The balanced random network model attracts considerable interest because it explains the irregular spiking activity at low rates and large membrane potential fluctuations exhibited by cortical neurons in vivo. In this article, we investigate to what extent this model is also compatible with the experimentally observed phenomenon of spike-timing-dependent plasticity (STDP). Confronted with the plethora of theoretical models for STDP available, we reexamine the experimental data. On this basis, we propose a novel STDP update rule, with a multiplicative dependence on the synaptic weight for depression, and a power law dependence for potentiation. We show that this rule, when implemented in large, balanced networks of realistic connectivity and sparseness, is compatible with the asynchronous irregular activity regime. The resultant equilibrium weight distribution is unimodal with fluctuating individual weight trajectories and does not exhibit development of structure. We investigate the robustness of our results with respect to the relative strength of depression. We introduce synchronous stimulation to a group of neurons and demonstrate that the decoupling of this group from the rest of the network is so severe that it cannot effectively control the spiking of other neurons, even those with the highest convergence from this group.