A chain of three-oscillator system was constructed with living biological oscillators of phasmodial slime mold, Physarum polycehalum and the oscillation patterns were analyzed by the symmetric Hopf bifurcation theory using group theory. Multi-stability of oscillation patterns was observed, even when the coupling strength was fixed. This suggests that the coupling strength is not an effective parameter to obtain a desired oscillation pattern among the multiple patterns. Here we propose a method to control oscillation patterns using resonance to external stimulus and demonstrate pattern switching induced by frequency resonance given to only one of oscillators in the system

Fujii, Teruo

Takamatsu, Atsuko

Tanaka, Reiko

Yamamoto, Takatoki

English

Atsuko Takamatsu

Crossref

Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
HTffiL ABSTRACT + LinKS 
CONTROL OF OSCILLATION PATTERNS 
IN A SYMMETRIC COUPLED BIOLOGICAL 
OSCILLATOR SYSTEM 
Atsuko Takamatsu*tl, Reiko Tanaka~§, Takatoki Yamamotot, and 
Teruo Fujiit 
*PRESTO, Japan Science and Technology Corporation (1ST), Japan 
t Underwater Technology Research Center, Institute of Industrial Science, The University of Tokyo, 
4-6-1, Komaba, Meguro-ku, Tokyo, 153-8505, Japan 
f Dept. of Applied Physics and Physico-Informatics, Keio University, Hiyoshi, Kohoku-ku, Yokohama, 
223-8522,Japan 
§Control and Dynamical Systems, California Institute ofTechnology, MC 107-81 
Pasadena, CA, 91125, USA 
Abstract. A chain of three-oscillator system was constructed with living biological oscillators of 
phasmodial slime mold, Physarum polycehalum and the oscillation patterns were analyzed by 
the symmetric Hopf bifurcation theory using group theory. Multi-stability of oscillation patterns 
was observed, even when the coupling strength was fixed. This suggests that the coupling 
strength is not an effective parameter to obtain a desired oscillation pattern among the multiple 
patterns. Here we propose a method to control oscillation patterns using resonance to external 
stimulus and demonstrate pattern switching induced by frequency resonance given to only one of 
oscillators in the system. 
INTRODUCTION 
Most of biological systems can be modeled with geometrical symmetries. For 
example, neuronal systems for control of animal locomotion consist of central nervous 
system and ganglions arranged symmetrically. Such systems could be modeled by a 
coupled oscillator system that has potential to show a variety of oscillation patterns, 
namely, a variety of locomotion patterns. One of the most popular parameters to 
control the patterns would be coupling strength between oscillators. It is, however, 
ineffective to obtain a desired pattern among multiple patterns if the system has multi-
stability. Here we propose a method to control the oscillation patterns by controlling 
the frequencies of a single oscillator in symmetric coupled oscillators. 
We constructed three-oscillator system in a chain with living biological oscillators 
of plasmodial slime mold, Physarum polycehalum. The plasmodial slime mold is an 
amoeboid multinucleated unicellular organism and shows various oscillatory 
phenomena in the concentration of intracellular chemicals such as A TP and Ca2+ that 
are considered to lead contraction/relaxation states in the cell, namely, thickness 
1 Electronic address: atsuko@iis. u-tokyo.ac.jp 
CP676, Experimental Chaos: 7'' Experimental Chaos Conference, 
edited by V. In, L. Kocarev, T. L. Carroll, B. J. Gluckman, S. Boccaletti, and J. Kurths 
© 2003 American Institute of Physics 0-7354-0145-4/03/$20.00 
230 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
oscillation of the cell, induced by complicated mechanochemical reactions among the 
