Stability of Equilibrium Points in Cellular Neural ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
891
Stability of Equilibrium Points in Cellular Neural Networks with Negative Slope Activation Function Qi Han
School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing, China Email: [email protected]
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
© 2014 ACADEMY PUBLISHER doi:10.4304/jcp.9.4.891-895
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: k2 ( i , r ) l2 ( j , r ) ⎧ = − + y t c y t ⎪ ij ( ) ∑ ∑ akl gi + k , j +l ( y ( t ) ) + ηij , ij ij ( ) k = k1 ( i , r ) l = l1 ( j , r ) ⎪ ⎨ k2 ( i , r ) l2 ( j , r ) ⎪ = η ∑ ∑ d kl ukl + vij , ij ⎪ k = k1 ( i , r ) l = l1 ( j , r ) ⎩ (1) where yij (t ) ∈ R denotes the states vector, cij is a positive parameter, r is positive integer denoting neighborhood radius, A = ( akl ) 2 r +1 × 2 r +1 ≠ 0 is intra( )( )
( )( 2r +1)×( 2r +1)
neuron connection weight matrix , D = d kl
is input cloning template, ukl is the input, vij is the bias, k1 ( i, r ) = max {1 − i, − r} , l1 ( j , r ) = min {1 − j , − r} ,
k2 ( i, r ) = min {Ν − i, r} , l2 ( j , r ) = max {Μ − j ,
and g (⋅) is the activation function defined by g ( y ) = − ( y +1 − y −1 ) 2 .
whose characteristic is shown in Fig. 1. g(y) 1 -1
1 y -1
Fig. 1 Piecewise linear function f ( x ( t ) ) —negative slope
r} ,
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JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
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 x = ( x1 , x2 , ", xn ) = y11 , y12,", y1M , ", y NM T
)
T
,
coefficient matrices A and D are obtained through the templates A and D , C = diag ( c1 " cn ) , the input vector U = ( u1 ,..., un )
T
V = ( v1 ,..., vn )
T
,
and
f ( x ) = ( g ( y1 ) ,..., g ( yn ) ) . The kth cell in Eq. (2) is
xi ( t ) = − ( ci + aii ) xi ( t ) + wi ;
if xi ( t ) ≥ 1 , we have f ( xi ( t ) ) = −1 . Therefore, when xi ( t ) ≥ 1 , Eq. (3) can be transformed as
xi ( t ) = −ci xi ( t ) − aii + wi ;
if xi ( t ) ≤ −1 , we have f ( xi ( t ) ) = 1 . Therefore, when xi ( t ) ≤ −1 , Eq. (3) can be transformed as
xi ( t ) = −ci xi ( t ) + aii + wi .
T
denoted by Ο k ( k = iN + j , where 1 ≤ i ≤ N , 1 ≤ j ≤ M , i denotes ith row and j denotes jth column of the CNN).
( )n×n , defined by (2), composed of
The matrix A = aij
template has the form ⎡ A1 ⎢A ⎢ 3 ⎢0 ⎢ ⎢0 ⎢# ⎢ ⎢0 ⎢0 ⎣
⎡ a00 ⎢a ⎢ 0, −1 ⎢ 0 A1 = ⎢ ⎢ # ⎢ 0 ⎢ ⎢⎣ 0
A2 A1 A3 0 #
0 A2 A1 A3 #
0 ... 0 ... A2 ... A1 ... # %
0 0 0 0 #
0 0
0 0
0 0
0 0
A1 A3
0
"
0
a01 " a00 "
0
a01 a00 a0, −1 # 0 0
a11 ⎡ a10 ⎢a ⎢ 1, −1 a10 ⎢ 0 a1, −1 A2 = ⎢ # ⎢ # ⎢ 0 0 ⎢ 0 ⎢⎣ 0 ⎡ a−1,0 a−1,1 ⎢a ⎢ −1, −1 a−1,0 ⎢ 0 a−1, −1 A3 = ⎢ # ⎢ # ⎢ 0 0 ⎢ 0 ⎣⎢ 0
0 0 0 0 #
⎤ ⎥ ⎥ ⎥ ⎥ , ⎥ ⎥ ⎥ A2 ⎥ A1 ⎥⎦ n×n
0 # a00
# 0
% "
0
" a0, −1
0
wi′ =
j =1, j ≠ i
( )
n
aij f x j + ∑ dij u j + vi .
