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Advances in Artificial Neural Systems
Volume 2014 (2014), Article ID 750532, 10 pages
Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales
School of Science, Jimei University, Xiamen 361021, China
Received 31 December 2013; Accepted 12 February 2014; Published 2 April 2014
Academic Editor: Songcan Chen
Copyright © 2014 Jinxiang Cai et al. 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.
We present stability analysis of delayed Wilson-Cowan networks on time scales. By applying the theory of calculus on time scales, the contraction mapping principle, and Lyapunov functional, new sufficient conditions are obtained to ensure the existence and exponential stability of periodic solution to the considered system. The obtained results are general and can be applied to discrete-time or continuous-time Wilson-Cowan networks.
The activity of a cortical column may be mathematically described through the model developed by Wilson and Cowan [1, 2]. Such a model consists of two nonlinear ordinary differential equations representing the interactions between two populations of neurons that are distinguished by the fact that their synapses are either excitatory or inhibitory . A comprehensive paper has been done by Destexhe and Sejnowski  which summarized all important development and theoretical results for Wilson-Cowan networks. Its extensive applications include pattern analysis and image processing . Theoretical results about the existence of asymptotic stable limit cycle and chaos have been reported in [5, 6]. Exponential stability of a unique almost periodic solution for delayed Wilson-Cowan type model has been reported in . However, few investigations are fixed on the periodicity of Wilson-Cowan model  and it is troublesome to study the stability and periodicity for continuous and discrete system with oscillatory coefficients, respectively. Therefore, it is significant to study Wilson-Cowan networks on time scales [9, 10] which can unify the continuous and discrete situations.
Motivated by recent results [11–13], we consider the following dynamic Wilson-Cowan networks on time scale : , where , represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant , respectively. and represent the function of the excitatory and inhibitory neurons with natural decay over time, respectively. and are related to the duration of the refractory period; and are positive scaling coefficients. , , , and are the strengths of connections between the populations. , are the external inputs to the excitatory and the inhibitory populations. is the response function of neuronal activity. , correspond to the transmission time-varying delays.
The main aim of this paper is to unify the discrete and continuous Wilson-Cowan networks with periodic coefficients and time-varying delays under one common framework and to obtain some generalized results to ensure the existence and exponential stability of periodic solution on time scales. The main technique is based on the theory of time scales, the contraction mapping principle, and the Lyapunov functional method.
Definition 1. A time scale is an arbitrary nonempty closed subset of the real set . The forward and backward jump operators , and the graininess are defined, respectively, by
These jump operators enable us to classify the point of a time scale as right-dense, right-scattered, left-dense, or left-scattered depending on whether The notation means that . Denote .
Definition 2. One can say that a time scale is periodic if there exists such that ; then ; the smallest positive number is called the period of the time scale.
Clearly, if is a -periodic time scale, then and . So, is a -periodic function.
Definition 3. Let be a periodic time scale with period . One can say that the function is periodic with period if there exists a natural number such that , for all and is the smallest number such that . If , one can say that is periodic with period if is the smallest positive number such that for all .
Definition 4 (Lakshmikantham and Vatsala ). For each , let be a neighborhood of . Then, one defines the generalized derivative (or Dini derivative), , to mean that, given , there exists a right neighborhood of such that
for each , , where .
In case is right-scattered and is continuous at , one gets
Definition 5. A function is called right-dense continuous provided that it is continuous at right-dense points of and the left-side limit exists (finite) at left-dense continuous functions on . The set of all right-dense continuous functions on is defined by .
Definition 6. A function is called a regressive function if and only if .
The set of all regressive and right-dense continuous functions is denoted by . Let . Next, we give the definition of the exponential function and list its useful properties.
Definition 7 (Bohner and Peterson ). If is a regressive function, then the generalized exponential function is defined by with the cylinder transformation
Lemma 9 (Bohner and Peterson ). If , then(i) and ;(ii);(iii), where ;(iv);(v);(vi);(vii);(viii).
Lemma 10 (contraction mapping principle ). If is a closed subset of a Banach space and is a contraction, then has a unique fixed point in .
For any -periodic function defined on , denote , , , and . Throughout this paper, we make the following assumptions:(), , , , , , , , , , , , , and are -periodic functions defined on , , .() is Lipschitz continuous; that is, , for all , and , .
For simplicity, take the following denotations:
Lemma 11. Suppose () holds; then is an -periodic solution of (1) if and only if is the solution of the following system:
Proof. Let be a solution of (1); we can rewrite (1) as follows:
which leads to
Multiplying both sides of the above equalities by and , respectively, we have
Integrating both sides of the above equalities from to and using and , we have Since and , , we obtain that The proof is completed.
3. Main Results
In this section, we prove the existence and uniqueness of the periodic solution to (1).
Theorem 12. Suppose ()-() hold and . Then (1) has a unique -periodic solution, where and .
Proof. Let with the norm ; then is a Banach space . Define
for . Note that
Let and . Obviously, is a closed nonempty subset of . Firstly, we prove that the mapping maps into itself. In fact, for any , we have
Similarly, we have
It follows from (23) and (24) that
Next, we prove that is a contraction mapping. For any , , we have Similarly, we have From (26) and (27), we can get Note that . Thus, is a contraction mapping. By the fixed point theorem in the Banach space, possesses a unique fixed point. The proof is completed.
Proof. It follows from Theorem 12 that (1) has an -periodic solution .
Let be any solution of (1); then we have which leads to For any , construct the Lyapunov functional , where Calculating along (1), we can get which leads to Note that We have From (34) and (36), we can get By assumption (), it follows that for . On the other hand, we have where , It is obvious that which means that Thus, we finally get Therefore, the unique periodic solution of (1) is globally exponentially stable. The proof is completed.
In this section, two numerical examples are shown to verify the effectiveness of the result obtained in the previous section. Consider the following Wilson-Cowan neural network with delays on time scale :
Case 1. Consider . Take . Obviously, , Take , , , and . We have . Let , . One can easily verify that It follows from Theorems 12 and 13 that (43) has a unique -periodic solution which is globally exponentially stable (see Figure 1).
Case 2. Consider . Equation (43) reduces to the following difference equation: for . Take . Obviously, , , , , , and . We have . Let , . If , , choosing , by simple calculation, we have