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Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales.

1. Introduction

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 [2]. A comprehensive paper has been done by Destexhe and Sejnowski [3] which summarized all important development and theoretical results for Wilson-Cowan networks. Its extensive applications include pattern analysis and image processing [4]. 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 [7]. However, few investigations are fixed on the periodicity of Wilson-Cowan model [8] 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 T:

[mathematical expression not reproducible], (1)

t [member of] T, where [X.sub.P] (t), [X.sub.N] (t) represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant t, respectively. [a.sub.P] (t) > 0 and [a.sub.N] (t) > 0 represent the function of the excitatory and inhibitory neurons with natural decay over time, respectively. [r.sub.P] (t) and [r.sub.N] (t) are related to the duration of the refractory period; [k.sub.P] (t) and [k.sub.N] (t) are positive scaling coefficients. [w.sub.P] (t), [w.sub.N] (t), [w.sub.P] (t), and [w.sub.N] (t) are the strengths of connections between the populations. [I.sub.P] (t), [I.sub.N] (t) are the external inputs to the excitatory and the inhibitory populations. G(*) is the response function of neuronal activity. [[tau].sub.P] (t), [[tau].sub.N] (t) 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.

2. Preliminaries

In this section, we give some definitions and lemmas on time scales which can be found in books [14,15].

Definition 1. A time scale T is an arbitrary nonempty closed subset of the real set R. The forward and backward jump operators [sigma], [rho] : T [right arrow] T and the graininess [mu]: T [right arrow] [R.sup.+] are defined, respectively, by

[sigma] (t) := inf {s > T}: [rho] (t) := sup {s [member of] T : s}, [mu] (t) := [sigma] (t) - t. (2)

These jump operators enable us to classify the point {t} of a time scale as right-dense, right-scattered, left-dense, or left-scattered depending on whether

[sigma](t) = t, [sigma](t)>t, [rho](t) = t, [rho](t)<t, respectively, for any t [member of] T. (3)

The notation [[a, b].sub.T] means that [[a, b].sub.T] := {t [member of] T : a [less than or equal to] t [less than or equal to] b}. Denote T := {t [member of] T : t [greater than or equal to] 0}.

Definition 2. One can say that a time scale T is periodic if there exists p > 0 such that t [member of] T; then t [+ or -] p [member of] T; the smallest positive number p is called the period of the time scale.

Clearly, if T is a p-periodic time scale, then [sigma] (t + np) = [sigma](t) + np and [mu] (t + np) = [mu](t). So, [mu](t) is a p-periodic function.

Definition 3. Let T ([not equal to] R) be a periodic time scale with period p. One can say that the function f : T [right arrow] R is periodic with period [omega] > 0 if there exists a natural number n such that [omega] = np, f(t + [omega]) = f(t) for all t [member of] T and [omega] is the smallest number such that f(t + [omega]) = f(t). If T = R, one can say that f is periodic with period [omega] > 0 if [omega] is the smallest positive number such that f(t + [omega]) = f(t) for all t [member of] R.

Definition 4 (Lakshmikantham and Vatsala [16]). For each t [member of] T, let N be a neighborhood of t. Then, one defines the generalized derivative (or Dini derivative), [D.sup.+][u.sup.[DELTA]] (t), to mean that, given [epsilon] > 0, there exists a right neighborhood N([epsilon]) [subset] N of t such that

u ([sigma] (t)) - u (s) / u (t, s) < [D.sup.+][u.sup.[DELTA]] (t) + [epsilon] (4)

for each s [member of] N([epsilon]), s > t, where [mu](t, s) = [sigma](t) - s.

In case t is right-scattered and u(t) is continuous at t, one gets

[D.sup.+][u.sup.[DELTA]] (t) = u([sigma] (t)) - u(t) / [sigma] (t) - t. (5)

Definition 5. A function f : T [right arrow] R is called right-dense continuous provided that it is continuous at right-dense points of T and the left-side limit exists (finite) at left-dense continuous functions on T. The set of all right-dense continuous functions on T is defined by [C.sub.rd] = [C.sub.rd] (T, R).

