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Neural Networks xx (2003) 1–6
www.elsevier.com/locate/neunet
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Controlling chaos in a chaotic neural network
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Guoguang Hea,*,1, Zhitong Caoa, Ping Zhua, Hisakazu Ogurab
a
Department of Physics, Zhejiang University, Hangzhou 310028, People’s Republic of China
Department of Human and Artificial Intelligent Systems, Faculty of Engineering, Fukui University, 3-9-1 Bunkyo, Fukui 910-8507, Japan
b
Received 18 May 2001; accepted 31 January 2003
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Abstract
The chaotic neural network constructed with chaotic neuron shows the associative memory function, but its memory searching process
cannot be stabilized in a stored state because of the chaotic motion of the network. In this paper, a pinning control method focused on the
chaotic neural network is proposed. The computer simulation proves that the chaos in the chaotic neural network can be controlled with this
method and the states of the network can converge in one of its stored patterns if the control strength and the pinning density are chosen
suitable. It is found that in general the threshold of the control strength of a controlled network is smaller at higher pinned density and the
chaos of the chaotic neural network can be controlled more easily if the pinning control is added to the variant neurons between the initial
pattern and the target pattern.
q 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Chaotic dynamic; Chaotic neural network; Controlling chaos; Pinning control method
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1. Introduction
Recently the chaotic neural network constructed with
chaotic neurons has received much attention because of its
complex dynamics and its potential application in the
associative dynamics, optimization and information processing, etc. (Adachi & Aihara, 1997; Aihara, Takabe, &
Toyoda, 1990; Tokuda, Nagashima, & Aihara, 1997).
Besides, the electrophysiological experiments of animals
have proved that chaos dynamics exists in real neurons and
neural networks play an important role in neuron activity
(Degn, Holden, & Olsen, 1987; Freeman, 1987; Tsuda,
1991). It is believed that the investigation of the dynamics
characters of chaotic neural networks is helpful to an
understanding of the memory rules of the brain.
In Adachi and Aihara (1997), the non-periodic associative dynamics of the chaotic neural network was studied in
detail. The network can retrieve the stored patterns, but they
appear nonperiodically since the network is in chaos. So it is
* Corresponding author. Tel./fax: þ 86-571-8884-1465.
E-mail address: [email protected], [email protected] (G.
He).
1
Address until October 2003: Ogura Laboratory, Department of Human
and Artificial Intelligent Systems, Faculty of Engineering, Fukui
University, 3-9-1, Bunkyo, Fukui 910-8507, Japan.
difficult to distinguish the stored patterns from each other in
the chaotic neural network (Kobori, Ikoda, & Nakayama,
1996), which limits the applications of the associative
memory function of the network. If one wants to realize the
information processing in the associative dynamics system,
the chaotic orbit of the network should be controlled in a
certain expected way and the network should be stabilized
in a stored pattern. The problem of controlling the chaos of
the chaotic neural network therefore is put forward.
Since the initial work on chaotic control was made by
Ott, Grebogi, and Yorke (1990) (OGY), much progress has
been achieved in this field. Several control methods were
put forward, such as chaos synchronization (Pecora &
Carroll, 1990), OPF (Hunt, 1991) and delayed controlling
feedback (Pyragas, 1992). But all the above methods
only suited to the low dimension chaotic systems. For a
high dimension spatial-temporal chaotic system, Hu proposed a control method, i.e. the chaotic pinning control
method, which has been used to successfully control the
chaos of a coupled map lattice systems in one dimension
(Hu & Qu, 1994).
The chaotic neural network shows the more complex
spatial-temporal chaotic dynamics compared to the
coupled map lattice system in which each lattice site of
coupled map lattice systems is only coupled to its nearest
0893-6080/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0893-6080(03)00055-8
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G. He et al. / Neural Networks xx (2003) 1–6
ones. Besides the neurons in the chaotic neural networks
are connected to each other in whole spatial, the delay
feedback of the networks is more complicated than the
coupled map lattice system. We applied Hu’s pinning
control method to the chaotic neural network and found
that the method was not succeeded in controlling the
chaos of the chaotic neural network. Several other
attempts have been made to control the chaos of the
network, but the target of the control are spatially uniform
or periodic states which are far from some states useful
for information processing (Adachi, 1995; Mizutani, Sato,
Uchiyama, & Sonehara, 1995). In this work, we propose a
pinning method focused on the chaotic neural network.
