Networks of nonlocally coupled leaky integrate-and-fire neurons exhibit a variety of complex collective behaviors, such as partial synchronization, frequency or amplitude chimeras, solitary states, and bump states. In particular, the bump states consist of one or many regions of asynchronous elements within a sea of subthreshold (quiescent) elements. The asynchronous domains travel in the network in a direction predetermined by the initial conditions. In the present study, we investigate the occurrence of bump states in networks of leaky integrate-and-fire neurons in two-dimensions using nonlocal toroidal connectivity, and we explore possible mechanisms for stabilizing the moving asynchronous domains. Our findings indicate that (I) incorporating a refractory period can effectively anchor the position of these domains in the network, and (II) the switching off of some randomly preselected nodes (i.e., making them permanently idle/inactive) can likewise contribute to stabilizing the positions of the asynchronous domains. In particular, in case II for large values of the coupling strength and a large percentage of idle elements, all nodes acquire different fixed (frozen) values in the quiescent region and oscillations cease throughout the network due to self-organization. For the special case of stationary bump states, we propose an analytical approach to predict their properties. This approach is based on the self-consistency argument and is valid for infinitely large networks. Case I is of particular biomedical interest in view of the importance of refractoriness for biological neurons, while case II can be biomedically relevant when designing therapeutic methods for stabilizing moving signals in the brain.