Mean-square stability of Riemann–Liouville fractional Hopfield’s graded response neural networks with random impulses

• Published in: Advances in difference equations Vol. 2021; no. 1; pp. 1 - 20
• Main Authors: Agarwal, R., Hristova, S., O’Regan, D., Kopanov, P.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 04.02.2021; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Difference and Functional Equations; Functional Analysis; Hopfield’s graded response neural network; Impulses; Impulses at random times; Initial conditions; Lyapunov functions; Mathematics; Mathematics and Statistics; Mean-square Mittag-Leffler stability in time; Neural networks; Neurons; Ordinary Differential Equations; Partial Differential Equations; Riemann–Liouville fractional derivative; Stability; Stochastic processes; 34F99; Lyapunov functions; 34A37; Riemann–Liouville fractional derivative; 34D20; Impulses at random times; Hopfield’s graded response neural network; 92B20; Mean-square Mittag-Leffler stability in time
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-021-03237-8

In this paper a model of Hopfield’s graded response neural network is investigated. A network whose neurons are subject to a certain impulsive state displacement at random times is considered. The model is set up and studied. The presence of random moments of impulses in the model leads to a change of the solutions to stochastic processes. Also, we use the Riemann–Liouville fractional derivative to model adequately the long-term memory and the nonlocality in the neural networks. We set up in an appropriate way both the initial conditions and the impulsive conditions at random moments. The application of the Riemann–Liouville fractional derivative leads to a new definition of the equilibrium point. We define mean-square Mittag-Leffler stability in time of the equilibrium point of the model and study this type of stability. Some sufficient conditions for this type of stability are obtained. The general case with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons is studied.