Research Papers

Complex Orthogonal Decomposition Applied to Nematode Posturing

[+] Author and Article Information
B. F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

P. W. Sternberg

e-mail: pws@caltech.edu

C. J. Cronin

e-mail: cjc@caltech.edu
Division of Biology,
California Institute of Technology,
Pasadena, CA 91125

C. A. Coppola

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received October 12, 2011; final manuscript received September 15, 2012; published online May 14, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 8(4), 041010 (May 14, 2013) (8 pages) Paper No: CND-11-1174; doi: 10.1115/1.4023548 History: Received October 12, 2011; Revised September 15, 2012

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.

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Grahic Jump Location
Fig. 1

Above is the captured worm, arbitrarily located and oriented. The head points down, and is indicated by a small dot. The coordinates are translated to the origin and rotated, as shown below, where y indicates the transverse deflection of the worm markers. The head (dot) points right.

Grahic Jump Location
Fig. 2

The COVs showing relative mean squared modal amplitudes

Grahic Jump Location
Fig. 3

Top: (dominant) mode one, with real (solid line) and imaginary (dashed line) parts. (The vertical axes are magnified compared to Fig. 1.) The head is at marker 1 (the circle) and points left. Bottom: mode one in the complex plane, where the head is marked with the circle.

Grahic Jump Location
Fig. 4

Top: dominant mode oscillation frequency, taken from the complex whirl rate, over time. Bottom: spectrum of the dominant modal coordinate, where dB is computed as 20 log q1(t), and where q1 is in mm.

Grahic Jump Location
Fig. 5

Mode two, with real (solid line) and imaginary (dashed line) parts. (The vertical axes are magnified compared to Fig. 1.) The circle indicates the head marker.

Grahic Jump Location
Fig. 6

Top: the heading angle θ(t) (lower isolated curve, in red) in radians; the scaled change in angle 10θ·(t) (black, overlapping), in radians, computed by finite differences; and the real part of the scaled moving-averaged second modal coordinate 10q2(t) (magenta/light, overlapping) in mm. Bottom: the product of the second modal coordinate and the change in heading angle. q2(t)θ·(t), in mm per second.

Grahic Jump Location
Fig. 7

(a) Mode three, with real (solid line) and imaginary (dashed line) parts. The circle indicates the head marker. (b) The upper dotted curve shows the instantaneous centroid heading speed in mm/s, and the lower solid curve shows the third modal coordinate history.

Grahic Jump Location
Fig. 8

The upper plot shows the temporal variation of the first-mode wavelength in worm lengths per cycle (top solid curve), centroid speed in mm/s (dashed curve), projected wave speed in mm/s (lower solid curve), and dominant modal coordinate frequency in Hz (dashed-dotted curve) from short-time decompositions with T=3τ. The lower plot shows a histogram of the short-time averaged wavelengths.

Grahic Jump Location
Fig. 9

The upper plot shows the temporal variation of the first-mode wavelength in worm lengths per cycle (top solid curve), centroid speed in mm/s (dashed curve), projected wave speed in mm/s (lower solid curve), and dominant modal coordinate frequency in Hz (dashed-dotted curve) from short-time decompositions with T=5τ. The lower plot shows a histogram of the short-time averaged wavelengths.




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