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In this paper we show that spike-time-dependent-plasticity (STDP), a powerful learning paradigm for spiking neural systems, can be implemented using a crossbar memristive array combined with neurons that asynchronously generate spikes of a given shape. Such spikes need to be sent back through the neurons input terminal. The shape of the spikes turns out to be very similar to the neural spikes observed in biology for real neurons. The STDP learning function obtained by combining such neurons with memristors is exactly that of the STDP learning function obtained from neurophysiological experiments on real synapses. Using this result, we propose memristive crossbar architectures capable of performing asynchronous STDP learning.