## Stability of Equilibrium Points in Cellular Neural

## Networks with Negative Slope Activation

## Function

Qi HanSchool of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing, China

Email: Hanqicq@163.com

Qian Xiong, Chao Liu, Jun Peng, Lepeng Song and Sijing Liu

School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing, China

**Abstract****—In the paper, the region of the number of **
**equilibrium points of every cell in cellular neural networks **
**with negative slope activation function is considered by the **
**relationship between parameters of cellular neural networks. **
**Some sufficient conditions are obtained by using the **
**relationship among connection weights. Three theorems and **
**a corollary are gotten by our new methods. Depending on **
**these sufficient conditions, inputs and outputs of a CNN, the **
**regions of the values of parameters can be obtained. Some **
**numerical simulations are presented to support the **
**effectiveness of the theoretical analysis. **

**Index Terms****—Cellular neural network; Equilibrium point; **
**negative slope activation function **

I. INTRODUCTION

Cellular neural networks (CNNs) were first introduced in 1988[1-2]. CNNs have extensively found application in various engineering fields, such as image processing, robotic and biological versions, higher brain functions, associative memories and so on [3-5].

It is easy to know that stability of CNNs play a
important role for the application of CNNs. There have
been abundant researches about stability of CNNs. Some
sufficient conditions for CNNs to be stable were obtained
by constructing Lyapunov Function [6-7], and these
conditions generally made equilibrium point global
asymptotically stable. However, some authors presented
some conditions which made equilibrium points locally
stable, and there generally were multiple equilibrium
points [8-11]. In [8-9], the region of the number of
equilibrium points of every cell in cellular neural
networks is researched, however, the activation functions
are the unity gain activation function and thresholding
activation function, respectively. Therefore, in the paper,
the region of the number of equilibrium points of every
cell in cellular neural networks with negative slope
activation function will be considered. If the activation
function of CNNs is *f x*

_{( )}

= −### (

*x*+ − −1

*x*1 2

### )

, we call the activation function as negative slope activation function.The remaining part of this paper is organized as

follows. In the next Section, some regions of the number of equilibrium points of CNNs are obtained. In Section III, some numerical simulations are given to verify the theoretical results. Some conclusions are finally drawn in Section IV.

II. MAIN RESULTS

Consider a two dimensional cellular neural networks defined by the following differential equations:

### ( )

### ( )

( ) ( )### ( )

### ( )

( ) ( )

2 2

1 1

2 2

1 1

, ,

, ( , ) ( , ) , ,

( , ) ( , )

,

,
*k* *i r* *l* *j r*

*ij* *ij* *ij* *kl* *i k j l* *ij*

*k k i r l l* *j r*

*k* *i r* *l* *j r*

*ij* *kl* *kl* *ij*

*k k i r l l* *j r*

*y* *t* *c y* *t* *a g* *y t*

*d u* *v*

η

η

+ +

= =

= =

⎧

= − + +

⎪ ⎪ ⎨ ⎪

= +

⎪ ⎩

### ∑ ∑

### ∑ ∑

(1)
where *yij*( )*t* ∈*R* denotes the states vector, *cij* is a

positive parameter, *r* is positive integer denoting
neighborhood radius, *A*=

### ( )

*akl*

_{(}

_{2 1 2 1}

_{r}_{+ ×}

_{) (}

_{r}_{+}

_{)}≠0 is

intra-neuron connection weight matrix , *D*=

### ( )

*dkl*

_{(}

_{2 1 2 1}

_{r}_{+ ×}

_{) (}

_{r}_{+}

_{)}

is input cloning template, *ukl* is the input, *vij* is the bias,

### ( )

### {

### }

1 , max 1 , ,

*k i r* = − −*i* *r* *k*_{2}

### ( )

*i r*, =min

### {

Ν −*i r*, ,

### }

### ( )

### {

### }

1 , min 1 , ,

*l* *j r* = − −*j* *r* *l*_{2}

### ( )

*j r*, =max

### {

Μ −*j*,

*r*

### }

, and*g*( )⋅ is the activation function defined by

### ( )

### (

1 1 2### )

*g y* = − *y*+ − −*y* .
whose characteristic is shown in Fig. 1.