(7)
j =1
(8)
n
ρi = ∑ dij u j j =1
and
ϑi =
(3) (4)
f ( xi ( t ) ) = − xi ( t ) .
Therefore, when −1 ≤ xi ( t ) ≤ 1 , Eq. (3) can be transformed as
j =1, j ≠ i
n
± aij + ∑ dij u j + vi ;
For simplicity, denote
j =1
In Eq. (3), if −1 ≤ xi ( t ) ≤ 1 ,
n
∑
can be transformed as β i = ( − aii + wi′ ) ci , t → ∞ .
where
n
(6)
When β i ≥ 1 , we have f ( βi ) = −1 , and the Eq. (5)
( )n×n is similar to A .
∑
xi ( t ) = ci + aii + wi = R2 . Suppose that β i is equilibrium point of system (3), then we have −ci β i + aii f ( β i ) + wi = 0 . (5)
where
0 ⎤ 0 ⎥⎥ 0 ⎥ , ⎥ # ⎥ a01 ⎥ ⎥ a00 ⎥⎦ M ×M
The definition of matrices D = dij
wi =
and if xi ( t ) = −1 , we have
can be transformed as β i = ( aii + wi′ ) ci , t → ∞ ,
0
System (2) can be written as xi = −ci xi + aii f ( xi ) + wi , i = 1, " , n
xi ( t ) = −ci − aii + wi = R1 ,
When β i ≤ −1 , we have f ( βi ) = 1 , and the Eq. (5)
0⎤ a11 " 0 0 ⎥⎥ a10 " 0 0⎥ and ⎥ # % # # ⎥ 0 " a10 a11 ⎥ ⎥ 0 " a1, −1 a10 ⎥⎦ M ×M 0 " 0 0 ⎤ a−1,1 " 0 0 ⎥⎥ a−1,0 " 0 0 ⎥ . ⎥ # % # # ⎥ 0 " a−1,0 a−1,1 ⎥ ⎥ 0 " a−1, −1 a−1,0 ⎦⎥ M ×M "
In Eq. (3), if xi ( t ) = 1 , we have
n
∑
j =1, j ≠ i
aij .
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 aii < −ci , (i) if vi > ρi + ϑi − 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 2δ−1 for cell Oi ;
(ii) if vi < aii + ci − ρi − ϑi , 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 2δ−1 ; (iii) if vi ≥ aii + ci + ϑi + ρi and vi ≤ −aii − ci − ϑi − ρi , then there exist positive stable equilibrium points, negative equilibrium and pseudo equilibrium points for cell Oi . The number of isolated equilibrium points is grater than or equal to 3 and less than or equal to 3δ ; the
© 2014 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
893
number of stable equilibrium points is greater than or equal to 2 and less than or equal to 2δ . Proof. (i) From aii < −ci , we have R1 > R2 . In terms of vi > ρi + ϑi − ci − aii , we have R2 > 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 2. dxi/dt
(iii) From vi ≥ aii + ci + ϑi + ρi and vi ≤ −aii − ci − ϑi − ρi , we have R1 > 0 and R2 < 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 4. dxi/dt
R1
R1
-1
w 1
w
xi
R2 R2
-1
1
Fig. 4 The x − x phase plane trajectory of Eq. (1), where R1 > 0 ,
xi
R2 < 0 and aii < −ci .
Fig. 2 The x − x phase plane trajectory of Eq. (3), where R2 > 0 and aii < −ci
Therefore there exist positive stable equilibrium points for cell Oi . From Eq. (7) and (8) and R1 > 0 , we know that the equilibrium point of Eq. (3) is x* = ( a00 + wi′ ) ci ≥ 1 . *
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 2δ−1 . (ii) From vi < aii + ci − ρi − ϑi , we have R1 < 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 3. dxi/dt
-1
1
R1
xi
w
R2 Fig. 3 The
x − x
phase plane trajectory of Eq. (1), where R1 < 0 and aii < −ci .
