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RESEARCH ARTICLE

Novel tracking function of moving target using chaotic dynamics

in a recurrent neural network model

Yongtao Li Æ Æ Shigetoshi Nara

Received: 15 May 2007/Revised: /Accepted: 14 September 2007/Published online: 9 October 2007

? Springer Science+Business Media B.V. 2007

Abstract

neural network model is applied to controlling an object to

track a moving target in two-dimensional space, which is

set as an ill-posed problem. The motion increments of the

object are determined by a group of motion functions

calculated in real time with firing states of the neurons in

the network. Several cyclic memory attractors that corre-

spond to several simple motions of the object in two-

dimensional space are embedded. Chaotic dynamics

introduced in the network causes corresponding complex

motions of the object in two-dimensional space. Adaptively

real-time switching of control parameter results in con-

strained chaos (chaotic itinerancy) in the state space of the

network and enables the object to track a moving tar-

get alonga certain trajectory

performance of tracking is evaluated by calculating the

success rate over 100 trials with respect to nine kinds of

trajectories along which the target moves respectively.

Computer experiments show that chaotic dynamics is

useful to track a moving target. To understand the relations

between these cases and chaotic dynamics, dynamical

structure of chaotic dynamics is investigated from

dynamical viewpoint.

Chaotic dynamics introduced in a recurrent

successfully.The

Keywords

Neural network

Chaotic dynamics ? Tracking ? Moving target ?

Introduction

Associated with rapid development of science and tech-

nology, great attentions on biological systems have been

paid because of excellent functions not only in information

processing, but also in well-regulated functioning and

controlling, which work quite adaptively in various envi-

ronments. Despite many attempts to understand the

mechanisms of biological systems, we have yet poor

understanding them.

In biological systems, well-regulated functioning and

controlling originate from the strongly nonlinear interac-

tion between local systems and total system. Therefore, it is

very difficult to understand and describe these systems

using the conventional methodologies based on reduc-

tionism, which means that a system is decomposed into

parts or elements. The conventional reductionism more or

less falls into two difficulties due to enormous complexity

originating from dynamics in systems with large but finite

degrees of freedom. One is ‘‘combinatorial explosion’’ and

the other is ‘‘divergence of algorithmic complexity’’. These

difficulties are not yet solved in spite of many efforts. On

the other hand, a novel idea based on functional viewpoint

was introduced to understand the mechanisms. It is a new

approach called ‘‘the methodology of complex dynamics’’,

which has been constructed in various fields of science and

engineering in the several decades associated with the

remarkable development of computers and simulation

methods. Especially, chaotic dynamics observed in bio-

logical systems including brains has attracted great interest

(Babloyantz and Destexhe 1986; Skarda and Freeman

1987). It is considered that chaotic dynamics would play

important roles in complex functioning and controlling of

biological systems including brains. From this viewpoint,

many dynamical models have been constructed for

Y. Li (&) ? S. Nara

Graduate School of Natural Science and Technology,

Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530,

Japan

e-mail: li@chaos.elec.okayama-u.ac.jp

123

Cogn Neurodyn (2008) 2:39–48

DOI 10.1007/s11571-007-9029-6

Page 2

approaching the mechanisms by means of large-scale

simulation or heuristic methods. Artificial neural networks

in which chaotic dynamics can be introduced has been

attracting great interests.

