IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.9 SEPTEMBER 2011

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LETTER

Multi-Layer Perceptron with Glial Network for Solving Two-SpiralProblem
Chihiro IKUTA†a) , Nonmember, Yoko UWATE†b) , and Yoshifumi NISHIO†c) , Members

SUMMARY
In this study, we propose a multi-layer perceptron with aglial network which is inspired from the features of glias in the brain. All
glias in the proposed network generate independent oscillations, and theoscillations propagate through the glial network with attenuation. We apply the proposed network to the two-spiral problem. Computer simulations
show that the proposed network gains a better performance than the conventional multi-layer perceptron.
key words: multi-layer perceptron, back propagation, glia, glial network,
two-spiral problem

1.

Introduction

A human’s brain is made by many nervous cells; neurons
and glias. The neurons have been extensively studied bymany researchers, because we can see that the neurons actively work in the brain. They transmit electrical signals
to other neurons in the network. The glias had not been noticed for a long time compared with the neurons, because wecould not observe the activity of the glias clearly. Currently,
we can research the activity of the glias by checking ions in
the brain. Some researchers discovered that the glias havesome ion receptors [1], and that the glias transmit signals toother glias and neurons using by waves of ions; calcium ion(Ca2+ ), glutamic acid (Glu), adenosine triphosphate (ATP),and so on [2], [3]. These ions are considered to be important
for activity of a nerve in the brain. Especially, Ca2+ density
propagates to wide range of the brain and aﬀects neurons’
membrane potentials. We consider that the biological behavior of the glias is applicable to artificial neural networks
to enhance their abilities.
Artificial neural networks are inspired by the featuresof the biological neurons and their networks. Now, various artificial neural networks are proposed for diﬀerent purposes and some of them have been utilized in real engineering systems. The Multi-Layer Perceptron (MLP) is the most
famous feed-forward neural network. The MLP can be applied to solve many diﬀerent kinds of tasks, for example,
a pattern recognition, a pattern classification, a data mining,and so on. In general, the MLP is trained by the Back Propagation (BP) algorithm [4]. In the BP algorithm, we calculate
the error between the teacher value and the output value, afManuscript received February 25, 2011.
Manuscript revised May 24, 2011.
†The authors are with Tokushima University, Tokushima-shi,770-8506 Japan.
a) E-mail: ikuta@ee.tokushima-u.ac.jpb) E-mail: uwate@ee.tokushima-u.ac.jpc) E-mail: nishio@ee.tokushima-u.ac.jp
DOI: 10.1587/transfun.E94.A.1864

ter that, this error is propagated backward in the network tomodify the weights between neurons. However, the BP algorithm often falls into a local minimum, because the BP
uses the steepest decent method. The MLP should escapeout from the local minimum to achieve a good performance.
In order to overcome this problem, some researchers proposed a method giving noise to the MLP [5], [6]. We noticethat the same eﬀect as noise can be realized by the glia’s
feature in the brain. We also consider that the artificial glial
network can make the position dependency of the neurons
in the MLP, which may give a novel feature to the MLP.
In this study, we propose the MLP with the glial network. The artificial glial networks are designed to shake thethreshold values of the neurons by random but correlated oscillation signals. Computer simulation results show that theproposed networks gain a better performance to solve theTwo-Spiral Problem (TSP) than the conventional MLP.
2.

Multi-Layer Perceptron with Glial Network

In this study, we consider the MLP with four layers (oneinput layer, two hidden layers, and one output layer) andwe assume that the glial network is connected to the second
hidden layer only. In the proposed network, each glia isconnected to one neuron and its neighbor glias.
The standard updating rule of the neuron in the MLP isgiven by Eq. (1).
⎛ n⎞⎜⎜⎜⎟⎟⎟yi (t) = f ⎜⎜⎝⎜ ωi j x j (t) − θi (t)⎟⎟⎠⎟ ,(1)j=1where y is the output of the neuron, x is the input of theneuron, ω is the weight between neurons, n is the number ofthe neurons in the prior layer, and f is the output function.
For the neurons in the second layer, we assume that theupdating rule can be given by Eq. (2).
⎛ n⎞⎜⎜⎜⎟⎟⎟(2)yi (t) = f ⎜⎜⎝⎜ ωi j x j (t) − θi (t) + αΨi (t)⎟⎟⎠⎟ ,j=1

where Ψ is the output of the glia and α controls the strengthof the eﬀect from the glia. We assume that the glias influence the thresholds of their corresponding neurons, because
the glias in the brain are reported to aﬀect to neurons’ membrane potentials by Ca2+ densities.
We use the sigmoidal function as the output function f .

c 2011 The Institute of Electronics, Information and Communication Engineers
Copyright LETTER1865

Fig. 1

3.