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 2δ−1 .
© 2014 ACADEMY PUBLISHER
Then, we can get our result. Corollary 1. In Eq. (6), choose initial state x ( 0 ) = 0 , and let aii < −ci , (i) if
vi − ρi − ϑi > 0 ,
and
vi ≤ −aii − ci − ϑi − ρi 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 2δ−1 ; (ii) if vi + ρi + ϑi < 0 and vi ≥ aii + ci + ϑi + ρi 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 2δ−1 . Theorem 2. In Eq. (6), when aii = −ci , (i) if vi > ρi + ϑi , 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 2δ−1 ; (ii) if vi < ρi + ϑi , 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 2δ−1 . Theorem 3. In Eq. (6), when aii > −ci ,
(i) if vi ≥ ϑi + ρi , there exist not more than 2δ−1 positive stable equilibrium points for cell Oi , if vi ≥ ci + aii + ϑi + ρi , stable equilibrium points are equal or greater than 1 for cell Oij ; (ii) if vi ≤ ϑi + ρi , there exist not more than 2δ−1 negative stable equilibrium points for cell Oi , if vi ≤ ci + aii − ϑi − ρi , stable equilibrium points are equal or less than -1 for cell Oi .
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JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
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: ⎛ a−1, −1 a−1,0 a−1,1 ⎞ ⎜ ⎟ A = ⎜ a0, −1 a00 a0,1 ⎟ . (9) ⎜a ⎟ ⎝ 1,−1 a1,0 a1,1 ⎠ Therefore, a CNN with 2 rows and 2 columns can be written as ⎧ x11 = −c11 x11 + a00 f ( x11 ( t ) ) + a0,1 f ( x12 ( t ) ) ⎪ ⎪ + a1,0 f ( x21 ( t ) ) + a11 f ( x22 ( t ) ) + η11 , ⎪ ⎪ x12 = −c12 x12 + a0, −1 f ( x11 ( t ) ) + a00 f ( x12 ( t ) ) ⎪ + a1, −1 f ( x21 ( t ) ) + a1,0 f ( x22 ( t ) ) + η12 , ⎪ ⎨ ⎪ x21 = −c21 x21 + a−1,0 f ( x11 ( t ) ) + a−1,1 f ( x12 ( t ) ) ⎪ + a00 f ( x21 ( t ) ) + a0,1 f ( x22 ( t ) ) + η21 , ⎪ ⎪ ⎪ x22 = −c22 x22 + a−1, −1 f ( x11 ( t ) ) + a−1,0 f ( x12 ( t ) ) ⎪ + a0.−1 f ( x21 ( t ) ) + a00 f ( x22 ( t ) ) + η22 , ⎪⎩ where ⎧η11 = d 00 u11 + d 0,1u12 + d1,0 u21 + d11u22 + v11 , ⎪ ⎪η12 = d 0, −1u11 + d 00 u12 + d1, −1u21 + d1,0 u22 + v12 , ⎨ ⎪η21 = d −1,0 u11 + d −1,1u12 + d 00 u21 + d 0,1u22 + v21 , ⎪η = d −1, −1u11 + d −1,0 u12 + d 0.−1u21 + d 00 u22 + v22 . ⎩ 22 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.
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)
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⎞ ⎜ ⎟ A = ⎜ 0.1 −5.9 0.1 ⎟ , c11 = c12 = c21 = c22 = 1 , and ⎜ 0.2 0.4 0.1 ⎟ ⎝ ⎠
(b)
⎛ −0.3 0.1 −0.1⎞ ⎜ ⎟ D = ⎜ 0.2 0.2 0.2 ⎟ . ⎜ 0.2 0.4 −0.1⎟ ⎝ ⎠
Then, from inputs and outputs of CNNs in Fig. 6, we can get 2.3 < v11 < 2.6, − 2.6 < v12 < −2.3, −3.6 < v21 < −1.3, 1.7 < v22 < 3.2 . Therefore , we choose v11 = 2.5, v12 = −2.4, v21 = −3, v22 = 3 . 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 © 2014 ACADEMY PUBLISHER
(c)
JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
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with thresholding activation function, Neural computing & Applications, Published online: 26 September 2012. [10] Q. Han, X. Liao and C. Li. Analysis of associative memories based on stability of cellular neural networks with time delay. Neural Computing & Applications, Published online: 20 January 2012. [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.
