## Qualitative Analysis of Dynamic Activity Patterns in Neural Networks

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Haihui Wu, "Pseudo Almost-Periodic Solution of Shunting Inhibitory Cellular Neural Networks with Delay", *Journal of Applied Mathematics*, vol. 2011, Article ID 510789, 14 pages, 2011. https://doi.org/10.1155/2011/510789

# Pseudo Almost-Periodic Solution of Shunting Inhibitory Cellular Neural Networks with Delay

**Academic Editor:**Ivanka Stamova

#### Abstract

Shunting inhibitory cellular neural networks are studied. Some sufficient criteria are obtained for the existence and uniqueness of pseudo almost-periodic solution of this system. Our results improve and generalize those of the previous studies. This is the first paper considering the pseudo almost-periodic SICNNs. Furthermore, several methods are applied to establish sufficient criteria for the globally exponential stability of this system. The approaches are based on constructing suitable Lyapunov functionals and the well-known Banach contraction mapping principle.

#### 1. Introduction

It is well known that the cellular neural networks (CNNs) are widely applied in signal processing, image processing, pattern recognition, and so on. The theoretical and applied studies of CNNs have been a new focus of studies worldwide (see [1–12]). Bouzerdoum and Pinter in [1] have introduced a new class of CNNs, namely, the shunting inhibitory CNNs (SICNNs). Shunting neural networks have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Recently, Chen and Cao [9] have studied the existence of almost-periodic solutions of the following system of SICNNs: where denotes the cell at the position of the lattice, the -neighborhood of is is the activity of the cell , is the external input to , the constant represents the passive decay rate of the cell activity, is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell , and the activation function is a positive continuous function representing the output or firing rate of the cell . Since studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic properties such as periodic oscillatory behavior, almost-periodic oscillatory properties, chaos, and bifurcation. To the best of our knowledge, few authors have studied almost-periodic solutions for SINNs with delays and variable coefficients, and most of them discuss the stability, periodic oscillation in the case of constant coefficients. In their paper, they investigated the existence and stability of periodic solutions of SINNs with delays and variable coefficients. They considered the SICNNs with delays and variable coefficients where, for each and are all continuous periodic functions and and is a positive constant.

In this paper, we consider the following more general SICNNs: By using the Lyapunov functional and contraction mapping, a set of criteria are established for the globally exponential stability, the existence, and uniqueness of pseudo almost-periodic solution for the SICNNs. This is the first paper considering the pseudo almost-periodic solution of SICNNs. Since the nature is full of all kinds of tiny perturbations, either the periodicity assumption or the almost-periodicity assumption is just approximation of some degree of the natural perturbations. A well-known extension of almost periodicity is the asymptotically almost periodicity, which was introduced by Frechet. In 1992, Zhang [13, 14] introduced a more general extension of the concept of asymptotically almost periodicity, the so-called pseudo almost periodicity, which has been widely applied in the theory of ODEs and PDEs. However, it is rarely applied in the theory of neural networks or mathematical biology. This paper is expected to establish criteria that provide much flexibility in the designing and training of neural networks and to shed some new light on the application of pseudo almost periodicity in neural networks, population dynamics, and the theory of differential equations.

Throughout this paper, we will use the notations , where is a bounded continuous function on .

In this paper, we always use unless otherwise stated.

In this paper, we always consider system (1.4) together with the following assumptions. are almost periodic on with , and and are pseudo almost periodic on with . is bounded, continuous and differentiable with for , where is constant. are bounded, and continuous, and there exist positive numbers such that , for all .

#### 2. Preliminaries and Basic Results of Pseudo Almost-Periodic Function

In this section, we explore the existence of pseudo almost-periodic solution of (1.4). First, we would like to recall some basic notations and results of almost periodicity and pseudo almost periodicity [15, 16] which will come into play later on.

Let be close and let (resp., ) denote the -algebra of bounded continuous complex-valued functions on (respectively, ) with supremum norm denoting the Euclidean norm in ; that is, , .

*Definition 2.1. *A function is called almost periodic if, for each , there exists an such that every interval of length contains a number with the property that .

*Definition 2.2. *A function is called almost periodic in , uniformly in , if, for each and any compact set of , there exists an such that every interval of length contains a number with the property that The number is called an -translation number of . Denote by the set of all such function.

Set

*Definition 2.3. *A function is called pseudo almost periodic (pseudo almost periodic in , uniformly in ), if , where and . The function and are called the almost periodic component and the erigodic perturbation, respectively, of the function . Denote by the set of all such functions .

Define It is trivial to show that is a Banach space with .

Let be a complex matrix-valued function with elements (entries) which are continuous on . We consider the homogeneous linear ODE and nonhomogeneous linear ODE as follows: where denotes an -column vector.

*Definition 2.4 (see [15, 16]). *The homogeneous linear ODE (2.2) is said to admit an exponential dichotomy if there exist a linear projection on and positive constants such that
where is a fundamental matrix of (2.2) with ; is the identity matrix.

*Definition 2.5 (see [15, 16]). *The matrix is said to be row dominant if there exists a number such that for all and .

