Chaotic dynamical system of Hopfield neural network influenced by neuron activation threshold and its image encryption

In the field of artificial neural networks, researchers often use the hyperbolic tangent function as an activation function to imitate the firing rules of biological neurons and to add nonlinear characteristics to neural networks. However, prior studies have neglected to consider the effect of the bias of the activation function, which represents the firing threshold of biological neurons, on the dynamical behaviors of neural networks. In this paper, we aim to study the influence of neuronal thresholds on dynamics of the Hopfield neural network (HNN). The bias of the activation function is used as the control parameter in this investigation. The proposed HNN model is analyzed through various methods, including phase portraits, 0-1 tests, Lyapunov exponent spectra, bifurcation diagrams, and bi-parameter dynamic maps. The results of the analysis illustrate that the firing threshold could induce a range of complex dynamical phenomena in the HNN, such as periodic attractors, chaotic attractors, and forward and reverse period doubling bifurcations. Furthermore, the hardware implementation of the proposed HNN model is successfully demonstrated through circuit simulations. These experiments confirm the consistency of the results obtained through numerical simulations. Finally, the potential application of the proposed HNN model is further explored by constructing an image encryption system. The results demonstrate that the chaotic attractor has good randomness properties and that its application in image encryption has a high level of security. This study may provide valuable insights into dynamics of the HNN model influenced by the neuron firing threshold and highlight the potential for practical applications of these models in engineering.