Fig. 1 Piecewise linear function *f x t*

### ( )

( ) —negative slope*g*(

*y*)

*y *
1

1 -1

Let *r*=1 and *n*= ×*N* *M*. If the system (1) has *N* rows
and *M *columns, then it can be put in vector form as

### ( )

*x*= −*Cx*+*Af x* +*DU*+*V* , (2)

where

### (

_{1 2}, , ,

### )

*T*

### (

_{11 12, , 1}, , ,

### )

*T*

*n* *M* *NM*

*x*= *x x* " *x* = *y* *y* _{"}*y* " *y* ,
coefficient matrices *A* and*D*are obtained through the

templates*A*and*D*, *C*=*diag c*

### (

_{1}"

*c*

_{n}### )

, the input vector### (

1,...,*n*

### )

*T*

*U* = *u* *u* , *V* =

### (

*v*

_{1},...,

*v*

_{n}### )

*T*and

### ( )

### (

### ( )

1 ,...,### ( )

*n*

### )

*T*

*f x* = *g y* *g y* . The* k*th cell in Eq. (2) is
denoted by Ο*k*(*k*=*iN*+ *j*, where 1≤ ≤*i* *N*, 1≤ ≤*j* *M* ,
*i* denotes *i*th row and *j* denotes *j*th column of the CNN).
The matrix *A*=

### ( )

*aij*

_{n n}_{×}, defined by (2), composed of template has the form

1 2

3 1 2

3 1 2

3 1

1 2

3 1

0 0 ... 0 0

0 ... 0 0

0 ... 0 0

0 0 ... 0 0

0 0 0 0 0

0 0 0 0 0 _{n n}

*A* *A*

*A* *A* *A*

*A* *A* *A*

*A* *A*

*A* *A*
*A* *A* _{×}

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ # # # # % # # , 00 01

0, 1 00 01

0, 1 00 1

00 01

0, 1 00

0 0 0

0 0

0 0 0

0 0 0

0 0 0 _{M M}

*a* *a*

*a* *a* *a*

*a* *a*
*A*
*a* *a*
*a* *a*
−
−
− _{×}
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
"
"
"
# # # % # #
"
"
,
10 11
1, 1 10 11

1, 1 10 2

10 11 1, 1 10

0 0 0

0 0

0 0 0

0 0 0

0 0 0

*M M*

*a* *a*

*a* *a* *a*

*a* *a*
*A*
*a* *a*
*a* *a*
−
−
− _{×}
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
"
"
"
# # # % # #
"
"
and
1,0 1,1
1, 1 1,0 1,1

1, 1 1,0 3

1,0 1,1

1, 1 1,0

0 0 0

0 0

0 0 0

0 0 0

0 0 0

*M M*

*a* *a*

*a* *a* *a*

*a* *a*
*A*
*a* *a*
*a* *a*
− −
− − − −
− − −
− −
− − − _{×}
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
"
"
"
# # # % # #
"
"
.

The definition of matrices*D*

### ( )

*d*

_{ij n n}×

= is similar to*A*.

System (2) can be written as

### ( )

, 1, ,*i* *i i* *ii* *i* *i*

*x* = −*c x* +*a f x* +*w i* = " *n* (3)

where

### ( )

1, 1

*n* *n*

*i* *ij* *j* *ij* *j* *i*

*j* *j i* *j*

*w* *a f x* *d u* *v*

= ≠ =

=

### ∑

+### ∑

+ . (4)In Eq. (3), if − ≤1 *x ti*

_{( )}

≤1 , *f x t*

### (

*i*

### ( )

### )

= −*x ti*

### ( )

. Therefore, when − ≤1*x ti*

_{( )}

≤1 , Eq. (3) can be
transformed as
### ( )

### (

### ) ( )

*i* *i* *ii* *i* *i*

*x t* = − *c* +*a* *x t* +*w* ;

if *x ti*

_{( )}

≥1, we have *f x t*

### (

*i*

### ( )

### )

= −1. Therefore, when### ( )

1*i*

*x t* ≥ , Eq. (3) can be transformed as

### ( )

### ( )

*i* *i i* *ii* *i*

*x t* = −*c x t* −*a* +*w* ;

if *x ti*

_{( )}

≤ −1, we have *f x t*

### (

*i*

### ( )

### )

=1. Therefore, when### ( )

1*i*

*x t* ≤ − , Eq. (3) can be transformed as

### ( )

### ( )

*i* *i i* *ii* *i*

*x t* = −*c x t* +*a* +*w*.