The feedback control in our pinning method is imposed
on the internal state of neurons while the feedback control
in Hu’s method is put to the output of the system. The
computer simulation proved that the chaos of the chaotic
neural network could be controlled by the pinning control
method and the output of the network could be stabilized
in a stored pattern. In Section 2, the chaotic neural
network used as a controlling object in the work is
described. Our pinning control method and its mathematical model are proposed in Section 3. In Section 4 are
given the computer simulation results and the discussion
about two control parameters, pinning distance and
strength of control. Conclusion and discussion are in
Section 5.
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2. Chaotic neural network model
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The chaotic neural network model used in the paper is
constructed with chaotic neurons by considering the spatiotemporal summation of both external inputs and feedback
inputs from other chaotic neurons (Adachi & Aihara, 1997;
Aihara et al., 1990). Fig. 1 shows the structure of the
neural network. The dynamics of the ith chaotic neuron in
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the chaotic neural network is described as follows:
2
M
t
N
t
X
X
X
X
ked Aj ðt 2 dÞ þ
wij
kfd xj ðt 2 dÞ
xi ðt þ 1Þ ¼f 4 vij
j¼1
2a
j¼1
d¼0
#
M
X
vij Aj ðtÞ þ ke ji ðtÞ
ð2Þ
j¼1
hi ðt þ 1Þ ¼
N
X
wij xj ðtÞ þ kf hi ðtÞ
ð3Þ
ui ; Qi ð1 2 kr Þ
ð4Þ
xi ðt þ 1Þ ¼ f {ji ðt þ 1Þ þ hi ðt þ 1Þ þ zi ðt þ 1Þ}
ð5Þ
where: ji ðt þ 1Þ; hi ðt þ 1Þ and zi ðt þ 1Þ are internal state
terms for external inputs, feedback inputs from the
constituent neurons in network and refractoriness,
respectively.
The chaotic neural network is constructed with 100
chaotic neurons in this paper. As the external input term Aj
is taken as a temporally constant in the chaotic neural
network (Adachi & Aihara, 1997), ji ðtÞ can be included in
the threshold. If we define a new threshold ai ; Eq. (5) can be
simplified as the following form with only two internal
states hi and zi :
100
X
wij xj ðtÞ
ð6Þ
ð7Þ
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wij ¼
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Fig. 1. Chaotic neural network model.
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n
n
X
p¼1
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ð2xpi 2 1Þð2xpj 2 1Þ
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ð8Þ
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We take the output function of the neuron f ðxÞ as Sigmoid
function with the steepness parameter 1; i.e. f ðyÞ ¼ 1={1 þ
expð2y=1Þ}; refractoriness function as gðxÞ ¼ x: The feedback weights are defined according to the following
symmetric auto-associative matrix of n binary patterns:
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zi ðt þ 1Þ ¼ kr zi ðtÞ 2 ag{xi ðtÞ} þ ai
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zi ðt þ 1Þ ¼ 2ag{xi ðtÞ} þ kr zi ðtÞ 2 ui
j¼1
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j¼1
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where Aj is the ith external stimulation, f ð·Þ and gð·Þ are
the output function and the refractory function of the
neuron, respectively, a is the refractory scaling parameter,
Qi is the threshold of the ith neuron, vij and wij are
synaptic weights to the ith constituent neuron from the jth
external input and from the jth constituent neuron,
respectively. A neuron does not receive a feedback from
itself, i.e. wii ¼ 0: ke ; kf and kr are the decay parameters
for the external inputs, the feedback inputs, and the
refractoriness, respectively.
