2009 International Workshop on Nonlinear Maps and their Applications
NOMA'09, Urbino, Italy, September 10-11, 2009 (paper id: 010)

Complex Pattern in a Chain of Coupled Maps Basedon Neuron Model with Space-Varying Coupling
Yoko Uwate

Yoshifumi Nishio

Ruedi Stoop

Institute of Neuroinfomatics,
University / ETH Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: yu001@ini.phys.ethz.ch

Dept. of Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Institute of Neuroinfomatics,
University / ETH Zurich,
Winterthurerstrasse 190,CH-8057 Zurich, Switzerland,
Email: ruedi@ini.phys.ethz.ch3Abstract— Biological neurons are able to exhibit spiking andbursting behavior. The interesting phenomena is ensembles ofneurons by synchronization. We study synchronization phenomena in a chain of 2-dimensional maps based on neuronal modelwith a space-varying coupling. We realize the space-varying
coupling by normal distribution function. In the case of the spacevarying coupling, wave propagation and complex pattern can beobserved.

2Xn+1

10Pk-1XpuI. I NTRODUCTION-2

Generally, complex dynamical phenomena can be observedin networks formed by many elements with nonlinearity.
Coupled Map Lattice (CML) has proposed by Kaneko andBunimovich [1]-[5], to use as general models for the complex high-dimensional dynamics, such as biological systems,
networks in DNA, economic activities, neural networks, andevolutions. We can observed the spatio-temporal patterns inCML. Moreover, coupled oscillatory systems can also produce interesting phase patterns, including wave propagation,clustering, and complex phase patterns [6]-[9]. It is very
important to make clear this mechanism of these spatiotemporal patterns for understanding complex patterns observedin natural science. Usually, the chaotic maps are used for CML
and many interesting spatio-temporal patterns were observed.
Recently, a discrete map for spiking-bursting neural behavior was proposed by Rulkov [10], [11]. Rulkov map in the formof a two-dimensional map can be useful for understanding
the dynamical mechanism of oscillators in the large scale
networks. And Rulkov map produce spiking-bursting behavior
such as real neurons. In this study, we use Rulkov maps
applying for CML to study synchronization in large scale
networks. Furthermore, we assume that the coupling strengthof between the neurons is not simple and the coupling strength
plays important role for whole system. We consider thatthe coupling strength is changed with space. Space-varying
coupling is realized by using normal distribution function.
In this study, we study synchronization in a chain of
coupled Rulkov maps with space-varying coupling. Whenthe space-varying coupling has positive value, irregular wave
propagation can be observed. While, in the case of the negative
space-varying coupling, complex patterns are observed.

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Xps

-2-1

0Xn

123Fig. 1. Rulkov map. The dashed line illustrates a superstable cycle Pk . The
stable and unstable fixed points of the map are indicated by xps and xpu ,respectively.

II. T WO C OUPLED RULKOV M APS
Consider the two coupled Rulkov maps [10] as following
equation.
xi,n+1 = f (xi,n , yi,n + βi,n ),(1)
yi,n+1 = yi,n − µ(xi,n + 1) + µσi + µσi,n ,α/(1 − xn ) + y,xn 6 0f (xn , y) =α + y,0 < xn < α + y and xn 6 0
−1,xn > α + y or xn−1 > 0,(2)where x and y are the fast and slow dynamical variables,respectively. The coupling between the cells is provided by
the current flowing from one cell to the other. This couplingis modeled by
βi,nσi,n= gβ e (xj,n − xi,n ),(3)= gσ e (xj,n − xi,n ),where g denotes the coupling strength. In the numericalsimulations the values of the coefficients are set to be equal:
β e = 1.0 and σ e = 1.0. The other parameters has thefollowing values: µ = 0.001, α = 5.0 and σ = 0.24. The
coupling between the maps is symmetrical, i.e., gji = gij = g.

- 116 -First, we investigate basic synchronization phenomena when
the normal coupling without changing the value of couplingis used. When two Rulkov maps are coupled with positive
value g = 0.029, two bursting waves are synchronized at thein-phase as shown in Fig. 2. While, introduction of negativecoupling g = −0.029, in this regime of synchronization shows
anti-phase (see. Fig. 3). For comparison between the in-phaseand the anti-phase states, the oscillation frequency of the antiphase is faster than the in-phase state.
When three maps are coupled with negative coupling, weobserve the three-phase synchronization as shown in Fig. 4.

1.5
10.5

X1,n0-0.5-1-1.5
1.5
10.5

X2,n0-0.5-1-1.5
1.5
10.5

X3,n1.5-10.5

X1,n0-0.5

1-1.5

00-0.5500

1000

15002000

n-1

Fig. 4.

