With inherent sparse spike-based coding and asynchronous event-driven computation, spiking neural network (SNN) is naturally suitable for processing event stream data of event cameras. In order to improve the feature extraction and classification performance of bio-inspired hierarchical SNNs, in this paper an event camera object recognition system based on biological synaptic plasticity is proposed. In our system input event streams were firstly segmented adaptively using spiking neuron potential to improve computational efficiency of the system. Multi-layer feature learning and classification are implemented by our bio-inspired hierarchical SNN with synaptic plasticity. After Gabor filter-based event-driven convolution layer which extracted primary visual features of event streams, we used a feature learning layer with unsupervised spiking timing dependent plasticity (STDP) rule to help the network extract frequent salient features, and a feature learning layer with reward-modulated STDP rule to help the network learn diagnostic features. The classification accuracies of the network proposed in this paper on the four benchmark event stream datasets were better than the existing bio-inspired hierarchical SNNs. Moreover, our method showed good classification ability for short event stream input data, and was robust to input event stream noise. The results show that our method can improve the feature extraction and classification performance of this kind of SNNs for event camera object recognition.