Over decade years ago, chaotic itinerancy was observed

in neural networks and was proposed as a universal

dynamical concept in high-dimensional dynamical sys-

tems. Artificial neural networks for chaotic itinerancy were

studied with great interests (Aihara et al. 1990; Tsuda

1991, 2001; Kaneko and Tsuda 2003; Fuji et al. 1996). As

one of those works, by Nara and Davis, chaotic dynamics

was introduced in a recurrent neural network model

(RNNM) consisting of binary neurons, and for investigat-

ing the functional aspects of chaos, they have applied

chaotic dynamics by means of numerical methods to

solving, for instance, a memory search task which is set in

an ill-posed context (Nara and Davis 1992, 1997; Nara

et al. 1993, 1995; Kuroiwa et al. 1999; Suemitsu and Nara

2003). In their papers, they proposed that chaotic itinerancy

could be potentially useful dynamics to solve complex

problem, such as ill-posed problems. Standing on this

viewpoint, auditory behaviour of the cricket shows a typ-

ical ill-posed problem in biological systems. Female

cricket can track towards directions of male position leaded

by calling song of male in dark fields with a large number

of obstacles (Huber and Thorson 1985). This behaviour

includes two ill-posed properties. One is that darkness and

noisy environments prevent female from accurate deciding

of directions of male positions, and the other is that a large

number of big obstacles in fields force female to solve two-

dimensional maze as one of ill-posed problems. Therefore,

in order to investigate the brain from functional aspects, we

try to construct a model to approach insect behaviours to

solve ill-posed problems. As one of functional examples,

chaotic dynamics introduced in a recurrent network model

was applied to solving a two-dimensional maze, which

is set as an ill-posed problem (Suemitsu and Nara 2004).

A simple coding method translating the neural states into

motion increments and a simple control algorithm adap-

tively switching a system parameter to produce chaotic

itinerant behaviours are proposed. The conclusions show

that chaotic itinerant behaviours can give better perfor-

mance to solving a two-dimensional maze than that of

random walk.

In order to further investigate functional aspects of

chaotic dynamics, it was applied to tracking a moving

target, which is set as another ill-posed problem. Generally

speaking, as an object is tracking an target that is moving

along a certain trajectory in two-dimensional space, it is an

ill-posed problem because there are many tracking results

with uncertainty. In conventional methods, the object is set

to obtain more precise information from the target as

possible, so as to successfully capture the moving target.

However, in our study, the object obtain only rough

information of the target and successfully capture the

moving target using chaotic dynamics in RNNM.

In the case of tracking a moving target, the first problem

is to realize two-dimensional motion control of the object.

In our model, we assume that the object moves with dis-

crete time steps. The firing state in the neural network is

transformed into two-dimensional motion increments by

the coding of motion functions, which will be illustrated in

the later section. In addition, several limit cycle attractors,

which are corresponding to the prototypical simple motions

of the object in two-dimensional space, are embedded in

the neural network. At a certain time, if the firing pattern

converges into an prototypical attractor, the object moves

in a monotonic direction of several directions in two-

dimensional space. If chaotic dynamics is introduced into

the network, the firing pattern could not converge into an

prototypical attractor, that is, attractors fall to ruin. At the

same time, the corresponding motion of the object is cha-

otic in two-dimensional space. By adaptive switching of a

certain system parameter, chaotic itinerancy generated in

the neural network results in complex two-dimensional

motions of the object in various environments. Considering

this point, we have proposed a simple control algorithm of

tracking a moving target, and quite good tracking perfor-

mances have been obtained, as will be stated in the later

section .

This paper is organized as follows. In the next section

we describe the network model and chaotic dynamics in it,

the control algorithm of tracking a moving target is illus-

trated in Sect. ‘‘Algorithm of tracking a moving target’’.

We discuss the results of computer simulation and evaluate

the performance of tracking a moving target in Sect.

‘‘Experimental results’’.

Chaotic dynamics in a recurrent neural network model

Our study works with a fully interconnected recurrent

neural network consisting of N neurons , which is shown as

Fig. 1. Its updating rule is defined by

Siðt þ 1Þ ¼ sgn

X

u?0;

u\0:

j2GiðrÞ

WijSjðtÞ

0

@

1

A

sgnðuÞ ¼

þ1

?1

?

ð1Þ

where Si(t) = ± 1 (i = 1 * N) represents the firing state of

a neuron specified by index i at time t. Wij is an

asymmetrical connection weight (synaptic weight) from

the neuron Sjto the neuron Si, where Wiiis taken to be 0.

Gi(r) means a connectivity configuration set of connectivity

40Cogn Neurodyn (2008) 2:39–48

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r (0\r\N) that is fan-in number for the neuron Si. At a

certain time t, the state of neurons in the network can be

represented as a N-dimensional state vector S(t), called as

state pattern. Time development of state pattern S(t)

depends on the connection weight matrix Wij and

connectivity r, therefore, in the case of full connectivity

r = N – 1, if Wij could be appropriately determined,

arbitrarily chosen state pattern nðtÞ would be multiple

stationary states in the development of S(t), which is

equivalent to storing memory states in the functional

context. In our study, Wij are determined by a kind of

orthogonalized learning method and taken as follows.

