Neuromorphic computing inspired by the human brain is highly desirable in the artificial intelligence age. Thus, it is essential to comprehensively investigate the neuromorphic characteristics of artificial synapses and neurons which are the unit cells in an artificial neural network (ANN). Memristors are considered ideal candidates to serve as artificial synapses and neurons in the ANN. Herein, two-terminal memristors based on two-dimensional (2D) MoSe nanosheets are fabricated, demonstrating analog resistive switching (RS) behaviors. Unlike the digital RS behaviors with a sharp transition between the two resistance states, the analog RS provides a series of tunable resistance states, which is more suitable for the realization of synaptic plasticity. Thus, the fabricated memristors successfully implement the synaptic functions, such as paired-pulse facilitation, long-term potentiation and long-term depression. The analog memristors can be utilized to construct the ANN for image recognition, leading to a high recognition accuracy of 92%. In addition, the synaptic memristors can emulate the “learning-forgetting” experience of the human brain. Furthermore, to demonstrate the ability of single neuron learning in our devices, the memristors are studied as artificial nociceptors to recognize noxious stimuli. Our research provides comprehensive investigations on the neuromorphic characteristics of artificial synapses and nociceptors, suggesting promising prospects for applications in neuromorphic computing based on 2D MoSe nanosheets.