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Discrete-time impulsive Hopfield neural networks with finite distributed delays. (English) Zbl 1122.92001

Introduction: A neural network is a network that performs computational tasks such as associative memory, pattern recognition, optimization, model identification, signal processing, etc., on a given pattern via interaction between a number of interconnected units characterized by simple functions. From the mathematical point of view, an artificial neural network corresponds to a nonlinear transformation of some inputs into certain outputs. Many types of neural networks have been proposed and studied in the literature and the Hopfield-type network has become an important one due to its potential for applications in various fields of daily life. The model proposed by J. J. Hopfield [Proc. Natl. Acad. Sci. 79, 2554–2558 (1982; Zbl 1369.92007)], also known as Hopfield’s graded response neural network, is based on an analogue circuit consisting of capacitors, resistors and amplifiers. More details about artificial neural networks can be found in Section 2.
Hopfield neural networks have found applications in a broad range of disciplines and have been studied both in continuous and discrete time cases by many researchers. Most neural networks can be classified as either continuous or discrete. In spite of this broad classification, there are many real world systems and natural processes that behave in a piecewise continuous style interlaced with instantaneous and abrupt changes (impulses). Periodic dynamics of the Hopfield neural networks is one of the realistic and attractive modellings for researchers. Signal transmission between the neurons causes time delays. Therefore the dynamics of Hopfield neural networks with discrete or distributed delays has a fundamental concern.
We introduce the discrete counterpart of a class of Hopfield neural networks with periodic impulses and finite distributed delays. Combining some ideas of H. Akça et al. [Dyn. Syst. Appl. 13, No. 1, 77–92 (2004; Zbl 1058.34007)] and X.Yang et al. [Phys. Lett. A 343, 108–116 (2005)], we obtain a sufficient condition for the existence and global exponential stability of a unique periodic solution of the discrete system considered.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
68T05 Learning and adaptive systems in artificial intelligence
34A37 Ordinary differential equations with impulses
94C99 Circuits, networks
34C25 Periodic solutions to ordinary differential equations
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