We can transform Eq. (1) into the following reduced and
simultaneous forms:
ji ðt þ 1Þ ¼
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ð1Þ
d¼0
hi ðt þ 1Þ ¼ kf hi ðtÞ þ
157
d¼0
krd g{xi ðt 2 dÞ} 2 Qi
xi ðt þ 1Þ ¼ f {hi ðt þ 1Þ þ zi ðt þ 1Þ}
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t
X
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ð9Þ
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Fig. 3. Sketch map of control methods. (a) Our control method, (b) Hu’s
control method.
Fig. 2. Four stored patterns.
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where xpi is the ith component of the pth binary pattern. In
this way, the binary patterns can be stored as basal memory
patterns. We use a picture composed of 10 £ 10 matrix to
show the stored patterns of the neural network constructed
with 100 neurons. A neuron with its output xi equal to 1 is
represented by a block ‘B’ while a neuron with its output xi
equal to 0 is denoted by a dot ‘·’. Four stored patterns
employed in this paper are shown in Fig. 2. When the decay
parameters of the network are set to certain values, the
network generates nonperiodic sequential patterns including
the stored ones as its output sequence. The states of the
network are usually described by the Hamming distance
between the output pattern of the network and one of the
four stored patterns, which is defined by
Hp ¼
100
X
lxi 2 xpi l
i¼1
for the pth stored pattern. When the network retrieves the
pth stored pattern or its reverse one exactly, the Hamming
distance will be 0 or 100.
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3. Pinning control method focused on the chaotic
neural network
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Each neuron in the chaotic neural network defined by
Eqs. (6) – (8) is coupled with others and evolve in nonlinear
way as time. If a part of the neurons in the network
are controlled, the controlling effect can be spread to the
entire neural network as time elapses because of its space
coupling and delay feedback. That is the main idea of the
pinning control. The dynamics of our chaotic neural
network is different from those of Hu’s coupled map lattice
system (Hu & Qu, 1994). First, any neuron is coupled to all
other neurons in the network while each site is only coupled
to its nearest ones in the coupled map lattice system, and
then the input of each neuron in our network is related to not
only its own history but also the history of other neurons
while only last outputs are feedback in Hu’s model. Our
simulation proved that the pinning control method used in
the coupled map lattice system was not suitable for our
network.
We propose a pinning control method focused on the
chaotic neural network. The chaotic neural network with
pinning control is described in the following way:
xi ðt þ 1Þ ¼ f {hi ðt þ 1Þ þ zi ðt þ 1Þ}
ð10Þ
hi ðt þ 1Þ ¼ kf hi ðtÞ þ
100
X
(
wij xj ðtÞ þ
j¼1
100=I
X
)
dðj 2 Il 2 1ÞKuj ðtÞ
l¼0
ð11Þ
uj ðtÞ ¼ xj ðtÞ 2 x~ j
ð12Þ
zi ðt þ 1Þ ¼ kr zi ðtÞ 2 ag{xi ðtÞ} þ ai
ð13Þ
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where dð·Þ is a pulse function which is used to distinguish
the pinned neurons from the unpinned ones. The jth neuron
is pinned if j 2 Il–1 ¼ 0: uj is controlling signal which is
constructed by comparing the last output and the expected
output ð~xj Þ for each pinned neuron. I is pinning distance, a
distance between two nearest pinned neurons, K is a control
strength.
The delay feedback is determined by the feedback input
of the internal state and the refractoriness in the chaotic
neural network. We therefore believe that the control aim
can be reached more easily by changing the internal state
than by adding control to its output. As shown in Fig. 3, the
feedback control in our pinning method is imposed on the
internal state of neurons while the feedback control in Hu’s
method is put to the output of the system (Hu & Qu, 1994).
The role of the feedback control on the network is carried
out through the output function of the neurons, which is
nonlinear in the chaotic neural network. Though we take a
linear feedback control signal in our control method, its
function is non-linear. In Hu’s pinning method, a non-linear
feedback control was considered.
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4. Computer simulation
Our computer simulation is made according to Eqs.
(10) – (13). We take a ¼ 10:0; kr ¼ 0:95; kf ¼ 0:20 and ai ¼
2:0 (i ¼ 1,2,…,100) in our simulation. The network without
control signal will be in chaos under the above parameters
and the stored patterns in Fig. 2 are retrieved nonperiodically. The initial state of the network is given in Fig. 4. The
fourth stored pattern (shown in Fig. 2d) is chosen as our
expected output pattern, i.e. control target. The Hamming
distance between the initial state and the expected output
pattern is 8. Without the pinning control, its Hamming
distances between the output of the network and the
controlling target at different times are displayed in Fig. 5.
