We study the behavior of a minimal model of synaptically sustained persistent activity that consists of two quadratic integrate-and-fire neurons mutually coupled via excitatory synapses. Importantly, each of the neurons is excitable, as opposed to an oscillator; hence when uncoupled it sits at a subthreshold rest state. When the constituent neurons are mutually coupled via sufficiently strong fast excitatory synapses, the system demonstrates bistability between a fixed point (quiescent background state) and a limit cycle (memory state with synaptically driven spiking activity). Previous work showed that this persistent activity can be stopped by an excitatory input that synchronizes the network. Here we analyzed how this persistent state reacts to partial synchronization. We considered three types of progressively more complex excitatory synaptic kernels: delta pulse, square, and exponential. The first two cases were treated analytically, and the latter case numerically. Using phase-plane methods, we characterized the shape of the region, such that all orbits starting within it correspond to infinite spike trains; this constitutes the persistent activity region. In the case of instant coupling, all such active orbits were neutrally stable; in the case of noninstant coupling, the activity region contained a unique stable limit cycle (so the activity region was the basin of attraction for the limit cycle). This limit cycle corresponded to purely antiphase spiking of two neurons. Increasing synchronization shifted the system toward the border of the activity region, eventually terminating spiking activity. We calculated three measures of robustness of the active state: width of the activity region in the phase plane, critical level of synchronization that can be tolerated by the persistent spiking activity, and speed of reconvergence to the limit cycle. Our analysis revealed that the self-sustained activity is more robust to synchronization when each individual neuron is closer to SNIC bifurcation (closer to being an intrinsic oscillator), the recurrent synaptic excitation is stronger, and the synaptic decay is slower, which is in agreement with the existing data on local circuits in the cortex that show sustained activity.