We explore the mechanism of synchronized bursting activity with frequency of about 10 Hz that appears in cortical tissues at low extracellular magnesium concentration [Mg²⁺]_o. We hypothesize that this activity is persistent, namely coexists with the quiescent state, and depends on slow NMDA conductances. To explore this hypothesis, we construct and investigate a conductance-based model of excitatory cortical networks. Population bursting activity can persist for physiological values of the NMDA decay time constant (~100 ms). Neurons are synchronized at the time scale of bursts, but not of single spikes. A reduced model of a cell coupled to itself can encompass most of this highly synchronized network behavior and is analyzed using the fast-slow method. Synchronized bursts appear for intermediate values of the NMDA conductance g_NMDA if NMDA conductances are not too fast. Regular spiking activity appears for larger g_NMDA. If the single cell is a conditional burster, persistent synchronized bursts become more robust. Weakly synchronized states appear for zero AMPA conductance g_AMPA. Enhancing g_AMPA increases both synchrony and the number of spikes within bursts, and decreases the bursting frequency. Too strong g_AMPA, however, prevents the activity because it enhances neuronal intrinsic adaptation. When [Mg²⁺]_o is increased, higher g_NMDA values are needed to maintain bursting activity. Bursting frequency decreases with [Mg²⁺]_o, and the network is silent with physiological [Mg²⁺]_o. Inhibition weakly decreases the bursting frequency if inhibitory cells receive enough NMDA-mediated excitation. This study explains the importance of conditional bursters in layer V in supporting epileptiform activity at low [Mg²⁺]_o.