In order to determine whether the network is in chaos, we
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G. He et al. / Neural Networks xx (2003) 1–6
as the control target. Fortunately, it was found that the
outputs of the controlled network converge in the fourth
stored pattern. Carrying out further simulations by taking
different initial states and control targets, we found that the
output of the controlled network always converge in the
reverse of the target with the pinning control method given
in Eqs. (10) –(13). If the state of every neuron of a target
pattern, x~ i ; is replaced by 1 2 x~ i ; the target pattern will
become its reverse. Obviously, the network will be
stabilized in the target patterns if one replaces the
controlling signals of Eq. 12 by the following equation:
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Fig. 4. Initial state.
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calculated the Lyapunov spectrum (Shimada & Nagashima,
1979), the largest Lyapunov exponent lm ¼ 0:000280:
Obviously, the system is in chaos. The detail duscussion
about chaotic dynamics of the neural network can be found
in Adachi and Aihara (1997).
When the pinning control is added to the network, its
Hamming distances at different times are shown in Fig. 6. In
the simulation, we take the pinning distance I ¼ 2 and the
control strength K ¼ 20:0: The control signals are added to
the system from 12,000 simulation step to 16,000 simulation
step. In Fig. 6, one can find that when the pinning control is
added to the system the network is stabilized to the reverse
of the control target whose Hamming distance from the
target is 100. The output of the stabilized network is given in
Fig. 7. The chaos control of the network is realized, but the
stable output of the controlled network is the reverse of the
stored patterns. Our control aim is to let the network be
stabilized in one of the stored patterns. We performed a
simulation by taking the reverse of the fourth stored pattern
uj ðtÞ ¼ xj ðtÞ 2 ð1 2 x~ j Þ
ð14Þ
In following simulation, we will replace Eq. (12) by Eq. (14)
and the network will be stabilized in our expect patterns.
From the simulation of I ¼ 2; we found that the chaos of
the network is controlled only when K is taken to be larger
than a threshold. To find out the correlation among the
pinning distance, the control strength and the control effect,
we make simulation under the different pinning distances
and control strengths. For a fixed pinning distance, the
control target can be reached only when the control strength
is larger than a threshold or is called minimal control
strength. If the control strength is smaller than the threshold,
the system is still in chaos. The thresholds of the control
strength for the different pinning distances are listed in case
A of Table 1. The data are obtained by using the state given
in Fig. 4 as the initial state and the fourth stored pattern as
the target pattern. It is observed from Table 1 that threshold
of the control strength is non-linear to the pinning distance.
The threshold of the control strength is the smallest when
the pinning distance is 3. By comparison with the expected
output, the variations in the initial pattern exist in neurons 6,
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Fig. 5. The Hamming distance between the orbit and the expected output pattern without the pinning control.
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Fig. 6. The Hamming distance between the orbit and the expected output pattern under the pinning control added from 12,000 step to 16,000 step.
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22, 23, 46, 52, 62, 72, 89. When the pinning distance I is 3,
there are three variant neurons pinned, which are neurons
22, 46 and 52, according to j ¼ nI þ 1: For I ¼ 2; there are
two variant neurons pinned while there is only one variant
neuron pinned for I ¼ 4 and I ¼ 5; and no variant neuron
pinned for I ¼ 6: For I ¼ 3; more variant neurons are
pinned than any other case. Besides, we found that the
threshold of the control strength is independent on when the
pinning is added. Based on the above results, we guess that
the threshold of the control strength may be related to the
pinned sites. The chaos of the chaotic neural network can be
controlled more easily if more variant neurons are pinned.
