Nonlinear Spring Model of Self-Organizing Mapand its Chaotic Behavior
Haruna MATSUSHITA∗ and Yoshifumi NISHIO∗∗ Department of Electrical and Electronic Engineering, Tokushima University

2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Telephone: +81–88–656–7470, Fax: +81–88–656–7471,
Email: {haruna, nishio}@ee.tokushima-u.ac.jp

Abstract— The Self-Organizing Map (SOM) is an unsupervisedneural network introduced by Kohonen and is a model simplifying self-organization process of the brain. However, SOM isstill far away from the realization of the brain mechanism. Inorder to realize more powerful and more flexible mechanism, it isimportant to propose new models of the brain mechanism and toinvestigate their behaviors. In this study, as the first step to realizea new nonlinear spring model of SOM, we propose a simple one
dimensional 2-neuron model connected by a nonlinear spring. We
investigate its behavior under a simple assumption where input
vectors are given to the model periodically. Computer simulated
results show that the neurons oscillate chaotically.

I. I NTRODUCTION
The Self-Organizing Map (SOM) is a subtype of artificial
neural networks. It is trained using unsupervised learning
to produce low dimensional representation of the training
samples while preserving the topological properties of theinput space. SOM is introduced by Kohonen in 1982 [1] andis a model 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 this study, as the first step to realize a new nonlinear
spring model of SOM, we propose a simple one dimensional 2neuron model connected by a nonlinear spring. We investigate
its behavior under a simple assumption where input vectors are
given to the model periodically.
In Section II, we explain the learning algorithm of SOM.
In Section III, we propose and explain the nonlinear springmodel of SOM in detail. The behaviors of the proposed model
are investigated by calculating Poincaré map and Lyapunov
exponents in Section IV. Computer simulated results show thatthe neurons oscillate chaotically.

to be from 0 to 1.
(SOM1) An input vector x j is inputted to all the neurons at
the same time in parallel.
(SOM2) Distances between x j and all the weight vectors arecalculated. The winner, denoted by c, is the neuron with theweight vector closest to the input vector x j ;c = arg min{wi − xj },(1)iwhere  ·  is the distance measure, Euclidean distance.
(SOM3) The weight vectors of the neurons are updated as;
wi (t + 1) = wi (t) + hc,i (t)(xj − wi (t)),

where t is the learning step. h c,i (t) is called the neighborhoodfunction and is described as a Gaussian function;ri − r c 2hc,i (t) = α(t) exp −,(3)2σ 2 (t)where r i − rc  is the distance between map nodes c and i on
the map grid, α(t) is the learning rate, and σ(t) corresponds
to the width of the neighborhood function. Both α(t) and σ(t)decrease with time as follows;α(t) = α(0)

α(T )
α(0)

t/T


, σ(t) = σ(0)σ(T )σ(0)t/T, (4)where T is the maximum number of the learning.
(SOM4) The steps from (SOM1) to (SOM3) are repeated for
all the input data.

II. S ELF -O RGANIZING M AP (SOM)
We explain the learning algorithm of the Self-Organizing
Map (SOM). SOM consists of m neurons located at a regular
low-dimensional grid, usually a 2-D grid. The basic SOM algorithm is iterative. Each neuron i has a d-dimensional weightvector w i = (wi1 , wi2 , · · · , wid ) (i = 1, 2, · · · , m). The initialvalues of all the weight vectors are given over the input spaceat random. The range of the elements of d-dimensional input
data xj = (xj1 , xj2 , · · · , xjd ) (j = 1, 2, · · · , N ) are assumed

1-4244-1474-1/07/$25.00 ©2007 IEEE.

(2)

1099

Winner c
Input vector
Fig. 1.

Winner c
Input vector

Learning process of Self-Organizing Map.

III. N ONLINEAR S PRING M ODEL OF SOM
We consider a simple one dimensional 2-neuron model as
the first step to realize the nonlinear spring model of SOM.
The model is shown in Fig. 2. The two neurons are assumed
to have the same mass m and to be connected by a nonlinear
spring with the natural length l whose restoring force F against
the variation x is represented by
F = −bx3

where t = 0 is the time when the input vector is given. The
shape of this function is shown in Fig. 4. Note that the otherneuron does not receive a direct effect from the input vector.

fˆ (t )

Input

(5)

Winnerwhere b denotes the stiffness of the spring.

Fig. 3.

Neuron 1

Neuron 2m v

m v

0l1Fig. 2.

