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We analyze extensively the temporal properties of the train of spikes emitted by a simple model neuron as a function of the statistics of the synaptic input. In particular we focus on the asynchronous case, in which the synaptic inputs are random and uncorrelated. We show that the NMDA component acts as a non-stationary input that varies on longer time scales than the inter-spike intervals. In the subthreshold regime, this can increase dramatically the coefficient of variability (bringing it beyond one). The analysis provides also simple guidelines for searching parameters that maximize irregularity.