Multi-scroll and coexisting attractors in a Hopfield neural network under electromagnetic induction and external stimuli

The application of external stimuli to biological neurons is a valuable tool for investigating neuronal properties, understanding neural circuitry, and developing therapeutic interventions for neurological disorders. In this article, we propose a discrete fractional Hopfield neural network model consisting of four neurons to explore the influence of external stimuli in the presence of electromagnetic induction and radiation. To incorporate the electromagnetic induction between connected neurons, we construct and employ a discrete fractional sine memristor. Additionally, we introduce a multi-level pulse function to the sine memristor element to examine the chaotic dynamics of the neural network model. The qualitative behavior of the network model is demonstrated through stability analysis and bifurcation diagrams showcasing chaos. The study also focuses on understanding the coexisting behavior of the neural network model in the presence and absence of external stimuli. Moreover, we investigate the generation of multi-scroll attractors by varying the level of the pulse function, which is introduced to electromagnetic induction. Numerical simulations reveal that increasing the level of the multi-pulse function doubles the number of scrolls in the attractors when external stimuli are present. The findings presented in this article contribute to our understanding of discrete fractional memristors and shed light on the dynamical behavior of neurons and their electrical activity in the brain. Innovation within the discrete fractional-order Hopfield neural networks realm entails the creation and utilization of fresh ideas, methodologies, and strategies that harness fractional-order dynamics to confront diverse hurdles and enhance the effectiveness of Hopfield networks. Discrete fractional-order Hopfield neural networks have the capacity to propel an array of applications forward, spanning artificial intelligence, machine learning, control systems, and optimization, showcasing their potential for substantial progress in various domains.