Problems of stability and convergence of dynamic processes in neurodynamics

Abstract

The aim of the article is to derive suficient conditions for stability and convergence of solutions to systems of differential equations that model neurodynamic processes, in particular the learning and operation dynamics of Hopfield neural networks.
Research methodology. The analysis of asymptotic stability and convergence is based on Lyapunov's second method. Appropriate Lyapunov functions are constructed for the considered models, estimates for the derivative along trajectories are obtained, and conditions are formulated under which trajectories approach stationary regimes.
Results of the research. The proposed approach extends the classical stability theory, which typically focuses on the stability of an individual trajectory (most often an equilibrium point), to the case of neurodynamic processes. Suficient conditions for asymptotic stability of equilibrium states and for convergence of solutions to these equilibria are derived for systems of difierential equations describing neurodynamic dynamics. For Hopfield network models, criteria are established that guarantee the decrease of a corresponding Lyapunov functional and, consequently, the convergence of learning and operating processes to stationary regimes. The obtained conditions are presented in a form suitable for verification in terms of model parameters and properties of the interconnection matrix. Practical significance. The derived stability and convergence criteria can be used in the design and tuning of Hopfield neural networks and related neurodynamic models to ensure guaranteed convergence of the learning process and stability of operating modes, as well as for parameter selection and validation in applied problems of information processing and optimization.