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A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays
This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this Letter, the analysis problem for the existence and stability of periodic solutions is investigated for a class of general discrete-time recurrent neural networks with time-varying delays. For the neural networks under study, a generalized activation function is considered, and the traditional assumptions on the boundedness, monotony and differentiability of the activation functions are removed. By employing the latest free-weighting matrix method, an appropriate LyapunovāKrasovskii functional is constructed and several sufficient conditions are established to ensure the existence, uniqueness, and globally exponential stability of the periodic solution for the addressed neural network. The conditions are dependent on both the lower bound and upper bound of the time-varying time delays. Furthermore, the conditions are expressed in terms of the linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.This work was supported in part by the National Natural Science Foundation of China under Grant 50608072, an International Joint Project sponsored by the Royal Society of the UK and the National Natural Science Foundation of China, and the Alexander von Humboldt Foundation of Germany
Oscillations in I/O monotone systems under negative feedback
Oscillatory behavior is a key property of many biological systems. The
Small-Gain Theorem (SGT) for input/output monotone systems provides a
sufficient condition for global asymptotic stability of an equilibrium and
hence its violation is a necessary condition for the existence of periodic
solutions. One advantage of the use of the monotone SGT technique is its
robustness with respect to all perturbations that preserve monotonicity and
stability properties of a very low-dimensional (in many interesting examples,
just one-dimensional) model reduction. This robustness makes the technique
useful in the analysis of molecular biological models in which there is large
uncertainty regarding the values of kinetic and other parameters. However,
verifying the conditions needed in order to apply the SGT is not always easy.
This paper provides an approach to the verification of the needed properties,
and illustrates the approach through an application to a classical model of
circadian oscillations, as a nontrivial ``case study,'' and also provides a
theorem in the converse direction of predicting oscillations when the SGT
conditions fail.Comment: Related work can be retrieved from second author's websit
Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument
We consider a new model for shunting inhibitory cellular neural networks,
retarded functional differential equations with piecewise constant argument.
The existence and exponential stability of almost periodic solutions are
investigated. An illustrative example is provided.Comment: 24 pages, 1 figur
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Discrete-time recurrent neural networks with time-varying delays: Exponential stability analysis
This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier LtdThis Letter is concerned with the analysis problem of exponential stability for a class of discrete-time recurrent neural networks (DRNNs) with time delays. The delay is of the time-varying nature, and the activation functions are assumed to be neither differentiable nor strict monotonic. Furthermore, the description of the activation functions is more general than the recently commonly used Lipschitz conditions. Under such mild conditions, we first prove the existence of the equilibrium point. Then, by employing a LyapunovāKrasovskii functional, a unified linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the DRNNs to be globally exponentially stable. It is shown that the delayed DRNNs are globally exponentially stable if a certain LMI is solvable, where the feasibility of such an LMI can be easily checked by using the numerically efficient Matlab LMI Toolbox. A simulation example is presented to show the usefulness of the derived LMI-based stability condition.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, the Alexander von Humboldt Foundation of Germany, the Natural Science Foundation of Jiangsu Education Committee of China (05KJB110154), the NSF of Jiangsu Province of China (BK2006064), and the National Natural Science Foundation of China (10471119)
Dynamics of neural systems with discrete and distributed time delays
In real-world systems, interactions between elements do not happen instantaneously, due to the time
required for a signal to propagate, reaction times of individual elements, and so forth. Moreover,
time delays are normally nonconstant and may vary with time. This means that it is vital to introduce
time delays in any realistic model of neural networks. In order to analyze the fundamental
properties of neural networks with time-delayed connections, we consider a system of two coupled
two-dimensional nonlinear delay differential equations. This model represents a neural network,
where one subsystem receives a delayed input from another subsystem. An exciting feature of the
model under consideration is the combination of both discrete and distributed delays, where distributed
time delays represent the neural feedback between the two subsystems, and the discrete
delays describe the neural interaction within each of the two subsystems. Stability properties are
investigated for different commonly used distribution kernels, and the results are compared to the
corresponding results on stability for networks with no distributed delays. It is shown how approximations
of the boundary of the stability region of a trivial equilibrium can be obtained analytically
for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used
to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical
findings
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