chemicals, proteins, and intracellular organelles. The thickness oscillation causes 
protoplasmic streaming inside tube structures of the plasmodial cell, which would 
relate to interactions between partial bodies of the cell. Thus, this system can be 
regarded as an assemble of nonlinear oscillators coupled by the tube structure. We 
have constructed two-oscillator systems and three-, four-, five-oscillator systems in 
rings by a cell patterning method and have shown that the symmetrically arranged 
oscillator-systems exhibit multiple oscillation patterns predicted by the symmetric 
Hopf bifurcation theory using geometrical symmetry proposed by Golubitsky and 
Stewart [1-4]. In order to obtain more information on the system, we investigated the 
responses to environmental change. As one of the results, we demonstrate pattern 
switching between different oscillation patterns induced by frequency resonance in 
order to control the multi-stable patterns. 
METHODS 
A three-oscillator system in a chain was constructed with a microfabricated 
structure of SU-8 (photoresist resin) as shown in Fig. 1(a),(b) by a method previously 
reported [1,3]. The structure consists of oscillator parts, where thickness oscillation is 
observed, and channel structures. The coupling strength between the oscillators can be 
controlled with the channel width in the microfabricated structure [1]. In this 
experiment, the channel width was fixed at 400pm that corresponds to strong coupling. 
In this condition, the system could show multiple oscillation patterns [2]. It is known 
that oscillation frequency in the plasmodium is affected by temperature change [5]. A 
chip for temperature control was fabricated by patterning a transparent resistant 
material, ITO (Indium Tin Oxide), on a glass substrate to be used as multi-channel 
sensors and heaters. Local stimuli in small areas (2 x2mm) can be applied to multiple 
points with small distances (4mm) [6, 7]. The plasmodial oscillator system was placed 
on this chip. Moreover, transparency of ITO material is useful for optical observation 
of cells. Stimuli were applied by periodic change of temperature to one of the 
oscillators located in the center (Oscillator 2; denoted as 02 in Fig. 1 (c)). 
Temperature of the other oscillators was maintained at 21 T. 
FIGURE 1. (a), (b) Experimental setup. (c) Three-oscillator system in a chain. 
231 
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OSCILLATION PATTERNS 
The system shows multiple oscillation patterns irrespective of existence of stimulus 
by temperature change. In Fig.2 (a), oscillator 1 (01) and 3 (03) show anti-phase 
oscillation while oscillator 2 (02) shows double frequency compared with the other 
oscillators (pattern A). In Fig.2 (b), adjacent oscillators are in anti-phase (pattern B). 
In Fig.2 (c), the frequencies of all oscillators are doubled (pattern C). As seen in the 
time series of thickness oscillation in Fig.2 (c), the frequencies are original before the 
time 200 (sec), and doubled after the time 200 (sec) (denoted with a gray box). In 
Fig.2 (d), the phases in adjacent oscillators are shifted by 90° then the phase wave 
propagates from one end to another end (pattern D). The observed patterns were 
dependent on the sample of the plasmodium even when the coupling strength was 
fixed, which implies that this system has multi-stability. In addition, spontaneous 
pattern switching was observed except between the pattern A and B. The most 
frequently observed pattern was A (41 %; in number of periods in 30 minutes 
observation of 33 samples), secondly C (13%) and D (12%), followed by B (7%), 
when the coupling strength is fixed at one parameter. The probability of the observed 
patterns depends on the coupling strength. 
