The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of Caenorhabditis elegans, an intensively studied nematode. The decomposition indicates that the worm has a multimodal posturing behavior, involving a dominant forward locomotion mode, a secondary, steering mode, and likely a mode for reverse motion. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode is estimated in the complex plane. The frequency is obtained from the complex modal coordinate's complex whirl rate of the complex modal coordinate, and from its fast Fourier transform. Short-time decompositions indicate the variation of the wavelength and frequency through the time record.

References

References
1.
Feeny
,
B. F.
,
2008
, “
A Complex Orthogonal Decomposition for Wave Motion Analysis
”,
J. Sound Vib.
,
310
(
1–2
), pp.
77
90
.10.1016/j.jsv.2007.07.047
2.
NIH-National Human Genome Research Institute
, “
International Genome Team Deciphers Genetic Instructions for a Complete Animal
,” ScienceDaily, 11 December 1998, Retrieved Feb. 6, 2008, http://www.sciencedaily.com/releases/1998/12/981211084627.htm
3.
Mendel
,
J. E.
,
Korswagen
,
H. C.
,
Liu
,
K. S.
,
Hajdu-Cronin
,
Y. M.
,
Simon
,
M. I.
,
Plasterk
,
R. H. A.
, and
Sternberg
,
P. W.
,
1995
, “
Participation of the Protein Go in Multiple Aspects of Behavior in C. elegans
,”
Science
,
267
, pp.
1652
1655
.10.1126/science.7886455
4.
Jorgensen
,
E. M.
,
2005
, “
GABA
,” WormBook, August 31:1–13, http://www.wormbook.org
5.
Bastiani
,
C.
, and
Mendel
,
J.
,
2006
, “
Heterotrimeric G Proteins in C. elegans
,” WormBook, 2006 Oct 13:1–25, http://www.wormbook.org
6.
Rand
,
J. B.
,
2007
, “
Acetylcholine
,” WormBook, 2007 Jan 30:1–21, http://www.wormbook.org
7.
Chase
,
D. L.
, and
Koelle
,
M. R.
,
2007
, “
Biogenic Amine Neurotransmitters in C. elegans
,” WormBook, 2007 Feb 20:1–15, http://www.wormbook.org
8.
Li
,
W.
,
Feng
,
Z.
,
Sternberg
,
P. W.
, and
Xu
,
X. Z. S.
,
2006
, “
A C. elegans Stretch-Sensitive Neuron Revealed by a Mechanosensitive TRP Channel Homologue
,”
Nature
,
440
, pp.
684
687
.10.1038/nature04538
9.
Van Buskirk
,
C.
, and
Sternberg
,
P. W.
,
2007
, “
EGF Signaling Induces Behavioral Quiescence in C. elegans
,”
Nature Neurosci.
,
10
, pp.
1300
1307
.10.1038/nn1981
10.
Karbowski
,
J.
,
Cronin
,
C. J.
,
Seah
,
A.
,
Mendel
,
J. E.
,
Cleary
,
D.
, and
Sternberg
,
P. W.
,
2006
, “
Conservation Rules, Their Breakdown, and Optimality in Caenorhabditis Sinusoidal Locomotion
,”
J. Theor. Biol.
,
242
, pp.
652
669
.10.1016/j.jtbi.2006.04.012
11.
Karbowski
,
J.
,
Schindelman
,
G.
,
Cronin
,
C. J.
,
Seah
,
A.
, and
Sternberg
,
P. W.
2008
, “
Systems Level Circuit Model of C. elegans Undulatory Locomotion: Mathematical Modeling and Molecular Genetics
,”
J. Comput. Neurosci.
,
24
, pp.
253
276
.10.1007/s10827-007-0054-6
12.
Boyle
,
J. H.
, and
Cohen
,
N.
,
2008
, “
Caenorhabditis elegans Body Wall Muscles Are Simple Actuators
,”
BioSystems
,
94
, pp.
170
181
.10.1016/j.biosystems.2008.05.025
13.
Boyle
,
J. H.
,
Bryden
,
J.
, and
Cohen
,
N.
,
2008
, “
An Integrated Neuro-Mechanical Model of C. elegans Forward Locomotion
,”
Lect. Notes Comput. Sci.
,
4984
, pp.
37
47
.10.1007/978-3-540-69158-7
14.
Haspel
,
G.
,
O'Donovan
,
M. J.
, and
Hart
,
A. C.
,
2010
, “
Motoneurons Dedicated to Either Forward or Backward Locomotion in the Nematode Caenorhabditis elegans
,”
J. Neurosci.
,
30
(
33
), pp.
11151
11156
.10.1523/JNEUROSCI.2244-10.2010
15.
Korta
,
J.
,
Clark
,
D. A.
,
Gabel
,
C. V.
,
Mahadevan
,
L.
, and
Samuel
,
A. D. T.
,
2007
, “
Mechanosensation and Mechanical Load Modulate the Locomotory Gait of Swimming C. elegans
,”
J. Exp. Biol.
,
210
, pp.
2383
2389
.10.1242/jeb.004572
16.
Pierce-Shimomura
,
J. T.
,
Chen
,
B. L.
,
Mun
,
J. J.
,
Ho
,
R.
,
Sarkis
,
R.
, and
McIntire
,
S. L.
,
2008
, “
Genetic Analysis of Crawling and Swimming Locomotory Patterns in C. elegans
,”
Proc. Natl. Acad. Sci.
,
105
(
52
), pp.
20982
20987
.10.1073/pnas.0810359105
17.
Berri
,
S.
,
Boyle
,
J. H.
,
Tassieri
,
M.
,
Hope
,
I. A.
, and
Cohen
,
N.
