LETTER Communicated by Rajesh Rao Optimal Spike-Timing-Dependent Plasticity for PreciseAction Potential Firing in Supervised Learning

In timing-based neural codes, neurons have to emit action potentials at precise moments in time. We use a supervised learning paradigm to derive a synaptic update rule that optimizes by gradient ascent the like-lihood of postsynaptic firing at one or several desired firing times. We find that the optimal strategy of up- and downregulating synaptic effi-cacies depends on the relative timing between presynaptic spike arrival and desired postsynaptic firing. If the presynaptic spike arrives before the desired postsynaptic spike timing, our optimal learning rule pre-dicts that the synapse should become potentiated. The dependence of the potentiation on spike timing directly reflects the time course of an ex-citatory postsynaptic potential. However, our approach gives no unique reason for synaptic depression under reversed spike timing. In fact, the presence and amplitude of depression of synaptic efficacies for reversed spike timing depend on how constraints are implemented in the op-timization problem. Two different constraints, control of postsynaptic rates and control of temporal locality, are studied. The relation of our results to spike-timing-dependent plasticity and reinforcement learning is discussed.