By the group theory, this system has Zz-symmetry that has a reflection denoted with 
f3 in Fig. 2 (e). The symmetric Hopf bifurcation predicts the patterns A and in the 
system with Zz-symmetry [4]. The pattern A keeps spatio-temporal symmetry under 
the action of reflection ,Bwith a phase shift of 180°. Double frequency of 02 in the 
pattern A is generated to keep the symmetry when the phase is shifted by 180°, since 
02 is located on the reflection axis. The pattern B keeps the symmetry under only the 
reflection ,8. The pattern D, however, dose not keep the symmetry under the reflection 
j3 with any phase shift. This oscillation pattern would be generated by D4-symmetry 
where a hidden oscillator is assumed, whose details will be reported elsewhere. 
Double frequency of all oscillators in the pattern C cannot be explained by Zz-
symmetry, which could be caused by system specific characteristics of the 
plasmodium. For simplicity, we focus on only the patterns A and B in this report and, 
in order to control the patterns A and B, we applied the external stimulus to the center 
oscillator to keep Zz-symmetry. 
PATTERN SWITCHING BY RESONANCE 
We applied the stimulus by change of temperature to the center oscillator 02 and 
changed the only one control parameter {if I(Q) from 0.5 to 2.0, where {if is an angular 
frequency of stimulus and t4J is an original angular frequency of 01 and 03 before 
stimulus ({QJ =UJJ~~, t4J ~w2 12in the pattern A, t4J ~UJJ~WrCQ3 in the pattern B). 
Fig. 3 shows an example of response to the periodic stimulus. Temperature was 
changed in a sinusoidal function form (To+ A sin( {if t+¢); To= 21.0°C, A= 1SC, ¢is 
an arbitrary initial phase) from the time around 1900 (sec) as shown in Fig. 3(a). The 
system showed the oscillation pattern A before starting the stimulus and the pattern B 
232 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
after several periods of stimulation. When the stimulus started, the double frequency 
of 02 was destroyed, then the dynamical change in phase difference between 01 and 
03 from 180° to 0° was observed at the time around 3500 (sec) in Fig. 3 (f). Switching 
from the pattern B to A was also observed under different condition of stimulus (data 
not shown). 
The mechanism of the switching is considered as follows. Each plasmodial 
oscillator shows resonance against external periodic stimulus which means the 
oscillator synchronize with external the stimulus to change its frequency. 
(a) 01 02 03 
~ 
110 ,-------.--,---.:::---, 
90 
20 
(c) 
-c-80 
::> 
.; 
]lioo 
Q) 
c 
""' () 
~60 
80 
0 
0 
(e) !~ 
100 200 300 
Time( sec) 
200 400 600 
Time( sec) 
~ 
m, (i.l2 o.b 
(b) 
100 
" 80 
~ 130 f=d~#=""'f=,..,.,..f~~ 
CiJ 
lll110 
~ 
~ 90 ~F=*-""'F-#-~ 
120 
100 
zo-~1~o~o-~20~o~~3~oo~47.o~o 
Time( sec) 
(d) 
200 
~-~~-_____L_~________L-
1 00 200 300 400 
Time( sec) 
FIGURE 2. (a-d) Schematic diagrams of oscillation patterns and thickness oscillation. Open circles are 
oscillators. Concentric circles are oscillators have double frequency. Bidirectional and unidirectional 
arrows mean that phase difference between oscillators is 180'and 90', respectively. (e) Z2 symmetry in 
a chain of three oscillators. ~ is a reflection. 
233 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
6 24 
~ 
:::!. 22 
~ 
:::l 
Cii 20 
~ 
~ 18 
I-
~ 
:::l 
cd en 
(/) 
(]) 
c 
:.:. 
() 
E 
1-
(rad/sec 
~ 360 
0) 
QJ 
:::, 270 
QJ 
u 
c 
QJ 180 
~ 
-o 90 QJ 
IF> 
"' £ 0 0.. 
1000 
• 
X 
xxxx x x 
X X X 
X X 
X • 
oe oe o 0 •o ._o •:!C •~ •>v eX0 exoexoexo x8 • 
• • • • • • • • • • • • • • 
__j__ ___ _____t ___ I • • 
1500 2000 2500 3000 3500 4000 
Time( sec) 
FIGURE 3. Pattern switching by periodic force. (a) Stimulus by temperature change. (b-d) Thickness 
