ORIGINAL ARTICLE

Analysis of associative memories based on stability of cellularneural networks with time delay

Qi Han • Xiaofeng Liao • Chuandong Li

Received: 29 October 2011 / Accepted: 5 January 2012 / Published online: 20 January 2012

� Springer-Verlag London Limited 2012

Abstract In the paper, associative memories based on

cellular neural networks with time delay are presented. In

some previous papers, the relationship between cloning

templates is closer and stronger. Therefore, some methods

are used to make the relationship loose. First, some theories

on stability of cellular neural networks are given. Then,

associative memories based on cellular neural networks are

given on the basis of these theories. In addition, a design

procedure of associative memories is introduced. Finally,

some examples are given to verify the theoretical results

and design procedures.

Keywords Cellular neural networks � Associative

memories � Cloning template � Time delay

1 Introduction

Cellular neural networks (CNNs) were first introduction

in 1988 [1, 2]. CNNs are well fit for very large-scale

integration implementations due to this local intercon-

nection property and have found many applications in a

variety of areas, such as image processing [3], pattern

recognition [4], medical diagnosis [5], and associative

memories [6]. In this paper, we mainly discuss the

application of CNNs with time delay in associative

memories. At its simplest, an associative memory is a

system that stores mappings from specific input patterns

to specific output patterns. That is to say, a system which

‘‘associates’’ two patterns is that when one of two patterns

is presented, the other can be reliably recalled. There

are two kinds of associative memory: auto-associative

memories and hetero-associative memories.

Since Liu and Michel [7] reported that CNNs are

effective as an associative memories medium, associative

memories have received a great deal of interest. Next, we

would introduce some papers about associative memories

according to the time sequence. Sparsely interconnected

neural networks for associative memories were presented

in [8], and sparse synthesis technique was applied to the

design of a class of CNNs. A design algorithm for CNNs

with space-invariant cloning template with applications

to associative memories was presented in [9]. A synthesis

procedure for associative memories using discrete-time

CNNs (DTCNNs) with learning and forgetting capabili-

ties is presented in [10]. A synthesis procedure of CNNs

for associative memories was introduced in [11], where

the method assured the global asymptotic stability of the

equilibrium point. DTCNNs with a globally asymptoti-

cally stable equilibrium point were designed to behave as

associative memories in [12]. In last 10 years, associative

memories were achieved by local stability of equilibrium

points of CNNs. In [13–15], the number of memory

patterns of CNNs, which were locally exponentially

stable was obtained, and associative memories based on

CNNs were designed. A design method for synthesizing

associative memories based on discrete-time recurrent

neural networks was presented in [16]. In [17], a new

design procedure for synthesizing associative memories

based on CNNs with time delays characterized by input

and output matrices was introduced. In addition, in

[18, 19], associative memories based on neural networks

are presented.

Q. Han (&) � X. Liao � C. Li

State Key Laboratory of Power Transmission Equipment

and System Security, College of Computer Science,

Chongqing University, Chongqing 400030, China

e-mail: yiding1981@yahoo.com.cn

123

Neural Comput & Applic (2013) 23:237–244

DOI 10.1007/s00521-012-0826-4

From the above introduction about associative memo-

ries, it is easy to know that stability of CNNs play an

important role in associative memories. In the paper, the

outputs of CNNs are thought as memory patterns of asso-

ciative memories. Only when CNNs are stable, the outputs

of CNNs are fixed. If a CNN are not stable, the outputs will

be not stable, and memory patterns will be not obtained by

the CNN. Therefore, we should get the conditions of sta-

bility of CNNs for achieving associative memories. There

have been abundant researches about stability of CNNs

[20]. Some sufficient conditions for CNNs to be stable

were obtained by constructing Lyapunov functional

[21–28], and these conditions generally made equilibrium

point global asymptotically stable. However, some authors

present some conditions that made equilibrium points

locally stable, and there generally were multiple equilib-

rium points [13–15, 29–31]. In addition, during hardware

implementation, time delays occur due to finite switching

speed of the amplifiers and communication time. Time

delay may lead to an oscillation and furthermore, to

instability of networks. Therefore, the study of stability of

CNNs with time delay is practically required. There have

existed many papers about CNNs with time delay [13–15,

17, 22–32].