(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. REFERENCES [1] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Transactions Circuits Systems, 35:1257-1272, 1988. [2] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Transactions Circuits Systems, 35:1273-1290 1988. [3] M. Itoh, L.O. Chua, Advanced image processing cellular neural networks, International Journal of Bifurcation and Chaos, 17: 1109-1150, 2007. [4] Z. G. Zeng, and J. Wang, Associative memories based on continuous-time cellular neural networks designed using space-invariant cloning templates, Neural Networks, 22: 651-657, 2009. [5] L. O. Chua and T. Roska, Cellular neural networks and visual computing, Cambridge: Cambridge University Press, 2002. [6] S. P. Xiao and X. M. Zhang, New globally asymptotic stability criteria for delayed cellular neural networks, IEEE Transactions on Circuits and Systems—II: Express Briefs, 56: 659-663, 2009. [7] C. J. Li, C. D. Li, X. F. Liao and T. W. Huang, Impulsive effects on stability of high-order BAM neural networks with time delays, Neurocomputing, 74: 1541-1550, 2011. [8] Q. Han, X. Liao, T. Weng, C. Li and H. Huang, Analysis on equilibrium points of cells in cellular neural networks described using cloning templates, Neurocomputing, 89: 106-113, 2012. [9] Q. Han, X. Liao, T. Weng, C. Li and Hongyu Huang, Analysis on equilibrium points of cellular neural networks
© 2014 ACADEMY PUBLISHER
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 Engineering, Chongqing University of Science and Technology.
891
Stability of Equilibrium Points in Cellular Neural Networks with Negative Slope Activation Function Qi Han
School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing, China Email: [email protected]
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
© 2014 ACADEMY PUBLISHER doi:10.4304/jcp.9.4.891-895
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: k2 ( i , r ) l2 ( j , r ) ⎧ = − + y t c y t ⎪ ij ( ) ∑ ∑ akl gi + k , j +l ( y ( t ) ) + ηij , ij ij ( ) k = k1 ( i , r ) l = l1 ( j , r ) ⎪ ⎨ k2 ( i , r ) l2 ( j , r ) ⎪ = η ∑ ∑ d kl ukl + vij , ij ⎪ k = k1 ( i , r ) l = l1 ( j , r ) ⎩ (1) where yij (t ) ∈ R denotes the states vector, cij is a positive parameter, r is positive integer denoting neighborhood radius, A = ( akl ) 2 r +1 × 2 r +1 ≠ 0 is intra( )( )
( )( 2r +1)×( 2r +1)
neuron connection weight matrix , D = d kl
is input cloning template, ukl is the input, vij is the bias, k1 ( i, r ) = max {1 − i, − r} , l1 ( j , r ) = min {1 − j , − r} ,
k2 ( i, r ) = min {Ν − i, r} , l2 ( j , r ) = max {Μ − j ,
and g (⋅) is the activation function defined by g ( y ) = − ( y +1 − y −1 ) 2 .
