Multistability for Delayed Neural Networks via Sequential Contracting

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.