2008 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'08, Gold Coast, Australia, March 6-8, 2008

Three-Neuron Nonlinear Spring Model of Self-Organizing Map
Haruna Matsushita and Yoshifumi Nishio
Tokushima University, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {haruna, nishio}@ee.tokushima-u.ac.jp

Abstract
In our previous research, as the first step to realize a new SelfOrganizing Map model, we have proposed a simple one dimensional 2-neuron model connected by a nonlinear spring.
This study proposes one dimensional 3-neuron model connected by a nonlinear spring in order to represent a relationship between the winner and its neighboring neurons in SOM
algorithm. Furthermore, we consider two kinds of forces byexternal input vectors and investigate their behaviors.
1. Introduction
The Self-Organizing Map (SOM) is a subtype of artificial
neural networks. It is trained using unsupervised learning toproduce low dimensional representation of the training samples while preserving the topological properties of the input
space. SOM is introduced by Kohonen in 1982 [1] and is amodel simplifying self-organization process of the brain.
However, SOM is still far away from the realization of thebrain mechanism. In order to realize more powerful and more
flexible mechanism, it is important to propose new models of
the brain mechanism and to investigate their behaviors.
In our previous research, as the first step to realize a new
nonlinear spring model of SOM, we have proposed a simple
one dimensional 2-neuron model connected by a nonlinear
spring [2]. We have investigated its behavior under a simple
assumption where input vectors are given to the model periodically.
In this study, we propose one dimensional 3-neuron model
connected by a nonlinear spring. In the SOM algorithm,
the neuron nearer to the winner can be updated more significantly. By increasing the number of neurons from two to
three, we represent a relationship between the winner and its
neighboring neurons. Furthermore, we consider two kinds offorces by external input vectors; a rectangular wave and a sine
curve.
In Section 2, we explain the 3-neuron nonlinear springmodel of SOM in detail. The behaviors of the proposed model
are investigated by calculating the projection of attractors andPoincaré map in Section 3. Furthermore, in order to investigate chaotic behavior in detail, one-parameter bifurcation di-agram and the largest Lyapunov exponent are calculated. We
investigate diﬀerent behaviors between two kinds of the external force. Computer simulated results show that the neurons
oscillate chaotically.
2. 3-Neuron Nonlinear Spring Model of SOM
This study proposes one dimensional 3-neuron SOM
model connected by a nonlinear spring. The model is shownin Fig. 1. The three neurons are assumed to have the same
mass m and to be connected by the nonlinear spring with thenatural length l, whose restoring force F against the variation
x is represented byF = −bx3(1)where b denotes the stiﬀness of each spring.
Neuron 1

Neuron 2bm v1

m v20b

Neuron 3lm v32lxˆ 2 , xˆ3

Figure 1: Three-neuron nonlinear spring model of SOM.

Without loss of generality, we fix the position of the Neuron 1 as the origin of the x-coordinate. The position of theNeuron 2 and the Neuron 3, denoted by x̂2 and x̂3 , and the velocities of the neurons (v1 , v2 and v3 ) are chosen as the statevariables. The motion equation of the model can be describedas- 315 - d x̂
2




dt



d x̂3





dt



 dv1m

dt




dv

2


m


dt




dv

 m 3
dt

=v2

=v3

=

−av1 + b( x̂2 − l)3=−av2 − b( x̂2 − l)3 + b( x̂3 − x̂2 − l)3=−av3 − b( x̂3 − x̂2 − l)3 ,(2)where a is the friction parameter. By changing the variables
and parameters;x̂2 − l = x2 ,x̂3 − 2l = x3 ,√√√bbb
v1 =y1 , v2 =y2 , v3 =y3 ,mm
m
√
mat=
τ,
k= √,bbmfˆ (t )

