Volume 2008, Article ID 648148,14pages doi:10.1155/2008/648148

*Research Article*

**New Results of a Class of Two-Neuron Networks**

**with Time-Varying Delays**

**Chuangxia Huang,1 _{Yigang He,}2_{and Lihong Huang}3**

*1 _{College of Mathematics and Computers, Changsha University of Science and Technology,}*

*Changsha, Hunan 410076, China*

*2 _{College of Electrical and Information Engineering, Hunan University, Changsha, Hunan 410082, China}*

*3 _{College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China}*

Correspondence should be addressed to Chuangxia Huang,cxiahuang@126.com

Received 19 September 2008; Accepted 4 December 2008

Recommended by Marta Garcia-Huidobro

With the help of the continuation theorem of the coincidence degree, a priori estimates, and diﬀerential inequalities, we make a further investigation of a class of planar systems, which is generalization of some existing neural networks under a time-varying environment. Without assuming the smoothness, monotonicity, and boundedness of the activation functions, a set of suﬃcient conditions is given for checking the existence of periodic solution and global exponential stability for such neural networks. The obtained results extend and improve some earlier publications.

Copyrightq2008 Chuangxia Huang 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.

**1. Introduction**

Neural networks are complex and large-scale nonlinear dynamics, while the dynamics of the delayed neural network are even richer and more complicated 1. To obtain a deep and clear understanding of the dynamics of neural networks, one of the usual ways is to investigate the delayed neural network models with two neurons, which can be described by diﬀerential systemssee2–8. It is hoped that, through discussing the dynamics of two-neuron networks, we can get some light for our understanding about the large networks. In

7, T´aboas considered the system of delay diﬀerential equations

˙

*x*1t −*x*1t *αf*1

*x*1t−*τ*, x2t−*τ*

*,*

˙

*x*2t −*x*2t *αf*2

*x*1t−*τ*, x2t−*τ*

*,* 1.1

*∂f*1

*∂x*20*,*0*/*0*,*

*∂f*2

*∂x*10*,*0*/*0*,* 1.2

and the negative feedback conditions:*x*2*f*1x1*, x*2*>*0,*x*2 */*0;*x*1*f*2x1*, x*2*<*0,*x*1 */*0. T´aboas

showed that there is an*α*0*>*0 such that for*α > α*0, there exists a nonconstant periodic solution

with period greater than 4. Further study on the global existence of periodic solutions to system1.1can be found in2,3. All together there is only one delay appearing in both equations. Ruan and Wei6investigated the existence of nonconstant periodic solutions of the following planar system with two delays

˙

*x*1t −*a*0*x*1t *a*1*f*1

*x*1t−*τ*1, x2t−*τ*2

*,*

˙

*x*2t −*b*0*x*2t *b*1*f*2

*x*1t−*τ*1, x2t−*τ*2

*,* 1.3

where*a*0 *>* 0, *b*0 *>* 0,*a*1 and*b*1 are constants, the functions*f*1 and*f*2 satisfy*fj* ∈ *C*3R2,
*fj*0*,*0 0,∂f*j/∂xj*0*,*0 0,*j* 1*,*2;*x*2*f*1x1*, x*2*/*0 for*x*2 */*0;*x*1*f*2x1*, x*2*/*0 for*x*1 */*0;
∂f1*/∂x*20*,*0*/*0,∂f2*/∂x*10*,*0*/*0.

Recently, Chen and Wu9considered the following system:

˙

*xt *−*μ*1*xt F*

*yt*−*τ*1

*I*1*,*

˙

*yt *−*μ*2*yt*−*G*

*xt*−*τ*2

*I*2*,*

1.4

where*μ*1and*μ*2are positive constants,*τ*1and*τ*2are nonnegative constants with*τ* :*τ*1 *τ*2*>*

0,*I*1and*I*2are constants, and*F*and*G*are bounded*C*1-functions with ˙*Ft>*0, and ˙*Gt>*0

for*t* ∈ *R*. By discussing a two-dimensional unstable manifold of the transformation of the
system1.4, they got some results about slowly oscillating periodic solutions.