,
2009
, “
Forward Locomotion of the Nematode C. elegans is Achieved Through Modulation of a Single Gait
,”
HFSP J.
,
3
(
3
), pp.
186
193
.10.2976/1.3082260
18.
Feng
,
Z.
,
Cronin
,
C. J.
,
Wittig
, Jr.,
J. H.
,
Sternberg
,
P. W.
, and
Schafer
,
W. R.
,
2004
, “
An Imaging System for Standardized Quantitative Analysis of C. elegans Behavior
,”
BMC Bioinformatics
,
5
, p.
115
.10.1186/1471-2105-5-115
19.
Cronin
,
C. J.
,
Mendel
,
J. E.
,
Muhktar
,
S.
,
Kim
,
Y.-M.
,
Stirbl
,
R. C.
,
Bruck
,
J.
, and
Sternberg
,
P. W.
,
2005
, “
An Automated System For Measuring Parameters of Nematode Sinusoidal Movement
,”
BMC-Genetics
,
6
, p.
5
.10.1186/1471-2156-6-5
20.
Lumley
,
J. L.
,
1970
,
Stochastic Tools in Turbulence
,
Academic Press
,
New York
.
21.
Berkooz
,
G.
,
Holmes
,
P.
, and
Lumley
,
J. L.
,
1993
, “
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
,”
Annu. Rev. Fluid Mech.
,
25
, pp.
539
575
.10.1146/annurev.fl.25.010193.002543
22.
Cusumano
,
J. P.
, and
Bai
,
B.-Y.
,
1993
, “
Period-Infinity Periodic Motions, Chaos, and Spatial Coherence in a 10 Degree of Freedom Impact Oscillator
,”
Chaos, Solitons, and Fractals
,
3
(
5
), pp.
515
535
.10.1016/0960-0779(93)90003-J
23.
Epureanu
,
B. I.
,
Tang
,
L. S.
, and
Paidoussis
,
M. P.
,
2004
, “
Exploiting Chaotic Dynamics for Detecting Parametric Variations in Aeroselastic Systems
,”
AIAA J.
,
42
(
4
), pp.
728
735
.10.2514/1.9556
24.
FitzSimons
,
P.
, and
Rui
,
C.
,
1993
, “
Determining Low Dimensional Models of Distributed Systems
,”
Advances in Robust and Nonlinear Control Systems
, ASME DSC-Vol.
53
, pp.
9
15
.
25.
Kerschen
,
G.
,
Golinval
,
J. C.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2005
, “
The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview
,”
Nonlinear Dynamics
,
41
(
1–3
), pp.
147
169
.10.1007/s11071-005-2803-2
26.
Yasuda
,
K.
, and
Kamiya
,
K.
,
1997
, “
Experimental Identification Technique of Nonlinear Beams in Time Domain
,”
ASME Design Engineering Technical Conferences
,
Sacramento, CA
, CD-ROM.
27.
Ma
,
X.
,
Azeez
,
M. A. F.
, and
Vakakis
,
A. F.
,
2000
, “
Nonlinear Normal Modes and Nonparametric System Identification of Nonlinear Oscillators
,”
Mech. Syst. Signal Proc.
,
14
(
1
), pp.
37
48
.10.1006/mssp.1999.1267
28.
Liang
,
Y. C.
,
Lee
,
H. P.
,
Lim
,
S. P.
,
Lin
,
W. Z.
,
Lee
,
K. H.
, and
Wu
,
C. G.
,
2002
, “
Proper Orthogonal Decomposition and its Applications—Part 1: Theory
,”
J. Sound Vib.
,
252
(
3
), pp.
527
544
.10.1006/jsvi.2001.4041
29.
Feeny
,
B. F.
, and
Kappagantu
,
R.
,
1998
, “
On the Physical Interpretation of Proper Orthogonal Modes in Vibrations
,”
J. Sound Vib.
,
211
(
4
), pp.
607
616
.10.1006/jsvi.1997.1386
30.
Feeny
,
B. F.
,
2002
, “
On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems
,”
ASME, J. Vib. Acoust.
,
124
(
1
), pp.
157
160
.10.1115/1.1421352
31.
Kerschen
,
G.
, and
Golinval
,
J. C.
,
2002
, “
Physical Interpretation of the Proper Orthogonal Modes Using the Singular Value Decomposition
,”
J. Sound Vib.
,
249
(
5
), pp.
849
865
.10.1006/jsvi.2001.3930
32.
Feeny
,
B. F.
, and
Liang
,
Y.
,
2003
, “
Interpreting Proper Orthogonal Modes in Randomly Excited Vibration Systems
,”
J. Sound Vib.
,
265
(
5
), pp.
953
966
.10.1016/S0022-460X(02)01265-8
33.
Feeny
,
B. F.
,
2012
, “
Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media
J. Acoust. Vib.
, (to be published).
34.
Meirovitch
,
L.
,
1997
,
Principles and Techniques in Vibrations
,
Prentice-Hall, NJ
.
35.
Ginsberg
,
J.
,
2001
,
Mechanical and Structural Vibrations
,
Wiley
,
New York
.
36.
Oppenheim
,
A. V.
, and
Schafer
,
R. W.
,
1989
,
Discrete-Time Signal Processing
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
37.
Noble
,
B.
, and
Daniel
,
J. W.
,
1977
,
Applied Linear Algebra
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
38.
Stiernagle
,
T.
,
2006
, “
Maintenance of C. elegans
,” Wormbook, The C. elegans Research Community, doi/10.1895/wormbook. 1.101.1, http://www.wormbook.org
39.
Newland
,
D. E.
,
1993
,
An Introduction to Random Vibrations, Spectral and Wavelet Analysis
,
3rd ed.
,
Longman Scientific and Technical
,
Singapore
.
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