B̂

B̂

(3)
0= y20tmπb 2mπbt(b)= y3= −ky1 + x23(4)= −ky2 − x23 + (x3 − x2 )3

as Eq. (5) or Eq. (6). The Neuron 2 always does not receive
the direct eﬀect from the input vector and is influenced onlyby the restoring force of the nonlinear spring. In this case, themotion equation is modified as= −ky3 − (x3 − x2 )3 ,

 dx2dτ

dx3

dτ

 dy1
dτdy2
dτ


dy3

dτ

√
fˆ(t) = B̂ sin

mt,b√ )
(m0≤t≤
π ,by2=y3=−ky1 + x23 − f (τ)=−ky2 − x23 + (x3 − x2 )3=−ky3 − (x3 − x2 )3 + f (τ − π),f (τ) =(6)where sign(·) is the signum function and t = 0 is the time
when the input vector is given. The shapes of these functions
are shown in Fig. 3. Note that the other neurons do not receivea direct eﬀect from the input vector.
Neuron 2=(7)where f (τ) isand a sine curve;

Neuron 1mπbFigure 3: Force by external input vector. (a) Rectangular wave described by Eq. (5). (b) Sine wave described byEq. (6).

Next, we model the learning process of the SOM by the external force by input vectors. When an input vector is given
to the 3-neuron model as Fig. 2, the winner, which is the neuron nearest to the input vector, is attracted to the input. In thisstudy, we consider two kinds of forces by the external inputvector; a rectangular wave;√ √ )(bm0≤t≤fˆ(t) = B̂sign sint ,π ,(5)mbfˆ (t )mπb 2(a)the normalized equations are given as dx2dτdx3dτ dy1dτdy2dτdy3dτfˆ (t ))B(sign (sin τ) + 12(8)when the external force is Eq. (5), and isf (τ) =B(sin τ+ | sin τ |) ,2(9)when the external force is Eq. (6), and
B=Neuron 3

B̂
.
b(10)

The shape of each f (τ) is shown in Fig. 4.
Input

Winner

3. Computer Simulation Results

Figure 2: Input vector and winnerIn this study, in order to investigate the simplest learning
process of 3-neuron model, we concentrate on the case thatthe input vectors are given to the right-hand side and lefthandside of the model alternately with the fixed frequency√b/m/(2π). Therefore, only the Neuron 1 or the Neuron 3
becomes the winner and is attracted to the input with the force

In this section, we show some computer calculation results
obtained by using Runge-Kutta method with time step δt =2π/500 for Eq. (7).
3.1. Attractors and Poincaré maps
The projection of attractors onto x2 –y2 and x3 –y3 plane,
when the external force is the rectangular wave as Eq. (8), are- 316 -f (τ )
10B

10π2π

3π

τ(1)

0(a)

f (τ )−10−10−3π2π

3πτ0−5−5B0

5y20y25−2−101x2

234−3510105500−2−1

0

1x2

234

5N→∞ N(1)y3−10−20

24x6−4802x3(a)(b)

1010886644220468(11)−2−4−4−6−6−8−202x2−10−441010886644220y3y30−2−10−4

j=1

where e( j) is a normalized basis.
Figures 7 and 8 show One-parameter bifurcation diagramof x2 and the largest Lyapunov exponents of the attractors, which are calculated using Shimada-Nagashima algorithm [5]. The control parameter is B and another parameterk is fixed as 0.15. The Lyapunov exponent takes positive values for a wide range of B in both Fig. 7(b) and Fig. 8(b).
Therefore, the nonlinear spring model can be said to generatechaos. We observe the bifurcation in detail. In case of using
the rectangular wave, complicated bifurcation property can−23−8(2)log |DT( j) · e( j)|,−10

Figure 5: Projection of attractors and their Poincaré maps
when force by external input vector is rectangular wave (8).
Fixed parameter k = 0.15 and B = 15. (a) Attractors.
(b) Poincaré maps. (1) x2 –y2 plane. (2) x3 –y3 plane.