oscillation in oscillator 1-3. (e) Angular frequency. Bars, fiJJed circles, crosses, open circles are those of 
stimulus, thickness change in OsciJJator 1, 2 and 3, respectively. Original frequencies: WJ=0.044±0.003, 
~q=0.083 ± 0.010, OJ.J=0.043 ± 0.002. Frequency of the stimulus: wr = 0.026. Frequencies during 
stimulus: WJ=0.040 ±0.001, tq=0.040 ±0.001, 0Jj=0.039 ±0.002. (f) Phase difference of osciJlation 
between osciJlator 1 and 3. 
234 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
In one oscillator system of the plasmodium, we have observed 1:1, 1:2, 1:3 (m: Ctf; m 
is angular frequency of 01 or 03 during stimulus) resonances and the resonances in 
other fractional number combination such as 3:2, 4:5, etc., and have found that its 
phase diagram has Arnol'd tongues [8]. If Ctf lm o is within the tongue, 02 resonates 
with a different frequency mfrom original mo. In the case of Fig. 3, m0 - 0.044±0.003 
and m- 0.039±0.002 and the system shows 3:2 resonance. It should be noted that the 
frequency was reduced to 86 ±9 (%) of the original in all the samples where switching 
from the patterns A to B was observed. The system showed the resonance in an 
appropriate ratio in most cases. If the stimulus was applied to increase the frequency, 
pattern switching occurs from A to B was not observed. On the other hand, the 
frequency increased when the switching from B to A. Consequently, it is concluded 
that the control between the pattern A and B can be achieved only by controlling the 
frequency of the center oscillator that affects on both ends of oscillators symmetrically. 
CONCLUSION 
We demonstrated the control of the oscillation patterns by using the resonance to 
external stimulus in the multi-stable system. The multi-stability of the system is 
commonly observed in other systems of coupled oscillators as well [9]. Thus, the 
method we demonstrated here could be generally effective to control such system if 
the oscillation pattern depends on the frequency. Furthermore, we observed the 
dynamical pattern switching from/to in-phase to/from anti-phase by taking notice of 
the oscillators in both ends. The anti-phase mode, for example, corresponds to walking 
mode in bipedal animal and in-phase mode corresponds to hopping mode. This 
method might also be useful as the model of the control of animal locomotion. 
Moreover, it should be stated the importance of consideration of the symmetry in the 
system which generally observed in nature [4, 10]. 
REFERENCES 
1. Takamatsu, A. , Fujii, T., Endo, I. Phys. Rev. Lett. 85 2026 (2000). 
2. Takamatsu, A. eta!., Phys. Rev. Lett. 87 078102 (2001). 
3. Takamatsu, A. and Fujii, T. "Construction of a living coupled oscillator system of plasmodial slime mold by a 
microfabricated structure" in Sensors Update Vol. 10, edited by Wiley-VCH, Weinheim, 2002, pp. 33-46. 
4. Golubitsky, M. and Stewart, I., The Symmetry perspective, Birkhiiuser Verlag, Basel, 2001. 
5. Matsumoto, K. eta!. 1. Theor. Bioi, 131 175 (1988). 
6. Yamamoto, T. Nojima, T. and Fujii, T. "PDMS-glass hybrid microreactor for cell-free protein analysis" (2002) 
submitted. 
7. Takamatsu, A. and Fujii, T., "Observation of response to stimuli in oscillating cells patterned by 
microfabricated structure" Proceedings of The Sixth International Symposium on Micro Total Analysis System 
(jlFASs), Nara, Japan, 2002, to appear. 
8. Takamatsu, A., Yamamoto, T. and Fujii, T., "Resonant pattern in a plasmodial slime mold by periodic force of 
temperature." (2002) in preparation. 
9. Yoshimoto, M, Yoshikawa, K. and Mori, Y. Phys. Rev. E 47 864 (1993). 
10. Tanaka, R., Iwata, Sand Shin, S. Trans. of the Society of Instrument and Control Engineers 33 441 (1997). 
235 