However, delays considered in all above systems1.1–1.4are constants. It is well known that the delays in artificial neural networks are usually time-varying, and they sometime vary violently with time due to the finite switching speed of amplifiers and faults in the electrical circuit. They slow down the transmission rate and tend to introduce some degree of instability in circuits. Therefore, fast response must be required in practical artificial neural-network designs. As pointed out by Gopalsamy and Sariyasa10,11, it would be of great interest to study the neural networks in periodic environments. On the other hand, drop the assumptions of continuous first derivative, monotonicity, and boundedness for the activation might be better candidates for some purposes, see 12. Motivated by above mentioned, in this paper, we continue to consider the following planar system:

˙

*x*1t −*a*1tx1t *b*1tf1

*x*1

*t*−*τ*11t

*, x*2

*t*−*τ*12t

*I*1t,

˙

*x*2t −*a*2tx2t *b*2tf2

*x*1

*t*−*τ*21t

*, x*2

*t*−*τ*22t

*I*2t,

1.5

where*ai*t∈*CR,*0*,*∞,*bi*t, I*i*t∈*CR, R, i*1*,*2, are periodic with a common period
*ω>*0,*fi*·*,*·∈*CR*2*, R*,*τij*t∈*CR,*0*,*∞, i,*j*1*,*2 being*ω*-periodic.

functions, a family of suﬃcient conditions are given for checking the existence of periodic solution and global exponential stability for such neural networks. Our results extend and improve some earlier publications. The remainder of this article is organized as follows. In

Section 2, the basic notations and assumptions are introduced. After giving the criteria for checking the existence of periodic solution and global exponential stability for the neuron networks in Section 3, one illustrative example and simulations are given inSection 4. We also conclude this paper inSection 5.

**2. Preliminaries**

In this section, we state some notations, definitions, and lemmas. Assume that nonlinear system1.5is supplemented with initial values of the type

*ui*t *φi*t, −*τ* ≤*t*≤0*,* *τ* max
1≤*i,j*≤2

*τij*t

*,* 2.1

in which*φi*t, i1*,*2 are continuous functions. Set*φ* φ1*, φ*2*, φ*3*T*, if*x*∗t x_{1}∗t, x∗_{2}t*T*

is a solution of system1.5, then we denote

_{φ}_{−}* _{x}*∗

_{}2

*i*1

sup

−*τ*≤*t*≤0

_{φ}_{i}_{t}_{−}* _{x}*∗

*i*t*.* 2.2

*Definition 2.1.* The solution*x*∗t x∗_{1}t, x_{2}∗t*T* is said to be globally exponentially stable, if
there exist*λ >*0 and*m*≥1 such that for any solution*xt x*1t, x2t*T* of1.5, one has

_{x}_{i}_{t}_{−}* _{x}*∗

*i*t≤*mφ*−*x*∗*e*−*λt* for*t*≥0*,* 2.3

where*λ*is called to be globally exponentially convergent rate.

To establish the main results of the model1.5, some of the following assumptions will be applied:

H1|*fi*x1*, x*2| ≤ *αi|x*1| *βi|x*2| *Mi* for allx1*, x*2*T* ∈*R*2*, i* 1*,*2, where*αi* ≥ 0*, β _{i}* ≥

0*, Mi>*0 are constants;

H2there exist constants*αi* ≥0*, βi* ≥ 0*,*such that|*fj*x1*, x*2−*fj*y1*, y*2| ≤*αi|x*1−*y*1|
*βi|x*2−*y*2|for anyx1*, x*2*T* ∈*R*2*,* y1*, y*2*T* ∈*R*2.

For*V* ∈*Ca,* ∞, R, one denotes

*D*−*V*t lim sup

*h*→0−

*V*t *h*−*V*t

*h* *,*

*D*−*V*t lim inf

*h*→0−

*V*t *h*−*V*t

*h* *.*

2.4

Consider an abstract equation in a Branch space *X* In this section, we use the
coincidence degree theory to obtain the existence of an*ω*-periodic solution to1.5. For the
sake of convenience, we briefly summarize the theory as below.

Let*X*and*Z*be normed spaces,*L*: Dom*L*⊂*X* →*Z*be a linear mapping, and*N*:*X* →
*Z*be a continuous mapping. The mapping*L*will be called a Fredholm mapping of index zero
if dim Ker*L*codim Im*L <*∞and Im*L*is closed in*Z*. If*L*is a Fredholm mapping of index
zero, then there exist continuous projectors*P*:*X* →*X*and*Q*:*Z* →*Z*such that Im*P* Ker*L*

and Im*L*Ker*Q*ImI−*Q*. It follows that*L*|Dom*L*∩Ker*P*:I−*P*X →Im*L*is invertible.
We denote the inverse of this map by*Kp*. IfΩis a bounded open subset of*X*, the mapping*N*

is called*L*-compact onΩif*QNΩ*is bounded and*Kp*I−*QN*:Ω →*X*is compact. Because

Im*Q*is isomorphic to Ker*L*, there exists an isomorphism*J*: Im*Q* →Ker*L*.

