As the third-generation neural network, the spiking neural network (SNN) has become one of the most promising neuromorphic computing paradigms to mimic brain neural networks over the past decade. The SNN shows many advantages in performing classification and recognition tasks in the artificial intelligence field. In the SNN, the communication between the pre-synapse neuron (PRE) and the post-synapse neuron (POST) is conducted by the synapse. The corresponding synaptic weights are dependent on both the spiking patterns of the PRE and the POST, which are updated by spike-timing-dependent plasticity (STDP) rules. The emergence and growing maturity of spintronic devices present a new approach for constructing the SNN. In the paper, a novel SNN is proposed, in which both the synapse and the neuron are mimicked with the spin transfer torque magnetic tunnel junction (STT-MTJ) device. The synaptic weight is presented by the conductance of the MTJ device. The mapping of the probabilistic spiking nature of the neuron to the stochastic switching behavior of the MTJ with thermal noise is presented based on the stochastic Landau-Lifshitz-Gilbert (LLG) equation. In this way, a simplified SNN is mimicked with the MTJ device. The function of the mimicked SNN is verified by a handwritten digit recognition task based on the MINIST database.