MLP with glial network.

Fig. 2

MLP with talking glial network.

Structure of Glial Networks

In this study, we propose two kinds of structures of the glialnetworks; the glial network and the talking glial network.
For both networks, we assume that the glias produce independent random oscillations and the oscillations propagate
through the glial network with attenuation. We also assume
that the boundary condition of the glial network is a ringcondition, namely the one end of the network is connectedto the other end of the network.

Fig. 3 Relationship between two diﬀerent glias of (a) MLP with glial
network and (b) MLP with talking glial network.

3.1 Glial Network
In the first model, the output of the glial network is definedas Eq. (3).
Ψi (t) =m

β|k| ψi+k (t − |k|),

(3)

k=−m

where ψ is the random oscillation in the range [−1, 1] generated by each glia, β is the attenuation parameter, and m is
the propagating range of the oscillations.
Figure 1 shows the structure of the MLP with the glial
network. In this model, the MLP does not aﬀect to the glialnetwork.

Fig. 4 Relationship between two diﬀerent glias at diﬀerent time steps of(a) MLP with glial network and (b) MLP with talking glial network.

3.2 Talking Glial Network

4.

In the second model, the output of the glial network is defined as Eq. (4).

Because the random oscillations generated in the glias propagate through the glial network, the output of the glial network has some correlation characteristics.
We investigated the relationship between the diﬀerent
positions of the outputs of the glial networks. The outputs
are recorded during the BP learning with the following network parameters; the numbers of the neurons of the MLP
are 2-20-20-1, the number of the glias is 20, the attenuation
parameter is β = 0.8, and the propagating range is m = 9.
Figure 3 shows the relationship between the outputs ofthe glial network at the positions of the 11th glia and the10th glia. They are not correlated for the both types ofMLPs, because the oscillations propagate through the glial
network with time delay.
In order to take the eﬀect of the propagation delay into
account, we investigated the similar relationship between

Ψi (t) = {yi (t − 1) − 0.5}m

β|k| ψi+k (t − |k|),

(4)

k=−m

where yi is the output of the neuron. Hence, in the talking
glial network, the output of the glial network is controlled
by the output of the MLP. This means that the neurons andthe glias influence stronger and more dynamic in the talking
glial network during the learning process than the case of theabove simpler glial network. We consider that this model isreflected better the relationship between the neurons and theglias in the real brain.
Figure 2 shows the structure of the MLP with the talking glial network.

Output of Glial Networks

IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.9 SEPTEMBER 2011

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Fig. 5 Time series of two diﬀerent glias’ positions of MLP with glialnetwork.

Fig. 7

Two-spiral problems with 130 points.
Table 1

(a)
(b)
(c)
(d)

Average0.03346
0.024100.02035
0.02809

Learning performance.
Minimum
0.000200.00009
0.00008
0.00011

Maximum
0.169700.129970.14149
0.16140

Std. Dev.
0.03776
0.02801
0.025130.03111

Fig. 6 Time series of two diﬀerent glias’ positions of MLP with talking
glial network.

the outputs but at the diﬀerent time steps. Figure 4 shows
the relationship between the outputs of the 11th position at
t+1 and the 10th position at t. In this case, we can clearly observe the correlation of the data. This eﬀect can be observed
from the time series of the outputs of the glial network. Although it is not clearly visible, Figs. 5 and 6 show that thetwo time series take correlated values within ±1 time step.
Further, we confirmed that the distance of the positions
corresponds to the propagation delay in the sense of the correlations. For example, the outputs of the 11th glia at t + 2
and the 9th glia at t are correlated, though the relationshipbetween them is smaller by the eﬀect of the attenuation.