Wij¼

X

L

l¼1

X

ljk ¼ 1...K;l ¼ 1...Lg is an attractor pattern

set, K is the number of memory patterns included in a cycle

and L is the number of memory cycles. nky

vector of nk

Kronecker’s delta. This method was confirmed to be

effective to avoid spurious attractors that affect L attractors

with K-step maps embedded in the network when con-

nectivity r = N (Nara and Davis 1992, 1997; Nara et al.

1993, 1995; Nara 2003; Suemitsu and Nara 2003).

In the case of full connectivity r = N – 1, as time

evolves, the state pattern S(t) converges into one of the

cyclic memory patterns. Therefore, the network can func-

tion as a conventional associative memory. If the state

pattern S(t) is one of the memory patterns, nk

output S(t + 1) will be the next memory pattern of the

cycle, nkþ1

l

: Even if the state pattern S(t) is near one of the

memory patterns, nk

l: the output sequence S(t + kK)(k =

1,2,3...) will converge to the memory pattern nk

words, for each memory pattern, there is a set of the state

patterns, called as memory basin Bl

memory basin Bl

(k = 1,2,3...) will converge to the memory pattern nk

K

k¼1

ðnkþ1

l

Þi? ðnk

lÞy

j

ð2Þ

where fnk

lis the conjugate

l0 ¼ dll0 ? dkk0; where d is

lwhich satisfies nky

l? nk0

l; then the next

l: In other

k. If S(t) is in the

k, then the output sequence S(t + kK)

l:

It is quite difficult to estimate basin volume accurately

because one must check the final state (limk!1SðkKÞÞ of all

initial state patterns (the total number is 2N), as requires an

enormous amounts of time. Therefore, a statistical method

is applied to estimating the approximate basin volume.

First, random initial state patterns are generated with a

sufficiently large amount so that they can cover the entire N-

dimensional state space uniformly. As state updating

develops, it is specified that the final state limk!1SðkKÞ of

each initial pattern would converge into a certain memory

attractor. The ratios between the number of initial state

patterns that converge into a certain memory attractor and

the total number of initial state patterns are taken. The rate

of convergence to each memory attractor is proportional to

the basin volume, and is regarded as the approximate basin

volume for each memory attractor. An actual example of

the basin volume is shown in Fig. 2. The basin volume

shows that almost all initial state patterns converge into one

of the memory attractors averagely, that is, there are mainly

the memory attractors in the whole state space.

Next, we continue to decrease connectivity r. When r is

large enough, r ^ N, memory attractors are stable, the

network can still function as a conventional associative

memory. When r becomes smaller and smaller, the basin

volume of all the memory attractors are becoming smaller

and smaller, in other words, more and more state patterns

gradually do not converge into a certain memory pattern

despite the network is updated for a long time, that is, each

basin vanishes and the attractor becomes unstable. So that

if the number of connectivity r becomes quite small, state

patterns do not converge into any memory pattern even if

the network is updated for a long time.

Fig. 1 Fully interconnected recurrent neural network model

0

0.01

0.02

0.03

0.04

0.05

0.06

05 1015202530

Basin volume

Memory pattern number

Fig. 2 Basin volume fraction (r = N – 1 = 399): The horizontal axis

represents memory pattern number (1–24). The basin number 25 shows

the volume fraction which corresponds to the initial patterns that

converged into cyclic output states with a period of six steps but not any

one of the memory attractors. The basin number 26 shows the volume

fraction which corresponds to the initial patterns which did not

converged, that means chaotically itinerant. The vertical axisrepresents

the ratio of each sample to the total number of samples. Alternative

hatching and nonhatching are used to show different cyclic attractors

Cogn Neurodyn (2008) 2:39–4841

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Since chaotic dynamics in the network depends on

system parameter—the connectivity r, in order to analyze

the destabilizing process in our model, we have calculated

a bifurcation diagram of overlap, where overlap means

one-dimensional projection of state pattern S(t) to a certain

reference pattern. Therefore, an overlap m(t) is defined by

mðtÞ ¼1

NSð0Þ ? SðtÞð3Þ

where S(0) is an initial pattern(reference pattern) and S(t) is

the state pattern at time step t. Because m(t) is a normalized

inner product, –1 £ m(t) £ 1. For connectivity from 1 to

N – 1, we have respectively calculated the corresponding

overlap m(t) as state pattern S(t) evolves for long time.