In Eq. (3), if *x ti*

_{( )}

=1, we have
### ( )

1*i* *i* *ii* *i*

*x t* = − −*c* *a* +*w* =*R* ,

and if *x ti*

_{( )}

= −1, we have
### ( )

2*i* *i* *ii* *i*

*x t* = +*c* *a* +*w* =*R* .

Suppose thatβ*i*is equilibrium point of system (3), then
we have

### ( )

0*i* *i* *ii* *i* *i*

*c*β *a f* β *w*

− + + = . (5)
Whenβ*i*≤ −1 , we have *f*

### ( )

β*i*=1, and the Eq. (5) can be transformed as

### (

### )

,*i* *aii* *wi* *c ti*

β = + ′ → ∞, (6) where

1, 1

*n* *n*

*i* *ij* *ij* *j* *i*

*j* *j i* *j*

*w* *a* *d u* *v*

= ≠ =

′ =

### ∑

± +### ∑

+ ; (7)Whenβ*i*≥1 , we have *f*

### ( )

β*i*= −1, and the Eq. (5) can be transformed as

### (

### )

,*i* *aii* *wi* *c ti*

β = − + ′ → ∞. (8) For simplicity, denote

1

*n*

*i* *ij* *j*

*j*
*d u*
ρ
=
=

### ∑

and 1,*n*

*i*

*ij*

*j* *j i*

*a*
ϑ

= ≠

=

### ∑

.**Note 1**. Let δ be the sum of a cell and the number of its

adjacent cells.

From above analysis about CNNs, we can get the following theorem.

**Theorem 1**. In Eq. (3), when *a _{ii}*< −

*c*,

_{i}(i) if *vi* >ρ ϑ*i*+ − −*i* *ci* *aii*, then there exist only positive
stable equilibrium points for cell *Oi*, and the number of
these points is grater than or equal to 1 and less than or
equal to _{2}δ−1_{ for cell }

*i*

*O*;

(ii) if *vi*<*aii*+ −*ci* ρ ϑ*i*− *i*, then there exist only negative
stable equilibrium points for cell *Oi*, and the number of
these points is grater than or equal to 1 and less than or
equal to _{2}δ−1_{; }

number of stable equilibrium points is greater than or
equal to2and less than or equal to 2δ_{. }

**Proof**. (i) From *a _{ii}*< −

*c*, we have

_{i}*R*

_{1}>

*R*

_{2}. In terms of

*i* *i* *i* *i* *ii*

*v* >ρ ϑ+ − −*c* *a* , we have *R*_{2}>0. The *x*−*x* phase

plane trajectory of Eq. (1) is shown in Fig. 2.

Fig. 2 The *x*−*x* phase plane trajectory of Eq. (3), where *R*_{2}>0

and *aii*< −*ci*

Therefore there exist positive stable equilibrium points
for cell *Oi*. From Eq. (7) and (8) and *R*1>0, we know
that the equilibrium point of Eq. (3) is

### (

### )

*

00 *i* *i* 1

*x* = *a* +*w*′ *c* ≥ .

Then, the number of different values of * _{x}**

_{ is equal to }

that of *wi*′. If *akl* =0, for all

### ( ) ( )

*k l*, ≠ 0,0 , then the number of different values of

*wij*′ is 1. If

*akl*≠0, for all

### ( ) ( )

*k l*, ≠ 0,0 , then the maximum value of the number of different values of

*wij*′ is 2δ−1. Furthermore, we know that the number of positive stable equilibrium points is grater than or equal to 1 and less than or equal to

_{2}δ−1

_{. }

(ii) From *vi* <*aii*+ − −*ci* ρ ϑ*i* *i* , we have *R*1<0 . The

*x*−*x* phase plane trajectory of Eq. (1) is shown in Fig. 3.

Fig. 3 The *x*−*x* phase plane trajectory of Eq. (1), where *R*1<0 and

*ii* *i*

*a* < −*c* .

There exist only negative stable equilibrium points for
cell *Oi*, and the number of these points is grater than or
equal to 1 and less than or equal to _{2}δ−1_{. }

(iii) From *vi* ≥*aii*+ + +*ci* ϑ ρ*i* *i* and

*i* *ii* *i* *i* *i*

*v* ≤ − − − −*a* *c* ϑ ρ , we have *R*_{1}>0 and *R*_{2}<0. The

*x*−*x* phase plane trajectory of Eq. (1) is shown in Fig. 4.

Fig. 4 The *x*−*x* phase plane trajectory of Eq. (1), where *R*_{1}>0,

2 0

*R* < and *aii*< −*ci*.

