Spontaneous waves are ubiquitous during early brain development and are hypothesized to drive the development of receptive fields (RFs). Different stages of spontaneous waves in the retina have been observed to coincide with the development of the retinotopic map, ON-OFF segregation, and orientation selectivity in the early visual pathway of mammals, and can be characterized by different activity patterns in the retina and downstream areas. Stage II waves, which occur in rodents right after birth, have been implicated in a possible synaptic pruning process, relating these stage II retinal waves to the refinement of the retinotopic map. However, the mechanisms underlying the activity-dependent effects of retinal waves on primary visual cortex (V1) are poorly understood. In this work, we build a biologically-constrained model of the development of thalamocortical synapses onto neurons in V1 driven by stage II retinal waves using a spike-timing dependent triplet learning rule. Using this model, together with a reduced rate-based model, we propose possible mechanisms underlying such a pruning process and predict how characteristics of the retinal waves may lead to different RF structures, including periodic RFs. We introduce gap junctions into the V1 network and show that such a coupling can serve to promote precise local retinotopy. Finally, we discuss how the spatial distribution of synaptic weights at the end of stage II may affect the emergence of orientation selectivity of V1 neurons during stage III waves. The mechanisms uncovered in this work may be useful in understanding synaptic structures that emerge across cortical regions during development.