Figure 3 shows the overlap m(t) as a function of connec-

tivity. In the case of large enough connectivity r, state

pattern S(t) at each K time step is same with S(0). With the

decrease of connectivity r, state pattern S(t) at each K time

step gradually becomes different to S(0), that is, cyclic

memory attractor becomes unstable. Finally, non-period

dynamics occurs, that is, cyclic memory attractors ruin.

In our previous papers, we confirmed that the non-period

dynamics in the network is chaotic wandering. In order to

investigate the dynamical structure, we calculated basin

visiting measures and it suggests that the trajectory can

pass the whole N-dimensional state space, that is, cyclic

memory attractors ruin due to a quite small connectivity

(Nara and Davis 1992, 1997; Nara et al. 1993, 1995; Nara

2003; Suemitsu and Nara 2003).

Motion control and memory patterns

Biological data show that the number of neurons in the

brain varies dramatically from species to species and the

human brain has about 100 billion (1011) neurons, but one

human has only over 600 muscles that function to produce

force and cause motion. These motions are controlled by

the neuron system. That is, the motions of relatively few

muscles are controlled by the activities of enormous neu-

rons. Therefore, the neural network consisting of N neurons

is used to realize two-dimensional motion control of an

object.

We confirmed that chaotic dynamics introduced in the

network does not so sensitively depend on the size of the

neuron number (Nara 2003). However, if N is too small,

chaotic dynamics can not occur; whereas if N is oversized,

it results in excessive computing time. Therefore, the

number of neurons is N = 400 in our actual computer

simulation. At a certain time, the state pattern in the net-

work is represented by 400-dimensional state vectors,

while the motion in two-dimensional space is only two-

dimensional vectors. Suppose that a moving object moves

from the position (px(t),py(t)) to (px(t + 1), py(t + 1)) with a

set of motion increments (Dfx(t), Dfy(t)). The state pattern

S(t) at time t is a 400-dimensional vector, so we must

transform it to two-dimensional motion increments by

coding. The coding relations are implemented by replacing

motion increments with a group of motion functions

(fx(S(t)), fy(S(t))). In 2-dimensional space, the actual

motion of the object is given by

pxðt þ 1Þ ¼ pxðtÞ þ fxðSðtÞÞð4Þ

pyðt þ 1Þ ¼ pyðtÞ þ fyðSðtÞÞ

where fx(S(t)), fy(S(t)) are the x-axis increment and y-axis

increment respectively, and they are calculated from firing

states of the neural network model and defined by

ð5Þ

fxðSðtÞÞ ¼4

NA ? CfyðSðtÞÞ ¼4

NB ? D

ð6Þ

where A, B, C, D are four independent N/4 dimensional

sub-space vectors of state pattern S(t). Therefore, after the

inner product between two independent sub-space vectors

is normalized by 4/N, motion functions range from –1 to

+1, that is,

?1 ? fxðSðtÞÞ?þ1

ð7Þ

?1 ? fyðSðtÞÞ?þ1

Referring to Eq. (4) and (5) and the definition of motion

functions, in our actual simulations, two-dimensional space

is digitized with a resolution 8/N = 0.02 due to the binary

neuron state ±1 and N = 400.

Next, let us consider the construction of memory

attractors corresponding to prototypical simple motions. It

is considerable that two-dimensional motion consists of

several prototypical simple motions. We take four types of

ð8Þ

-1

-0.5

0

0.5

1

0 50 100 150 200 250

connectivity r

300 350 400

m(t)

Fig. 3 Bifurcation diagram with respect connectivity r: The hori-

zontal axis represents connectivity r (0–399). The vertical axis

represents the long-time behaviours of overlap m(t) at K-step

mappings

42Cogn Neurodyn (2008) 2:39–48

123

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motion that one object moves toward (+ 1, + 1), (–1, + 1),

