A two-neuron HETUF-memristive hopfield neural network and its application in image encryption

Memristive Hopfield neural networks (MHNN) integrate the non-volatile memory of memristors and feedback dynamics of neural networks, showing great potential in chaotic information security. Nevertheless, most existing MHNNs adopt the conventional [Formula: see text] activation function, which suffers from vanishing gradients and low dynamical complexity. Besides, studies on activation function optimization for Hopfield neural networks (HNN) are still insufficient. Therefore, a novel hybrid exponential Tanh unit function (HETUF) activation function is proposed in this work. Based on the HETUF activation function, a compact two-neuron HETUF-memristive Hopfield neural network (HETUF-MHNN) is constructed. Dynamical analyses and quantitative comparisons with the tanh-MHNN are conducted by using phase portraits, 0-1 test, Lyapunov exponents and bifurcation diagrams. A new phenomenon named derived chaotic decay with time-dependent attenuation is discovered, and its essential difference from traditional multi-stability is clarified. Additionally, a coordinate-constrained attractor capturing strategy is presented, and the corresponding chaotic sequences also pass the NIST test. Finally, a low-complexity spatial domain HETUF-MHNN-based image encryption scheme is designed and validated.