2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Synchronization of 2-dimensional Maps Based on Neuron Modelwith Time-Varying Coupling
Yoko Uwate† , Yosifhimi Nishio‡ and Ruedi Stoop††University / ETH Zurich,
Winterthurerstrasse 190,CH-8057 Zurich, Switzerland,
Email: yu001@ini.phys.ethz.ch‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: nishio@ee.tokushima-u.ac.jp

3Abstract2

Biological neurons are able to exhibit spiking and bursting
behavior. The interesting phenomena is ensembles of neurons by synchronization. In this study, we investigate complex patterns observed in coupled 2-dimensional maps basedon neuronal model with time-varying coupling.

Xn+1

10Pk-1Xpu

1. Introduction-2

Generally, complex dynamical phenomena can be observed in networks formed by many elements with nonlinearity. Coupled Map Lattice (CML) has proposed by Kanekoand Bunimovich [1]-[5], to use as general models for thecomplex high-dimensional dynamics, such as biological systems, networks in DNA, economic activities, neural networks, and evolutions. We can observed the spatio-temporal
patterns in CML. It is very important to make clear this mechanism of the spatio-temporal patterns for understanding complex patterns observed in 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 [6], [7]. Rulkov map (see.
Fig. 1) in the form of a two-dimensional map can be usefulfor understanding the dynamical mechanism of oscillators in
the large scale networks. And Rulkov map produce spikingbursting behavior like real neurons. In this study, we consider
that Rulkov maps are used for CML. Furthermore, we assume
that the coupling strength of between the neurons is not simple and the coupling strength plays important role for wholesystem. We consider that the coupling strength is changed
with time. Time-varying coupling is realized by switching thepositive and negative values, periodically. In this study, weinvestigate synchronization phenomena observed in two coupled Rulkov maps with time-varying coupling, and demonstrate complex patterns observed in a chain of maps.

-3-3

Xps

-2-10Xn

123

Figure 1: Rulkov map. The dashed line illustrates a superstable cycle Pk . The stable and unstable fixed points of themap are indicated by x ps and x pu , respectively.
2. Two Coupled Rulkov Maps
Consider the two coupled Rulkov maps [6] as followingequation.
xi,n+1
yi,n+1= f (xi,n , yi,n + βi,n ),= yi,n − µ(xi,n + 1) + µσi + µσi,n ,α/(1 − xn ) + y,




α + y,f (xn , y) =  −1,(1)xn 6 00 < xn < α + y and xn 6 0
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 thecurrent flowing from one cell to the other. This coupling ismodeled by- 197 -βi,nσi,n= gβe (x j,n − xi,n ),e= gσ (x j,n − xi,n ),(3)where g denotes the coupling strength. In the numerical simulations the values of the coefficients are set to be equal:βe = 1.0 and σe = 1.0. The other parameters has the following values: µ = 0.001, α = 5.0 and σ = 0.24. The coupling
between the maps is symmetrical, i.e., g ji = gi j = g.

1.5
10.5

X1,n0-0.5-1-1.51.5
12.1. Normal Coupling

0.5

X2,n0-0.5

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

-1-1.5
1.5
10.5

X3,n0-0.5-1-1.50500

1000

1500

2000

nFigure 4: Three-phase synchronization (g = −0.029).
2.2. Time-Varying Coupling

1.5

Figure 5 shows the characteristics of the time-varying coupling. The value of the time-varying coupling switches positive and negative periodically and p denotes the switching
period.

10.5

X1,n0-0.5-1-1.5
1.5
10.04

0.5

0.02
gX2,n 0-0.5

0

n-0.02-1-0.04

-1.5
0500

1000

1500

p2000

nFigure 5: Characteristics of time-varying coupling.
Figure 2: In-phase synchronization (g = 0.029).

1.510.5

X1,n 0-0.5-1-1.5
1.5
10.5

X2,n0-0.5-1-1.5
0500

1000

1500

2000

nFigure 3: Anti-phase synchronization (g = −0.029).

By using the time-varying coupling, we observe the synchronization switching between in-phase and anti-phase independence on the coupling strength (when the switching period is set to p = 1000, p = 200) as shown in Figs. 6 and7. Upper figure shows the two wave forms generated fromtwo coupled Rulkov maps and lower figure shows the characteristics of the time-varying coupling. When the couplingstrength has positive value, the two maps are synchronized inin-phase. While, in the case of that the coupling value is negative, two maps are synchronized in anti-phase. By switching of the value of the time-varying coupling, synchronization
states also switches between the in-phase and the anti-phasestates.
Next we calculated the frequency of bursting part when theiteration time is set to n = 1000000 as shown in Fig. 8. From
this result we can confirm that in the case of normal positive
coupling, only three peaks exist. In contrast, there are manypeaks by using negative coupling. When we use the time-- 198 -

2

2000

1.5

g=0.029g= - 0.029
g=0.029, -0.029 (period: 500)
g=0.029, -0.029 (period: 100)x1, x2

1
0.5
0

1500
frequency

-0.5
-1
-1.5
-2

1000

0.04
g0.02

500

0-0.02
-0.04
10000

11000

12000

13000

14000

15000

0

n050

100

150

200

250

300

350

400

length of burst part

Figure 6: Switching of in-phase and anti-phase synchronization (p = 1000).

