In recent years, memristors have shown great potential in building artificial neural networks. Many researchers have worked on improving and innovating the design of memristors. The bifunctional memristor proposed in this paper provides new ideas for establishing artificial neural networks. And a novel second-order memristor Hindmarsh–Rose neuron (SOM-HR) is constructed on this basis. The SOM-HR model can well simulate the dynamical behavior of the memristive autapse HR neuron under electromagnetic radiation (EMR), and many types of firing patterns, coexisting firing patterns and state transition are found. In addition, the correctness of the numerical simulation is verified by circuit simulation and digital signal processor (DSP) implementation. Finally, the chaotic sequences generated by this model are used for image encryption, which guarantees the security of the encryption scheme. The results of the study provide a reference for construction and application of complex neural network. •A second-order flux-controlled memristor is designed.•An analog circuit for the memristor was constructed.•Biological neuron modeling based on bifunctional memristor•The dynamical behavior of the new neural model is analyzed.•The neuron model was implemented by analog circuit and DSP platform respectively.