Definition 6. A function p : T [right arrow] T is called a regressive function if and only if 1 + p (t) [mu] (t) [not equal to] 0.

The set of all regressive and right-dense continuous functions is denoted by R. Let [R.sup.+] := {p [member of] [C.sub.rd] : 1 + p(t)[mu](t) > 0 for all t [member of] T}. Next, we give the definition of the exponential function and list its useful properties.

Definition 7 (Bohner and Peterson [14]). If p [member of] [C.sub.rd] is a regressive function, then the generalized exponential function [e.sub.p] (t, s) is defined by

[e.sub.p] (t, s) = exp {[[integral].sup.t.sub.s] [[xi].sub.[mu]([tau])] (p ([tau])) [DELTA][tau]}, s, t [member of] T, (6)

with the cylinder transformation

[mathematical expression not reproducible]. (7)

Definition 8. The periodic solution

[Z.sup.*] (t) = [([X.sup.*.sub.P] (t), [X.sup.*.sub.N] (t)).sup.T] (8)

of (1) is said to be globally exponentially stable if there exists a positive constant e and N = N([epsilon]) > 0 such that all solutions

Z (t) = [([X.sub.P] (t), [X.sub.N] (t)).sup.T] (9)

of (1) satisfy

[mathematical expression not reproducible]. (10)

Lemma 9 (Bohner and Peterson [15]). If p, q [member of] R, then

(i) [e.sub.0] (t, s) [equivalent to] 1 and [e.sub.p] (t, t) [equivalent to] 1;

(ii) [e.sub.p] ([sigma] (t), s) = (1 + [mu] (t) p (t)) [e.sub.p] (t, s);

(iii) 1/[e.sub.p] (t, s) = [e.sub.[??]p] (t, s), where [??]p(t) = -p(t)/(1 + [mu] (t) p (t));

(iv) [e.sub.p] (t, s) = 1/[e.sub.p] (s, t) = [e.sub.[??]p] (s, t);

(v) [e.sub.p] (t, s) [e.sub.p] (s, r) = [e.sub.p] (t, r);

(vi) [e.sub.p] (t, s) [e.sub.q] (t, s) = [e.sub.p[direct sum]q] (t, s);

(vii) [e.sub.p] (t, s) / [e.sub.q] (t, s) = [e.sub.p[??]q] (t, s);

(viii) [(1/[e.sub.p] (*, s)).sup.[DELTA]] = -p(t)/[e.sup.[sigma].sub.p] (*, s).

Lemma 10 (contraction mapping principle [17]). If [OMEGA] is a closed subset of a Banach space X and F : [OMEGA] [right arrow] [OMEGA] is a contraction, then F has a unique fixed point in [OMEGA].

For any [omega]-periodic function V defined on T, denote [bar.V] = [max.sub.t[member of][0,[omega]]], [V.bar] = [min.sub.t[member of][0,[omega]]] V (t), [absolute value of ([bar.V])] = [max.sub.t[member of][0,[omega]]] [absolute value of (V (t))], and [absolute value of ([V.bar])] = [min.sub.t[member of][0,[omega]]] [absolute value of (V (t))]. Throughout this paper, we make the following assumptions:

([A.sub.1]) [k.sub.P] (t), [k.sub.N] (t), [r.sub.P] (t), [r.sub.N] (t), [w.sup.1.sub.P](t), [w.sup.2.sub.P] (t), [w.sup.1.sub.N] (t), [w.sup.2.sub.N] (t), [a.sub.P] (t), [a.sub.N] (t), [[tau].sub.P] (t), [[tau].sub.N] (t), [I.sub.P] (t), and [I.sub.N] (t) are [omega]- periodic functions defined on T, -[a.sub.P] (t), -[a.sub.N] (t) [member of] [R.sup.+].

([A.sub.2]) G(*) : R [right arrow] R is Lipschitz continuous; that is, [absolute value of (G(u) - G(v))] [less than or equal to] L[absolute value of (u-v)], for all u, v [member of] R, and G(0) = 0, [sup.sub.v[member of]R] [absolute value of (G(v))] [less than or equal to] M.

