Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays

doi:10.1016/j.neunet.2007.07.007

Neural Networks

Volume 20, Issue 10, December 2007, Pages 1067-1080

https://doi.org/10.1016/j.neunet.2007.07.007 Get rights and content

Abstract

The paper presents theoretical results on the global exponential periodicity and global exponential stability of a class of recurrent neural networks with various general activation functions and time-varying delays. The general activation functions include monotone nondecreasing functions, globally Lipschitz continuous and monotone nondecreasing functions, semi-Lipschitz continuous mixed monotone functions, and Lipschitz continuous functions. For each class of activation functions, testable algebraic criteria for ascertaining global exponential periodicity and global exponential stability of a class of recurrent neural networks are derived by using the comparison principle and the theory of monotone operator. Furthermore, the rate of exponential convergence and bounds of attractive domain of periodic oscillations or equilibrium points are also estimated. The convergence analysis based on the generalization of activation functions widens the application scope for the model design of neural networks. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of neural networks.

Introduction

Periodic oscillation in recurrent neural networks is a very interesting dynamic behavior as many biological and cognitive activities (e.g., heartbeat, respiration, mastication, locomotion, and memorization) exhibit periodicity. Persistent oscillation, such as limit cycles, represents a common feature of neural firing patterns produced by dynamic interplay between cellular and synaptic mechanisms. Stimulus-evoked oscillatory synchronization was observed in many biological neural systems including the cerebral cortex of mammals and the brain of insects. In addition, periodic oscillations in recurrent neural networks have found many applications such as associative memories (Nishikawa, Lai, & Hoppensteadt, 2004), pattern recognition (Wang, 1995), machine learning (Ruiz, Owens, & Townley, 1998), and robot motion control (Jin & Zacksenhouse, 2003).

In both biological and artificial neural systems, time delays due to integration and communication are ubiquitous, and often become a source of instability. The time delays in electronic neural networks are usually time-varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. They slow down the transmission rate and tend to introduce some degree of instability in circuits. It was also known that time delays can cause oscillations in neurodynamics (Belair et al., 1996, Gopalsamy and Leung, 1996). Therefore, it is important to investigate the periodicity of recurrent neural networks with time delays.

In recent years, studies of the periodic oscillation of various neural networks are reported in Cao and Wang, 2000, Cao and Wang, 2002, Chen and Wang (2004), Chen, Wang, and Liu (2000), Dong (2002), Fang and Li (2000), Guo and Huang (2003), Guo, Huang, Dai, and Zhang (2003), Jiang, Li, and Teng (2003), Liu and Liao (2004), Liu, Chen, Cao, and Huang (2003), Liu, Chen, and Huang (2004), Rehim, Jiang, and Teng (2004), Townley et al. (2000), Wang and Zou (2004), Yang and Dillon (1994). Any equilibrium point can be viewed as a special case of periodic solution with an arbitrary period or zero amplitude. In this sense, the analysis of periodic oscillation of neural networks is more general than the stability analysis of equilibrium points. There are numerous results on the stability analysis of recurrent neural networks. For example, the results on the existence, uniqueness, and global stability of the equilibria of the recurrent neural network model (2) with and without time delay are discussed in Arik and Tavsanoglu (2000), Cao and Wang (2003), Chen and Wang (2004), Civalleri, Gilli, and Pandolfi (1993), Cohen and Grossberg (1983), Grossberg, 1968a, Grossberg, 1968b, Grossberg, 1969a, Grossberg, 1969b, Grossberg (1971), Grossberg (1988), Hirsch (1989), Hopfield (1984), Hu and Wang (2002b), Liang, X. and Wang, J. (2000), Liang and Wang (2001), Liang and Si (2001), Liao and Wang (2000), Liao and Wang (2003), Takahashi (2000), Wang and Zou (2002), Xia and Wang (2001b), Zeng, Wang, and Liao (2003) and the references cited therein.

In conducting a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the characteristics of activation function as well as network parameters. When neural networks are designed for problem solving, it is desirable for their activation functions to be general. In many electronic circuits, amplifiers, which may generally have continuous input–output functions, can be adopted. To facilitate the design of neural networks, it is important that the neural networks with general activation functions are studied. The generalization of activation functions will provide a wider scope for neural network designs and applications.

