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Picture Archiving and Communication System (PACS) is an important technology for the transmission of medical images and related information. Using the PACS technology, medical images can be transmitted and shared over local area networks, wide area networks. However, ensuring transmission security continues to be crucial challenges in the field of modern medicine. In this paper, a hyperbolic memristor model with different hysteresis loops for different input signals is proposed, which has both locally active and passive properties under different parameters. Based on the memristor model, a four-neuron Hopfield neural network (HNN) is introduced, incorporating a memristor as a replacement for one of the synaptic weights, while subjecting one of the four neurons to electromagnetic radiation. The effects of the memristor parameters and coupling weights on neural network are studied, which reveals the existence of coexistence behavior in neural network. In addition, an equivalent circuit of HNN is implemented to demonstrate the accuracy of the dynamical analysis. Finally, a medical image encryption approach with bi-directional dynamic permutation and multi-directional dynamic diffusion process is proposed. The experimental findings demonstrate that the encryption scheme has strong robustness, which can be applied in PACS to enhance the security of medical images during transmission.