It is believed that energy efficiency is an important constraint in brain evolution. As synaptic transmission dominates energy consumption, energy can be saved by ensuring that only a few synapses are active. It is therefore likely that the formation of sparse codes and sparse connectivity are fundamental objectives of synaptic plasticity. In this work we study how sparse connectivity can result from a synaptic learning rule of excitatory synapses. Information is maximised when potentiation and depression are balanced according to the mean presynaptic activity level and the resulting fraction of zero-weight synapses is around 50%. However, an imbalance towards depression increases the fraction of zero-weight synapses without significantly affecting performance. We show that imbalanced plasticity corresponds to imposing a regularising constraint on the L1-norm of the synaptic weight vector, a procedure that is well-known to induce sparseness. Imbalanced plasticity is biophysically plausible and leads to more efficient synaptic configurations than a previously suggested approach that prunes synapses after learning. Our framework gives a novel interpretation to the high fraction of silent synapses found in brain regions like the cerebellum.