Figure 8: Histogram of length of bursting part.
150

2140

1.5average length of burst partx1, x210.5
0-0.5-1-1.5-20.04g0.02

130120
110
100
90080

-0.02
-0.04
10000

11000

12000

13000

14000

15000

n700100

200

300

400

500

600

700

800

9001000 1100

period of TVC

Figure 7: Switching of in-phase and anti-phase synchronization (p = 200).

Figure 9: Average length of bursting part.

where x and y are the fast and slow dynamical variables, revarying coupling, several peaks of bursting part are observed. spectively. µ = 10−3 and σi are the parameters of the indiFigure 9 show the simulation result of the average length of vidual map and g is the coupling. The function f () has thebursting part by changing period of the time-varying cou- following form:
pling. We confirmed two peaks of the period of time-varyingα/(1 − xn ) + yn , xn 6 0,coupling in this figure. We confirm that the average length of
bursting part decrease, when the period of time-varying cou

0 < xn < α + yn α + yn ,pling becomes large.
f (xn , yn ) = andxn−1 6 0,
 −1,xn > α + yn or xn−1 > 0,
3. Synchronization in a Chain of Coupled Maps(5)
This
model
is
amodificationof
themodelpresentedinIn this section, we investigate synchronization in a chain ofSec.
2.
In
thissimulations,
we
take
α=3.5
and
σis
set
i
coupled maps:
for randomly distributed in the interval [0.15:0.16]. The simulation results of the space-time plot by changing period of
xi,n+1 = f (xi,n xi,n−1 , yi,n )
time-varying coupling are shown in Figs. 10-12. The hori1+ g(xi+1,n − 2xi,n + xi−1,n ),zontal axis is iteration time n and the vertical axis is space i.
2When the time varying coupling is set to positive, the wave
yi,n+1 = yi,n − µ(xi,n + 1) + µσi
propagation can be observed. While, in the case of negative1
coupling, we confirm the random patterns are produced. By+µ g(xi+1,n − 2xi,n + xi−1,n ),decreasing the period of time-varying coupling, the length of2
i = 1, ..., N,(4) each region of wave propagation and random pattern becomes- 199 -

100

100

180
0.5
70
06050

-0.5

40

1.5-130

90

-1.5

120

80
0.5

100070i (space)

1.5

90i (space)shorter. Figure 13 shows the simulation result of the spacetime plot when the coupling is set to g = ±0.4 and the periodof time-varying coupling is p = 100.
We consider that the interesting complex patterns can beobserved by switching coupling strength.

60-2-2.5
050

100

200

300

400

600

700

800

900

1000

n (time)

-0.5

40500

-1Figure 12: Space-time plots of xi . (g = ±0.2, p = 20).

30-1.520100-21049003-2.50100

200

300

400

500

600

700

800

900

1000

802n (time)

70i (space)

Figure 10: Space-time plots of xi . (g = ±0.2, p = 500).

160
5004010030

1.5-22090

1-310800.5070i (space)-1-40060
50

100

200

300

400

500

600

700

800

900

1000

n (time)-0.5

40

Figure 13: Space-time plots of xi . (g = ±0.4, p = 100).

-130-1.5
20-2.5[3] K. Kaneko, “Pattern Dynamics in SpatiotemporalChaos,” Physica D, vol.34, pp.1–41, 1989.

Figure 11: Space-time plots of xi . (g = ±0.2, p = 100).

[4] K. Kaneko, “Spatiotemporal Chaos in One- and TwoDimensional Coupled Map Lattice,” Physica D, vol.37,
pp.60–82, 1989.

-2

1000100

200300400500600

7008009001000

n (time)4. Conclusions
In this study, we have investigated synchronization phenomena observed in two coupled Rulkov maps with timevarying coupling, and demonstrated complex patterns observed in a chain of maps. In the future works, we investigatesynchronization of maps by the other coupling methods suchas the chemical and diffusion coupling.
References[5] K. Kaneko, “Simulating Physics with Coupled Map Lattice - Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos,” Formation, Dynamics, and Statistics of Patterns, World Sci., pp.1–52,
1990.
[6] N. F. Rulkov, “Modeling of Spiking-Bursting Neural
Behavior using Two-dimensional Map,” Physical Rev.,E, vol. 65, 041922, 2002.
[7] N. F. Rulkov, I. Timofeev and M. Bazhenov, “Oscillations in Large-Scale Cortical Networks: Map-BasedModel,” Journal of Computational Neuroscience vol. 17
pp. 203–223, 2004.

[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattice,” Prog. Theor. Phys., vol.75, no.5,pp.1033–1044, 1985.
[2] L. A. Bunimovich and Ya. G. Sinai, “Spacetime Chaosin Coupled Map Lattices,” Nonlineality 1, no.4, pp.491–516, 1988.

- 200 -