the critical line is sensitive to the distribution of synaptic times and that the stationary state can be made stable with appropriate distributions. The analysis also shows that the oscillationThis paper studies the dynamics of a network of sparsely connected inhibitory integrate-and-fire neurons in a regime where individual neurons fire irregularly and at low rates. In the limit as the number of neurons $ N \rightarrow \infty $, the network exhibits a sharp transition between a stationary and an oscillatory global activity regime, where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is mainly controlled by synaptic times, but also depends on the characteristics of the external input. In large but finite networks, global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of system parameters in these two different regimes. The results are in good agreement with numerical simulations. The paper analyzes a sparsely connected network of identical inhibitory integrate-and-fire neurons to understand the coexistence of individual neurons with low firing rates and fast collective oscillations. The analysis shows that the global oscillation only appears above a well-defined parameter threshold. A linear stability analysis shows that the time-independent solution becomes unstable when the strength of recurrent inhibition exceeds a critical level. When this critical level is reached, the stationary solution becomes unstable and an oscillatory solution develops via a Hopf bifurcation. The time scale of the period of the corresponding global oscillations is set by a synaptic time, independently of the firing rate of individual neurons, but the period's precise value also depends on the characteristics of the external input. The analysis is then pushed to higher orders, leading to a reduced evolution equation describing the network's collective dynamics. The effects of the finite size of the network are also discussed. It is shown that having a large but finite number of neurons gives a small stochastic component to the collective evolution equation. As a result, cross-correlations in a finite network present damped oscillations both above and below the critical inhibition level. Below the critical level, the noise controls the oscillation amplitude, which decreases as the number of neurons increases. Above the critical level, the main effect of the noise is to produce a phase diffusion of the global oscillation. An increase in the number of neurons results in an increase of the global oscillation coherence time and a reduced damping in average cross-correlations. The effect of some simplifying assumptions is studied. The effect of allowing variability in synaptic times and number of synaptic connections from neuron to neuron is discussed. The effect of introducing a more detailed description of postsynaptic currents into the model is also considered. The technical aspects of the computations are detailed in several appendices. The paper also discusses the effect of inhomogeneous synaptic times and connectivity on the instability line in the plane $ (\mu_{ext}, \sigma_{ext}) $. The results show that the critical line is sensitive to the distribution of synaptic times and that the stationary state can be made stable with appropriate distributions. The analysis also shows that the oscillation