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Abstract
The nematode Caenorhabditis elegans (C. elegans) is a model organism, commonly studied due to ease of maintenance and comparative simplicity of its neurological structure. We investigate C. elegans' locomotion using dynamic diffraction and nonlinear dynamics. Observed diffraction intensity time-series relate to the net electric field diffracted from all points of the worm at any point in the far-field diffraction pattern [1]. Consequently, key features of locomotion can be recovered by analyzing the intensity time-series. We found significant markers of low-dimensional chaos which prove that C. elegans locomotion satisfies the chaos criteria outlined by David Feldman [2]: determinism, aperiodic orbits, bounded orbits, and sensitive dependence on initial conditions. To prove that C. elegans locomotion meets the aforementioned criteria, we use nonlinear analysis - e.g., Takens (1981) embedding [3], mutual information (MI), lag plots, false nearest neighbors (FNN), largest Lyapunov exponent (LLE), correlation dimension, recurrence plots, and surrogate data analysis - to characterize and analyze the time-series.
First, we take the Fourier transform (FT) of the time-series and observe a broad frequency spectrum, which provides evidence that the criteria of aperiodic orbits is satisfied. Second, we generate lag plots of our locomotion data and explain how the plots satisfy all four of the chaos criteria. Third, our time-series fulfills the criteria for low-dimensional chaos with a typical positive LLE value (base e) around 1.39 ± 0.02 s−1, at optimal embedding dimension n=4, indicating sensitive dependence on initial conditions. Next, our calculated non-integer average correlation dimension of ~ 2.08 ± 0.24 means that our data may have characteristics of both 2D and 3D space, which would indicate possible chaotic dynamics. Furthermore, the correlation dimension value stabilizes with increasing embedding dimension, indicating deterministic dynamics. We also visualize the locomotion dynamics using recurrence plots because the resulting plots can prove our data satisfies determinism, aperiodic orbits, and sensitive dependence on initial conditions. As a final measure to test for chaos, we also use surrogate data analysis to prove that our time-series is nonlinear. All results provide strong evidence that C. elegans locomotion is indeed chaotic.
First, we take the Fourier transform (FT) of the time-series and observe a broad frequency spectrum, which provides evidence that the criteria of aperiodic orbits is satisfied. Second, we generate lag plots of our locomotion data and explain how the plots satisfy all four of the chaos criteria. Third, our time-series fulfills the criteria for low-dimensional chaos with a typical positive LLE value (base e) around 1.39 ± 0.02 s−1, at optimal embedding dimension n=4, indicating sensitive dependence on initial conditions. Next, our calculated non-integer average correlation dimension of ~ 2.08 ± 0.24 means that our data may have characteristics of both 2D and 3D space, which would indicate possible chaotic dynamics. Furthermore, the correlation dimension value stabilizes with increasing embedding dimension, indicating deterministic dynamics. We also visualize the locomotion dynamics using recurrence plots because the resulting plots can prove our data satisfies determinism, aperiodic orbits, and sensitive dependence on initial conditions. As a final measure to test for chaos, we also use surrogate data analysis to prove that our time-series is nonlinear. All results provide strong evidence that C. elegans locomotion is indeed chaotic.