(–1, –1), (+ 1, –1) in two-dimensional space, as prototyp-

ical simple motions. Under these situations, four groups of

attractor patterns, which are corresponding to the proto-

typical simple motions of the object in two-dimensional

space, are embedded in the neural network by means of

embedding of associative memory introduced in the pre-

vious section. Each group of attractor patterns includes six

patterns that are corresponding to one prototypical simple

motion. Each group is a cyclic memory, or a limit cycle

attractor in 400-dimensional state space. We take nk

(l = 1, 2, 3, 4 and k = 1, 2, …, 6) as the attractor pattern

that is k pattern in l group (see Fig. 4). Therefore, in our

actual simulation, L = 4, K = 6. In the present simulation,

we directly employed K = 6 because it is an optimized

selection after one of the authors and his collaborators had

done a number of simulations for various K (3, 5, 6, 10, for

instance). All of the results show that , if K is too small, it is

difficult to avoid spurious attractors, on the other hand,

quite large K can not also give stronger attraction. The

corresponding relations between attractor patterns and

prototypical simple motions are shown as follows.

l

ðfxðnk

ðfxðnk

ðfxðnk

ðfxðnk

When connectivity r is sufficiently large, one random

initial pattern converges into one of four limit cycle

attractors as time evolves. The corresponding motion of the

object in 2-dimensional space becomes monotonic, and a

simulation example is shown in Fig. 5. On the other hand,

when connectivity r is quite small, chaotic dynamics is

observed in the network with the development of time. At

the same time, the corresponding motion of the object is

chaotic, and Fig. 6 shows an simulation example of chaotic

1Þ; fyðnk

2Þ; fyðnk

3Þ; fyðnk

4Þ; fyðnk

1ÞÞ ¼ ðþ1;þ1Þ

2ÞÞ ¼ ð?1;þ1Þ

3ÞÞ ¼ ð?1;?1Þ

4ÞÞ ¼ ðþ1;?1Þ

motion in two-dimensional space. Therefore, when the

network evolves, monotonic motion and chaotic motion

can be switched by switching the connectivity r.

Algorithm of tracking a moving target

Now we want to discuss how to realize motion control so

as to track a moving target. In our study, we suppose that

an object is tracking a target that is moving along a certain

trajectory in two-dimensional space, and the object can

obtain the rough directional information D1(t) of the

moving target. At a certain time t, the present position of

the object is assumed at the point (px(t), py(t)). This point is

taken as the origin point and two-dimensional space can be

divided into four quadrants. If the target is moving in the

first quadrant, D1(t) = 1. Therefore, if the target is moving

in the nth quadrant, D1(t) = n (n = 1,2,3,4), which is called

global target direction for it is a rough directional

information.

Fig. 4 Memory attractor patterns: Pattern (1–24) includes l = 4

groups of cyclic memory consisting k = 6 patterns. Each cyclic

memory corresponds to a prototypical simple motion

-1

0

1

2

3

4

5

6

-10123456

START

r=399

Fig. 5 An example of monotonic motion: When associative network

state (r = 399) occurs in the state space, the object performs

monotonic motion (+ 1, + 1) after some updating steps, from start

point (0, 0) in two-dimensional space

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2.5-2 -1.5-1 -0.50 0.51

START

r=30

Fig. 6 An example of chaotic motion: When chaotic network state

(r = 40) occurs in the network, correspondingly, the object moves

chaotically from start point (0,0) in two-dimensional space

Cogn Neurodyn (2008) 2:39–4843

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Next, we also suppose that the object can also know

another directional information D2(t), which means which

quadrant the moving object has moved toward from time

t – 1 to t, that is, in the previous step. The direction D2(t) is

called global motion direction, and defined as

D2ðtÞ ¼

1

2

3

4

ðcxðtÞ ¼ þ1 and cyðtÞ ¼ þ1Þ

ðcxðtÞ ¼ ?1 and cyðtÞ ¼ þ1Þ

ðcxðtÞ ¼ ?1 and cyðtÞ ¼ ?1Þ

ðcxðtÞ ¼ þ1 and cyðtÞ ¼ ?1Þ

8

>

>

>

>

:

<

ð9Þ

where cx(t) and cy(t) are given as

cxðtÞ ¼pxðtÞ ? pxðt ? 1Þ

jpxðtÞ ? pxðt ? 1Þj

ð10Þ

cyðtÞ ¼pyðtÞ ? pyðt ? 1Þ

jpyðtÞ ? pyðt ? 1Þj

Now we know that global target direction D1(t) and

global motion direction D2(t) are time-dependent variables.

If the network can get feedback signals from these two

directions in real time, the connectivity r also becomes a

time-dependent variable r(t) and is determined by global

target direction D1(t) and global motion direction D2(t).

