We investigate the occurrence of quasisynchronous events in a random network of excitatory leaky integrate-and-fire neurons equipped with short-term plasticity. The dynamics is analyzed by monitoring both the evolution of global synaptic variables and, on a microscopic ground, the interspike intervals of the individual neurons. We find that quasisynchronous events are the result of a mixture of synchronized and unsynchronized motion, analogously to the emergence of synchronization in the Kuramoto model. In the present context, disorder is due to the random structure of the network and thereby vanishes for a diverging network size N (i.e., in the thermodynamic limit), when statistical fluctuations become negligible. Remarkably, the fraction of asynchronous neurons remains strictly larger than zero for arbitrarily large N. This is due to the presence of a robust homoclinic cycle in the self-generated synchronous dynamics. The nontrivial large-N behavior is confirmed by the anomalous scaling of the maximum Lyapunov exponent, which is strictly positive in a finite network and decreases as N−0.27. Finally, we have checked the robustness of this dynamical phase with respect to the addition of noise, applied to either the reset potential or the leaky current.