In previous papers, the researches for associative mem-

ories based on CNNs are not very comprehensive. For

example, bias vectors were computed by one of all memory

patterns in [7–9]; the relations between cloning templates

were stronger and closer in [13–17]. Thus, we give a new

method to achieve associative memories. The relations

between cloning templates are weaken through returning to

zero for initial states of all cells of a CNN. Our methods can

be used in auto-associative memories and hetero-associative

memories. The design procedures of associative memories

based on CNNs are given on the basis of some new theories.

The remaining parts of this paper are organized as

follows. In Sect. 2, a class of CNNs with time delay is

given. In Sect. 3, the main results are shown. First, a

theorem and its corollary are obtained. Then, some

methods of designing associative memories are given.

A design procedure on associative memories is given. In

Sect. 5, some examples are given to verify the theoretical

results and design procedures. Some conclusions are

finally drawn in Sect. 6.

2 Preliminaries

Consider CNNs whose cells are arranged on a rectan-

gular array composed of N rows and M columns, where

CNNs are defined by the following delay differential

equations:

_yij tð Þ ¼ ��cijyij tð Þ þXk2 i;rð Þ

k¼k1 i;rð Þ

Xl2 j;rð Þ

l¼l1 j;rð Þ�aklgiþk;jþl y tð Þð Þ

þXk2 i;rð Þ

k¼k1 i;rð Þ

Xl2 j;rð Þ

l¼l1 j;rð Þ

�bklgiþk;jþl y t� sð Þð Þ þ gij; t� 0;

gij ¼Xk2 i;rð Þ

k¼k1 i;rð Þ

Xl2 j;rð Þ

l¼l1 j;rð Þ

�dkl�ukl þ �vij;

y tð Þ ¼ u tð Þ; �s� t\0;

8>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

ð1Þ

where yijðtÞ 2 R denotes the states vector, �cij is a positive

parameter, r is positive integer denoting neighborhood

radius, �A ¼ ð�aklÞð2rþ1Þ�ð2rþ1Þ 6¼ 0 is the feedback cloning

template, �B ¼ ð�bklÞð2rþ1Þ�ð2rþ1Þ is the delay feedback cloning

template, s\? is the time delay, �D ¼ ð�dklÞð2rþ1Þ�ð2rþ1Þ is

input cloning template, �ukl is the input, �vij is the

bias, k1ði; rÞ ¼ max f1� i;�rg; k2ði; rÞ ¼ minfN � i; rg;l1ðj; rÞ ¼ maxf1� j;�rg; l2ðj; rÞ ¼ minfM � j; rg; and g(�)is the activation function defined by

g yð Þ ¼ yþ 1j j � y� 1j jð Þ=2:

Let r = 1, then we denote the expression of template �A

and �B as follows

�A ¼�a�1;�1 �a�1;0 �a�1;1

�a0;�1 �a0;0 �a0;1

�a1;�1 �a1;0 �a1;1

2

64

3

75 and

�B ¼�b�1;�1

�b�1;0�b�1;1

�b0;�1�b0;0

�b0;1

�b1;�1�b1;0

�b1;1

2

64

3

75:

