The distribution of synaptic efficacies in neocortex has an approximately lognormal shape. Many weak synaptic connections coexist with few very strong connections such that only 20% of synapses contribute 50% of total synaptic strength. Furthermore, recent evidence shows that weak connections fluctuate strongly while the few strong connections are relatively stable, suggesting them as a physiological basis for long-lasting memories. It remains unclear, however, through what mechanisms these properties of cortical networks arise. Here we show that lognormal-like synaptic weight distributions and the characteristic pattern of synapse stability can be parsimoniously explained as a consequence of network selforganization. We simulated a simple self-organizing recurrent neural network model (SORN) composed of binary threshold units. The network receives no external input or noise but self-organizes its connectivity structure solely through different forms of plasticity. Across a wide range of parameters, the network produces lognormal-like synaptic weight distributions and faithfully reproduces experimental data on synapse stability as a function of synaptic efficacy. Overall, our results suggest that the fundamental structural and dynamic properties of cortical networks arise from the self-organizing forces induced by different forms of plasticity.