The basis of attraction of a periodic oscillation or equilibrium point depends on the parameters and activation function of the neural network. It is also highly desirable to obtain the bounds of the attractive domain of a periodic oscillation or equilibrium point. However, it has not been fully explored in the existing studies. Therefore, the prior estimation of the attractors of periodic oscillations and equilibrium points of neural networks with time-varying delays deserves in-depth investigations.

Consider the model of a class of recurrent neural networks (DRNNs) with varying-time delays of the form $\frac{d x_{i} (t)}{d t} = - d_{i} x_{i} (t) + g_{i} (\sum_{j = 1}^{n} w_{i j} x_{j} (t - τ_{i j} (t)) + u_{i}), i = 1, \dots, n;$ where $x_{i} (t)$ denotes the state variable associated with the $i$ th neuron, $d_{i} > 0$ is a constant, $W = (w_{i j}) \in R^{n \times n}$ is a connection weight matrix, $u = {(u_{1}, \dots, u_{n})}^{T}$ denotes the external input vector that can be periodic or constant, $τ_{i j} (t)$ is a non-negative function representing the finite speed of the axonal signal transmission, and $g_{i} (\cdot)$ is an activation function for $i = 1, 2, \dots, n$ .

When $τ_{i j} (t) \equiv 0$ , the special case of model (1) becomes the recurrent neural networks (RNNs), which can be found in Grossberg (1988), Hirsch (1989) and applied in many fields such as optimization and robotics; e.g., Liang, X.B. and Wang, J. (2000), Liang and Wang (2001), Liang and Si (2001), Xia and Wang (2000), Xia and Wang (2001a), Xia, Leung, and Wang (2002), Xia and Wang, 2004a, Xia and Wang, 2004b, Zhang, Wang, and Xu (2002), Zhang, Wang, and Xia (2003). It includes a number of models from neurobiology, population biology, evolutionary theory with the appropriate assumptions. Due to the finite speeds of switching and transmission of signals, time delays unavoidably exist in a working network and thus should be incorporated into the model. If $τ_{i j} (t) \equiv τ_{i} (t), W D = D W$ and $W$ is nonsingular, where $D = diag (d_{1}, \dots, d_{n})$ , by using $y_{i} (t) = \sum_{j = 1}^{n} w_{i j} x_{j} (t) + u_{i} (i = 1, \dots, n)$ , DRNN model (1) can be transformed to the additive model of DRNN (Grossberg, 1988) $\frac{d y_{i} (t)}{d t} = - d_{i} y_{i} (t) + \sum_{j = 1}^{n} w_{i j} g_{j} (y_{j} (t - τ_{i j} (t))) + J_{i}, i = 1, \dots, n,$ where $J_{i} = d_{i} u_{i}$ . The dynamics of such an additive DRNN model have been widely investigated in recent years and many important results have been obtained. All of these existing results for model (2) can be applied to the model (1) when $τ_{i j} (t) \equiv τ_{i} (t), D W = W D, D = diag (d_{1}, \dots, d_{n})$ , and $W$ is nonsingular. However, when $τ_{i j} (t) \neq τ_{i} (t), D W \neq W D$ or $W$ is singular, whether or not the existing results for DRNN model (2) are effective for DRNN model (1) needs to be explored since the two models are not equivalent in general. Thus, it is important for DRNN model (1) to be analyzed directly.

In this paper, we consider DRNN model (1) with an activation function belonging to any of four classes of the continuous functions including monotone nondecreasing functions, globally Lipschitz continuous and monotone nondecreasing functions, semi-Lipschitz continuous mixed monotone functions, and Lipschitz continuous functions. All activation functions are possibly unbounded functions. For various activation functions, we will establish conditions for ascertaining the convergence of DRNN model (1) to unique periodic oscillation trajectories or equilibria within an estimated attractor at an estimated convergence rate.

