2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

Hopfield Neural Network with Glial Network
Chihiro Ikuta† , Yoko Uwate† , Yoshifumi Nishio† and Guoan Yang‡†Tokushima University
2–1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: {ikuta, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract‡

Xi’an Jiaotong University
No.28 Xianning West Road,
Xi’an City, Shaanxi Province, China
Email: gayang@mail.xjtu.edu.cn

minimum. We often add a noise to the Hopfield NN for escaping from the local minimum [7][8]. We added the glial
network to the Hopfield NN. In the Hopfield NN with glial
network, the glias are arranged in a grid-like formation. Theglias are connected with the neurons one by one. When theglia is excited, the glia generates the pulse. After that, thepulse propagates to neighboring glias. We show that the performance of the proposed Hopfield NN by solving the TSP.
Moreover, we confirm that the proposed Hopfield NN has abetter performance than the conventional Hopfield NN.

A glia is a nervous cell existing in a brain. This cell has important functions for the higher brain function. We have proposed a glial network for an artificial network from functionsof the biological glia. In this study, we propose a Hopfield
Neural Network (Hopfield NN) with glial network. In theHopfield NN with glial network, we connect the glias with theneurons one by one. The glia generates a pulse when the glia
is excited by the output of the connecting neuron. This pulse
aﬀects neuron’s threshold and neighboring glias. By solving
Traveling Salesman Problem, we confirm that the proposed
2. Hopfield NN with Glial Network
Hopfield NN has a better performance than the conventional
Hopfield NN.
The Hopfield NN was proposed by J.J. Hopfield. Thisnetwork has complete and symmetric connections which isshown in Fig. 1. (a) is an example of connections of Hopfield
1. Introduction
NN. In this study, neurons are arranged in the grid-like forNeural networks have been investigated by many re- mation (b). If we give any constraint conditions to the Hopsearchers, because the neuron composes the brain. The glia field NN, the Hopfield NN can decrease the energy. Thus, theis one of nervous cells existing in the brain. For a long time, Hopfield NN is applied to solve the optimization problem.
functions of this cell had been regarded as the support cell of However, the Hopfield NN has the local minimum problem.
the neuron. However, some researchers found its new func- Because this network uses the steepest descent method. Thistions [1]. It can transmit signals by the ions concentration method can decrease the energy, however, the solution is ofwave and aﬀect neurons [2][3]. Currently, this cell is consid- ten trapped into a local minimum.
ered to important cell for the brain functions. In the previous study, we proposed some artificial network models of theglia [4][5]. We added the glia network models to Multi-Layer
Perceptron (MLP). In this model, the glias generate pulses
when the glias are excited by a stimulus of connecting neurons. The pulse propagates to the connecting neurons and the(a)(b)
neighboring glias. We showed that the learning performanceof the MLP and some abilities are improved by the glia netFigure 1: The Hopfield NN.
work model. The glial network is one dimension structure.
In this study, we extend the glial network to two dimension 2.1. Hopfield NN for Solving TSPstructure. We propose the Hopfield Neural Network (HopThe TSP is a famous optimization problem. This task has
field NN) with glial network. The conventional Hopfield NN
is proposed by J.J. Hopfield in 1982 [6]. An energy function N × N cities in the two dimensional structure. We should visit
is decreased by using the steepest descent method. It can be all cities, then, we can not visit same city. We want to findapplied to an associative memory, Travel Salesman Problem the shortest tour. The Hopfield NN is known to be able to(TSP), and so on. The Hopfield NN is trapped into a local solve the TSP. We can obtain the weight parameters from the- 369 -constraint conditions. If we solve TSP by using the Hopfield where u is an internal state of neuron, and x is an output orNN, the energy function is obtained by Eq. (1).
input of neurons. w is decided by Eq. (2). Thus, the w is notchanged during a simulation. When we give the initial valueNN}A ∑{∑to the neurons, the neurons update by themselves.
2E=(xi j (t) − 1)
(1)
The proposed neuron updating rule is defined by Eq. (5)2 ij}
B ∑{∑(xi j (t) − 1)22 ji
N++ui, j (t + 1) =N N N{}D ∑∑∑di j xik (t) xi,k+1 (t) + x j,k−1 (t) ,2 i j k=−Aδim (1 − δ jn ) − Bδ j n(1 − δim )−Ddim (δn, j+1 + δn, j−1 ),wi j,mn xmn (t) + hi j + ϕ(t),(5)

m=1,n=1where E is an energy function, x is an output of the neuron, dis distance between the city and the city, and A, B and D areconstant number. We can obtain the weight of connections
for solving the TSP from the energy function. The weight ofconnections is calculated by Eq. (2).
wi jmnN∑Nwhere phi is a glial pulse. In Eq. (5), we give the glias’ pulses
ϕ to the neurons’ thresholds. The glias support the Hopfield
NN for escaping from the local minimum. The sigmoid isused for an activating function. It is described by Eq. (6).
xi j (t + 1) =1
1 + e−ui j (t+1).

