Lag-Synchronization and Global $$\psi $$-Mittag-Leffler Stability of Generalized Fractional Neural Networks

Fractional-order neural networks with impulsive effects and delays provide realistic models of complex neural dynamics, yet the synchronization behavior of such systems, particularly under generalized delay structures, remains insufficiently understood, limiting their applicability to areas such as secure communication and signal processing. To address this, we model a class of impulsive \(\psi \) -Hilfer fractional-order neural networks incorporating both simple and continuous time-varying delays. Using the Banach fixed point theorem, we establish the existence and uniqueness of solutions, and using Lyapunov’s direct method combined with the Razumikhin approach, we derive global \(\psi \) -Mittag-Leffler stability criteria. Sufficient algebraic conditions are obtained ensuring both the global \(\psi \) -ML stability of the drive system and the global \(\psi \) -ML lag-synchronization between the drive and response systems under impulsive control, and a numerical example confirms the theoretical predictions. Unlike prior studies restricted to Caputo or Riemann-Liouville operators and a single delay type, this work unifies simple and continuous time-varying delays under the general \(\psi \) -Hilfer framework, which recovers existing fractional operators as special cases, thereby extending and generalizing previously reported stability and synchronization results. Moreover, in contrast to prior models in which all states are processed through a single, common activation function, the proposed model assigns the simple delay and the continuous time-varying delay to two distinct activation functions, allowing each delay type to be captured through its own nonlinearity and thereby providing a more flexible and realistic representation of neuronal signal transmission.