whose characteristic is shown in Fig. 1. g(y) 1 -1
1 y -1
Fig. 1 Piecewise linear function f ( x ( t ) ) —negative slope
r} ,
892
JOURNAL OF COMPUTERS, VOL. 9, NO. 4, APRIL 2014
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 x = ( x1 , x2 , ", xn ) = y11 , y12,", y1M , ", y NM T
)
T
,
coefficient matrices A and D are obtained through the templates A and D , C = diag ( c1 " cn ) , the input vector U = ( u1 ,..., un )
T
V = ( v1 ,..., vn )
T
,
and
f ( x ) = ( g ( y1 ) ,..., g ( yn ) ) . The kth cell in Eq. (2) is
xi ( t ) = − ( ci + aii ) xi ( t ) + wi ;
if xi ( t ) ≥ 1 , we have f ( xi ( t ) ) = −1 . Therefore, when xi ( t ) ≥ 1 , Eq. (3) can be transformed as
xi ( t ) = −ci xi ( t ) − aii + wi ;
if xi ( t ) ≤ −1 , we have f ( xi ( t ) ) = 1 . Therefore, when xi ( t ) ≤ −1 , Eq. (3) can be transformed as
xi ( t ) = −ci xi ( t ) + aii + wi .
T
denoted by Ο k ( k = iN + j , where 1 ≤ i ≤ N , 1 ≤ j ≤ M , i denotes ith row and j denotes jth column of the CNN).
( )n×n , defined by (2), composed of
The matrix A = aij
template has the form ⎡ A1 ⎢A ⎢ 3 ⎢0 ⎢ ⎢0 ⎢# ⎢ ⎢0 ⎢0 ⎣
⎡ a00 ⎢a ⎢ 0, −1 ⎢ 0 A1 = ⎢ ⎢ # ⎢ 0 ⎢ ⎢⎣ 0
A2 A1 A3 0 #
0 A2 A1 A3 #
0 ... 0 ... A2 ... A1 ... # %
0 0 0 0 #
0 0
0 0
0 0
0 0
A1 A3
0
"
0
a01 " a00 "
0
a01 a00 a0, −1 # 0 0
a11 ⎡ a10 ⎢a ⎢ 1, −1 a10 ⎢ 0 a1, −1 A2 = ⎢ # ⎢ # ⎢ 0 0 ⎢ 0 ⎢⎣ 0 ⎡ a−1,0 a−1,1 ⎢a ⎢ −1, −1 a−1,0 ⎢ 0 a−1, −1 A3 = ⎢ # ⎢ # ⎢ 0 0 ⎢ 0 ⎣⎢ 0
0 0 0 0 #
⎤ ⎥ ⎥ ⎥ ⎥ , ⎥ ⎥ ⎥ A2 ⎥ A1 ⎥⎦ n×n
0 # a00
# 0
% "
0
" a0, −1
0
wi′ =
j =1, j ≠ i
( )
n
aij f x j + ∑ dij u j + vi .
(7)
j =1
(8)
n
ρi = ∑ dij u j j =1
and
ϑi =
(3) (4)
f ( xi ( t ) ) = − xi ( t ) .
Therefore, when −1 ≤ xi ( t ) ≤ 1 , Eq. (3) can be transformed as
j =1, j ≠ i
n
± aij + ∑ dij u j + vi ;
For simplicity, denote
j =1
In Eq. (3), if −1 ≤ xi ( t ) ≤ 1 ,
n
∑
can be transformed as β i = ( − aii + wi′ ) ci , t → ∞ .
where
n
(6)
When β i ≥ 1 , we have f ( βi ) = −1 , and the Eq. (5)
( )n×n is similar to A .
∑
xi ( t ) = ci + aii + wi = R2 . Suppose that β i is equilibrium point of system (3), then we have −ci β i + aii f ( β i ) + wi = 0 . (5)
where
0 ⎤ 0 ⎥⎥ 0 ⎥ , ⎥ # ⎥ a01 ⎥ ⎥ a00 ⎥⎦ M ×M
The definition of matrices D = dij
wi =
and if xi ( t ) = −1 , we have
can be transformed as β i = ( aii + wi′ ) ci , t → ∞ ,
0
System (2) can be written as xi = −ci xi + aii f ( xi ) + wi , i = 1, " , n
xi ( t ) = −ci − aii + wi = R1 ,
When β i ≤ −1 , we have f ( βi ) = 1 , and the Eq. (5)
0⎤ a11 " 0 0 ⎥⎥ a10 " 0 0⎥ and ⎥ # % # # ⎥ 0 " a10 a11 ⎥ ⎥ 0 " a1, −1 a10 ⎥⎦ M ×M 0 " 0 0 ⎤ a−1,1 " 0 0 ⎥⎥ a−1,0 " 0 0 ⎥ . ⎥ # % # # ⎥ 0 " a−1,0 a−1,1 ⎥ ⎥ 0 " a−1, −1 a−1,0 ⎦⎥ M ×M "
In Eq. (3), if xi ( t ) = 1 , we have
n
∑
j =1, j ≠ i
aij .
