Study Establishes Stability and Lag-Synchronization in Generalized Fractional Neural Networks

Researchers have established algebraic criteria ensuring global ψ-Mittag-Leffler stability and lag-synchronization in a generalized class of impulsive fractional-order neural networks. The mathematical investigation modeled impulsive ψ-Hilfer fractional-order neural networks incorporating both simple delays and continuous time-varying delays. Using the Banach fixed point theorem, the authors established the existence and uniqueness of solutions for the modeled systems. In addition, by combining Lyapunov’s direct method with the Razumikhin approach, the study derived global ψ-Mittag-Leffler stability criteria and obtained sufficient algebraic conditions that guarantee the stability of the drive system as well as lag-synchronization between drive and response systems under impulsive control.

A numerical example confirmed the theoretical predictions. The researchers contextualized the model within fractional-order neural networks subject to impulsive effects and delays, which provide realistic representations of complex neural dynamics. Prior to this work, synchronization behavior in such systems under generalized delay structures remained insufficiently understood, which limited their applicability to domains such as secure communication and signal processing.

Unlike earlier studies restricted to Caputo or Riemann-Liouville operators and a single delay type, this research unifies simple and continuous time-varying delays under the general ψ-Hilfer framework. Because the ψ-Hilfer operator recovers existing fractional operators as special cases, the resulting criteria extend and generalize previously reported stability and synchronization outcomes. Furthermore, whereas prior models routed all states through a single common activation function, the proposed formulation assigns the simple delay and continuous time-varying delay to two distinct activation functions, allowing each delay type to be captured through its own nonlinearity and providing a more flexible representation of neuronal signal transmission.

Subject of Research: Stability and lag-synchronization in generalized fractional-order neural networks

Article Title: Lag-Synchronization and Global \(\psi \)-Mittag-Leffler Stability of Generalized Fractional Neural Networks

Article References: Ahmad, A., Zada, A., Hussain, S., & Samei, M. E. (2026). Lag-Synchronization and Global $$\psi $$-Mittag-Leffler Stability of Generalized Fractional Neural Networks. Neural Processing Letters. https://doi.org/10.1007/s11063-026-11893-8

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DOI: 10.1007/s11063-026-11893-8

Keywords: fractional-order neural networks, lag-synchronization, Mittag-Leffler stability, psi-Hilfer operator, impulsive control, time-varying delays, Banach fixed point theorem, Lyapunov direct method, Global, Mittag-Leffler, Stability, Generalized