Lemma 2.6 (see [15, 16]). *If is a bounded, continuous, and row-dominant matrix function on , and there exists such that . Then (2.2) has a fundamental matrix solution satisfying
**
where is a positive constant, and with being a identity matrix. **For , suppose that for all . Define by .*

Lemma 2.7 (see [15, 16]). *Assume that the function is continuous in uniformly in for all compact subsets and such that Then .**It is obvious that if satisfies a Lipschitz condition; that is, there is an such that **
then is continuous in uniformly in . Obviously, if is uniformly continuous in and such that , then .*

Lemma 2.8 (see [15, 16]). *Assume that is an almost-periodic matrix function and . If (2.2) satisfies an exponential dichotomy, then (2.3) has unique pseudo almost periodic solution reading
**
and satisfying , where is a fundamental matrix solution of (2.2).*

*Definition 2.9. *System (1.4) is said to be globally exponentially stable (GES), if for any two solutions and of (1.4), there exist positive numbers and such that
where and denoting the solution of (1.4) through and respectively. Here is called the Lyapunov exponent of (1.4).

#### 3. Existence and Stability of Pseudo Almost-Periodic Solution

Theorem 3.1. *Assume that ()–() hold and **Then (1.4) has a unique pseudo almost-periodic solution, say , satisfying .*

*Proof. *For any , consider
Since , from Lemmas 2.6, 2.7, and 2.8, it follows that (3.2) has a unique pseudo almost-periodic solution, which is given by
Define the mapping by

Let , where .

Clearly, is closed and convex in . Note that
Therefore, for any , we have
Now, we will show that maps into itself. In fact, for any , by using , we have
For any , it follows from that
Since , is a contraction mapping. Therefore, there exists a unique fixed point such that That is, system (1.4) has a unique pseudo almost-periodic solution with .

Now we go ahead with the GES of (1.4). The approaches involve constructing suitable Lyapunov functions and application of a generalized Halanay's delay differential inequality. We will stop here to see our first criteria for the globally exponential stability of (1.4), which is delay dependent.

Theorem 3.2. * In addition to ()–(), if one further assumes that**
or**
Then there exists a unique pseudo-almost periodic solution of system (1.4) and all other solutions converge exponentially to the (pseudo) almost-periodic attractor.*

*Proof. *By Theorem (3.2), there exists a unique pseudo almost-periodic solution, namely, . Let be any other solution of (1.4) through . Assume that is satisfied and consider the auxiliary functions defined on as follows:
From()–(), one can easily show that is well defined and is continuous. From , it follows that as it follows that there exists an such that . Let Then we have

Consider the Lyapunov functional defined by
Calculating the upper-right derivative of and using the inequality , one has
where is defined by
From the above, we have and
Thus, it follows that there exists a positive constant such that
which implies that (1.4) is GES.

Now we assume that is satisfied. By carrying out similar arguments as above, one can easily show that there exists an such that
Consider the Lyapunov function
Similar to the above arguments, calculating the upper-right derivative produces
where
Then we have
Note that

Then there exists a positive constant such that
The proof is complete.

#### References

- A. Bouzerdoum and R. B. Pinter, “Shunting inhibitory cellular neural networks: derivation and stability analysis,”
*IEEE Transactions on Circuits and Systems I*, vol. 40, no. 3, pp. 215–221, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Liu, Z. You, and L. Cao, “On the almost periodic solution of generalized shunting inhibitory cellular neural networks with continuously distributed delays,”
*Physics Letters A*, vol. 360, no. 1, pp. 122–130, 2006. View at: Publisher Site | Google Scholar - T. Zhou, Y. Liu, and A. Chen, “Almost periodic solution for shunting inhibitory cellular neural networks with time-varying delays and variable coefficients,”
*Neural Processing Letters*, vol. 23, no. 3, pp. 243–255, 2006. View at: Publisher Site | Google Scholar - L. Chen and H. Zhao, “Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients,”
*Chaos, Solitons & Fractals*, vol. 35, no. 2, pp. 351–357, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Chen and J. Cao, “Almost periodic solution of shunting inhibitory CNNs with delays,”
*Physics Letters A*, vol. 298, no. 2-3, pp. 161–170, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Li, C. Liu, and L. Zhu, “Global exponential stability of periodic solution for shunting inhibitory CNNs with delays,”
*Physics Letters A*, vol. 337, no. 1-2, pp. 46–54, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Huang and J. Cao, “Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay,”
*Physics Letters A*, vol. 314, no. 3, pp. 222–231, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Liu and L. Huang, “Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,”
*Chaos, Solitons & Fractals*, vol. 31, no. 1, pp. 211–217, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Chen and J. Cao, “Almost periodic solution of shunting inhibitory CNNs with delays,”
*Physics Letters A*, vol. 298, no. 2-3, pp. 161–170, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Xia, J. Cao, and Z. Huang, “Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1599–1607, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Zhou, B. Xiao, Y. Yu, and L. Peng, “Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,”
*Chaos, Solitons & Fractals*, vol. 34, no. 3, pp. 860–866, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Liu, “Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions,”
*Journal of Computational and Applied Mathematics*, vol. 203, no. 1, pp. 159–168, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Y. Zhang,
*Pseudo almost periodic functions and their applications*, Ph.D. thesis, University of Western Ontario, Ontario, Canada, 1992. - C. Y. Zhang, “Pseudo-almost-periodic solutions of some differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 181, no. 1, pp. 62–76, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Y. He,
*Almost Periodic Differential Equations*, Higher Education Publishing House, Beijing, China, 1992. - A. M. Fink,
*Almost Periodic Differential Equations*, vol. 377 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1974. View at: MathSciNet

#### Copyright

Copyright © 2011 Haihui Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.