Excitatory synaptic connections in the adult neocortex consist of multiple synaptic contacts, almost exclusively formed on dendritic spines. Changes of dendritic spine shape and volume, a correlate of synaptic strength, can be tracked in vivo for weeks. Here, we present a combined model of spike-timing dependent dendritic spine plasticity and turnover that explains the steady state multi-contact configuration of synapses in adult neocortical networks. In this model, many presynaptic neurons compete to make strong synaptic connections onto postsynaptic neurons, while the synaptic contacts comprising each connection cooperate via postsynaptic firing. We demonstrate that the model is consistent with experimentally observed long-term dendritic spine dynamics under steady-state and lesion induced conditions, and show that cooperation of multiple synaptic contacts is crucial for stable, long-term synaptic memories. In simulations of a simplified network of barrel cortex, our plasticity rule reproduces whisker-trimming induced rewiring of thalamo-cortical and recurrent synaptic connectivity on realistic time scales.