Therefore, a simple control algorithm of tracking a moving

target is proposed and shown in Fig. 7, where RLis a

sufficiently large connectivity and RS is a quite small

connectivity that can lead to chaotic dynamics in the neural

network. Adaptive switching of connectivity is the core

idea of the algorithm. If global motion direction and global

target direction is coincident, that is, D2(t) = D1(t), the

network is updated with sufficiently large connectivity

r(t) = RL; otherwise, if global motion direction and global

target direction is not coincident, the network is updated

with quite small connectivity r(t) = RS. When the synaptic

connectivity

r(t) isdetermined

directions, D1(t – 1) and D2(t – 1), the motion increments

of the object are calculated from the state pattern of the

network updated with r(t). The new motion causes the next

D1(t) and

D2(t), andproduces

ð11Þ

by comparing two

thenextsynaptic

connectivity r(t + 1). By repeating this process, the

synaptic connectivity r(t) is adaptively switching between

RL and RS, the object is alternatively implementing

monotonic motion and chaotic motion in two-dimensional

space.

In closing this section, one point we must mention is

that we have to use an engineering approach to switch

the parameter r in computer simulation experiments

because we started from heuristic approach using a sim-

ple model to apply chaotic dynamics to complex

problems which includes ill-posed property. However, as

the future scope, we will develop it to investigate bio-

logical mechanism of advanced functions or controls in

real biological systems.

Experimental results

Generally speaking, it is difficult to give mathematical

proof that our method always produces correct solutions in

tracking of arbitrarily moving target. Necessarily, we must

rely on computer experiments and showing typical prop-

erties connected to universal effectiveness of chaos in

biological systems. Therefore, at the start point, some

simple trajectories along which the target moves should be

set. In future, more complex orbits of moving target will be

investigated. In order to simplify our investigation, we

have taken nine kinds of trajectories that include one cir-

cular trajectory and eight linear trajectories, shown in

Fig. 8.

Suppose that the initial position of the object is the

origin (0,0) of two-dimensional space. The distance d

between initial position of the object and that of the target

is a constant value. Therefore, at the beginning of tracking,

the object is at the circular center of the circular trajectory

and the other eight linear trajectories are tangential to the

circular trajectory along a certain angle a, where the angle

is defined by the x axis. The tangential angle a = np/

4(n = 1,2,…,8), so we number the eight linear trajectories

as LTn.

Fig. 7 Control algorithm of tracking a moving target: By judging

whether global target direction D1(t) coincides with global motion

direction D2(t) or not, adaptive switching of connectivity r between

RSand RLresults in chaotic dynamics or attractor’s dynamics in state

space. Correspondingly, the object is adaptively tracking a moving

target in two-dimensional space

Fig. 8 Trajectories of moving target: one is circular and eight are

linear(LT1– LT8)

44 Cogn Neurodyn (2008) 2:39–48

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Next, let us consider the velocity of the target. In

computer simulation, the object moves one step per dis-

crete time step, at the same time, the target also moves one

step with a certain step length SL that represents the

velocity of the target. The motion increments of the object

ranges from –1 to 1 (see Eq. (7) and (8)), so the step length

SL is taken with an interval 0.01 from 0.01 to 1 up to 100

different velocities. Because velocity is a relative quantity,

so SL = 0.01 is a slower target velocity and SL = 1 is a

faster target velocity relative to the object.

Now, let us look at a simulation of tracking a moving

target using the algorithm proposed above, shown in

Fig. 9. When an target is moving along a circular tra-

jectory at a certain velocity, the object captured the target

at a certain point of the circular trajectory, which is a

successful capture to a circular trajectory. Another simu-

lation of tracking a target that moves along a linear

trajectory is shown in Fig. 10, which is a successful

capture to a linear trajectory.

Performance evaluation

To show the performance of tracking a moving target, we

have evaluated the success rate of tracking a moving target

that moves along one of nine trajectories. However, even

though tracking a same target trajectory, the performance of

trackingdepends notonlyonsynapticconnectivityr, but also

on target velocity or target step length SL. Therefore, when

we evaluate the success rate of tracking, a pair of parameters,

that is, one of connectivity r (1 £ r £ 60) and one of target

velocity SL (0.01 £ T £ 1.0), is taken. Because we take 100

different target velocity with a same interval 0.01, we have

C100

60 pairs of parameters. We have evaluated the success rate

of tracking a circle trajectory, shown as Fig. 11. From the

simulation results, we can know that the success rate of

tracking a circle trajectory with chaotic dynamics is signifi-

cantly high, and that, the success rate highly depends on

synaptic connectivity r and the velocity of the target.

