平成23年度 電気関係学会四国支部連合大会

181-18

Effect of Real Biological Chaos
on Solving Ability of Neural Network
Kana Kurata

Yoko Uwate
Yoshifumi Nishio(Tokushima University)

1. Introduction

4. Simulation results

In the optimization problems, many algorithms pouring chaotic oscillations to the Neural Networks (NN) havebeen proposed in order to avoid the local minimum problems [1]. We have also investigated various methods to exploit chaotic features to enhance the ability of the neuralnetworks. However, in the past studies, only mathematical abstract models, e.g. the logistic map and the cubicmap, are considered as chaotic source.
In this study, in order to investigate the effect of chaotic
oscillations of real biological signals, we use chaotic pulse
waves obtained by real biological experiments and pourthe sampled chaotic data to the Hopfield NN solving Traveling Salesman Problems (TSP). By computer simulations, we investigate the effect of the chaotic pulse wave.
We also compare the results with the case of random noise.

We carry out computer simulations for 10 times of 100
iterations and recorded the minimum and the average values of the tour length. The simulated results of 4 subjects(A, B, C, D) are summarized in Figs. 2(a) and (b). We
compared the results with the result of random noise case.

2. Hopfield NN solving TSP
In this study, we choose 2 problems “bayg29” and“att48” from TSPLIB [2].
For solving N -element TSP by Hopfield NN, N × N
neurons are required. The state of N × N neurons are
asynchronously updated due to the following difference
equation:0NXxi (t + 1) = f @!
ωim;jn xjm (t) − θim (t) + βzim (t)(1)23500

135000
Sum Length of TSP

Sum Length of TSP

22500

21500

3. Chaotic pulse wave
We obtained pulse waves from real biological experiments. The sampling frequency of the data is 200Hz andmeasurement time is 5 minutes. The time waveform ofthe pulse wave and the corresponding reconstructed attractor are shown in Figs. 1 (a) and (b), respectively. We
calculated the averaged Lyapunov exponent as 4.5032 andhence we can say this pulse wave is chaotic.
2700

Relative voltage

2500
2300
2100
1900
1700
1500
11001

2001

3001

4001

12000019500
1

2

Random3
A(b)

Figure 1: Chaotic pulse wave. (a) Example of pulse waves.
(b) Example of recomstructed attractors of pulse wave.

In order to pour this chaotic pulse wave to the Hopfield
NN, we normalize the wave and use only one of every
30 samples. Further, in this study, we fix the amplitude
parameter β=0.7.

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4
B

C

5

6
Trial times

D115000
1
random

2

(a)

A

3
B

Trial timesCD(b)

Figure 2: Simulated results. (a) Simulated results of bay29.
(b) Simulated results of att48.

The results are summarized in Table 1. From this table,
we can confirm that the chaotic pulse wave gains better
performance than the random noise.

Table 1: Simulated results.
NoiseA
B
C
D

bay29
22025
21068
20853
20767
20920

Err.
143%
132%
130%
129%
131%

att48
132448126859126624
127897125090

Err.
295%
278%
278%
282%
273%

5. Conclusions
We have investigated the effect of the chaotic pulse wave
poured in the Hopfield NN solving TSP. By carrying out
computer simulations for the problem, we have confirmedthat the chaotic pulse wave had an effect to avoid local
minimum problems and achieved a performance to findgood solutions of TSP.
In this study, we could not obtain an enough results toconclude on the relationship between the performance ofthe neural networks and the stress added to the subject.
That is our important future research topic.

5001

Time (s)(a)

125000

20500

j,n=1

where f is sigmoidal function. zim is additional noise,
namely chaotic pulse wave in this study.

130000

References[1] Y. Hayakawa and Y. Sawada, “Effects of chaotic noiseon the performance of a neural network model for optimization problems,” Physical Review E, vol.51, no.4,
pp.2693–2696, 1995.
[2] “TSPLIB,” http://www.iwr.uni-heidelberg.de/groups/
comopt/software/TSPLIB95/