LetΩ⊂*Rn*_{be open and bounded,}_{f}_{∈}* _{C}*1

_{Ω, R}

*n*

_{}

_{∩}

_{CΩ, R}n_{}

_{and}

_{y}_{∈}

_{R}n_{\}

_{f∂Ω}_{∪}

_{S}*f*,

that is,*y*is a regular value of*f*. Here,*Sf* {*x*∈Ω:*Jf*x 0}, the critical set of*f*, and*Jf* is

the Jacobian of*f*at*x*. Then, the degree deg{*f,*Ω, y}is defined by

deg{*f,*Ω, y}
*x*∈*f*−1_{}_{y}_{}

sgn*Jf*x, 2.5

with the agreement that the above sum is zero if*f*−1_{y } _{∅}_{. For more details about the}

degree theory, one refers to the book of Deimling14.

Now, with the above notation, we are ready to state the continuation theorem.

**Lemma 2.2**Continuation theorem13, page 40. *LetLbe a Fredholm mapping of index zero and*

*letNbeL-compact on*Ω*. Suppose that*

a*for eachλ*∈0*,*1*, every solutionxofLxλNxis such thatx/*∈*∂Ω;*
b*QNx /*0*for eachx*∈*∂Ω*∩Ker*Land*

deg*JQN,* Ω∩Ker*L,* 0*/*0*.* 2.6

*Then, the equationLx* *Nx* *has at least one solution lying in*Dom*L*∩Ω*. For more details about*
*degree theory, we refer to the book by Deimling [14].*

For the simplicity of presentation, in the remaining part of this paper, we also introduce the following notation:

*ai*min

*ai*t

*,* *bij*max*bij*t*,* *Ii*max*Ii*t*,*

*D*_{}*a*1−*b*1*α*1 −*b*1*β*1
−*b*2*α*2 *a*2−*b*2*β*2

*,*

*D*1

*I*1 *b*1*M*1 −*b*1*β*1
*I*2 *b*2*M*2 *a*2−*b*2*β*2

*,* *D*2

*a*1−*b*1*α*1 *I*1 *b*1*M*1
−*b*2*α*2 *I*2 *b*2*M*2
*.*

**3. Main results**

**Theorem 3.1.** *SupposeH*1*holds andD*1*/D >*0*,D*2*/D >* 0*, then system*1.5*has aω-periodic*

*solution.*

*Proof.* Take*X*{*ut x*1t, x2t*T* ∈*CR, R*2:*ut ut* *ω*for*t*∈*R*}and denote

|*xi|* max

*t*∈0,ω|*xi*t|*,* *i*1*,*2; *u*0max*i*1,2|*xi|.* 3.1

Equipped with the norm·0,*X*is a Banach space. For any*ut*∈*X*, because of the periodicity,
it is easy to check that

*t* −→ −*a*1txt *b*1tf1

*xt*−*τ*11t

*, yt*−*τ*12t

*I*1t
−*a*2tyt *b*2tf2

*xt*−*τ*21t

*, yt*−*τ*22t

*I*2t

∈*X.* 3.2

Let

*L*: Dom*Lu*∈*X*:*u*∈*C*1* _{R, R}*2

_{}

_{u}_{ −→}

_{u}_{∈}

_{X,}*P* :*Xu* −→*u*∈*X,* *Q*:*X* *x* −→*x*∈*X,* 3.3

where for any*γ* γ1*, γ*2*T* ∈*R*2, we identify it as the constant function in*X*with the value

vector*γ* γ1*, γ*2*T*. Define*N*:*X* →*X*given by

Nut −*a*1tx1t *b*1tf1

*x*1

*t*−*τ*11t

*, x*2

*t*−*τ*12t

*I*1t
−*a*2tx2t *b*2tf2

*x*1

*t*−*τ*21t

*, x*2

*t*−*τ*22t

*I*2t

∈*X.* 3.4

Then, system1.5can be reduced to the operator equation*LuNu*. It is easy to see that