Choose n = N 9 M, then system (1) can be put in

vector form as

_x tð Þ ¼ �Cx tð Þ þ Af x tð Þð Þ þ Bf x t � sð Þð Þ þ DU þ V ; ð2Þ

where x = (x1, x2, …, xn)T = (y11, y12, …, y1M, …, yNM)T,

coefficient matrices A, B, and D are obtained through the

templates �A, �B and �D, C = diag(c1, …, cn), the input

vector U = (u1, …, un)T, the bias vector V = (v1, …, vn)T

and activation function f(x) = (g(y1), …, g(yn))T. The kth

cell in (2) is denoted by Ok (k = iN ? j, where 1 B i B N,

1 B j B M, i denotes ith row and j denotes jth column of

the CNN). The matrix A = (aij)n9n, defined by (2),

composed of template has the form

238 Neural Comput & Applic (2013) 23:237–244

123

A1 A2 0 0 . . . 0 0

A3 A1 A2 0 . . . 0 0

0 A3 A1 A2 . . . 0 0

0 0 A3 A1 . . . 0 0

..

. ... ..

. ... . .

. ... ..

.

0 0 0 0 0 A1 A2

0 0 0 0 0 A3 A1

26666666664

37777777775

n�n

;

A1 ¼

�a00 �a01 0 � � � 0 0

�a0;�1 �a00 �a01 � � � 0 0

0 �a0;�1 �a00 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � �a00 �a01

0 0 0 � � � �a0;�1 �a00

266666664

377777775

M�M

;

A2 ¼

�a10 �a11 0 � � � 0 0

�a1;�1 �a10 �a11 � � � 0 0

0 �a1;�1 �a10 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � �a10 �a11

0 0 0 � � � �a1;�1 �a10

266666664

377777775

M�M

and

A3 ¼

�a�1;0 �a�1;1 0 � � � 0 0

�a�1;�1 �a�1;0 �a�1;1 � � � 0 0

0 �a�1;�1 �a�1;0 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � �a�1;0 �a�1;1

0 0 0 � � � �a�1;�1 �a�1;0

2

66666664

3

77777775

M�M

:

The definition of matrices B = (bij)n9n and D = (dij)n9n

is similar to A.

Let a ¼ a1; a2; . . .; anð ÞT2 !n ¼ xi 2 Rnjxi ¼ 1 or xi ¼f�1; i ¼ 1; 2; . . .; ng; C að Þ ¼ x2Rnjxiai[1;i¼f 1;2;...;ng:Then, for xðtÞ2Cða0Þ;xðt�sÞ2Cða00Þ, the (2) can be

rewritten as

_xðtÞ ¼ �CxðtÞ þ Aa0 þ Ba00 þ DU þ V : ð3Þ

If b is an equilibrium point of (3), then we have

b ¼ C�1ððAþ BÞaþ DU þ VÞ 2 C að Þ; ð4Þ

where a [ Tn.

Lemma 1 [7] Suppose a = (a1, a2, …, an)T [ Tn. If

b = (b1, b2, …, bn)T = C-1((A ? B)a ? DU ? V) [C(a), then b is an asymptotically stable equilibrium point

of (2).

Proof Equation (3) has a unique equilibrium point at

xe = C-1((A ? B)a ? DU ? V) and xe = b [ C(a) by

assumption. Therefore, this equilibrium is also asymptoti-

cally stable, since (3) has all its n eigenvalues at -ci, i =

1, 2, …, n. h

3 Main result

In this section, we will give some theories about stability of

CNNs with time delay firstly. Then, some methods are

obtained on the basis of these theories for associative

memories based on CNNs.

3.1 Stability of CNNs with time delay

The (3) can be rewritten as

_xi ¼ �cixi þXn

j¼1

aija0j þXn

j¼1

bija00j þ

Xn

j¼1

dijuj þ vi: ð5Þ

Theorem 1 In (5), let xi (0) = 0.

(i) IfPn

j¼1 ðaij þ bijÞa þPn

j¼1 dijuj þ vi [ ci; a 2 !n,

then the (5) converges to a positive stable equilib-

rium point, and the positive equilibrium point is

bigger than 1.

(ii) IfPn

j¼1 ðaij þ bijÞaþPn

j¼1 dijuj þ vi\� ci; a 2 !n,

then the (5) converges to a negative stable equilib-

rium point, and the negative equilibrium point is less

than negative 1.

