Multimode function multistability for Cohen-Grossberg neural networks with mixed time delays.

Area of Science:

• Computational Neuroscience
• Dynamical Systems Theory
• Artificial Intelligence

Background:

• Cohen-Grossberg neural networks (CGNNs) are fundamental models in neuroscience and AI.
• Understanding stability in CGNNs with time delays is crucial for reliable network function.
• Existing stability concepts may not fully capture the complex behaviors of advanced neural networks.

Purpose of the Study:

• To introduce and define multimode function multistability for CGNNs with mixed time delays.
• To develop novel criteria for achieving various types of multistability in these networks.
• To generalize existing stability concepts like multiple exponential, polynomial, logarithmic, and asymptotic stability.

Main Methods:

• Derivation of upper and lower boundary functions based on the CGNN model and activation functions.
• Division of the state space into multiple regions using the zeros of boundary functions.
• Application of proof by contradiction, function continuity, Brouwer's fixed point theorem, and Lyapunov stability theorem.

Main Results:

• Established criteria for achieving multimode function multistability in CGNNs with mixed time delays.
• Demonstrated that different types of multistability (exponential, polynomial, logarithmic, asymptotic) can be achieved by selecting appropriate functions P(t).
• The proposed criteria are shown to be more general than existing results through numerical examples.

Conclusions:

• The concept of multimode function multistability offers a comprehensive framework for analyzing CGNNs.
• The derived criteria provide a robust method for guaranteeing diverse stability behaviors in complex neural networks.
• This work advances the theoretical understanding and practical application of stability analysis in neural network models.