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 aii < −ci , (i) if vi > ρi + ϑi − 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 2δ−1 for cell Oi ;
(ii) if vi < aii + ci − ρi − ϑi , 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 2δ−1 ; (iii) if vi ≥ aii + ci + ϑi + ρi and vi ≤ −aii − ci − ϑi − ρi , then there exist positive stable equilibrium points, negative equilibrium and pseudo equilibrium points for cell Oi . The number of isolated equilibrium points is grater than or equal to 3 and less than or equal to 3δ ; the
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number of stable equilibrium points is greater than or equal to 2 and less than or equal to 2δ . Proof. (i) From aii < −ci , we have R1 > R2 . In terms of vi > ρi + ϑi − ci − aii , we have R2 > 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 2. dxi/dt
(iii) From vi ≥ aii + ci + ϑi + ρi and vi ≤ −aii − ci − ϑi − ρi , we have R1 > 0 and R2 < 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 4. dxi/dt
R1
R1
-1
w 1
w
xi
R2 R2
-1
1
Fig. 4 The x − x phase plane trajectory of Eq. (1), where R1 > 0 ,
xi
R2 < 0 and aii < −ci .
Fig. 2 The x − x phase plane trajectory of Eq. (3), where R2 > 0 and aii < −ci
Therefore there exist positive stable equilibrium points for cell Oi . From Eq. (7) and (8) and R1 > 0 , we know that the equilibrium point of Eq. (3) is x* = ( a00 + wi′ ) ci ≥ 1 . *
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 2δ−1 . (ii) From vi < aii + ci − ρi − ϑi , we have R1 < 0 . The x − x phase plane trajectory of Eq. (1) is shown in Fig. 3. dxi/dt
-1
1
R1
xi
w
R2 Fig. 3 The
x − x
phase plane trajectory of Eq. (1), where R1 < 0 and aii < −ci .
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 2δ−1 .
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Then, we can get our result. Corollary 1. In Eq. (6), choose initial state x ( 0 ) = 0 , and let aii < −ci , (i) if
vi − ρi − ϑi > 0 ,
and
vi ≤ −aii − ci − ϑi − ρi 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 2δ−1 ; (ii) if vi + ρi + ϑi < 0 and vi ≥ aii + ci + ϑi + ρi 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 2δ−1 . Theorem 2. In Eq. (6), when aii = −ci , (i) if vi > ρi + ϑi , 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 2δ−1 ; (ii) if vi < ρi + ϑi , 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 2δ−1 . Theorem 3. In Eq. (6), when aii > −ci ,
(i) if vi ≥ ϑi + ρi , there exist not more than 2δ−1 positive stable equilibrium points for cell Oi , if vi ≥ ci + aii + ϑi + ρi , stable equilibrium points are equal or greater than 1 for cell Oij ; (ii) if vi ≤ ϑi + ρi , there exist not more than 2δ−1 negative stable equilibrium points for cell Oi , if vi ≤ ci + aii − ϑi − ρi , stable equilibrium points are equal or less than -1 for cell Oi .
