Examining the evolution of phase-space elements for C.elegans locomotion

TL;DR

The Lyapunov spectrum of C.elegans locomotion is estimated, which offers an insight into how volume elements evolve in the phase space of the underlying dynamical system and will be significant for the potential creation of future mathematical or computational models.

Abstract

The Caenorhabditis Elegans (C.elegans) nematodes have long been a model organism for quantitative behavioral analysis, due to their tractable nervous system and well-characterized genetics. In particular, dynamic diffraction has been a successful method of studying said microorganisms due to its low level of noise and the ability to simultaneously study multiple degrees of freedom of their neuromuscular system through their locomotion. In this study, we estimate the Lyapunov spectrum of C.elegans locomotion, which offers an insight into how volume elements evolve in the phase space of the underlying dynamical system. For that, we used the Sano-Sawada algorithm to estimate the spectra from the trajectories reconstructed using the Takens embedding procedure. In total, two positive and one negative exponents were calculated and verified to be non-spurious through investigations of their stability for different sets of parameters. Those exponents have values of 0.860, 0.389, and -3.451 respectively. The presence of two positive exponents indicates that C.elegans locomotion is hyperchaotic, while the total sum being negative indicates that the system is dissipative and non-Hamiltonian. Those are key observations for the underlying system and will be significant for the potential creation of future mathematical or computational models.

Figures (5)

• Figure 1: Lag plot (bottom) of a time series (top) used in the analysis below, visualized in a projected 3-dimensional phase space using the Takens procedure. The changing color represents evolution in time. Selecting $d=3$ seems to resolve most of the attractor, but according to the results from the FNN percentage jenny, a $d$ of at least 4 is required to properly resolve the trajectories. The axes of the figure are unmarked because they are in arbitrary units (AU), just like the pre-embedding time series.
• Figure 2: The process of selecting a neighborhood of radius $\epsilon$ around a particular point $\mathbf{x}_i$ on the reconstructed trajectory. The points $\mathbf{x}_a,\mathbf{x}_b$ within this neighborhood are considered neighboring points. Once an iteration has been completed, these points are mapped to $\mathbf{x}_{a+1},\mathbf{x}_{b+1}$ respectively. By analyzing the initial and final separation vectors from the trajectory point to its neighbors, we can identify the linear operator $A_j$ that corresponds to each neighboring point.
• Figure 3: Sample graph for one dataset. The different colored lines stand for different exponents. New exponents form as $d$ is increased.
• Figure 4: The estimates of $\lambda_1$ and $\lambda_2$ for a sample time series are plotted for a range of different values of $\epsilon$. For both exponents, a plateau is visibly formed in the middle of the graph, corresponding to a region where the estimates can be considered robust.
• Figure 5: Range of $\epsilon$ values for which the graphs of each dataset's positive exponents plateau. Most of the Lyapunov exponents plateau in similar regions, at around $2-5\%$ of their respective trajectories.