Control of Oscillation Patterns in a Symmetric Coupled Biological Oscillator System

Caltech Authors - Main

Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
HTffiL ABSTRACT + LinKS 
CONTROL OF OSCILLATION PATTERNS 
IN A SYMMETRIC COUPLED BIOLOGICAL 
OSCILLATOR SYSTEM 
Atsuko Takamatsu*tl, Reiko Tanaka~§, Takatoki Yamamotot, and 
Teruo Fujiit 
*PRESTO, Japan Science and Technology Corporation (1ST), Japan 
t Underwater Technology Research Center, Institute of Industrial Science, The University of Tokyo, 
4-6-1, Komaba, Meguro-ku, Tokyo, 153-8505, Japan 
f Dept. of Applied Physics and Physico-Informatics, Keio University, Hiyoshi, Kohoku-ku, Yokohama, 
223-8522,Japan 
§Control and Dynamical Systems, California Institute ofTechnology, MC 107-81 
Pasadena, CA, 91125, USA 
Abstract. A chain of three-oscillator system was constructed with living biological oscillators of 
phasmodial slime mold, Physarum polycehalum and the oscillation patterns were analyzed by 
the symmetric Hopf bifurcation theory using group theory. Multi-stability of oscillation patterns 
was observed, even when the coupling strength was fixed. This suggests that the coupling 
strength is not an effective parameter to obtain a desired oscillation pattern among the multiple 
patterns. Here we propose a method to control oscillation patterns using resonance to external 
stimulus and demonstrate pattern switching induced by frequency resonance given to only one of 
oscillators in the system. 
INTRODUCTION 
Most of biological systems can be modeled with geometrical symmetries. For 
example, neuronal systems for control of animal locomotion consist of central nervous 
system and ganglions arranged symmetrically. Such systems could be modeled by a 
coupled oscillator system that has potential to show a variety of oscillation patterns, 
namely, a variety of locomotion patterns. One of the most popular parameters to 
control the patterns would be coupling strength between oscillators. It is, however, 
ineffective to obtain a desired pattern among multiple patterns if the system has multi-
stability. Here we propose a method to control the oscillation patterns by controlling 
the frequencies of a single oscillator in symmetric coupled oscillators. 
We constructed three-oscillator system in a chain with living biological oscillators 
of plasmodial slime mold, Physarum polycehalum. The plasmodial slime mold is an 
amoeboid multinucleated unicellular organism and shows various oscillatory 
phenomena in the concentration of intracellular chemicals such as A TP and Ca2+ that 
are considered to lead contraction/relaxation states in the cell, namely, thickness 
1 Electronic address: atsuko@iis. u-tokyo.ac.jp 
CP676, Experimental Chaos: 7'' Experimental Chaos Conference, 
edited by V. In, L. Kocarev, T. L. Carroll, B. J. Gluckman, S. Boccaletti, and J. Kurths 
© 2003 American Institute of Physics 0-7354-0145-4/03/$20.00 
230 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
oscillation of the cell, induced by complicated mechanochemical reactions among the 
chemicals, proteins, and intracellular organelles. The thickness oscillation causes 
protoplasmic streaming inside tube structures of the plasmodial cell, which would 
relate to interactions between partial bodies of the cell. Thus, this system can be 
regarded as an assemble of nonlinear oscillators coupled by the tube structure. We 
have constructed two-oscillator systems and three-, four-, five-oscillator systems in 
rings by a cell patterning method and have shown that the symmetrically arranged 
oscillator-systems exhibit multiple oscillation patterns predicted by the symmetric 
Hopf bifurcation theory using geometrical symmetry proposed by Golubitsky and 
Stewart [1-4]. In order to obtain more information on the system, we investigated the 
responses to environmental change. As one of the results, we demonstrate pattern 
switching between different oscillation patterns induced by frequency resonance in 
order to control the multi-stable patterns. 
METHODS 
A three-oscillator system in a chain was constructed with a microfabricated 
structure of SU-8 (photoresist resin) as shown in Fig. 1(a),(b) by a method previously 
reported [1,3]. The structure consists of oscillator parts, where thickness oscillation is 
observed, and channel structures. The coupling strength between the oscillators can be 
controlled with the channel width in the microfabricated structure [1]. In this 
experiment, the channel width was fixed at 400pm that corresponds to strong coupling. 
In this condition, the system could show multiple oscillation patterns [2]. It is known 