Ker*LR*2_{,}

Im*L*{*x*∈*X* :*x*0}*,* which is closed in*X,*

dim Ker*L*codim Im*L*2*<*∞*,*

3.5

and*P*,*Q*are continuous projectors such that

Im*P* Ker*L,* Ker*Q*Im*L*ImI−*Q.* 3.6

It follows that*L*is a Fredholm mapping of index zero. Furthermore, the generalized inverse

to*LKp*: Im*L* →Ker*P*∩Dom*L*is given by

*Kp*u

t

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

*t*

0

*x*1sd*s*− 1
*ω*

*ω*

0

*s*

0

*x*1vd*v*d*s*

_{t}

0

*x*2sd*s*−

1

*ω*

_{ω}

0

_{s}

0

*x*2vd*v*d*s*
⎞
⎟
⎟
⎟
⎟

Thus,
QNut
⎛
⎜
⎜
⎜
⎜
⎝
1
*ω*
*ω*
0

−*a*1sx1s *b*1sf1

*x*1

*s*−*τ*11s

*, x*2

*s*−*τ*12s

*I*1s
d*s*
1
*ω*
*ω*
0

−*a*2sx2s *b*2sf2

*x*1

*s*−*τ*21s

*, x*2

*s*−*τ*22s

*I*2s
d*s*
⎞
⎟
⎟
⎟
⎟
⎠*,*

*Kp*I−*QNu*
t
⎛
⎜
⎜
⎜
⎜
⎝
*t*
0

−*a*1sx1s *b*1sf1

*x*1

*s*−*τ*11s

*, x*2

*s*−*τ*12s

*I*1s

d*s*

*t*

0

−*a*2sx2s *b*2sf2

*x*1

*s*−*τ*21s

*, x*2

*s*−*τ*22s

*I*2s
d*s*
⎞
⎟
⎟
⎟
⎟
⎠
−
⎛
⎜
⎜
⎜
⎜
⎝
1
*ω*
*ω*
0
*s*
0

−*a*1vx1v *b*1vf1

*x*1

*v*−*τ*11v

*, x*2

*v*−*τ*12v

*I*1v

d*v*d*s*

1
*ω*
* _{ω}*
0

*0*

_{s}−*a*2vx2v *b*2vf2

*x*1

*v*−*τ*21v

*, x*2

*v*−*τ*22v

*I*2v

d*v*d*s*

⎞
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎝
1
2−
*t*
*ω*
*ω*
0

−*a*1sx1s *b*1sf1

*x*1

*s*−*τ*11s

*, x*2

*s*−*τ*12s

*I*1s
d*s*
1
2−
*t*
*ω*
*ω*
0

−*a*2sx2s *b*2sf2

*x*1

*s*−*τ*21s

*, x*2

*s*−*τ*22s

*I*2s
d*s*
⎞
⎟
⎟
⎟
⎟
⎠*.*
3.8

Clearly,*QN*and*Kp*I−*QN*are continuous. For any bounded open subsetΩ⊂*X*,*QNΩ*

is obviously bounded. Moreover, applying the Arzela-Ascoli theorem, one can easily show
that*Kp*I−*QNΩ*is compact. Note that*Kp*I−*QN*is a compact operator and *QNΩ*

is bounded, therefore,*N* is*L*-compact onΩ with any bounded open subsetΩ ⊂ *X*. Since
Im*Q* Ker*L*, we take the isomorphism*J* of Im*Q* onto Ker*L* to be the identity mapping.
Corresponding to equation*LuλNu, λ*∈0*,*1, we have

˙

*x*1t *λ*

−*a*1tx1t *b*1tf1

*x*1t−*τ*11t

*, x*2t−*τ*12t

*I*1t

*,*

˙

*x*2t *λ*

−*a*2tx2t *b*2tf2

*x*1

*t*−*τ*21t

*, x*2

*t*−*τ*22t

*I*2t

*.* 3.9

Now we reach the position to search for an appropriate open bounded subset Ω for the
application of the Lemma 2.2. Assume that *u* *ut* ∈ *X* is a solution of system 3.9 for
some*λ* ∈0*,*1. Then, the components*xi*t i 1*,*2of*ut*are continuously diﬀerentiable.

Thus, there exists*ti*∈0*, ω*such that|*xi*t*i*|max*t*∈0,ω|*xi*t|. Hence, ˙*xi*t*i* 0. This implies