For simplicity, take the following denotations:

[mathematical expression not reproducible]. (11)

Lemma 11. Suppose ([A.sub.1]) holds; then Z(t) is an w-periodic solution of (1) if and only if Z(t) is the solution of the following system:

[mathematical expression not reproducible]. (12)

Proof. Let Z(t) = [([X.sub.P] (t), [X.sub.N] (t)).sup.T] be a solution of (1); we can rewrite (1) as follows:

[mathematical expression not reproducible], (13)

which leads to

[mathematical expression not reproducible]. (14)

Multiplying both sides of the above equalities by [mathematical expression not reproducible], respectively, we have

[mathematical expression not reproducible]. (15)

Integrating both sides of the above equalities from t to t + [omega] and using [X.sub.P] (t + [omega]) = [X.sub.P] (t) and [X.sub.N] (t + [omega]) = [X.sub.N] (t), we have

[mathematical expression not reproducible]. (16)

Since

[mathematical expression not reproducible] (17)

and [a.sub.P] (t + [omega]) = [a.sub.P] (t), [a.sub.N] (t + [omega]) = [a.sub.N] (t), we obtain that

[mathematical expression not reproducible]. (18)

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 ([A.sub.1])-([A.sub.2]) hold and max {[alpha], W} < 1. Then (1) has a unique [omega]-periodic solution, where

[mathematical expression not reproducible] (19)

and [alpha] := max {[[alpha].sub.1], [[alpha].sub.2]}.

Proof. Let X = {Z(t) = ([z.sub.P] (t), [z.sub.N](t)) | Z [member of] [C.sub.rd] (T, [R.sup.2]), Z (t + [omega]) = Z(t)} with the norm [parallel]Z[parallel] = [sup.sub.t[member of]T] {[absolute value of ([z.sub.P] (t))] + [absolute value of ([z.sub.N] (t))]}; then X is a Banach space [14]. Define

F : X [right arrow] X, (FZ)(t) = ([(FZ).sub.p] (t), [(FZ).sub.N] (t)), (20)

where Z(t) = ([z.sub.p] P(t), [z.sub.N] (t)) [member of] X and

[mathematical expression not reproducible] (21)

for t [member of] T. Note that

[mathematical expression not reproducible]. (22)

Let [OMEGA] = [Z(t) | Z [member of] X, [parallel]Z[parallel] [less than or equal to] I/(1 - W)} and [beta] := I/(1 - W). Obviously, [OMEGA] is a closed nonempty subset of X. Firstly, we prove that the mapping F maps [OMEGA] into itself. In fact, for any Z(t) [member of] [OMEGA], we have

[mathematical expression not reproducible]. (23)

Similarly, we have

[mathematical expression not reproducible]. (24)

It follows from (23) and (24) that

[parallel]FZ[parallel] [less than or equal to] [alpha]I + [alpha]W [parallel]Z[parallel] [less than or equal to] I / 1 - W. (25)

Hence, FZ [member of] [OMEGA].

Next, we prove that F is a contraction mapping. For any Z(t) = ([z.sub.P] (t), [z.sub.N] (t)) [member of] [OMEGA], Z' (t) = ([z'.sub.P] (t), [z'.sub.N] (t)) [member of] [OMEGA], we have

[mathematical expression not reproducible] (26)

Similarly, we have

[mathematical expression not reproducible]. (27)

From (26) and (27), we can get

[parallel] (FZ) - (FZ') [parallel] [less than or equal to] [alpha]W [parallel]Z - Z'[parallel]. (28)

Note that [alpha]W < 1. Thus, F is a contraction mapping. By the fixed point theorem in the Banach space, F possesses a unique fixed point. The proof is completed.

Theorem 13. Under the conditions of Theorem 12, suppose further the following.

([A.sub.3]) There exist some constants [epsilon] > 0, [xi] > 0, [xi]' > 0 such that

[mathematical expression not reproducible]; (29)

then the periodic solution of (1) is globally exponentially stable.