that oscillation frequency in the plasmodium is affected by temperature change [5]. A 
chip for temperature control was fabricated by patterning a transparent resistant 
material, ITO (Indium Tin Oxide), on a glass substrate to be used as multi-channel 
sensors and heaters. Local stimuli in small areas (2 x2mm) can be applied to multiple 
points with small distances (4mm) [6, 7]. The plasmodial oscillator system was placed 
on this chip. Moreover, transparency of ITO material is useful for optical observation 
of cells. Stimuli were applied by periodic change of temperature to one of the 
oscillators located in the center (Oscillator 2; denoted as 02 in Fig. 1 (c)). 
Temperature of the other oscillators was maintained at 21 T. 
FIGURE 1. (a), (b) Experimental setup. (c) Three-oscillator system in a chain. 
231 
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OSCILLATION PATTERNS 
The system shows multiple oscillation patterns irrespective of existence of stimulus 
by temperature change. In Fig.2 (a), oscillator 1 (01) and 3 (03) show anti-phase 
oscillation while oscillator 2 (02) shows double frequency compared with the other 
oscillators (pattern A). In Fig.2 (b), adjacent oscillators are in anti-phase (pattern B). 
In Fig.2 (c), the frequencies of all oscillators are doubled (pattern C). As seen in the 
time series of thickness oscillation in Fig.2 (c), the frequencies are original before the 
time 200 (sec), and doubled after the time 200 (sec) (denoted with a gray box). In 
Fig.2 (d), the phases in adjacent oscillators are shifted by 90° then the phase wave 
propagates from one end to another end (pattern D). The observed patterns were 
dependent on the sample of the plasmodium even when the coupling strength was 
fixed, which implies that this system has multi-stability. In addition, spontaneous 
pattern switching was observed except between the pattern A and B. The most 
frequently observed pattern was A (41 %; in number of periods in 30 minutes 
observation of 33 samples), secondly C (13%) and D (12%), followed by B (7%), 
when the coupling strength is fixed at one parameter. The probability of the observed 
patterns depends on the coupling strength. 
By the group theory, this system has Zz-symmetry that has a reflection denoted with 
f3 in Fig. 2 (e). The symmetric Hopf bifurcation predicts the patterns A and in the 
system with Zz-symmetry [4]. The pattern A keeps spatio-temporal symmetry under 
the action of reflection ,Bwith a phase shift of 180°. Double frequency of 02 in the 
pattern A is generated to keep the symmetry when the phase is shifted by 180°, since 
02 is located on the reflection axis. The pattern B keeps the symmetry under only the 
reflection ,8. The pattern D, however, dose not keep the symmetry under the reflection 
j3 with any phase shift. This oscillation pattern would be generated by D4-symmetry 
where a hidden oscillator is assumed, whose details will be reported elsewhere. 
Double frequency of all oscillators in the pattern C cannot be explained by Zz-
symmetry, which could be caused by system specific characteristics of the 
plasmodium. For simplicity, we focus on only the patterns A and B in this report and, 
in order to control the patterns A and B, we applied the external stimulus to the center 
oscillator to keep Zz-symmetry. 
PATTERN SWITCHING BY RESONANCE 
We applied the stimulus by change of temperature to the center oscillator 02 and 
changed the only one control parameter {if I(Q) from 0.5 to 2.0, where {if is an angular 
frequency of stimulus and t4J is an original angular frequency of 01 and 03 before 
stimulus ({QJ =UJJ~~, t4J ~w2 12in the pattern A, t4J ~UJJ~WrCQ3 in the pattern B). 
Fig. 3 shows an example of response to the periodic stimulus. Temperature was 
changed in a sinusoidal function form (To+ A sin( {if t+¢); To= 21.0°C, A= 1SC, ¢is 
an arbitrary initial phase) from the time around 1900 (sec) as shown in Fig. 3(a). The 
system showed the oscillation pattern A before starting the stimulus and the pattern B 
232 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
after several periods of stimulation. When the stimulus started, the double frequency 
of 02 was destroyed, then the dynamical change in phase difference between 01 and 
03 from 180° to 0° was observed at the time around 3500 (sec) in Fig. 3 (f). Switching 
from the pattern B to A was also observed under different condition of stimulus (data 
not shown). 
The mechanism of the switching is considered as follows. Each plasmodial 
oscillator shows resonance against external periodic stimulus which means the 