Proof In (5), there exists a unique equilibrium point

bi ¼Xn

j¼1

ðaij þ bijÞaj þXn

j¼1

dijuj þ vi

!,ci: ð6Þ

(i) IfPn

j¼1 ðaij þ bijÞaj þPn

j¼1 dijuj þ vi [ ci and

xi (0) = 0, we have bi [ 1 in (6). Therefore, whenPnj¼1 ðaij þ bijÞaj þ

Pnj¼1 dijuj þ vi [ ci, then the

(5) converges to a positive stable equilibrium point

by Lemma 1, and the positive equilibrium point is

bigger than 1.

(ii) IfPn

j¼1 ðaij þ bijÞaj þPn

j¼1 dijuj þ vi\� ci and

xi (0) = 0, we have bi \ 1 in (6). Therefore, whenPnj¼1 ðaij þ bijÞaj þ

Pnj¼1 dijuj þ vi\� ci, then the

(5) converge to a negative stable equilibrium point

by Lemma 1, and the negative equilibrium point is

less than negative 1.

If we choose vi = 0 in Theorem 1, then we can get the

following corollary.

Corollary 1 In (5), let xi (0) = 0 and vi = 0

(i) IfPn

j¼1 ðaij þ bijÞaj þPn

j¼1 dijuj [ ci, then the (5)

converges to a positive equilibrium point, and the

positive equilibrium point is bigger than 1.

Neural Comput & Applic (2013) 23:237–244 239

123

(ii) IfPn

j¼1 ðaij þ bijÞaj þPn

j¼1 dijuj\� ci, then the (5)

converges to a negative equilibrium point, and the

negative equilibrium point is less than negative 1. h

Remark 1 In the Theorem 1 and its corollary, though the

initial states of a CNN have to return to zero, our methods

does not strictly limit the relations between cloning tem-

plates of the CNN. For example,Pn

j¼1 aij þ bij

�� ��� �\ci\1

andP1

i¼�1

P1i¼�1 �aij þ �bij

�� ��� �\1 in [15, 16]; however, our

theories do not have these limitations.

3.2 Notations

Suppose that there exists a set of memory patterns, and

these memory patterns can be written as a matrix C = (a1,

a2, …, am), where ai ¼ ai1; a

i2; . . .; ai

n

� �T2 !n and ai is a set

of outputs (a memory pattern) of all cells of a CNN.

Therefore, C has n rows and m columns. Choose a set of

input patterns, and these input patterns can be written as a

matrix U = (U1, U2, …, Um), where U is corresponding to

the set of memory patterns, Ui ¼ ui1; u

i2; . . .; ui

n

� �T, and ui

j

denotes an input data of jth cell of a CNN in ith memory

pattern. Note that when the inputs of a CNN is Ui, the

memory pattern is ai, namely, if Ui is a set inputs of a

CNN, then ai is a set of outputs of the CNN corresponding

to Ui, where we can use uij; a

ij

� �to describe the

relationship.

Then, we divide all cells of a CNN into three small sets

on the basis of the memory patterns C. When all outputs of

a cell of the CNN in different memory pattern ai are 1, then

we classify the cell as set P = {Oa, Ob, …}, where Oa

denotes a-th cell in (2). When all outputs of a cell of the

CNN in different memory pattern ai are -1, then we

classify the cell as set Q = {Oc, Od, …}. When all outputs

of a cell of the CNN in different ai are not same, namely,

the outputs can be 1 and -1, then we classify the cell as set

R = {Oe, Of, …}. Let Oj = j. |P|, |Q| and |R| denote the

number of elements in sets P, Q, and R, respectively.

Let P1 ¼ Oa;P2 ¼ Ob; . . .;Pi ¼ Oj; . . .; P Pj j ¼ Oc; where

0 B i B |P|, 1 B j B n, a \ b \ j \ c, and Oj [ P. Both

definitions of Qi and Ri are similar with that of Pi,

respectively.