In order to evaluate the chaotic behavior of the model indetail, we calculate the largest Lyapunov exponent of the attractors. By using the Jacobian Matrix DT obtained by integrating the variational equations of Eq. (7), we can calculate
the largest Lyapunov exponent [4];λ1 = lim−5−4

3.2. Bifurcation Diagram and Lyapunov Exponent

N1 ∑−5y2shown in Fig. 5(a). We can confirm that the orbits of both
attractors look chaos.
In order to investigate the chaotic behavior of the modelin detail, we define the Poincaré section as τ = 2nπ and plot
the discrete data on the Poincaré section onto x2 –y2 and x3 –y3 . The Poincaré maps are shown in Fig 5(b). We can seethat the Poincaré maps are folded have the shape like strange
attractors [3]. We can estimate the attractors are chaos fromthe shape of Poincaré maps.
Similarly, the projection of attractors in case of using thesine curve as Eq. (9) and their Poincaré map are shown in
Fig. 6. The attractors look more complex chaos than Fig. 5(a).
The Poincaré maps are also folded and have the shape likestrange attractors.

(2)

y2

Figure 4: Force by periodic external input vector. (a) Forcedescribed by Eq. (8). (b) Force described by Eq. (9).

y3(b)−2−4−4−6−6−8−8−10−10−4−20x32(a)46802x2

40−2−6−2−6−4−20x32468(b)Figure 6: Projection of attractors and their Poincaré maps
when force by external input vector is sine curve (9). Fixedparameter k = 0.15 and B = 15. (a) Attractors. (b) Poincarémaps. (1) x2 –y2 plane. (2) x3 –y3 plane.

- 317 -

be confirmed from Fig. 7(a). For B < 11.64, chaotic attractors can be observed for almost parameter values, and we canconfirm the wide chaos region. On the other hand, in case ofusing the sine wave, period doubling bifurcation can be confirmed from Fig. 8(a). We can confirm more and wider chaosregion in smaller value of B than Fig. 7(a) For B < 9.57,
chaotic attractors can be observed for almost parameter values.

432x2

1
0−1

3−2

2

−3012

4

6

8

10121416101214

16x2

B
0.8

00.6
−1

0.4
0.2

−30

λ1−202468101214−0.2

16−0.4B
0.8−0.6
00.6
0.42468BFigure 8: Bifurcation diagram and largest Lyapunov exponent
changing B for k = 0.15 when external force is sine curve (9).

λ1

0.2
0−0.2

References−0.4−0.6
02468
B101214[1] T. Kohonen, Self-organizing Maps, Berlin, Springer,
1995.

16Figure 7: Bifurcation diagram and largest Lyapunov exponent
changing B for k = 0.15 when external force is rectangularwave (8).

[2] H. Matsushita and Y. Nishio, “Nonlinear Spring Modelof Self-Organizing Map and its Chaotic Behavior,” Proc.
of ICCCAS, vol. 2, pp. 1099–1102, 2007.
[3] Y. Ueda, The Road to Chaos, Santa Cruz, Aerial Press,
1992.

4. Conclusions[4] H. Kawakami, “Bifurcation of Periodic Responses inForced Dynamic Nonlinear Circuits: Computation of Bifurcation Values of the System Parameters,” IEEE Trans.
Circuits and Systems, vol. CAS–31, No. 3, pp. 248–260,
1984.

In this study, we have proposed one dimensional 3-neuron
nonlinear spring model of SOM. By increasing the numberof neurons from two to three, we have modeled the relationship between the winner and its neighboring neurons. Furthermore, we have considered two kinds of forces by external [5] I. Shimada and T. Nagashima, “A Numerical Approach
input vectors; a rectangular wave and a sine curve. In order
to Ergodic Problem of Dissipative Dynamical Systems,”
to investigate chaotic behavior in detail, one-parameter bifurProgress of Theoretical Physics, vol. 61, No. 6, pp. 1605–
cation diagram and the largest Lyapunov exponent have be1616, 1979.
calculated. We have investigated diﬀerent behaviors between
two kinds of the external force and have confirmed that theneurons oscillate chaotically.

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