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[Display omitted] Spiking neural network, inspired by the human brain, consisting of spiking neurons and plastic synapses, is a promising solution for highly efficient data processing in neuromorphic computing. Recently, memristor-based neurons and synapses are becoming intriguing candidates to build spiking neural networks in hardware, owing to the close resemblance between their device dynamics and the biological counterparts. However, the functionalities of memristor-based neurons are currently very limited, and a hardware demonstration of fully memristor-based spiking neural networks supporting in-situ learning is very challenging. Here, a hybrid spiking neuron combining a memristor with simple digital circuits is designed and implemented in hardware to enhance neuron functions. The hybrid neuron with memristive dynamics not only realizes the basic leaky integrate-and-fire neuron function but also enables the in-situ tuning of the connected synaptic weights. Finally, a fully hardware spiking neural network with the hybrid neurons and memristive synapses is experimentally demonstrated for the first time, and in-situ Hebbian learning is achieved with this network. This work opens up a way towards the implementation of spiking neurons, supporting in-situ learning for future neuromorphic computing systems. [ABSTRACT FROM AUTHOR]