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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: ⎛ a−1, −1 a−1,0 a−1,1 ⎞ ⎜ ⎟ A = ⎜ a0, −1 a00 a0,1 ⎟ . (9) ⎜a ⎟ ⎝ 1,−1 a1,0 a1,1 ⎠ Therefore, a CNN with 2 rows and 2 columns can be written as ⎧ x11 = −c11 x11 + a00 f ( x11 ( t ) ) + a0,1 f ( x12 ( t ) ) ⎪ ⎪ + a1,0 f ( x21 ( t ) ) + a11 f ( x22 ( t ) ) + η11 , ⎪ ⎪ x12 = −c12 x12 + a0, −1 f ( x11 ( t ) ) + a00 f ( x12 ( t ) ) ⎪ + a1, −1 f ( x21 ( t ) ) + a1,0 f ( x22 ( t ) ) + η12 , ⎪ ⎨ ⎪ x21 = −c21 x21 + a−1,0 f ( x11 ( t ) ) + a−1,1 f ( x12 ( t ) ) ⎪ + a00 f ( x21 ( t ) ) + a0,1 f ( x22 ( t ) ) + η21 , ⎪ ⎪ ⎪ x22 = −c22 x22 + a−1, −1 f ( x11 ( t ) ) + a−1,0 f ( x12 ( t ) ) ⎪ + a0.−1 f ( x21 ( t ) ) + a00 f ( x22 ( t ) ) + η22 , ⎪⎩ where ⎧η11 = d 00 u11 + d 0,1u12 + d1,0 u21 + d11u22 + v11 , ⎪ ⎪η12 = d 0, −1u11 + d 00 u12 + d1, −1u21 + d1,0 u22 + v12 , ⎨ ⎪η21 = d −1,0 u11 + d −1,1u12 + d 00 u21 + d 0,1u22 + v21 , ⎪η = d −1, −1u11 + d −1,0 u12 + d 0.−1u21 + d 00 u22 + v22 . ⎩ 22 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.
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)
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⎞ ⎜ ⎟ A = ⎜ 0.1 −5.9 0.1 ⎟ , c11 = c12 = c21 = c22 = 1 , and ⎜ 0.2 0.4 0.1 ⎟ ⎝ ⎠
(b)
⎛ −0.3 0.1 −0.1⎞ ⎜ ⎟ D = ⎜ 0.2 0.2 0.2 ⎟ . ⎜ 0.2 0.4 −0.1⎟ ⎝ ⎠
Then, from inputs and outputs of CNNs in Fig. 6, we can get 2.3 < v11 < 2.6, − 2.6 < v12 < −2.3, −3.6 < v21 < −1.3, 1.7 < v22 < 3.2 . Therefore , we choose v11 = 2.5, v12 = −2.4, v21 = −3, v22 = 3 . 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 © 2014 ACADEMY PUBLISHER
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with thresholding activation function, Neural computing & Applications, Published online: 26 September 2012. [10] Q. Han, X. Liao and C. Li. Analysis of associative memories based on stability of cellular neural networks with time delay. Neural Computing & Applications, Published online: 20 January 2012. [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.
(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. REFERENCES [1] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Transactions Circuits Systems, 35:1257-1272, 1988. [2] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Transactions Circuits Systems, 35:1273-1290 1988. [3] M. Itoh, L.O. Chua, Advanced image processing cellular neural networks, International Journal of Bifurcation and Chaos, 17: 1109-1150, 2007. [4] Z. G. Zeng, and J. Wang, Associative memories based on continuous-time cellular neural networks designed using space-invariant cloning templates, Neural Networks, 22: 651-657, 2009. [5] L. O. Chua and T. Roska, Cellular neural networks and visual computing, Cambridge: Cambridge University Press, 2002. [6] S. P. Xiao and X. M. Zhang, New globally asymptotic stability criteria for delayed cellular neural networks, IEEE Transactions on Circuits and Systems—II: Express Briefs, 56: 659-663, 2009. [7] C. J. Li, C. D. Li, X. F. Liao and T. W. Huang, Impulsive effects on stability of high-order BAM neural networks with time delays, Neurocomputing, 74: 1541-1550, 2011. [8] Q. Han, X. Liao, T. Weng, C. Li and H. Huang, Analysis on equilibrium points of cells in cellular neural networks described using cloning templates, Neurocomputing, 89: 106-113, 2012. [9] Q. Han, X. Liao, T. Weng, C. Li and Hongyu Huang, Analysis on equilibrium points of cellular neural networks
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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 Engineering, Chongqing University of Science and Technology.