Let

K ¼

k1 0 � � � 0

0 k2 � � � 0

..

. ... . .

. ...

0 0 � � � kn

0

BBB@

1

CCCA

n�n

;

where, ki [ 0.

Let

D ¼ DR1;DR2

; . . .;DR Rj j

� �T

; A ¼ AR1;AR2

; . . .;AR Rj j

� �T

;

B ¼ BR1;BR2

; . . .;BR Rj j

� �T

and K ¼ KR1;KR2

; . . .;KR Rj j

� �T

;

where HRidenotes Rith row of matrix H.

Let

LD ¼ d�1;�1; d�1;0; d�1;1; d0;�1; d0;0; d0;1; d1;�1; d1;0; d1;1

� �;

LA ¼ a�1;�1; a�1;0; a�1;1; a0;�1; a0;0; a0;1; a1;�1; a1;0; a1;1

� �;

LB ¼ b�1;�1; b�1;0; b�1;1; b0;�1; b0;0; b0;1; b1;�1; b1;0; b1;1

� �;

K0 ¼

K 0 � � � 0

0 K � � � 0

..

. ... . .

. ...

0 0 0 K

0

BB@

1

CCA

NMmð Þ� NMmð Þ

;

C0 ¼ a1ð ÞT; a2ð ÞT; . . .; amð ÞT� �T

:

Choose l 2 f1; 2; . . .;mg; q 2 f1; 2; . . .; ng and k 2 fR1;

R2; . . .;R Rj jg. Next, we introduce some symbols as

follows.Let

U0 ¼ ððAþ BÞa1ÞTððAþ BÞa2ÞT; . . .; ððAþ BÞamÞT� �T

;

U0 ¼ U0R1;U0R2

; . . .;U0R Rj j;U0NMþR1

; . . .;U0l�NMþRk; . . .;U0m�NMþR Rj j

� �T

;

Nlq ¼

0 ulq�1ð ÞMþ1 ul

q�1ð ÞMþ2

ulq�1ð ÞMþ1 ul

q�1ð ÞMþ2 ulq�1ð ÞMþ3

ulq�1ð ÞMþ2 ul

q�1ð ÞMþ3 ulq�1ð ÞMþ4

..

. ... ..

.

ulqM�2 ul

qM�1 ulqM

ulqM�1 ul

qM 0

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

M�3

;

Nl ¼

0 Nl1 Nl

2

Nl1 Nl

2 Nl3

Nl2 Nl

3 Nl4

..

. ... ..

.

NlN�2 Nl

N�1 NlN

NlN�1 Nl

N 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

NMð Þ�9

; N ¼

N1

N2

..

.

Nm

0BBBBBBBBB@

1CCCCCCCCCA

NMmð Þ�9

;

N ¼ NR1;NR2

; . . .;NR Rj j ;NNMþR1; . . .;Nl�NMþRj

; . . .;Nm�NMþRk3

� �T

;

X0 ¼ XR1;XR2

; . . .;XR Rj j ;XNMþR1; . . .;Xl�NMþRk

; . . .;Xm�NMþRk3

� �T

;

D ¼ K0C0;

D ¼ DR1;DR2

; . . .;DR Rj j ;DNMþR1; . . .;Dl�NMþRk

; . . .;Dm�NMþRk3

� �T

;

n0q lð Þ ¼ cq �Xn

j¼1

aqjalj �Xn

j¼1

dqjulj:

3.3 Associative memories

In order to get parameters A, B, C, D, and V for a CNN with time

delay, Corollary 1 is used to achieve associative memories.

