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2014 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Neural Networks with Discontinuous/Impact Activations

verfasst von: Marat Akhmet, Enes Yılmaz

Verlag: Springer New York

Buchreihe : Nonlinear Systems and Complexity

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents as its main subject new models in mathematical neuroscience. A wide range of neural networks models with discontinuities are discussed, including impulsive differential equations, differential equations with piecewise constant arguments, and models of mixed type. These models involve discontinuities, which are natural because huge velocities and short distances are usually observed in devices modeling the networks. A discussion of the models, appropriate for the proposed applications, is also provided.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The dynamics of artificial neural networks is one of the most applicable and attractive objects for the mathematical foundations of neuroscience. In the last decades, recurrent neural networks (RNNs), Cohen–Grossberg neural networks (Hopfield neural networks as a special version), and cellular neural networks (CNNs) have been deeply investigated by using various types of difference and differential equations due to their extensive applications in classification of patterns, associative memories, image processing, artificial intelligence, signal processing, optimization problems, and other areas [58, 76, 78–81, 83, 84, 88, 104, 109, 117, 118, 132, 138, 141, 143, 144, 177, 184, 185, 189, 193, 194, 200, 275, 286, 298, 303].
Marat Akhmet, Enes Yılmaz
Chapter 2. Differential Equations with Piecewise Constant Argument of Generalized Type
Abstract
Differential equations with piecewise constant argument (EPCA) were proposed for investigations in [63, 91] by founders of the theory, K. Cook, S. Busenberg, J. Wiener, and S. Shah. They are named as differential EPCA. In the last three decades, many interesting results have been obtained, and applications have been realized in this theory. Existence and uniqueness of solutions, oscillations and stability, integral manifolds and periodic solutions, and many other questions of the theory have been intensively discussed. Besides the mathematical analysis, various models in biology, mechanics, and electronics were developed by using these systems. The founders proposed that the method of investigation of these equations is based on a reduction to discrete systems. That is, only values of solutions at moments, which are integers or multiples of integers, were discussed. Moreover, systems must be linear with respect to the values of solutions, if the argument is not deviated. It reduces the theoretical depth of the investigations as well as the number of real-world problems, which can be modeled by using these equations.
Marat Akhmet, Enes Yılmaz
Chapter 3. Impulsive Differential Equations
Abstract
Let \(\mathbb{R},\, \mathbb{N}\), and \(\mathbb{Z}\) be the sets of all real numbers, natural numbers, and integers, respectively. Denote by \(\theta =\{\theta _{i}\}\) a strictly increasing sequence of real numbers such that the set \(\mathcal{A}\) of indexes i is an interval in \(\mathbb{Z}.\) The sequence θ is a B−sequence, if one of the following alternatives is valid:
Marat Akhmet, Enes Yılmaz
Chapter 4. Periodic Motions and Equilibria of Neural Networks with Piecewise Constant Argument
Abstract
In this chapter we consider Hopfield-type neural networks systems with piecewise constant argument of generalized type. Sufficient conditions for the existence of a unique equilibrium and a periodic solution are obtained. The stability of these solutions is investigated.
Marat Akhmet, Enes Yılmaz
Chapter 5. Equilibria of Neural Networks with Impact Activations and Piecewise Constant Argument
Abstract
In this chapter we introduce two different types of impulsive neural networks with piecewise constant argument of generalized type called (θ, θ)−type neural networks and (θ, τ)−type neural networks, respectively. For these types, sufficient conditions for existence of a unique equilibrium are obtained, existence and uniqueness of solutions and an equivalence lemma for such systems are established, and stability criterion for the equilibrium based on linear approximation is proposed.
Marat Akhmet, Enes Yılmaz
Chapter 6. Periodic Motions of Neural Networks with Impact Activations and Piecewise Constant Argument
Abstract
In this chapter we derive some sufficient conditions for the existence and stability of periodic solutions for each (θ, θ)-type neural networks and (θ, τ)-type neural networks, respectively. Examples with numerical simulations are given to illustrate our results.
Marat Akhmet, Enes Yılmaz
Chapter 7. The Method of Lyapunov Functions: RNNs
Abstract
In this chapter, we apply the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type to a model of RNNs. The model involves alternating argument. Sufficient conditions are obtained for global exponential stability of the equilibrium point. Examples with numerical simulations are presented to illustrate the results.
Marat Akhmet, Enes Yılmaz
Chapter 8. The Lyapunov–Razumikhin Method: CNNs
Abstract
In this chapter, by using the concept of differential equations with piecewise constant arguments of generalized type [13–15, 18], the model of cellular neural networks (CNNs) [79, 80] is developed. Lyapunov–Razumikhin technique is applied to find sufficient conditions for uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.
Marat Akhmet, Enes Yılmaz
Backmatter
Metadaten
Titel
Neural Networks with Discontinuous/Impact Activations
verfasst von
Marat Akhmet
Enes Yılmaz
Copyright-Jahr
2014
Verlag
Springer New York
Electronic ISBN
978-1-4614-8566-7
Print ISBN
978-1-4614-8565-0
DOI
https://doi.org/10.1007/978-1-4614-8566-7

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