Then, we can get our result.

**Corollary 1. **In Eq. (6), choose initial state*x*

### ( )

0 =0, andlet*aii*< −*ci*,

(i) if

0

*i* *i* *i*

*v* − ρ − ϑ > ,

and

*i* *ii* *i* *i* *i*

*v* ≤ − − − −*a* *c* ϑ ρ

then there exist positive stable equilibrium points, and the
number of these points is grater than or equal to 1 and
less than or equal to _{2}δ−1_{; }

(ii) if

0

*i* *i* *i*

*v* + ρ + ϑ <

and

*i* *ii* *i* *i* *i*

*v* ≥*a* + + +*c* ϑ ρ

then there exist negative stable equilibrium points, and
the number of these points is grater than or equal to 1 and
less than or equal to _{2}δ−1_{. }

**Theorem 2**. In Eq. (6), when *a _{ii}*= −

*c*,

_{i}(i) if *vi* > ρ + ϑ*i* *i*, then there exists only positive stable
equilibrium points for cell *Oi*, and the number of these
points is grater than or equal to 1 and less than or equal to

1

2δ− _{; }

(ii) if *vi*< ρ + ϑ*i* *i*, then there exists only negative stable
equilibrium points for cell *Oi*, and the number of these
points is grater than or equal to 1 and less than or equal to

1

2δ− _{. }

**Theorem 3**. In Eq. (6), when*a _{ii}*> −

*c*,

_{i}(i) if *vi*≥ ϑ + ρ*i* *i*, there exist not more than 2δ−1 positive
stable equilibrium points for cell *Oi* , if

*i* *i* *ii* *i* *i*

*v* ≥ +*c* *a* + ϑ + ρ , stable equilibrium points are equal

or greater than 1 for cell *Oij*;

(ii) if *vi* ≤ ϑ + ρ*i* *i* , there exist not more than

1

2δ−

negative stable equilibrium points for cell *Oi* , if

*i* *i* *ii* *i* *i*

*v* ≤ +*c* *a* − ϑ − ρ , stable equilibrium points are equal

or less than -1 for cell *Oi*.
*dx*i/*dt*

*x*i

*R*1

1 -1

*R*2

w

*dx*i/*dt*

*x*i

*R*1

1 -1

*R*2

w
*dx*i/*dt*

*x*i

*R*1

1 -1

*R*2

III. NUMERICAL EXAMPLE

In this section, some numerical simulations are given
to verify the theoretical results. Consider a cellular neural
network, and its cloning template *A* is as follows:

1, 1 1,0 1,1 0, 1 00 0,1 1, 1 1,0 1,1

*a* *a* *a*

*A* *a* *a* *a*

*a* *a* *a*

− − − −

− −

⎛ ⎞

⎜ ⎟

=⎜ ⎟

⎜ ⎟

⎝ ⎠

. (9)

Therefore, a CNN with 2 rows and 2 columns can be written as

### ( )

### (

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

11 11 11 00 11 0,1 12 1,0 21 11 22 11 12 12 12 0, 1 11 00 12 1, 1 21 1,0 22 12 21 21 21 1,0 11 1,1 12

00 21 0,1 22 21 22

,

,

,

*x* *c x* *a f x* *t* *a* *f x* *t*

*a* *f x* *t* *a f x* *t*

*x* *c x* *a* *f x* *t* *a f x* *t*

*a* *f x* *t* *a* *f x* *t*

*x* *c x* *a* *f x* *t* *a* *f x* *t*

*a f x* *t* *a* *f x* *t*

*x*

−

−

− −

= − + +

+ + + η

= − + +

+ + + η

= − + +

+ + + η

=

### (

### ( )

### )

### (

### ( )

### )

### ( )

### (

### )

### (

### ( )

### )

22 22 1, 1 11 1,0 12 0. 1 21 00 22 22,

*c x* *a* *f x* *t* *a* *f x* *t*

*a* *f x* *t* *a f x* *t*

− − −

−

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪

− + +

⎪

⎪_{ +} _{+} _{+ η}

⎪⎩

where

11 00 11 0,1 12 1,0 21 11 22 11 12 0, 1 11 00 12 1, 1 21 1,0 22 12

21 1,0 11 1,1 12 00 21 0,1 22 21 22 1, 1 11 1,0 12 0. 1 21 00 22 22

, , ,

.