240 Neural Comput & Applic (2013) 23:237–244

123

Let ki [ max1� i� n cif g. We choose

U0 ¼ 0:5D; ð7Þ

and

Aþ B� �

Cþ DU ¼ KC: ð8Þ

Equation (7) shows that the sign ofPn

j¼1 ðaij þ bijÞaj of a

cell Oi is equal to the sign of output of the cell. Equation

(8) shows that sign ofPn

j¼1 ðaij þ bijÞaj þPn

j¼1 dijuj of a

cell Oi of a CNN is equal to the sign of output of the cell.

By Corollary 1, we know that sign ofPn

j¼1 ðaij þ bijÞaj þPnj¼1 dijuj of a cell of a CNN with time delay play

important role for the sign of output of the cell.

However, it is difficult to be obtain A; B and D through

(7) and (8). Therefore, (7) and (8) are needed to transform.

Equation (7) can be transformed as

X0 � ðLAþ LBÞ ¼ 0:5D; ð9Þ

Therefore,

LAþ LB ¼ 0:5pinvðX0ÞD: ð10Þ

where pinv(�) denotes pseudo inverse of a matrix.

Equation (8) can be transformed as

N � LD ¼ D� U0: ð11Þ

Therefore,

LD ¼ pinv N� �

D� X0 � ðLAþ LBÞ� �

: ð12Þ

Remark 2 When matrices X0 or N are irreversible, the

value of LA ? LB or LD is an approximate value.

Next, we discuss how to get bias vi for a cell in sets

P and Q.

Choose that vþ 2 R is the bias of all cells in set P, and

v� 2 R is the bias of all cells in set Q. Then, the regions of

vþ and v� can be get by Theorem 1.

If ai = 1 in (5), we have viðlÞ[ ci �Pn

j¼1

ðaij þ bijÞalj �Pn

j¼1 dijulj ¼ n0i lð Þ:

If ai = -1 in (5), we have vi lð Þ\ci �Pn

j¼1

ðaij þ bijÞalj �Pn

j¼1 dijulj ¼ n0i lð Þ:

Thus, we choose

vþ � max1� l�m;i2P

n0i lð Þ� ��� �� ð13Þ

as a bias of all cells in set P, and

v� � � max1� l�m;i2Q

n0

i lð Þn o���

��� ð14Þ

as a bias of all cells in set Q.

Remark 3 Note that cloning template A, B, and D are

computed by the cells in set R, which can reduce the effect

from the cells in set P and Q.

4 Design procedure of a CNN with time delay

In this section, we give a design procedure of parameters of

a CNN with time delay on the basis of the above theories.

Eleven steps are given as follows.

Step 1. Give a set of memory patterns a1, a2, …, am for

the CNN, where ai is a set of outputs of all cells of the

CNN, and m is the number of patterns. Let matrix

C = (a1, a2, …, am). Denote an input matrix

U = (U1, U2, …, Um), which is corresponding to the

matrix C.

Step 2. Determine time delay s.

Step 3. Divide all cells of the CNN into three sets P, Q,

and R. If all outputs of a cell in all memory patterns are

1, the cell will be classified as set P. If all outputs of a

cell in all memory patterns are -1, the cell will be

classified as set Q. If the outputs of a cell in all memory

patterns are 1 and -1, the cell will be classified as set R.

Step 4. Let biases vi (i [ R) of all cells in set R are zero.

Step 5. Determine all constants �cijð1� i�N; 1� j�MÞ.Obtain coefficient matrix C.

Step 6. Determine matrix K such that ki[max1�i�n cif gand get matrix K0.Step 7. In set R, compute cloning template �Aþ �B from

(10). Obtain coefficient matrix A ? B.

Step 8. In set R, compute cloning template �D from (12).

Obtain coefficient matrix D.

Step 9. Compute n0i lð Þði 2 P; 1� l�mÞ in set P. Choose

vþ[ max1� l�m;i2P n0i lð Þ� ��� �� in terms of (13), and bias

vi of all cells in set P is equal to vþ.

Step 10. Compute n0i lð Þði 2 Q; 1� l�mÞ in set Q. Choose

v�\�max1� l�m;i2Q n0i lð Þ� ��� �� in terms of (14), and bias

vi of all cells in set Q is equal to v�.

