Predefined-Time Synchronization for FitzHugh-Nagumo Neural Networks With External Disturbances and Multiple Time Scales: An Adaptive Control Approach

The problem of predefined-time (PdT) synchronization of FitzHugh-Nagumo (FHN) neural networks is investigated. The FHN neural networks have shown proficiency in accurately representing the essential characteristics of neurons and effectively simulating the structural dynamics and functioning of the brain. The investigation of synchronization in FHN neural networks will enhance our comprehension of how neurons behave dynamically and promote their valuable utilization in the field of artificial intelligence. In this study, the successful integration of adaptive control and PdT control techniques not only achieves synchronization within a predetermined time but also eliminates the need to preselect controlling gains. The work develops new PdT stability theorems for singular perturbation systems and synchronizing controllers that exclude linear terms. Consequently, a set of sufficient conditions for achieving this synchronization goal is provided. Furthermore, it mathematically demonstrates the implementability of an upper bound for the settling time by parameterizing the predefined time as a controller parameter, thereby eliminating the need for explicit recalculations. Moreover, the control gains can be adjusted adaptively, enhancing the flexibility of control parameter tuning. Finally, several numerical simulations are presented to validate the effectiveness of the designed adaptive PdT control strategies. Note to Practitioners—The study of neural network-based synchronization in nonlinear systems is rapidly emerging as an important tool in many fields, including automation, telecommunications, computer science, mathematics, physics, and biology. Among the neural network architectures, FitzHugh-Nagumo (FHN) neural networks stand out for their ability to closely model the behaviors and dynamics of real neurons. This capability is not just theoretical; it has significant practical applications in enhancing the understanding of brain functions and improving the performance of artificial intelligence systems. In particular, neural networks with different time scales have better self-learning and generalization ability. Our innovative approach is the integration of adaptive control with predefined-time control techniques. This combination not only guarantees synchronization within a specific time frame but also dynamically adjusts these control parameters, enhancing the system’s adaptability to changes and disturbances without requiring prior knowledge of specific control gains. In practical applications, achieving efficient synchronization can significantly enhance AI systems, making them more efficient and reliable. For instance, in autonomous vehicles, better synchronization can improve decision-making processes, while in robotics, it can lead to more advanced and responsive automations.