平成20年度電気関係学会四国支部連合大会

371

17-2

Nonlinear Spring Model of Self-Organizing Map
Arranged in Rectangular Grid and its Chaotic Behaviors
Haruna MATSUSHITA(Tokushima University)

1. Introduction
The Self-Organizing Map (SOM) is a subtype of artiﬁcial neural networks [1] and is a model simplifying
self-organization process of the brain. However, SOM
is still far away from the realization of the brain mechanism. In order to realize more powerful and more
ﬂexible mechanism, it is important to propose newmodels of the brain mechanism and to investigate theirbehaviors.
This study proposes N × M SOM model connectedby the nonlinear spring. The proposed model consistsof N × M neurons located at a rectangular grid.

2. Nonlinear Spring Model of SOM
Arranged in N × M
In this study, we propose the nonlinear spring modelof SOM with a rectangular structure. The proposedmodel consists of neurons located at N × M rectangular grid, and the neurons are connected by the nonlinear spring. The model is shown in Fig. 1.
The input vectors are given to the 4 corners of themodel, and the neuron nearest to the input becomesa winner. Therefore, the Neuron 11, 1M , N 1 or N M ,becomes the winner, and it is attracted to the input.
The other neurons always do not receive the direct
eﬀect from the input vector and are inﬂuenced onlyby the restoring force of the nonlinear spring from theneighboring neurons. The input pattern is that theinput vectors are given to the corner of the model near
the Neuron 11, 1M , N M and N 1 in rotation.

Yoshifumi NISHIO(Tokushima University)
grid and the initial states of the velocities to 0.
First, we consider the 2 × 2 model (N = M = 2).
The projections of attractor onto x12 –vx12 and itsPoincaré map are shown in Fig. 2. We can conﬁrm
that the orbit of attractor looks chaos. Poincaré map
is folded and has the shape like strange attractors.
2
500−2−5−4−1046

48

50

52

54−64849(a)50515253(b)Figure 2: Projection of attractor onto x12 –vx12 plane andits Poincaré map of 2×2 with B = 15, l = 50 and k = 0.31.
(a) Attractor. (b) Poincaré map.

Furthermore, we consider the nonlinear springmodel with 5 × 5 neurons (N = M = 5). The projections of attractor onto x53 –vx53 and its Poincaré map
are shown in Fig. 3. We can see that the attractor andPoincaré map are more complex than the 2 × 2 model.
0.6

0.3
0.4

0.25
0.2

0.2

0.15
0.1

0yij v y( N − 1)l

N1vxN 1v y N 2 vxN2v y Nj v xNjv y NM v xNM

0.05
0−0.2

−0.05
−0.1

−0.4
4.2

(i − 1)lv yi1 v xi1v y i 2 v xi 2v yij v xijv yiM v xiMv y21 v x21v y22 v x22v y 2 j v x2 jv y 2 M v x2 Mv y11 v x11v y12 v x12v y1 j v x1 jv y1M v x1M02l( j − 1)l

4.4

4.6

4.8

5(a)

5.2

5.4

4.2

4.4

4.6

4.8

5(b)

Figure 3: Projection of attractor onto x53 –vx53 plane and2l0( M − 1)l xij

Figure 1: Nonlinear spring model of SOM with N × M
rectangular shape.

its Poincaré map of 5×5 with B = 0.89, l = 2 and k = 0.15.
(a) Attractor. (b) Poincaré map.

4. Conclusions
We have proposed the SOM model whose neurons
are arranged in 2-dimensional array and connected by
the nonlinear spring. We have considered the models
arranged 2 × 2 and 5 × 5 array and have investigatedtheir chaotic behaviors.

3. Computer Simulation Results
We show computer calculation results obtained byusing Runge-Kutta method. We set the initial statesof the positions to its own physical location on the 2-D

Reference[1] T. Kohonen, Self-organizing Maps, Berlin, Springer, vol. 30,
1995.