Step 11. Synthesize the CNN with the connection

weight matrices A, B, C, D, time delay s, and bias

vector V.

5 Numerical examples

In this section, we will give some numerical simulations to

verify the theoretical results in this paper.

Example 1 Consider the same example introduced in

[16]. The inputs and the output patterns of a CNN are

represented by two pairs of (5 9 5)-pixel images showed

Neural Comput & Applic (2013) 23:237–244 241

123

in Fig. 1 (black pixel = 1, white pixel = -1), where the

inputs of the CNN compose the word ‘‘MO’’ in Fig. 1a, and

the patterns to be memorized to constitute the word ‘‘LS’’

in Fig. 1b.

We design all parameters of a CNN with time delay for

associative memories on the basis of design procedure in

Sect. 4.

Step 1. In terms of Fig. 1, we get memory patterns

ð�1; 1;�1; . . .;�1ÞT; ð�1; 1; 1; . . .;�1ÞT and input pat-

terns 1;�1;�1; . . .; 1ð ÞT; �1; 1; 1; . . .; 1ð ÞT. Let matrix

C ¼ ðð�1; 1;�1; . . .;�1ÞT; ð�1; 1; 1; . . .;�1ÞTÞ and

U ¼ ðð1;�1;�1; . . .; 1ÞT; ð�1; 1; 1; . . .; 1ÞTÞStep 2. Choose s = 1.

Step 3. From memory patterns, all cells of the CNN can

be divided into three sets, P = {O2, O7, O12, O22, O23,

O24}, Q = {O1, O5, O6, O8, O9, O10, O11, O15, O16, O18,

O20, O21, O25}, R = {O3, O4, O13, O14, O17, O19}.

Step 4. Choose bias of all cells in set R is equal to zero,

namely, v3 = v4 = v13 = v14 = v17 = v19 = 0.

Step 5. Let �cij ¼ 1; 1� i�N; 1� j�M; then we can

obtain C = diag(1, 1, …, 1)n9n.

Step 6. Let K = diag(3, 3, …, 3)n9n, then we have

K0 ¼ diagð3; 3; . . .; 3Þnm�nm.

Step 7. From (10), we get LA ? LB = (0, 0, 0, 0, 1.5, 0,

0, 0, 0)T. Then, we have matrix A ? B = diag(1.5)n9n.

Step 8. From (12), we get LD ¼ ð�1:1761;�2:2045;

�0:4602; 1:2386; 0:4602; 0:0909; 0:2898; 0:2443;

0:1875ÞT. Then, we can obtain matrix D.

Step 9. Let vþ ¼ 8 in set P. Therefore, v2 = v7 =

v12 = v22 = v23 = v24 = 8.

Step 10. Let v� ¼ �8 in set Q. Therefore,

v1 = v5 = v6 = v8 = v9 = v10 = v11 = v15 = v16 =

v18 = v20 = v21 = v25 = -8.

Step 11. Synthesize the CNN with A, B, C, D, and V.

Note that the �aii þ �bii [�cij in the example; however, in

previous paper [15],Pn

j¼1 �aij þ �bij

�� ��\�cij must be satisfied.

Therefore, in the paper, though the initial states of a CNN

have to return to zero, we reduce many limitations for the

relationships between cloning templates.

Through the above eleven steps, we can get a CNN that

achieve associative memories for ‘‘MO’’ to ‘‘LS’’. When

the inputs of the CNN are ‘‘M’’, we can get time response

curves of the CNN in Fig. 2. In Fig. 2, we find that states of

all cells will be stable after a time. When states of all cells

are stable, the value of equilibrium point is

x� ¼ ð�8:6986; 10:5338;�3:0567;�2:9658;�9:7440;

� 10:6360; 12:8292;�5:0225;�7:0339;�8:8747;

� 11:7383; 7:3066;�2:5226;�2:4657;�13:7041;

� 10:8178; 2:6135;�12:5792; 2:8862;�11:3519;

� 10:8747; 11:6758; 11:5508; 10:8122;�11:3065ÞT:

Therefore, we know that all outputs of the CNN

corresponding to the equilibrium point are

� 1; 1;�1;�1;�1;�1; 1;�1;�1;�1;�1; 1;�1;�1;

� 1;�1; 1;�1;�1;�1;�1; 1; 1; 1;�1;

where the outputs of the CNN are same with ‘‘L’’.