oscillator synchronize with external the stimulus to change its frequency. 
(a) 01 02 03 
~ 
110 ,-------.--,---.:::---, 
90 
20 
(c) 
-c-80 
::> 
.; 
]lioo 
Q) 
c 
""' () ~60 
80 
0 
0 
(e) !~ 
100 200 300 
Time( sec) 
200 400 600 
Time( sec) 
~ 
m, (i.l2 o.b 
(b) 
100 
" 80 ~ 130 f=d~#=""'f=,..,.,..f~~ 
CiJ 
lll110 
~ 
~ 90 ~F=*-""'F-#-~ 
120 
100 
zo-~1~o~o-~20~o~~3~oo~47.o~o 
Time( sec) 
(d) 
200 
~-~~-_____L_~________L-
1 00 200 300 400 
Time( sec) 
FIGURE 2. (a-d) Schematic diagrams of oscillation patterns and thickness oscillation. Open circles are 
oscillators. Concentric circles are oscillators have double frequency. Bidirectional and unidirectional 
arrows mean that phase difference between oscillators is 180'and 90', respectively. (e) Z2 symmetry in 
a chain of three oscillators. ~ is a reflection. 
233 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
6 24 
~ 
:::!. 22 
~ 
:::l 
Cii 20 
~ 
~ 18 
I-
~ 
:::l 
cd 
en (/) 
(]) 
c 
:.:. 
() 
E 
1-
(rad/sec 
~ 360 0) 
QJ 
:::, 270 
QJ 
u 
c 
QJ 180 ~ 
-o 90 QJ 
IF> 
"' £ 0 0.. 
1000 
• 
X 
xxxx x x 
X X X 
X X 
X • 
oe oe o 0 •o ._o •:!C •~ •>v eX0 exoexoexo x8 
• 
• • • • • • • • 
• • • 
• • • 
__j__ ___ _____t ___ I • • 
1500 2000 2500 3000 3500 4000 
Time( sec) 
FIGURE 3. Pattern switching by periodic force. (a) Stimulus by temperature change. (b-d) Thickness 
oscillation in oscillator 1-3. (e) Angular frequency. Bars, fiJJed circles, crosses, open circles are those of 
stimulus, thickness change in OsciJJator 1, 2 and 3, respectively. Original frequencies: WJ=0.044±0.003, 
~q=0.083 ± 0.010, OJ.J=0.043 ± 0.002. Frequency of the stimulus: wr = 0.026. Frequencies during 
stimulus: WJ=0.040 ±0.001, tq=0.040 ±0.001, 0Jj=0.039 ±0.002. (f) Phase difference of osciJlation 
between osciJlator 1 and 3. 
234 
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
In one oscillator system of the plasmodium, we have observed 1:1, 1:2, 1:3 (m: Ctf; m 
is angular frequency of 01 or 03 during stimulus) resonances and the resonances in 
other fractional number combination such as 3:2, 4:5, etc., and have found that its 
phase diagram has Arnol'd tongues [8]. If Ctf lm o is within the tongue, 02 resonates 
with a different frequency mfrom original mo. In the case of Fig. 3, m0 - 0.044±0.003 
and m- 0.039±0.002 and the system shows 3:2 resonance. It should be noted that the 
frequency was reduced to 86 ±9 (%) of the original in all the samples where switching 
from the patterns A to B was observed. The system showed the resonance in an 
appropriate ratio in most cases. If the stimulus was applied to increase the frequency, 
pattern switching occurs from A to B was not observed. On the other hand, the 
frequency increased when the switching from B to A. Consequently, it is concluded 
that the control between the pattern A and B can be achieved only by controlling the 
frequency of the center oscillator that affects on both ends of oscillators symmetrically. 
CONCLUSION 
We demonstrated the control of the oscillation patterns by using the resonance to 
external stimulus in the multi-stable system. The multi-stability of the system is 
commonly observed in other systems of coupled oscillators as well [9]. Thus, the 
method we demonstrated here could be generally effective to control such system if 
the oscillation pattern depends on the frequency. Furthermore, we observed the 
dynamical pattern switching from/to in-phase to/from anti-phase by taking notice of 
the oscillators in both ends. The anti-phase mode, for example, corresponds to walking 
mode in bipedal animal and in-phase mode corresponds to hopping mode. This 
method might also be useful as the model of the control of animal locomotion. 
Moreover, it should be stated the importance of consideration of the symmetry in the 
system which generally observed in nature [4, 10]. 
REFERENCES 
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235 


Construction of a living coupled oscillator system of plasmodial slime mold by a microfabricated structure&quot; in Sensors Update Vol. 10, edited by Wiley-VCH,

Observation of response to stimuli in oscillating cells patterned by microfabricated structure&quot;

PDMS-glass hybrid microreactor for cell-free protein analysis&quot;

Resonant pattern in a plasmodial slime mold by periodic force of temperature.&quot;

The Symmetry perspective, Birkhiiuser Verlag,

http://authors.library.caltech.edu/27480/1/TAKaipcp03.pdf

Control of Oscillation Patterns in a Symmetric Coupled Biological Oscillator System

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