(6)

2.3. Glial Network(2)

The glia has relationships with neighboring glias and theconnecting neurons. If the neuron has a large output, theconnected glia is excited. We choose some neurons whichhi j = A + B,
(3)have large output in the network. The number of neurons towhere w is the weight of connections of neurons, δ is Kro- be chosen is K. In one simulation, K is not changed. Thenecker s delta and h is a threshold of the neuron.
glia has a period of inactivity (T ) when the glia generate thepulse.. During this period, the glia can not excite. When theglia is excited by the connecting neuron, the glia generate a2.2. Neuron Updating Rulepulse. This pulse propagates to the threshold of the connectIn this study, we propose the Hopfield NN with glial net- ing neurons and the neighboring glias. The glial network has
work shown in Fig. 2. The glial network is inspired from the a propagation range of the pulse (R). If R is 2, the glial eﬀectfeature of the biological glia. The glia is the nervous cell ex- influence the states of the glias in two neighborhoods. We
isting in the brain. This cell transmits signals to the glias and define the output of the glia in Eq. (7).
the neurons by ions concentration. We consider that the glia
aﬀect good influence to the artificial neural network. In theϕ(t + 1) = βϕ(t),(7)proposed method, the glias are connected with the neurons
one by one. The excited glias generate the pulse and aﬀect the where β is an attenuated parameter. We use β between 0 andconnecting neurons’ thresholds. Moreover, the pulse propa- 1. Figure 3 is an example of the generation pulses. We use
gates to the neighboring glias. We consider that the pulses the 16 × 16 glias, K = 8 and R = 2. First, the excitation glias
give the energy to the Hopfield NN, and that the Hopfield NN generate the pulses at some positions. Second, the neighboring glias of first excitation glias are excited. At t = 3, we canescapes out from the local minimum.
Glial Network
see that the pulses are generated like square.
1610.914161614141212

10

108

8

660.8
120.7

100.6
0.5

80.4
60.3

40.2
0.1

22468101214Figure 2: The Hopfield NN with glial network.
The neurons connect with all other neurons in the Hopfield
NN. The neuron updating rule is described by Eq. (4).
wi j,mn xmn (t) + hi j ,224(4)m=1,n=1- 370 -

68

10

12

14

16

2

16

16

14

14

121212

10

10

10888

6664

4224681012

14

1668101214(c) t=3.

1424(b) t=2.

16(d) t=4.
N∑01642(a) t=1.

Hopfield Networkui, j (t + 1) =44224681012(e) t=5.

1416246810(f) t=6.

Figure 3: Pulse propagation.

12141616

3. Simulation Results
In this section, we show the simulation results of the Hopfield NNs. We use the three diﬀerent TSPs (ulysses16, att48and gr96). We compare three Hopfield NNs which are theconventional Hopfield NN, the Hopfield NN with random
noise and the Hopfield NN with glial network. The Hopfield NN with random noise is given uniformed random noiseto every neuron’s threshold. We use four kinds of measures
to be obtained by 100 trials for the evaluation of the performance. They are an average distance (Ave.), a minimum distance (Min.), a maximum distance (Max.) and a standard deviation (Std. Dev.). We show a ratio of the results of the proposed Hopfield NN to the results of the conventional Hopfield
NN.

125

0.9
0.8

20
0.7

15

0.6

100.5

50.4
0.3

00.2

−5−1032

0.1

34

36

38

40

42

000.2

(a) Optimum tour.

3.1. Ulysees16
First, we show the results of ulysses16 in Table 1. TheHopfield NN with random noise is the best of all in the average of distance. However, the Hopfield NN with glial network and the Hopfield NN with random noise obtain the optimal solution. The conventional Hopfield NN is the worst ofall. We consider that the conventional Hopfield NN is trapped
into the local minimum. From this result, the Hopfield NN
with glial network can escape from the local minimum. Figure 4 shows that we obtained each local optimum solution
from three Hopfield NNs.
Table 1: Statistic results as ulysses16.
Ave.
Min.
Max.
Std. Dev.
Random 0.7505 0.8052 0.7859
0.4648Glia
0.7805 0.8052 0.8131
0.5425

30

0.6

0.8

10.8

10.8

10.8

1(b) Conventional.

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1
00

0.4

0.1
0.2

0.4

0.6

0.8

1

000.2

(c) Random.

0.4

0.6

(d) Glial network.

Figure 4: Ulysses16.

6000

10.9

5000
0.8

0.7

4000

0.6

3000

0.5

0.4

3.2. Att48

2000
0.3Next, we show the results of att48. Table 2 is the statistic
results. In this table, the tendency in the measures is similar
to ulysses16. We consider that the Hopfield NN with glial
network has high ability of searching local optimal solution.
In this case, the Hopfield NN with glial network obtains thebest tours of all. However, all Hopfield NN cannot obtain
the optimal solution. In this case, the Hopfield NN with glial
network obtains the best tours of all. Figure 5 shows that
we obtained each local optimum solution from three Hopfield
NNs.