Proof. It follows from Theorem 12 that (1) has an w-periodic solution [Z.sup.*] = [([X.sup.*.sub.P] (t), [X.sup.*.sub.N] (t)).sup.T].

Let Z(t) = [([X.sub.P] (t), [X.sub.N] (t)).sup.T] be any solution of (1); then we have

[mathematical expression not reproducible], (30)

which leads to

[mathematical expression not reproducible]. (31)

For any [alpha] [member of] [[-[[tau].sub.0], 0].sub.T], construct the Lyapunov functional V(t) = [V.sub.1] (t) + [V.sub.2] (t) + [V.sub.3] (t) + [V.sub.4] (t), where

[mathematical expression not reproducible]. (32)

Calculating [D.sup.+]V[(t).sup.[DELTA]] along (1), we can get

[mathematical expression not reproducible], (33)

which leads to

[mathematical expression not reproducible]. (34)

Note that

[mathematical expression not reproducible]. (35)

We have

[mathematical expression not reproducible]. (36)

From (34) and (36), we can get

[mathematical expression not reproducible]. (37)

By assumption ([A.sub.3]), it follows that V(t) [less than or equal to] V(0) for t [member of] [T.sup.+]. On the other hand, we have

[mathematical expression not reproducible], (38)

where [GAMMA]([epsilon]) = max {[[DELTA].sub.1], [[DELTA].sub.2]},

[mathematical expression not reproducible]. (39)

It is obvious that

[mathematical expression not reproducible], (40)

which means that

min {[xi], [xi]'} [e.sub.[epsilon]] (t, [alpha]) ([absolute value of ([X.sub.P] (t) - [X.sup.*.sub.P] (t))] + [absolute value of ([X.sub.N] (t) - [X.sup.*.sub.N] (t))]) [less than or equal to] V (0). (41)

Thus, we finally get

[mathematical expression not reproducible]. (42)

Therefore, the unique periodic solution of (1) is globally exponentially stable. The proof is completed.

4. Examples

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 T :

[mathematical expression not reproducible]. (43)

Case 1. Consider T = R. Take [([a.sub.P] (t), [a.sub.N] (t)).sup.T] = [(2 + sin (t), 2 + cos (t)).sup.T]. Obviously, [[a.sub.P].bar] = [[a.sub.N].bar] = 1,

[mathematical expression not reproducible]. (44)

Take [([I.sub.P] (t), [I.sub.N] (t)).sup.T] = [(-1 + sin (t), cos(t)).sup.T], [k.sub.P] (t) = [k.sub.N] (t) = [r.sub.P] (t) = [r.sub.N] (t) = 0.01, [w.sup.1.sub.P](t) = [w.sup.1.sub.N](t) = [w.sup.2.sub.P] (t) = [w.sup.2.sub.N] (t) = 0.1, and G(x) = (1/2)([absolute value of (x + 1)] - [absolute value of (x - 1)]). We have L = 1. Let [xi] = 1, [xi]' = 2. One can easily verify that

[mathematical expression not reproducible]. (45)

It follows from Theorems 12 and 13 that (43) has a unique 2[pi]-periodic solution which is globally exponentially stable (see Figure 1).

Case 2. Consider T = Z. Equation (43) reduces to the following difference equation:

[mathematical expression not reproducible], (46)

for n [member of] [Z.sup.+.sub.0]. Take [([a.sub.P] (n), [a.sub.N] (n)).sup.T] = [(1/2, 1/2).sup.T]. Obviously, [mathematical expression not reproducible]. We have L = 1. Let [xi] = 1, [xi]' = 2. If T = Z, ([mu] (t) = 1), choosing [omega] = 6, by simple calculation, we have

[mathematical expression not reproducible]. (47)

It follows from Theorems 12 and 13 that (46) has a unique 6-periodic solution which is globally exponentially stable (see Figure 2).