*a*1
*t*1
*x*1

*t*1*b*1
*t*1
*f*1
*x*1

*t*1−*τ*1

*t*1

*, x*2

*t*1−*τ*2

*t*1

*I*1

*t*1*,*

_{a}_{2}

*t*2

*x*2

*t*2*b*2
*t*2
*f*2
*x*1

*t*2−*τ*3

*t*2

*, x*2

*t*2−*τ*4

*t*2

*I*1

*t*2*.*

Since

*fi*

*x*1*, x*2≤*αix*1 *βix*2 *Mi* for*i*1*,*2*,* 3.11

we get

_{x}_{1}

*t*1≤

*α*1*b*1*x*1

*t*1−*τ*11

*t*1
*a*1

*β*1*b*1*x*2

*t*1−*τ*12

*t*1
*a*1

*b*1*M*1 *I*1
*a*1

*,*

*x*2

*t*2≤

*α*2*b*2*x*1

*t*2−*τ*21

*t*2
*a*2

*β*2*b*2*x*2

*t*2−*τ*22

*t*2
*a*2

*b*2*M*2 *I*2
*a*2

*.*

3.12

Set*k*1*D*1*/D*,*k*2*D*2*/D*, we find that

*k*1
*α*1*b*1

*a*1 *k*1
*β*1*b*1

*a*1 *k*2

*b*1*M*1 *I*1
*a*1 *,*

*k*2
*α*2*b*2

*a*2
*k*1

*β*2*b*2
*a*2

*k*2

*b*2*M*2 *I*2
*a*2

*.*

3.13

Now, we choose a constant number*d >*1 and take

Ω *x*1*, x*2

*T* _{∈}

*R*2; *xi< dki* for*i*1*,*2

*,* 3.14

where*k*1*D*1*/D >*0,*k*2*D*2*/D >*0. We will show thatΩsatisfies all the requirements given

inLemma 2.2. In fact, we will prove that if*xi*t−*τij*t∈Ωthen*xi*t∈Ωfor*i, j* 1*,*2*, t*≥0.

Therefore, it means that *u* *ut*is uniformly bounded with respect to*λ* when the initial
value function belongs toΩ. It follows from3.12and3.13that

_{x}_{i}_{} *αibi*
*ai*

_{x}_{1}

*ti*−*τi1*t
*βibi*

*ai*

_{x}_{1}

*ti*−*τi2*t *biMi* *Ii*
*ai*

≤ *αibi*
*ai* *dk*1

*βibi*
*ai* *dk*2

*biMi* *Ii*
*ai* *< d*

_{α}

*ibi*
*ai* *k*1

*βibi*
*ai* *k*2

*biMi* *Ii*
*ai*

*,*

3.15

*i*1*,*2. Therefore,

_{x}_{i}

0*< dki* for*i*1*,*2*.* 3.16

Clearly,*dki*,*i* 1*,*2 are independent of *λ*. It is easy to see that there are no*λ* ∈ 0*,*1and
*u* ∈ *∂Ω*such that*Lu* *λNu*. If*u* x1*, x*2*T* ∈ *∂Ω*∩Ker*L* *∂Ω*∩*R*2, then*u*is a constant

vector in*R*2_{with}_{|}_{x}_{i|}_{}_{dk}

*i*for*i*1*,*2. Note that*QNuJQNu*, we have

*QNu*
⎛
⎜
⎜
⎜
⎜
⎝

−*x*11
*ω*

*ω*

0

*a*1td*t* *f*1

*x*1*, x*2

1

*ω*

*ω*

0

*b*1td*t* 1
*ω*

*ω*

0
*I*1td*t*

−*x*2

1

*ω*

_{ω}

0

*a*2td*t* *f*2

*x*1*, x*2

1

*ω*

_{ω}

0
*b*2td*t*

1

*ω*

_{ω}

0
*I*2td*t*

⎞ ⎟ ⎟ ⎟ ⎟

We claim that

_{QNu}

*i>*0 *u*∈*∂Ω*∩Ker*L∂Ω*∩*R*3*, i*1*,*2*.* 3.18

Contrarily, suppose that there exists some*i*such that|QNu* _{i}*|0, that is,

*xi*

1

*ω*

_{ω}

0

*ai*td*tfi*

*x*1*, x*2

1

*ω*

_{ω}

0
*bi*td*t*

1

*ω*

_{ω}

0

*Ii*td*t.* 3.19

So, we have

*dki*|*xi| ≤*

1

*ai*

*fi*

*x*1*, x*2

1

*ω*

_{ω}

0
*bi*td*t*

1

*ω*

_{ω}

0
*Ii*td*t*

≤ *αibi*
*ai*

*dk*1
*βibi*

*ai*
*dk*2

*biMi* *Ii*
*ai*

*< d*

*αibi*
*ai*

*k*1
*βibi*

*ai*
*k*2

*biMi* *Ii*
*ai*

*dki,*

3.20

this is a contradiction. Therefore,3.17holds, and hence,

*QNu /*0*,* for any *u*∈*∂Ω*∩Ker*L∂Ω*∩*R*2_{.}_{}_{3.21}_{}

Now, consider the homotopy*F*:Ω∩Ker*L*×0*,*1 → Ω∩Ker*L*, defined by

*Fu, μ *−*μ*diag

1

*ω*

*ω*

0

*a*1td*t,*

1

*ω*

*ω*

0
*a*2td*t*

*u* 1−*μQNu,* 3.22

where*u*∈Ω∩Ker*L* Ω∩*R*2_{and}_{μ}_{∈}_{}_{0}_{,}_{1}_{}_{. When}_{u}_{∈}_{∂Ω}_{∩}_{Ker}_{L}_{}_{∂Ω}_{∩}* _{R}*2