0.2

1000

0.10002000

4000

6000

8000

0.2

(a) Optimum tour.
1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.6

0.1

000

0.2

0.4

0.6

0.8

100.2

(c) Random.

0.4

0.6

(d) Glial network.

Figure 5: Att48.

- 371 -

0.4(b) Conventional.

1

0.1

Table 2: Statistic results as att48.
Ave.
Min.
Max.
Std. Dev.
Random 0.8388 0.9036 0.9162
0.7259Glia0.9219 0.8883 1.1555
3.280003.3. Gr96
Finally, we show the results of solving gr96. Table 3 isthe statistic results. In the average of distance , the Hopfeild
NN with random noise is better than the proposed Hopfield
NN. We consider that the Hopfield NN with random noise
can search local optimum solution over wider range than theproposed Hopfield NN. Because in the Hopfield NN with random noise, the random noise is inputted to every neuron. In
the proposed Hopfield NN, the glial eﬀects are locally. Thus,
the proposed Hopfield NN cannot decrease the network energy than the Hopfield NN with random noise. However, theminimum distance of the Hopfield NN with glial network is
the shortest of all. From this result, we consider that the proposed Hopfield NN has the high local search performance.
Figure 6 shows that we obtained each local optimum solution
from three Hopfield NNs.
Table 3: Statistic results as gr96
Ave.
Min.
Max.
Std. Dev.
Random 0.9428 0.9842 0.9038
1.0821Glia
0.9767 0.9508 1.0192
0.9105

60

0.8

0.7

0.6

[4] C. Ikuta, Y. Uwate, and Y. Nishio, “Chaos Glial Network
Connected to Multi-Layer Perceptron for Solving TwoSpiral Problem,” Proc. ISCAS’10, pp. 1369-1371, May
2010.

0.5

00.4

0.3

−20

0.2

0.1

−40−40[1] P.G. Haydon, “Glia: Listening and Talking to theSynapse,” Nature Reviews Neuroscience, vol. 2, pp. 844847, 2001.

[3] S. Ozawa, “Role of Glutamate Transporters in Excitatory
Synapses in Cerebellar Purkinje Cells,” Brain and Nerve,
vol. 59, pp. 669-676, 2007.

120References[2] S. Koizumi, M. Tsuda, Y. Shigemoto-Nogami and K. Inoue, “Dynamic Inhibition of Excitatory Synaptic Transmission by Astrocyte-Derived ATP in Hippocampal Cultures,” Proc. National Academy of Science of U.S.A,
vol. 100, pp. 11023-11028, 2003.

0.940

old and neighboring glias. The glial network give the energy
to the Hopfield NN for escaping from local minimum. Fromsolving the TSP, we confirm that the proposed Hopfield NN isbetter performance than the conventional Hopfield NN. In all
simulations, the Hopfiled NN with glial network has the best
tours. However, the average of distance of proposed Hopfield NN is worse than the Hopfield NN with random noise.
From these results, the proposed Hopfield NN has high local
searching ability. We consider that the glial network gives thelocal energy to the Hopfield NN. In the future works, we will
confirm that the glial eﬀect is how to influence the Hopfield
NN. Moreover, we will find the eﬀective parameters of theglial network for solving the TSP.

0−20020400(a) Optimum tour.

0.2

0.4

0.6

0.8

1

[5] C. Ikuta, Y. Uwate, and Y. Nishio, “Performance and Features of Multi-Layer Perceptron with Impulse Glial Network,” Proc. IJCNN’11, pp. 2536-2541, Jun. 2011.

(b) Conventional.

110.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

[6] J.J. Hopfield, “Neural networks and physical systems
with emergent collective computational abilities”, Proc.
of the National Academy of Sciences of the USA, vol.79
no. 8, pp. 2554-2558, Apr. 1982.

0.1

0000.2

0.4

0.6

0.8

10(c) Random.

0.2

0.4

0.6(d) Glial network.

Figure 6: Gr96.

0.8

1[7] Y. Hayakawa, A. Marumoto, and Y. Sawada, “Eﬀects ofthe chaotic noise on the performance of a neural networkmodel for optimization problems,” Phys. Rev. E, vol. 51,no. 4, pp. 2693-2696, Apr. 1995.

[8] Y. Uwate, Y. Nishio, T. Ueta, T. Kawabe, and T. Ikeguchi,“Performance of Chaos and Burst Noise Injected to theHopfield NN for Quadratic Assignment Problems,” IEICE Transactions on Fundamentals, vol. E87-A, no. 4,
In this study, we have proposed the Hopfield NN with glial
pp. 937-943, Apr. 2004.
network which is inspired from the functions of the biologicalglia. We added the glial network to the Hopfield NN. The excited glia generates a pulse, this pulse aﬀect neuron’s thresh4. Conclusion- 372 -