Fractional-order stochastic delayed neural networks with impulses: mean square finite-time contractive synchronization.

This article presents a novel framework for mean square finite-time synchronization (MSFTSn) and mean square finite-time contractive synchronization (MSFTCSn) of fractional-order stochastic delayed neural networks (FOSDNNs) subject to hybrid control. The proposed hybrid control strategy is designed to guarantee synchronization of the error system within a finite time horizon. By combining continuous feedback with impulsive regulation, the hybrid mechanism effectively suppresses stochastic disturbances and compensates for time-delay effects, which significantly improves convergence rate and enhances contractive stability. The analytical approach integrates stochastic analysis with Lyapunov-based methods, the fractional Gronwall inequality, and an improved Razumikhin framework to establish novel synchronization criteria. In addition, a rigorous foundation is developed to address discontinuous neuron activation functions through set-valued map theory. Unlike integer-order models, the Caputo fractional derivative embeds past error trajectories, thereby capturing memory and hereditary properties of neural systems. This leads to a more realistic neural representation and reinforces the synchronization results. Theoretical findings demonstrate that hybrid control extends the range of stabilizing parameters beyond standard feedback schemes. Finally, numerical simulations are presented to validate the effectiveness and robustness of the proposed strategy, confirming its strong applicability in realistic neural network models.