_{and}

_{μ}_{∈}

_{}

_{0}

_{,}_{1}

_{}

_{,}

*u*x1

*, x*2

*T*is a constant vector in

*R*2with|

*xi|dki*for

*i*1

*,*2. Thus

_{Fu, μ}

0max_{i}_{}_{1,2}

−*xi*1

*ω*

*ω*

0

*ai*td*t* 1−*μfi*

*x*1*, x*2

1

*ω*

*ω*

0
*bi*td*t*

*.* 3.23

We claim that

*Fu, μ/* 0 u, μ∈∂Ω∩Ker*L*×0*,*1. 3.24

Contrarily, suppose that*Fu, μ*00, then,

*xi*

1

*ω*

_{ω}

0

*ai*td*t* 1−*μfi*

*x*1*, x*2

1

*ω*

_{ω}

0

Thus,

*dki*|*xi| ≤*

1−*μ*

*ai*
*bifi*

*x*1*, x*2

*< d*

*αibi*
*ai*

*k*1
*βibi*

*ai*

*k*2 *biMi* *Ii*
*ai*

*dki.* 3.26

This is impossible. Thus,3.22holds. From the property of invariance under a homotopy, it follows that

deg*JQN,*Ω∩Ker*L,*0

deg*F*·*,*0,Ω∩Ker*L,*0

deg*F*·*,*1,Ω∩Ker*L,*0

sgn

−1
*ω*

_{ω}

0

*a*1td*t* 0

0 −1

*ω*

_{ω}

0
*a*2td*t*

sgn

1

*ω*

_{ω}

0
*a*1td*t*

1

*ω*

_{ω}

0
*a*2td*t*

*/*
0*.*

3.27

We have shown thatΩsatisfies all the assumptions of Lemma 2.2. Hence, *Lu* *Nu*has at
least one*ω*-periodic solution on Dom*L*∩Ω. This completes the proof.

**Corollary 3.2.** *Suppose that there exist positive constantsMisuch that*|*fi*x1*, x*2| ≤*Mifori*1*,*2*.*

*Then system*1.5*has at least anω-periodic solution.*

*Proof.* As|*fi*x1*, x*2| ≤*Mi*i1*,*2*,*implies that*αi, βi* 0, hence, the conditions in Theorem

2.1 are all satisfied.

*Remark 3.3.* From Corollary 3.2, we can find that the condition*ai* *> bi* in4, Theorem 2.1

can be dropped out. Therefore, Theorem 3.1is greatly generalized results to 4, Theorem
2.1. Furthermore, we should point out that theTheorem 3.1is diﬀerent from the the existing
work in5when without assuming the boundedness, monotonicity, and diﬀerentiability of
activation functions. In fact, the explicit presence of*ω*in Theorem 2.1see5may impose
a very strict constraint on the coeﬃcients of1.5 e.g., when*ω*is very large or small. Since
our results are presented independent of*ω*, it is more convenient to design a neural network
with delays.

**Theorem 3.4.** *Suppose thatfi, i*1*,*2*satisfy the hypothesesH2. IfD*1*/D >*0*,D*2*/D >*0*, then*

*system*1.5*has exactly one periodic solutionxt* *. Moreover, it is globally exponentially stable.*

*Proof.* From H2, we can conclude H1 is true. It is obvious that all the hypotheses in

Theorem 2.1 hold with*Mi* |*fi*0*,*0|*, i* 1*,*2*.*Thus, system1.5has at least one periodic

calculation of the upper left derivative of|*xi*t−*xi*t|along the solutions of system1.5

leads to

*D* *xi*t−*xi*t

*D*−sgn*xi*t−*xi*t

*xi*t−*xi*t

≤ −*ai*t*x*1t−*x*1t *bi*t
×*fi*

*x*1

*t*−*τi1*t

*,x*2

*t*−*τi2*t

−*fi*

*x*1

*t*−*τi1*t

*, x*2

*t*−*τi2*t
≤ −*ai*t*x*1t−*x*1t *bi*t

×*αix*1

*t*−*τi1*t

−*x*1

*t*−*τi1*t *βix*2

*t*−*τi2*t

−*x*2

*t*−*τi2*t*,*

3.28

where*D*−denotes the upper left derivative,*i*1*,*2. Let*zi*t |*xi*t−*xi*t|. Then,3.27can

