In this article, we propose a novel chaotic neuron circuit with memristive neural synapses, construct an architecture of memristive chaotic neural network (MCNN) and implement associative memory application of bipolar images. The proposed neuron circuit mainly consists of synapse module and neuron module with chaotic dynamics characteristics. The synapse module is composed of memristors which represent synaptic weights. The neuron module employs voltage feedback operational amplifiers to accomplish integral operation and output function. MCNN utilizes a memristor crossbar array to perform matrix operations and can process the information in parallel. In addition, the proposed circuit of MCNN can accomplish continuous recursive operations and meet different applications due to the programmability of the memristor. The ex-situ method is utilized to train the memristor crossbar array. Furthermore, the associative memory applications of bipolar images are carried out based on the constructed circuits of MCNN with three and nine neurons. The simulation results in PSPICE software testify the functions of the MCNN circuit.