In order to observe the performance clearly, we have

taken the data of certain connectivities, and plot them in

two-dimensional coordinates, shown as Fig. 12. Compar-

ing these figures, we can see a novel performance, when

the target velocity becomes faster, the success rate has a

upward tendency, such as r = 51. In other words, when the

chaotic dynamics is not too strong, it seems useful to

tracking a faster target.

Certainly, the performance of tracking a moving tar-

get also depends on the target trajectory. In this paper,

because linear target trajectories is too much, we only show

two of them in Fig. 13. From the results of computer

experiment, we know the following two points. First, the

success rate decreases rapidly as the target velocity

increases. Second, comparing these success rate of the

linear trajectories with that of the circular trajectory, we are

sure that tracking a moving target of circular trajectory has

better performance than that of linear trajectory. However,

to some linear trajectories, quite excellent performance was

observed, such as Fig. 13b.

-20

-15

-10

-5

0

5

10

15

20

-20 -15-10-505 10 15 20

Object

Target

Capture

Fig. 9 An example of tracking a target that is moving along a

circular trajectory with the simple algorithm. The object captured the

moving target at the intersection point

-20

-15

-10

-5

0

5

10

15

20

-20 -15-10 -505 10 15 20

Object

Target

Capture

Fig. 10 An example of tracking a target that is moving along a linear

trajectory with the simple algorithm. The object captured the moving

target at the intersection point

0

10

20

Connectivity

30

40

50

60

0

20

40

60

80

100

0

0.2

0.4

0.6

0.8

1

Success Rate

Target Velocity

x10-2

Fig. 11 Success rate of tracking a moving target along circle

trajectory: Over 100 random initial patterns, the rate of successfully

capturing the moving target within 600 steps is estimated as the

success rate. The positive orientation obeys the right-hand rule. The

vertical axis represents success rate, and two axes in the horizontal

plane represents connectivity r and target velocity SL, respectively

Cogn Neurodyn (2008) 2:39–4845

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Discussion

In order to show the relations between the above cases and

chaotic dynamics, from dynamical viewpoint, we have

investigated dynamical structure of chaotic dynamics. For

small connectivities from 1 to 60, the network takes chaotic

wandering. During this wandering, we have taken a sta-

tistics of continuously staying time in a certain basin

(Suemitsu and Nara 2004) and evaluated the distribution

p(l,l) which is defined by

pðl;lÞ ¼fthe number of l jSðtÞ 2 blin s?t?s þ l

and Sðs ? 1Þ 62 bl

and Sðs þ l þ 1Þ 62 bl;ljl 2 ½1;L?g

ð12Þ

bl¼

X

X

K

k¼1

Bk

l

ð13Þ

T ¼

l

lpðl;lÞð14Þ

where l is the length of continuously staying time steps in

each attractor basin, and p(l,l) represents a distribution of

continuously staying l steps in attractor basin L = l

within T steps. In our actual simulation, T = 105. To

different connectivity r = 15 and r = 50, the distribution

p(l,l) are shown in Fig. 14a and b. In these figures, dif-

ferent basins are marked with different colors and

symbols. From the results, we know, with increase of the

connectivity, continuously staying time l becomes longer

and longer.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Success rate

Target Velocity ( x0.01)

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Success rate

Target Velocity ( x0.01)

(a)

r

16: downward tendency

(b)

r

51: upward tendency

Fig. 12 Success rates drawn

from Fig. 11. We take the data

of a certain connectivity and

show them in two dimension

diagram. The horizontal axis

represents target velocity from

0.01 to 1.0, and the vertical axis

represents success rate. With the

increase of target velocity,

(a) r = 16: downward tendency;

(b) r = 51: upward tendency

0

10 20 30 40 50 60

Connectivity

0 20

40

60

80

100

0

0.2

0.4

0.6

0.8

1

Success Rate

Target Velocity

x10-2

0

10 20 30 40 50 60

Connectivity

moving target along linear trajectory LT6

0 20

40

60

80

100

0

0.2

0.4

0.6

0.8

1

Success Rate

Target Velocity

x10-2

(a)

moving target along linear trajectory LT2

(b)

AA

Fig. 13 Success rates of

tracking a moving target along

different linear trajectories.