be transformed into

*D*−*zi*t≤ −*aizi*t *biαi* sup
*t*−*τ*≤*s*≤*t*

*z*1s *biβi* sup
*t*−*τ*≤*s*≤*t*

*z*2s. 3.29

From*D*1*/D >*0,*D*2*/D >*0 and3.13, we have

*k*1*>*
*b*1*α*1

*a*1 *k*1
*b*1*β*1

*a*1 *k*2*,*

*k*2*>*
*b*2*α*2

*a*2
*k*1

*b*2*β*2
*a*2

*k*2*.*

3.30

Then, there exists a constant*σ >*0 such that

−*k*1*a*1 *k*1*b*1*α*1 *k*2*b*1*β*1*< σ,*
−*k*2*a*2 *k*1*b*2*α*2 *k*2*b*2*β*2*< σ.*

3.31

Thus, we can choose a constant 0*< λ*1 such that

*λki*

−*kiai*

*k*1*biαi* *k*2*biβi*

*eλτ<*0*,* *i*1*,*2*.* 3.32

Now, we choose a constant*N*1 such that

*Nkie*−*λt>*1*,* ∀*t*∈−*τ,*0, i1*,*2*.* 3.33

Set

*Yi*t *Nki*

sup

−*τ*≤*s*≤0

*z*1s sup

−*τ*≤*s*≤0

*z*2s *ε*

for∀*ε >*0. From3.32and3.34, we obtain

*D*−*Yi*t*>*

−*kiai*

*k*1*biαi* *k*2*biβi*

*eλτN*

_{}_{2}

*j*1

sup

−*τ*≤*s*≤0

*zj*s *ε*

*e*−*λt*

−*aiYi*t *biαi* sup
*t*−*τ*≤*s*≤*t*

*Y*1s *biβi* sup
*t*−*τ*≤*s*≤*t*

*Y*2s.

3.35

In view of3.33and3.34, we have

*Yi*t *>*
2

*j*1

sup

−*τ*≤*s*≤0

*zj*s *ε > zi*t, ∀*t*∈−*τ,*0. 3.36

We claim that

*zi*t*< Yi*t, ∀*t >*0*, i*1*,*2*.* 3.37

Contrarily, there must exist*i*∈ {1*,*2}and*ti>*0 such that

*zi*t*i* *Yi*t*i*, *zi*t*< Yi*t ∀*t*∈−*τ, ti*, i1*,*2*.* 3.38

It follows that

0≤*D*−*zi*

*ti*

−*Yi*

*ti*

≤lim sup

*h*→0−

*zi*

*ti* *h*

−*zi*

*ti*

*h* −lim inf*h*→0−

*Yi*

*ti* *h*

−*Yi*

*ti*

*h*

*D*−*zi*

*ti*

−*D*−*Yi*

*ti*

*.*

3.39

From3.29,3.35, and3.38, we obtain

*D*−*zi*

*ti*

≤ −*aizi*

*ti*

*biαi* sup
*ti*−*τ*≤*s*≤*ti*

*z*1s *biβi* sup
*ti*−*τ*≤*s*≤*ti*

*z*2s*< D*−*Yi*

*ti*

*,* _{}_{3.40}_{}

which contradicts3.39. Hence,3.37holds. Letting*ε* → 0 and*m* max1≤*i*≤2{*Nki* 1},

from3.34and3.37, we have

*xi*t−*xi*t*zi*t≤*Nki*

_{}_{2}

*j*1

sup

−*τ*≤*s*≤0

*zj*s *ε*

*e*−*λt*

≤*mφ*−*xe*−*λt,* for*t*≥0*, i*1*,*2*.*

3.41

0 50 100 150
Time*t*

−0*.*6

−0*.*4

−0*.*2

0
0*.*2
0*.*4
0*.*6
0*.*8
1
1*.*2
1*.*4

Solution

*x*1

*,x*

2

**Figure 1:**Numerical solution*x*1t, x2tof system4.1, where*τ*11t 5*, τ*12t 2*, τ*21t 3*, τ*22t 6,

*x*1*s* −0*.*2*, x*2*s* 1*.*4 for*s*∈−6*,*0.

*Remark 3.5.* From Theorem 3.4, it is easy to see that our results independent of*ω* and *τ*,
we have dropped out the condition b of Theorem 3.1 in 4 and the condition: *ai* *>*
α*i* *βi*|*bi|ea*

∗

*iτ* of Theorem 3.1 in5. Therefore, it is generalized the corresponding results

in4,5.