*d u* *d u* *d u* *d u* *v*

*d* *u* *d u* *d* *u* *d u* *v*

*d* *u* *d* *u* *d u* *d u* *v*

*d* *u* *d* *u* *d* *u* *d u* *v*

− −

− −

− − − −

η = + + + +

⎧ ⎪

η = + + + +

⎪ ⎨

η = + + + +

⎪

⎪_{η =} _{+} _{+} _{+} _{+}

⎩

In Fig. 5 (a) and (b), inputs and outputs of CNNs are shown, respectively, where the white lattice stands for -1 and black for 1.

Fig. 5 Inputs of CNNs are shown in (a), and outputs are shown in (b).

We are based on Corollary 1 to design a CNN. Choose

0.5 0.3 0.1

0.1 5.9 0.1

0.2 0.4 0.1

*A*

− − −

⎛ ⎞

⎜ ⎟

=_{⎜} − _{⎟}

⎜ ⎟

⎝ ⎠

, *c*11= =*c*12 *c*21=*c*22=1, and

0.3 0.1 0.1

0.2 0.2 0.2

0.2 0.4 0.1

*D*

− −

⎛ ⎞

⎜ ⎟

=_{⎜} _{⎟}

⎜ _{−} ⎟

⎝ ⎠

.

Then, from inputs and outputs of CNNs in Fig. 6, we can get

11 12

2.3<*v* <2.6, − 2.6 <*v* < −2.3,

21 1.3, 1.7 22 3.2

*v* *v*

−3.6 < < − < < . Therefore , we choose

11 2.5, 12 2.4, 21 3, 22 3

*v* = *v* = − *v* = − *v* = .

Furthermore, a CNN is obtained. Fig. 6 shows the number of equilibrium points, where all parameters of the CNN are based on Corollary 1 (iii), and 100 random

initial states are used. We can find that the number of equilibrium points is accord with Corollary 1.

IV. CONCLUSIONS

In the paper, the region of the number of equilibrium
points of cellular neural networks was considered by the
relationship between parameters of cellular neural
networks with negative slope activation function. We find
that there are no more than 3δ_{isolated equilibrium points }

or 2δ_{ equilibrium points located in saturation regions for }

a cell in a CNN. Finally, numerical simulations were presented to verify the theoretical results.

(a)

(b)

(d)

Fig. 6 The number of equilibrium points of every cell CNNs are shown,

where the number of different equilibrium points of every cell can be known.

ACKNOWLEDGMENT

This work was supported in part by Research Project of Chongqing University of Science and Technology(CK2013B15, CK2011Z17), in part by Teaching & Research Program of Chongqing Education Committee (KJ131401, KJ131416, KJ121505, KJzh11221), in part by the National Natural Science Foundation of China (61170249, 61003247), in part by the Natural Science Foundation project of CQCSTC ( cstc2011pt-gc70007, 2010BB2284, cstc2011jjA80022, cstc2011jjA40005, and in part by the First Batch of Supporting Program for University Excellent Talents in Chongqing.

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[11]Q. Han, X. Liao, T. Huang, J. Peng, C. Li and H. Huang. Analysis and design of associative memories based on stability of cellular neural networks. Neurocomputing, 97: 192-200, 2012.

**Qi Han** received the B.S. degree in

Computer Science and Technology from Shandong University (Weihai), China, in 2005. He received M.S. degree in Computer Software and Theory from Chongqing University, China, in 2009. He received PhD degree in Computer Science and Technology at Chongqing University of China in 2012. Now he is a Lecturer with School of Electrical and Information Engineering, Chongqing University of Science and Technology. His current research interest covers chaos control and synchronization, cellular automata, neural network, associative memories.

**Qian Xiong** received the B.S. degree and

the M.S. degree in computer science from Chongqing University, Chongqing, China, in 2003 and 2006 respectively. She is currently a Lecturer with the School of Electrical and Information Engineering, Chongqing University of Science and Technology. Her research interests include artificial intelligence, web application and bilingual teaching.

**Chao Liu** received his Ph. D degree from Chongqing

Universtiy in 2012. Now he is a Lecturer with School of Electrical and Information Engineering, Chongqing University of Science and Technology, People's Republic of China. His current research interests include impulsive systems, switched systems and neural networks.

**Jun Peng** is a professor with School of Electrical and

Information Engineering, Chongqing University of Science and Technology.

**Lepeng Song** is professor with School of Electrical and

Information Engineering, Chongqing University of Science and Technology.

**Sijing Liu** is assistant with School of Electrical and Information