The remainder of the paper is organized in six sections. Section 2 provides preliminary informations. Sections 3 Monotone activation functions, 4 Non-monotone activation functions present the criteria for the global exponential periodicity of DRNN model (1) with monotone activation function and non-monotone activation function, respectively. As special cases of the results in Sections 3 Monotone activation functions, 4 Non-monotone activation functions, Section 5 discusses the conditions of global exponential stability of equilibrium point of DRNN model (1) with constant inputs. In Section 6, two numerical examples are given along with simulation results. Finally, Section 7 concludes the paper.

Section snippets

Preliminaries

In this section, we will introduce some definitions and notations which will be needed later.

In the following definitions, let $L, M, L M, M M$ and $S M$ denote the set of the Lipschitz continuous functions, the set of the monotone nondecreasing functions, the set of the Lipschitz continuous and the set of the monotone nondecreasing functions, the set of the mixed monotone functions, and the set of the semi-Lipschitz continuous and mixed monotone functions, respectively.

Definition 1

Let $g (x) = {(g_{1} (x_{1}), \dots, g_{n} (x_{n}))}^{T}$ for $x \in$

Monotone activation functions

In this section, we consider DRNN model (1) with activation function $g \in M$ or $g \in L M$ . First, DRNN model (1) can be embedded into the following $2 n$ -dimensional coupled system $i = 1, \dots, n$ ; ${\begin{cases} {\dot{η}}_{i} (t) = - d_{i} η_{i} (t) + g_{i} (w_{i i} η_{i} (t) + \sum_{j = 1, j \neq i}^{n} [w_{i j}^{+} η_{j} (t - τ_{i j} (t)) - w_{i j}^{-} ξ_{j} (t - τ_{i j} (t))] + u_{i} (t)), \\ {\dot{ξ}}_{i} (t) = - d_{i} ξ_{i} (t) + g_{i} (w_{i i} ξ_{i} (t) + \sum_{j = 1, j \neq i}^{n} [w_{i j}^{+} ξ_{j} (t - τ_{i j} (t)) - w_{i j}^{-} η_{j} (t - τ_{i j} (t))] + u_{i} (t)), \end{cases}$ with the initial states $η (s) = φ (s), ξ (s) = ψ (s), s \in [- τ, 0], φ, ψ \in C_{τ} .$ Let ${(η^{T} (t, φ, ψ), ξ^{T} (t, φ, ψ))}^{T}$ and ${(η_{t}^{T} (φ, ψ), ξ_{t}^{T} (φ, ψ))}^{T}$ denote solutions of system (10) in the product

Non-monotone activation functions

In this section, we consider DRNN model (1) with its activation functions $g \in S M$ or $g \in L$ .

Let $g \in M M$ and let $γ (u, v) = {(γ_{1} (η_{1}, ξ_{1}), \dots, γ_{n} (η_{n}, ξ_{n}))}^{T}$ be a mixed monotone representation of $g (x)$ . We embed the model (1) into the following $2 n$ -dimensional coupled system, $i = 1, \dots, n$ ; ${\begin{matrix} {\dot{η}}_{i} (t) = - d_{i} η_{i} (t) + γ_{i} (w_{i i} η_{i} (t) + h_{i} (η_{t}, ξ_{t}) + u_{i} (t), w_{i i} η_{i} (t) + h_{i} (ξ_{t}, η_{t}) + u_{i} (t)) \\ {\dot{ξ}}_{i} (t) = - d_{i} ξ_{i} (t) + γ_{i} (w_{i i} ξ_{i} (t) + h_{i} (ξ_{t}, η_{t}) + u_{i} (t), w_{i i} ξ_{i} (t) + h_{i} (η_{t}, ξ_{t}) + u_{i} (t)) \end{matrix}$ where $h_{i} (η_{t}, ξ_{t}) = \sum_{j = 1, j \neq i}^{n} w_{i j}^{+} η_{j} (t - τ_{i j} (t)) - \sum_{j = 1, j \neq i}^{n} w_{i j}^{-} ξ_{j} (t - τ_{i j} (t)) .$ Similar to Lemma 1, Lemma 2,

Global exponential stability

In this section, we consider the global exponential stability of DRNN model (1) with constant inputs; i.e., $\frac{d x_{i} (t)}{d t} = - d_{i} x_{i} (t) + g_{i} (w_{i i} x_{i} (t) + \sum_{j = 1, j \neq i}^{n} w_{i j} x_{j} (t - τ_{i j} (t)) + u_{i}), i = 1, \dots, n;$ where $u = {(u_{1}, \dots, u_{n})}^{T} \in R^{n}$ is a constant input vector, $τ_{i j} (t) (i, j = 1, \dots, n)$ are any non-negative bounded functions.

