Cluster synchronization of delayed coupled neural networks: Delay-dependent distributed impulsive control.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Computational Neuroscience

Background:

• Coupled Neural Networks (CNNs) are fundamental models for understanding complex brain functions.
• Time-varying delays in CNNs introduce significant challenges to achieving coordinated behavior like cluster synchronization.
• Existing control methods often struggle with the dynamic and delayed nature of these networks.

Purpose of the Study:

• To investigate and achieve cluster synchronization (CS) in coupled neural networks (CNNs) with time-varying delays.
• To develop novel delay-dependent distributed impulsive control strategies for enhanced network coordination.
• To address synchronization issues in CNNs with both fixed and switching coupling topologies.

Main Methods:

• Development of a new Halanay-like inequality incorporating delayed impulses.
• Application of Lyapunov theory combined with the proposed differential inequality.
• Design of delay-dependent distributed impulsive controllers tailored for fixed and switching topologies.

Main Results:

• Sufficient conditions for achieving cluster synchronization in delayed CNNs with fixed coupling topology were established.
• Sufficient conditions for achieving cluster synchronization in delayed CNNs with switching coupling topology were derived.
• Effectiveness of the designed controllers was validated through numerical simulations on CNNs with fixed and switching couplings.

Conclusions:

• The proposed delay-dependent distributed impulsive control method effectively achieves cluster synchronization in coupled neural networks with time-varying delays.
• The developed theoretical framework and controllers are applicable to networks with both static and dynamic coupling structures.
• This research offers a robust approach for controlling complex dynamical systems with inherent delays and topological changes.