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Estimates of the processes convergence in one model of Hopfield’s neurodynamics. (English) Zbl 1389.92002
Summary: We propose results of a mathematical model of Hopfield’s neurodynamics. The common “electrical interpretation” is used, in which, using the Kirchhoff law, the dynamics model is written as a system of ordinary quasilinear differential equations. We consider the system without delay, which has an indicated linear section of the diagonal form. Unlike in the known work by S. Haykin [Neural networks. A comprehensive foundation. 2nd ed. New York, NY: IEEE (1999; Zbl 0934.68076)], the matrix of the linear part is not single. The nonlinear functions satisfy the Lipschitz conditions with different sufficiently small constants. First of all, a stationary point of the system of dynamics equations is found. It corresponds to the steady state of the neural network. By replacing the type of “parallel transfer to the origin of coordinates”, the stability of the stationary point is reduced to the study of the stability of the zero state of equilibrium, which allows us to use traditional methods for studying the stability of dynamical systems. To study the stability, the second Lyapunov method with a quadratic function is used. If the linear part of the system “precedes the Lipschitz constant”, the state of equilibrium of the system is globally asymptotically stable. The dependences between the coefficients of the linear part and the Lipschitz constants, in which the asymptotic stability of the equilibrium state is guaranteed, are calculated.
92B20 Neural networks for/in biological studies, artificial life and related topics
68T05 Learning and adaptive systems in artificial intelligence