Optimal Spike-Timing Dependent Plasticity for Precise Action Potential Firing in Supervised Learing

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 via gradient ascent the likelihood of postsynaptic firing at one or several desired firing times. We find that the optimal strategy of up and down regulating synaptic efficacies 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 predicts that the synapse should become potentiated. The dependence of the potentiation on spike timing directly reflects the time course of an excitatory 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 depends on how constraints are implemented in the optimization problem. Two different constraints, i.e., control of postsynaptic rates or control of temporal locality, are studied. The relation of our results to Spike-Timing Dependent Plasticity (STDP) and reinforcement learning is discussed.