**4. Example and numerical simulation**

In this section, we will give a example to illustrate our results. Using the method of numerical simulation in15, we will find that the theoretical conclusions are in excellent agreement with the numerically observed behavior.

*Example 4.1.* Consider a two-neuron networks with delays as follows:

˙

*x*1t −*a*1tx1t *b*1tf1

*x*1t−*τ*11t

*, x*2

*t*−*τ*12t

*I*1t,

˙

*x*2t −*a*2tx2t *b*2tf2

*x*1

*t*−*τ*21t

*, x*2

*t*−*τ*22t

*I*2t,

4.1

where*a*1t 49*/*8 cos *t*,*a*2t 15*/*2 sin*t*,*b*1t 1 2 cos*t*,*b*2t 1−sin*t*,*I*1t sin*t*,
*I*2t 2 cos*t*,*f*1x1*, x*2 cos1*/*4*x*1 1*/*8|*x*1| sin*x*2−1*/*2|*x*2| 1,*f*2x1*, x*2 sin*x*1−
1*/*4*x*1 cos1*/*2x2 1*/*4|*x*2| 2, and 0 ≤ *τij*t *<* ∞ are any 2*π*-periodic continuous

functions for*i, j*1*,*2.

Obviously, the requirements of smoothness, monotonicity, and boundedness on the
activation functions are relaxed in our model. By some simple calculations, we obtain*α*1

−0*.*5 0 0*.*5 1

*x*1

0
0*.*2
0*.*4
0*.*6
0*.*8
1
1*.*2
1*.*4

*x*2

**Figure 2:**Phase trajectories of system4.1.

**5. Conclusion**

In this paper, we have derived suﬃcient algebraic conditions in terms of the parameters of the connection and activation functions for periodic solutions and global exponential stability of a two-neuron networks with time-varying delays. The obtained results extend and improve some earlier publications, which are all independent of the delays and the periodωand may be highly important significance in some applied fields.

**Acknowledgments**

This work was supported in part by the High-Tech Research and Development Program of China under Grant no. 2006AA04A104, China Postdoctoral Science Foundation under Grants no. 20070410300 and no. 200801336, the Foundation of Chinese Society for Electrical Engineering, and the Hunan Provincial Natural Science Foundation of China under Grant no. 07JJ4001.

**References**

1 J. Wu,*Introduction to Neural Dynamics and Signal Transmission Delay, vol. 6 ofde Gruyter Series in*
*Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001.*

2 M. Z. Baptistini and P. Z. T´aboas, “On the existence and global bifurcation of periodic solutions to
planar diﬀerential delay equations,”*Journal of Diﬀerential Equations, vol. 127, no. 2, pp. 391–425, 1996.*

3 S. M. S. Godoy and J. G. dos Reis, “Stability and existence of periodic solutions of a functional
diﬀerential equation,”*Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 381–398,*
1996.

4 C. Huang and L. Huang, “Existence and global exponential stability of periodic solution of
two-neuron networks with time-varying delays,”*Applied Mathematics Letters, vol. 19, no. 2, pp. 126–134,*
2006.

5 C. Huang, L. Huang, and T. Yi, “Dynamics analysis of a class of planar systems with time-varying
delays,”*Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1233–1242, 2006.*

7 P. T´aboas, “Periodic solutions of a planar delay equation,”*Proceedings of the Royal Society of Edinburgh.*
*Section A, vol. 116, no. 1-2, pp. 85–101, 1990.*

8 K. Liu and G. Yang, “Strict stability criteria for impulsive functional diﬀerential systems,”*Journal of*
*Inequalities and Applications, vol. 2008, Article ID 243863, 8 pages, 2008.*

9 Y. Chen and J. Wu, “Slowly oscillating periodic solutions for a delayed frustrated network of two
neurons,”*Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 188–208, 2001.*

10 K. Gopalsamy and S. Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic
environments in isolated neurons,”*IEEE Transactions on Neural Networks, vol. 13, no. 3, pp. 551–563,*
2002.

11 K. Gopalsamy and S. Sariyasa, “Time delays and stimulus dependent pattern formation in periodic
environments in isolated neurons—II,”*Dynamics of Continuous, Discrete & Impulsive Systems. Series B,*
vol. 9, no. 1, pp. 39–58, 2002.

12 M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear
and quadratic programming problems,”*IEEE Transactions on Circuits and Systems. I, vol. 42, no. 7, pp.*
354–366, 1995.

13 R. E. Gaines and J. L. Mawhin,*Coincidence Degree, and Nonlinear Diﬀerential Equations, vol. 568 of*
*Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.*

14 K. Deimling,*Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.*