(a) A moving target along linear

trajectory LT2; (b) A moving

target along linear trajectory

LT6

Fig. 14 The log plot of the frequency distribution of continuously

staying time l: The horizontal axis represents continuously staying

time steps l in a certain basin l during long time chaotic wandering,

and the vertical axis represents the accumulative number p(l,l) of the

same staying time steps l in a certain basin l. continuously staying

time steps l becomes long with the increase of connectivity r. (a)

r = 15: l is shorter; (b) r = 50: l is longer

46Cogn Neurodyn (2008) 2:39–48

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Referring to those novel performances talked in previ-

ous section, let us try to consider the reason. First, in the

case of slower target velocity, a decreasing success rate

with the increase of connectivity r is observed from both

circular target trajectory and linear ones. This point shows

that chaotic dynamics localized in a certain basin for too

much time is not better to track a slower target.

Second, in the case of faster target velocity, it seems

useful to track a faster target when chaotic dynamics is not

too strong. Computer experiments show that, when the

target moves quickly, the action of the object is always

chaotic so as to track the target. In past experiments, we

know that motion increments of chaotic motion is very

short. Therefore, shorter motion increments and faster

target velocity result in not good tracking performance.

However, when continuously staying time l in a certain

basin becomes longer, the object can move toward a certain

direction for l steps. This is useful to track the faster target

for the object. Therefore, when connectivity becomes a

little large (r = 50 or so), success rate arises following the

increase of target velocity, such as the case shown in

Fig. 12.

Third, we try to explain the reason why success rate of

tracking a moving target along a linear trajectory decreases

rapidly when the target velocity increases a little. In this

case, faster target velocity results in more chaotic motions

of the object. At the same time, the target has moved too

far away from the object along a linear trajectory. There-

fore, the success rates become worse. Generally speaking,

chaotic dynamics is not always useful to solve an ill-posed

problem. However, better performance can be often

observed using chaotic dynamics to solve an ill-posed

problem. As an issue for future study, a functional aspect of

chaotic dynamics still has context dependence.

Finally, let us consider the approach in robot navigation.

There are many approaches in robot navigation. As an

approach using dynamical neural network, a simple

mechanism—dynamical neural Smitt trigger was applied to

a small neural network controlling the behaviour of a

autonomous miniatur robot (Hu ¨lse and Pasemann 2002).

On the other hand, our model is a recurrent neural network

with N neurons, that is, a large neural network. Recent

works about brain-machine interface and the parietal lobe

suggested that, in cortical area, the ‘‘message’’ defining a

given hand movement is widely disseminated (Wessberg

et al. 2000; Nicolelis 2001). Therefore, the difference

between our novel approach and the Smitt trigger approach

in robot navigation are quite big. Our approach emphasizes

the whole state of neurons, but the Smitt trigger pays

attention to the interaction between a few of neurons.

Furthermore, our approach has a huge reservoir of redun-

dancy and results in great robustness. Generally speaking,

methods in robot navigation often fall into enormous

computing complexity. However, our approach proposed a

simple adaptive control algorithm.

Summary

We proposed a simple method to tracking a moving target

using chaotic dynamics in a recurrent neural network

model. Although chaotic dynamics could not always solve

all complex problems with better performance, better

results often were often observed on using chaotic

dynamics to solve certain ill-posed problems, such as

tracking a moving target and solving mazes (Suemitsu and

Nara 2004). From results of the computer simulation, we

can state the following several points.

•

A simple method to tracking a moving target was

proposed

Chaotic dynamics is quite efficient to track a target that

is moving along a circular trajectory.

Performance of tracking a moving target of a linear

trajectory is not better than that of a circular trajectory,

however, to some linear trajectories, excellent perfor-

mance was observed.

The length of continuously staying time steps becomes

long with the increase of synaptic connectivity r that

can lead chaotic dynamics in the network.

Continuously longer staying time in a certain basin

seems useful to track a faster target.

•

•

•

•

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