5. Conclusion Remarks

In this paper, we studied the stability of delayed Wilson-Cowan networks on periodic time scales and obtained some more generalized results to ensure the existence, uniqueness, and global exponential stability of the periodic solution. These results can give a significant insight into the complex dynamical structure of Wilson-Cowan type model. The conditions are easily checked in practice by simple algebraic methods.

http://dx.doi.org/ 10.1155/2014/750532

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11101187 and 11361010), the Foundation for Young Professors of Jimei University, the Excellent Youth Foundation of Fujian Province (2012J06001 and NCETFJ JA11144), and the Foundation of Fujian Higher Education (JA10184 and JA11154).

References

[1] H. R. Wilson and J. D. Cowan, "Excitatory and inhibitory interactions in localized populations of model neurons," Biophysical Journal, vol. 12, no. 1, pp. 1-24,1972.

[2] H. R. Wilson and J. D. Cowan, "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue," Kybernetik, vol. 13, no. 2, pp. 55-80,1973.

[3] A. Destexhe and T. J. Sejnowski, "The Wilson-Cowan model, 36 years later," Biological Cybernetics, vol. 101, no. 1, pp. 1-2, 2009.

[4] K. Mantere, J. Parkkinen, T. Jaaskelainen, and M. M. Gupta, "Wilson-Cowan neural-network model in image processing," Journal of Mathematical Imaging and Vision, vol. 2, no. 2-3, pp. 251-259, 1992.

[5] C. van Vreeswijk and H. Sompolinsky, "Chaos in neuronal networks with balanced excitatory and inhibitory activity," Science, vol. 274, no. 5293, pp. 1724-1726,1996.

[6] L. H. A. Monteiro, M. A. Bussab, and J. G. Berlinck, "Analytical results on a Wilson-Cowan neuronal network modified model," Journal of Theoretical Biology, vol. 219, no. 1, pp. 83-91, 2002.

[7] S. Xie and Z. Huang, "Almost periodic solution for Wilson-Cowan type model with time-varying delays," Discrete Dynamics in Nature and Society, vol. 2013, Article ID 683091, 7 pages, 2013.

[8] V. W. Noonburg, D. Benardete, and B. Pollina, "A periodically forced Wilson-Cowan system," SIAM Journal on Applied Mathematics, vol. 63, no. 5, pp. 1585-1603, 2003.

[9] S. Hilger, "Analynis on measure chains-a unified approach to continuous and discrete calculus," Results in Mathematics, vol. 18, pp. 18-56, 1990.

[10] S. Hilger, "Differential and difference calculus--unified!," Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 5, pp. 2683-2694, 1997.

[11] A. Chen and F. Chen, "Periodic solution to BAM neural network with delays on time scales," Neurocomputing, vol. 73, no. 1-3, pp. 274-282, 2009.

[12] Y. Li, X. Chen, and L. Zhao, "Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales," Neurocomputing, vol. 72, no. 7-9, pp. 1621-1630, 2009.

[13] Z. Huang, Y. N. Raffoul, and C. Cheng, "Scale-limited activating sets and multiperiodicity for threshold-linear networks on time scales," IEEE Transactions on Cybernetics, vol. 44, no. 4, pp. 488499, 2014.

[14] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, Mass, USA, 2001.

[15] M. Bohner and A. Peterson, Advance in Dynamic Equations on Time Scales, Birkhauser, Boston, Mass, USA, 2003.

[16] V. Lakshmikantham and A. S. Vatsala, "Hybrid systems on time scales," Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 227-235, 2002.

[17] A. Ruffing and M. Simon, "Corresponding Banach spaces on time scales," Journal of Computational and Applied Mathematics, vol. 179, no. 1-2, pp. 313-326, 2005.

Jinxiang Cai, Zhenkun Huang, and Honghua Bin

School of Science, Jimei University, Xiamen 361021, China

Correspondence should be addressed to Zhenkun Huang; hzk974226@jmu.edu.cn

Received 31 December 2013; Accepted 12 February 2014; Published 2 April 2014

Academic Editor: Songcan Chen

Caption: FIGURE 1: Globally exponentially stable periodic solution of (43).

Caption: FIGURE 2: Globally exponentially stable periodic solution of (46).
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Author:Cai, Jinxiang; Huang, Zhenkun; Bin, Honghua
Publication:Advances in Artificial Neural Systems
Date:Jan 1, 2014
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