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A methodology to realize spike-timing dependent plasticity and Hebbian learning in a neural network through the usage of memristive synapses is presented. Memristors act as a modulating synapse interconnection between neurons; plasticity is accomplished through adjusting the memristance via current spikes based on the relative timings of pre-synaptic and post-synaptic neuron spikes. The learning plasticity presented is continuous, asynchronous and deterministic. A CMOS implementation is presented along with SPICE simulations validating the methodology and design.