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Multistability for Delayed Neural Networks via Sequential Contracting.

Chang-Yuan Cheng, Kuang-Hui Lin, Chih-Wen Shih

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    This study reveals novel multistability in delayed neural networks by using geometric structures. A sequential contracting method proves global convergence to multiple equilibrium points.

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    Area of Science:

    • Computational Neuroscience
    • Dynamical Systems Theory

    Background:

    • Delayed neural networks are crucial for modeling complex biological processes.
    • Understanding multistability is key to comprehending neural computation and dynamics.

    Purpose of the Study:

    • To explore new multistability scenarios in general delayed neural network systems.
    • To leverage geometric structures for analyzing system dynamics and equilibria.
    • To develop criteria for predicting the number of equilibria based on geometric configurations.

    Main Methods:

    • Exploitation of geometric structures embedded within the system's equations.
    • Application of a novel sequential contracting approach for convergence analysis.
    • Formulation accommodating both smooth sigmoidal and piecewise-linear activation functions.

    Main Results:

    • Disparate numbers of equilibria are derived from different geometric configurations.
    • The sequential contracting method confirms global convergence to multiple equilibrium points.
    • The analytical framework is validated through several illustrative numerical examples.

    Conclusions:

    • Geometric analysis provides a powerful tool for understanding multistability in delayed neural networks.
    • The sequential contracting method offers a robust approach for proving global convergence.
    • This work advances the theoretical understanding of complex neural system dynamics.