To demonstrate our conjecture, we calculated the
thresholds of the control strength of the controlled
networks with the same pinned density and the different
pinned sites. Beside the neurons pinned according to j ¼
nI þ 1; we added the pinning control to other three
variant neurons for case B and other three un-variant
neurons for case C. The way of pinning these neurons is
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the same as described in Eqs. (11) and (14). Their
thresholds of the control strength are also listed in
Table 1. The initial state and target pattern in both case
B and C are the same as in case A. The pinned density
is same in both case B and case C, but the pinned sites
are different in both cases. In case B, more variant
neurons are pinned. As we expected, the threshold of the
control strength of case B is smaller than that of case C
at the same pinning density except I ¼ 3 in which the
threshold values for both cases are the same.
The results of Table 1 are obtained by taking the state
shown in Fig. 4 as the initial state. In Table 2, we listed
the thresholds of the control strength in different initial
states. In our simulation, the nearest stored pattern of an
initial state is chosen as the control target. From Tables 1
and 2, one can find that in general the threshold value of
the control strength is dependent on both the pinning
density and the pinning site. The threshold of the control
strength of a controlled network is smaller at
higher pinned density. At the same pinned density,
the control aim of the chaotic neural network can be
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540
541
542
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546
547
548
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494
550
Table 1
The correlation between pinning distance and control strength
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Pinning distance I
497
498
Minimal control strength K min
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502
503
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527
Fig. 7. Output of the stabilized network.
A
B
C
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2
3
4
5
6
20
16
19
18
17
17
48
30
40
80
39
61
59
35
59
Note: case A: the control is described by Eqs. (11) and (14); case B: besides
the neurons pinned as in case A, other three variant neurons are pinned.
Case C: besides the neurons pinned as in case A, other three un-variant
neurons are pinned.
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Table 2
The correlation between pinning distance and control strength under different initial states and pinning site
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Pinning distance I
565
Minimal control strength K min
Initial state
566
567
Initial state
568
569
570
Initial state
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572
617
618
2
3
4
5
6
619
620
A
B
C
12
11
12
34
30
34
39
27
30
46
46
33
117
78
117
621
A
B
C
14
14
14
22
24
24
78
47
39
78
59
78
NC
NC
NC
624
A
B
C
14
14
14
34
34
34
30
27
30
77
58
77
NC
NC
NC
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574
577
578
579
580
Acknowledgements
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5. Conclusion and discussion
References
In this paper, a pinning control method focused on the
chaos control of a chaotic neural network has been
proposed. Our computer simulation proves that the chaos
in the chaotic neural network can be controlled with our
pinning control method. The chaotic neural network can
be stabilized in a stored pattern if the control strength is
larger than its threshold and the pinning density is not too
small. Our study on the correlation among the threshold
value of the control strength, the pinning density and the
pinning site demonstrates that the threshold of the control
strength is dependent on both the pinning density and the
pinning site. In general, the threshold of the control
strength of a controlled network is smaller at higher
pinning density and the chaos control of the chaotic neural
network can be reached more easily if more variant
neurons are pinned. But some exceptions exist in Tables 1
and 2, which indicates a more complex correlation among
the threshold of the control strength, the pinning density
and the pinning site.
This work is the first step of our study on the chaos
control of the chaotic neural network. Since the stored
pattern as the control target is chosen previously, the control
method cannot be applied to information processing
directly. Our next work will be focused on finding the
method to search the control targets automatically and
realizing effective information processing using the chaotic
neural network.
Adachi, M. (1995). Controlling a simple chaotic neural network using
response to perturbation. Proceedings of NOLTA’95, 989 –992.
Adachi, M., & Aihara, K. (1997). Associative dynamics in a chaotic neural
network. Neural Networks, 10, 83 –98.
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584
586
626
633
The authors appreciated the helpful discussion with Dr.
J. Kuroiwa. This work is supported by the National Natural
Science Foundation of China (Grant No. 50077019)
582
585
625
631
reached more easily if more variant neurons are pinned.
Besides, one can observe that the chaos of the
network cannot be controlled if the pinning density is
too small.
581
583
623
629
Note: case A, case B and case C are as the same as described in Table 1. NC means that the network cannot be controlled.
575
576
622
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
613
669
614
670
615
671
616
672
NN 1731—29/3/2003—12:48—SHYLAJA—65591— MODEL 5