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Three-phase synchronization (g = −0.029).

1.5
1

individual map and g is the coupling. The function f () has
the following form:

0.5

X2,n 0-0.5-1α/(1 − xn ) + yn , xn ≤ 0,
 α+y ,
0 < xn < α + ynnf (xn , yn ) =and xn−1 ≤ 0,
−1,xn ≥ α + yn or xn−1 > 0,

-1.5
0500

1000

1500

2000

nFig. 2.

In-phase synchronization (g = 0.029).

(5)

1.5
1

map-1

0.5

X1,n 0
map-2-0.5

map-8-1-1.5

map-3

1.5
1

map-7

0.5

X2,nmap-4

0-0.5

map-6

-1

map-5

-1.5
0500

1000

1500

2000

Fig. 5.

nFig. 3.

Conceptual chain of coupled maps. (N = 8)

Anti-phase synchronization (g = −0.029).

III. S YNCHRONIZATION IN A C HAIN OF C OUPLED M APS
We consider a chain of coupled maps (see. Fig. 5):
xi,n+1 =yi,n+1 =f (xi,n xi,n−1 , yi,n )1+ g(xi+1,n − 2xi,n + xi−1,n ),2yi,n − µ(xi,n + 1) + µσi1+µ g(xi+1,n − 2xi,n + xi−1,n ),2
i = 1, ..., N,

In this simulations, we take α = 3.5 and σi is set for
randomly distributed in the interval [0.15:0.16]. The numberof coupled map is set to 100.
A. Normal Coupling(4)

where x and y are the fast and slow dynamical variables,respectively. µ = 10−3 and σi are the parameters of theFor comparison, we show synchronization of chain of theRulkov maps when the coupling strength is constant withspace. The simulation results of the space-time plot of positive
and negative constant coupling are shown in Figs. 6, 7,respectively. The horizontal axis is iteration time n and thevertical axis is space i. The wave propagation can be observed
when the space-varying coupling has positive value. In the caseof negative coupling, complex pattern is generated.

- 117 -(a) Example 1.
Fig. 6.

Space-time plots of xi with positive constant coupling. (g = 0.38)(b) Example 2.
Fig. 7. Space-time plots of xi with negative constant coupling. (g = −0.38)

B. Space-Varying Coupling
In this study, the space-varying coupling is realized by using
a normal distribution function. The function for a normal
distribution is given by the formula1(z − µ)2(6)exp(−),
2σ 22πσwhere µ is the mean, σ is the standard deviation. One exampleof space-varying coupling is shown in Fig. 8f (z) = √(c) Example 3.

0.44

Fig. 9. Examples of regular wave propagation with irregular wave propagation. (µ = 50, σ = 10)

0.435g(coupling strength)

0.43
0.425
0.42of the maps are shown in Fig. 9. From these figures, we can seethat irregular wave propagation can be occurred on the wave
propagation. The every form of the irregular wave propagation
is different by the initial condition of the maps.

0.415
0.41
0.405
0.4

010

20304050

60

7080

90

100i(space)

Fig. 8.

Space-varying coupling. (µ = 50, σ = 10)First, we consider that the space-varying coupling has positive value. The simulation results by changing initial condition

Next, we investigate synchronization of maps when thespace-varying coupling has negative value. In this case, wechange σ of the normal distribution function to study influenceof form of the space-varying coupling, Figure 10 shows thespace-varying coupling with negative value by changing σ.
The space-time plots are shown in Fig. 11. We confirmed thatthe high density of waves when coupling is high amplitude.

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IV. C ONCLUSIONS

(a)
(b)
(c)

-0.33

In this study, we have studied synchronization in a chainof coupled Rulkov maps with space-varying coupling. We
have realized the space-varying coupling by using normal
distribution function. When the space-varying coupling has
positive value, irregular wave propagation could be observed.
While, in the case of the negative space-varying coupling,
complex patterns were observed.
In the future works, we study these complex phenomena
by applying statistical analysis, and we investigate synchronization of maps by the other coupling methods such as thechemical and diffusion coupling.

g(coupling strength)

-0.34
-0.35
-0.36
-0.37
-0.38
-0.39
-0.4

0

10

20

30

40

50

60

70

80

90

100

i(space)Fig. 10. Space-varying coupling with negative value by changing σ. (a) σ =5, (b) σ = 10, (c) σ = 20.

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[10] N. F. Rulkov, “Modeling of Spiking-Bursting Neural Behavior using
Two-dimensional Map,” Physical Rev., E, vol. 65, 041922, 2002.
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(a) σ = 5.

(b) σ = 10.

(c) σ = 20.
Fig. 11.

Complex patterns with space-varying coupling (negative value).

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