Input vector and winner.

fˆ (t )

2x̂B̂

Two-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 the Neuron
2 (x̂) and the velocities of the neurons (v 1 and v2 ) are chosen
as the state variables. The motion equation of the model canbe described as⎧
dx̂
⎪
⎪
= v2
⎪
⎪dt
⎪
⎨
dv1(6)= −av1 + b(x̂ − l)3m⎪dt⎪⎪⎪⎪⎩ m dv2 = −av2 − b(x̂ − l)3dtwhere a is the friction parameter. By changing the variables
and parameters;bby1 , v2 =y2 ,x̂ − l = x, v1 =mm
(7)am
τ, k = √t=,bbmthe normalized equations are given as
⎧dx
⎪
⎪
= y2
⎪
⎪
⎪
⎨ dτ
dy1(8)= −ky1 + x3
⎪dτ
⎪
⎪
⎪
⎪
⎩ dy2 = −ky2 − x3 .
dτNext, we model the learning process of the SOM by theexternal force by input vectors. When an input vector is given
to the 2-neuron model as Fig. 3, the winner which is the neuron
nearer to the input vector is attracted to the input with thefollowing force;bmˆ(9)t0≤t<πf (t) = B̂ sinmbmπb 20Fig. 4.

tmπbForce by external input vector.

In this study, in order to investigate the simplest learning
process of the 2-neuron model, we concentrate on the casethat input vectors are given to the right-hand side and the lefthand side of the model alternately with the fixed frequency
b/m/(2π). In this case, the motion equation is modified as⎧dx⎪⎪= y2⎪⎪⎪⎨ dτdy1(10)= −ky1 + x3 − f (τ )⎪dτ⎪⎪⎪ dy2⎪⎩= −ky2 − x3 − f (τ − π),dτwhereB(11)f (τ ) = (sin τ + | sin τ |)2andB̂
B= .
(12)
bThe shape of f (τ ) is shown in Fig. 5.

1100

f (τ )
B

π0Fig. 5.

2π3πForce by periodic external input vector.

τ10IV. S IMULATION R ESULTS

8In this section, we show some computer calculation results
obtained by using Runge-Kutta method with time step δt =2π/100 for Eq. 10.
Figure 6 shows time series of x, namely the position ofNeuron 2 in Fig. 2. Figure 6(a) is obtained for the parameters
k = 0.27 and B = 8.5, while Figure 6(b) is for k = 0.1
and B = 12. Although we can see some periodicity in thewaveform in Fig. 6(a), the movement is completely nonperiodic in Fig. 6(b). The attractors corresponding to the timeseries are shown in Fig. 7. We can confirm that the orbits ofthe solution look chaos.

64y220−2−4−6−8−10−3−2−1012341234x(a)410386241y2x200−2−1−4−2−30−6−8500

1000

1500

2000

τ2500

3000

3500

4000

−10−3−2−10x(a)4(b)

3Fig. 7. Attractor of nonlinear spring model. (a) k = 0.27, B = 8.5. (b) k =0.1, B = 12.

2
1xFigure 9 shows the calculated largest Lyapunov exponents.
Because the Lyapunov exponents are positive for both cases,
we can say that the attractors are chaos.

0−1V. C ONCLUSIONS−2−30500

1000

1500

2000τ2500

3000

3500

4000(b)Fig. 6. Time series of nonlinear spring model. (a) k = 0.27, B = 8.5.
(b) k = 0.1, B = 12.

In this study, 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.
We have investigated its behavior under a simple assumption
where input vectors are given to the model periodically. We
have confirmed that the neurons oscillate chaotically.
R EFERENCES

In order to investigate the chaotic behavior of the modelin detail, we define the Poincaré section as τ = 2nπ andplot the discrete data on the Poincaré section onto x − y 2plane. Figure 8 shows the Poincaré map corresponding to theattractors in Fig. 7. We can see that the Poincaré map is foldedand has the shape like strange attractors [2].
In order to confirm the generation of chaos, we calculate theLyapunov exponents of the attractors using the algorithm of
Shimada-Nagashima [3]. The Jacobian matrix are obtained bycalculating the variational equations of Eq. (10) together [4].

[1] T. Kohonen, Self-organizing Maps, Berlin, Springer, vol. 30, 1995.
[2] Y. Ueda, The Road to Chaos, Santa Cruz, Aerial Press, 1992.
[3] I. Shimada and T. Nagashima, “A Numerical Approach to Ergodic
Problem of Dissipative Dynamical Systems,” Progress of TheoreticalPhysics, vol. 61, No. 6, pp. 1605-1616, 1979.
[4] H. Kawakami, “Bifurcation of Periodic Responses in Forced 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.

1101

103.58362.5421.5

10

λy2

2
1−20−6−0.5−8−10−3λ1 = 0.2257

0.5−4−2−10123−1 0104101x(a)103.5

8362.54103410221.5

λ1 = 0.8388

λ1y2

210

iteration n(a)01−20.5−40−6−0.5−8−10−3−2−101

23−1 0104101102103104

iteration n(b)x(b)
Fig. 8. Poincaré map of nonlinear spring model. (a) k = 0.27, B = 8.5.
(b) k = 0.1, B = 12.

Fig. 9. Lyapunov exponent of attractor. (a) k = 0.27, B = 8.5. (b) k =0.1, B = 12.

1102