Neuromorphic computing has shown great advantages towards cognitive tasks with high speed and remarkable energy efficiency. Memristor is considered as one of the most promising candidates for the electronic synapse of the neuromorphic computing system due to its scalability, power efficiency and capability to simulate biological behaviors. Several memristor-based hardware demonstrations have been explored to achieve the capacity of unsupervised learning with the spike-rate-dependent plasticity (SRDP) learning rule. However, the learning capacity is limited and few of the memristor-based hardware demonstrations have explored the online unsupervised learning at the network level with an SRDP algorithm. Here, we construct a memristor-based hardware system and demonstrate the online unsupervised learning of SRDP networks. The neuromorphic system consists of multiple memristor arrays as the synapse and the discrete CMOS circuit unit as the neuron. Unsupervised learning and online weight update of 10 MNIST handwritten digits are realized by the constructed SRDP networks, and the recognition accuracy is above 90% with 20% device variation. This work paves the way towards the realization of large-scale and efficient networks for more complex tasks.