Fig. 8

Example of learning curves of four MLPs.

during each trial. We calculate an average error, a minimum error, a maximum error, and the standard deviation(Std. Dev.). We use the Mean Square Error (MSE) for theerror function given by Eq. (5).
1 (ti − Oi )2 ,N i=1N

5.

Simulation Result

We show the performance of the proposed network by solving the Two-Spiral Problem (TSP). The TSP is a famous
benchmark of classification problems for the artificial neural networks [7], [8]. In this study, we prepare 130 points
belonging to the two spirals as shown in Fig. 7 and train theMLPs by the BP learning to give the correct spiral to whicha given point belongs, when the x and y coordinates of thepoint are inputted. This classification problem is known asa relatively diﬃcult one and usually can not be solved withthe standard three-layer MLP.
We compare the performance of four diﬀerent MLPs;(a) the conventional MLP, (b) the MLP with the glial network, (c) the MLP with the talking glial network, and (d) theMLP with the completely independent random noise. The
number of the trials is 200 and the MLPs learn 100000 times

MSE =(5)where N is the number of the learning points and t is theteacher value.
The simulated results are summarized in Table 1. In thetable, (a), (b), (c), and (d) are the results of the conventional
MLP, the MLP with the glial network, the MLP with talking
glial network, and the MLP with random noise, respectively.
We can see that the MLP with the talking glial network gives
the best learning performance for the average error and theminimum error. The conventional MLP is the worst for all
the measures.
Figure 8 shows an example of the learning curves by
the MLPs. Before around 10000 times, all MLPs’ learning
curves are similar, however, the error curves are changed
after that. The learning speed of the conventional MLP becomes very slow after around MSE = 0.03 at 50000 times

LETTER

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pability to the MLP.
6.

Fig. 9 Classification results of TSP with 130 points. (a) Conventional
MLP. (b) MLP with glial network. (c) MLP with talking glia network.
(d) MLP with random noise.

and the ability escaping out from local minimums seems tobe weak. The MLP with the random noise has better abilityescaping out from local minimums but the learning speed
is slow. The learning speed of the MLP with the glial network is very fast, but in this example this MLP does notlearn further after around 60000 iterations. We should notethat the learning curve of the MLP with the talking glial
network cannot be expressed by a simple word. The curve
has both slowly-learning part (10000-30000) and quicklylearning part (50000-60000), and especially many oscillating behaviors. We consider that this is because the MLP
with the talking glial network has more complex structure
than the others. This complex behavior on the learning curve
seems to work better for learning.
Finally, we show the classification results of the TSP inFig. 9. The generation of clear spirals also indicate the generalization capability of the MLPs. In this simulations, wegive the coordinates (x and y) to the trained MLPs and evaluate the outputs. We use the same initial conditions as Fig. 8for learning of the MLPs in order to obtain similar error tothe average MS E listed in Table 1. We can see that the resultof the MLP with the talking glial network is the best. Because, the spirals obtained by the MLP with the talking glial
network has only one discontinuity. The others have twoor more discontinuities in the spirals. Moreover, the spirals
obtained by the MLP with the talking glial network have almost constant thickness compared with the others and arealso smoother than the others.
From Figs. 8 and 9, the talking glial network gives abetter learning performance and a better generalization ca-Conclusions

In this study, we have proposed the MLP with the glial network which was inspired from the features of glia. First
type was the MLP with the glial network and the second
one is the MLP with the talking glial network. All glias
in the proposed network generated independent oscillations,and the oscillations propagated through the glial networkwith attenuation. We confirmed that the outputs of the glial
network were correlated between the neighboring positions.
We investigated the learning performance of the proposed
networks by solving the TSP with 130 points. Computer
simulated results showed that the proposed networks gained
a better performance than the conventional MLP and theMLP with simple random noise. Further, we confirmed thatthe MLP with the talking glial network was better than theMLP with the simple glial network.
The important novel feature of the proposed networks
is to give the MLPs the position dependency of the neurons, which is usually ignored in artificial neural networks
because of the simplicity. It is true that more detailed investigation is needed in order to claim the eﬀect of such a nonuniform feature on the learning performance. Real brains
must be, of course, non-uniform and hence we believe that akind of individuality will give wider chance to realize more
intelligent artificial neural networks.
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