When the inputs of the CNN are ‘‘O’’, we can get time

response curves of the CNN in Fig. 3. Then the value of

equilibrium point is

x� ¼ ð�9:9258; 8:5793; 3:1476; 2:9430;�7:8463;

� 8:1816; 6:9089;�12:7951;�15:2950;�13:7041;

� 28:1816; 7:8293; 3:5339; 2:4657;�11:3519;

� 8:1816;�2:7953;�6:6020; 3:5339;�10:7724;

� 8:1247; 8:2441; 12:7780; 14:2098;�8:8293ÞT:

Fig. 2 When the inputs of the CNN are ‘‘M’’, time response curves of

all cells of a CNN are shownFig. 1 a Inputs of a CNN, b outputs of the CNN or memory patterns

242 Neural Comput & Applic (2013) 23:237–244

123

Therefore, we know that all outputs of the CNN

corresponding to the equilibrium point are

� 1; 1; 1; 1;�1;�1; 1;�1;�1;�1;�1; 1; 1; 1;�1;�1;�1;

� 1; 1;�1;�1; 1; 1; 1;�1;

where the outputs of the CNN are same with ‘‘S’’.

Example 2 No matter what the inputs of a CNN are, if the

set of outputs of the CNN is always ‘‘M’’, we can only use

bias of the CNN to achieve the associative memories.

Choose

�A ¼�0:1 �0:1 �0:10:1 0:3 0:1�0:1 �0:1 �0:1

24

35; �B ¼

0:1 0:1 0:1�0:1 0:2 �0:10:1 0:1 0:1

24

35;

�D ¼0:1 �0:15 0:10:1 0:1 �0:10:1 �0:15 0:1

24

35; s ¼ 1

v1 ¼ v5 ¼ v6 ¼ v7 ¼ v9 ¼ v10 ¼ v11 ¼ v13 ¼ v15

¼ v16 ¼ v20 ¼ v21 ¼ v25 ¼ 4 and

v2 ¼ v3 ¼ v4 ¼ v8 ¼ v12 ¼ v14 ¼ v17 ¼ v18 ¼ v19 ¼ v22

¼ v23 ¼ v24 ¼ �4:

Then, when the initial states of CNN are 0, we can get a CNN

by the above parameters. No matter what the inputs of the

CNN are, the set of the outputs of the CNN is always ‘‘M’’.

6 Conclusions

In the paper, some new methods about associative

memories based on CNNs with time delay are given. First,

a theorem and a corollary are obtained, where the initial

states of CNNs are zero. It is easy to return to zero for a

CNN in applications. In addition, in order to achieve

associative memories, we get some broad conditions for

design cloning templates of CNNs. Then, some new

methods for associative memories are given based on the

theorem and corollary, and a design procedure is obtained.

Finally, some examples are given to show that our methods

are effective and useful.

Acknowledgments This work was supported in part by the project

of graduate innovation of Chongqing University under Grant

200909C1011, in part by the National Natural Science Foundation of

China under Grant 60973114, Grant 61170249 and Grant 61003247,

in part by the Natural Science Foundation project of CQCSTC under

Grant 2009BA2024, and in part by the State Key Laboratory of Power

Transmission Equipment & System Security and New Technology,

Chongqing University, under Grant 2007DA10512711206, in part by

Teaching & Research Program of Chongqing Education Committee

(KJ110401).

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