When animals are asleep, their brain is far from dormant. During slow wave sleep, as well as under certain types of anesthesia, the neocortex shows spontaneous patterns of synchronized activity in which neurons alternate between periods of silence, referred to as 'Down' phases and periods of tonic network activity, so-called 'Up' phases. The mechanisms underlying these Up-Down transitions remain unclear. We have previously reported that spontaneous activity in somatosensory cortex of urethane-anesthetized rats show that Up and Down durations are very irregular and exhibit positive serial correlations: we find mean durations of 200-500 ms, coefficients of variation of about 0.7 and serial correlation coefficients of 0.1-0.2. Furthermore, the mean firing rate of the excitatory cells across the Up phase show no significant decay. We aim to reproduce the experimentally observed data by studying a spiking network model. Specifically, we consider an all-to-all coupled network of excitatory and inhibitory leaky integrate-and-fire neurons driven by noisy external inputs, and include a spike-frequency adaptation mechanism for the excitatory neurons. The network operates in a novel bistable regime in which firing rates are kept low in the Up-state via powerful self-inhibition of excitatory neurons through a strong EI-IE loop. Importantly, in order to match our in-vivo data, the external noise must be strong to trigger Down to Up transitions and the adaptation sufficiently weak, but not zero, to introduce serial correlations. Our findings provide the basis for a new mechanism underlying Up-Down alternations.