Firstly, note that system (62) has the same equilibrium as the following constant delayed system $\frac{d x_{i} (t)}{d t} = - d_{i} x_{i} (t) + g_{i} (w_{i i} x_{i} (t) + \sum_{j = 1, j \neq i}^{n} w_{i j} x_{j} (t - τ_{i j}) + u_{i}), i = 1, \dots, n .$ Secondly, when $u$ and $τ_{i j}$ are all constants (i.e.,

Illustrative examples

In this section, we discuss two numerical examples to illustrate the results.

Example 1

Consider a DRNN model (1) with $n = 2$ , where the activation functions $g_{1} (σ) = g_{2} (σ) = σ^{\frac{1}{3}}$ . Note that $g_{i} (\cdot) \in M (i = 1, 2)$ . Let $W ≔ (w_{i j}) = (\begin{matrix} - 1 & 0.5 \\ 1 & - 2 \end{matrix}) .$ According to Theorem 1, DRNN model (1) is globally exponentially convergent to a unique $ω$ -periodic solution $x^{*} (t)$ for any $d_{i} > 0 (i = 1, 2)$ , all non-negative $ω$ -periodic time delays $τ_{i j} (t), i, j = 1, 2$ and any $ω$ -periodic oscillation input vector $u (t) = {(u_{1} (t), u_{2} (t))}^{T}$ . Specifically, let $d_{1} = d_{2} = 1, u_{1} (t) = u$

Concluding remarks

The paper presents analytical results on global exponential periodicity and global exponential stability of a general class of recurrent neural networks with time-varying delays and various activation functions (e.g., monotone nondecreasing functions, globally Lipschitz continuous and monotone nondecreasing functions, semi-Lipschitz continuous mixed monotone functions, and Lipschitz continuous functions). By using the comparison principle, the theory of monotone operator, algebraic criteria are

Acknowledgement

The first author is grateful for the financial support from the Education Department of Hubei Province (Z200622002), China.

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^☆: This work was supported by the Natural Science Foundation of China under Grant 60274007.

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Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays☆

Abstract

Introduction

Section snippets

Preliminaries

Monotone activation functions

Non-monotone activation functions

Global exponential stability

Illustrative examples

Concluding remarks

Acknowledgement

Physics Letters A

Physics Letters A

Physica D

Journal of Mathematical Analysis and Applications

Neural Networks

Physics Letters A

Neural Networks

Physics Letters A

Physics Letters A

Journal of Mathematical Analysis and Applications

Neural Networks

Journal of Differential Equations

Neural Networks

Neural Networks

On the global asymptotic stability of delayed cellular neural networks

IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications

Frustration, stability and delay-induced oscillation in a neural network model

SIAM Journal on Applied Mathematics

Periodic oscillatory solution of bidirectional associative memory networks with delays

Physical Review E

Exponential stability and periodic oscillatory solution in BAM networks with delays

IEEE Transactions on Neural Networks

Global asymptotic stability of a general class of recurrent neural networks with time-varying delays

IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications

Mixed monotone semiflows and stability for functional differential equations

Acta Mathematica Sinica

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IEEE Transactions on Neural Networks

On stability of cellular neural networks with delay

IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications

Absolute stability and global pattern formation and parallel memory storage by competitive neural networks

IEEE Transactions on Systems, Man and Cybernetics

Global exponential stability and periodic solutions of cellular neural networks with delay

Physical Review E

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On learning and energy–entropy dependence in recurrent and nonrecurrent signed networks

Journal of Statistical Physics

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Proceedings of the National Academy of Sciences