Multichannel measurements of C. elegans largest Lyapunov exponents using optical diffraction.

Dynamic diffraction (DOD) is a form of microscopy that allows the dynamic tracking of changing shapes in a 1D time series. DOD can capture the locomotion of a nematode while swimming freely in a 3D space, allowing the locomotion of the worm to more closely mimic natural behavior than in some other laboratory environments. More importantly, we are able to see markers of chaos as DOD covers dynamics on multiple length scales. This work introduces a multichannel method to measure the dynamic complexity of microscopic organisms. We show that parameters associated with chaos, such as the largest Lyapunov exponent (LLE), the mean frequency, mutual information (MI), and the embedding dimension, are independent of the specific point sampled in the diffraction pattern, thus demonstrating experimentally the consistency of our dynamic parameters sampled at various locations (channels) in the associated optical far-field pattern.

Dynamic Markers for Chaotic Motion in C. elegans.

We describe the locomotion of Caenorhabditis elegans (C. elegans) using nonlinear dynamics. C. elegans is a commonly studied model organism based on ease of maintenance and simple neurological structure. In contrast to traditional microscopic techniques, which require constraining motion to a 2D microscope slide, dynamic diffraction allows the observation of locomotion in 3D as a time series of the intensity at a single point in the diffraction pattern. The electric field at any point in the far-field diffraction pattern is the result of a superposition of the electric fields bending around the worm. As a result, key features of the motion can be recovered by analyzing the intensity time series. One can now apply modern nonlinear techniques; embedding and recurrence plots, providing valuable insight for visualizing and comparing data sets. We found significant markers of low-dimensional chaos. Next, we implemented a minimal biomimetic simulation of the central pattern generator of C. elegans with FitzHugh-Nagumo neurons, which exhibits undulatory oscillations similar to those of the real C. elegans. Finally, we briefly describe the construction of a biomimetic version of the Izquierdo and Beer robotic worm using Keener's implementation of the Nagumo et al. circuit.

Chaotic markers in dynamic diffraction.

In a dynamic far-field diffraction experiment, we calculate the largest Lyapunov exponent of a time series obtained from the optical fluctuations in a dynamic diffraction pattern. The time series is used to characterize the locomotory predictability of an oversampled microscopic species. We use a live nematode, Caenorhabditis elegans, as a model organism to demonstrate our method. The time series is derived from the intensity at one point in the diffraction pattern. This single time series displays chaotic markers in the locomotion of the Caenorhabditis elegans by reconstructing the multidimensional phase space. The average largest Lyapunov exponent (base e) associated with the dynamic diffraction of 10 adult wildtype (N2) Caenorhabditis elegans is 1.27±0.03s-1.

Fourier-Based Diffraction Analysis of Live Caenorhabditis elegans.

This manuscript describes how to classify nematodes using temporal far-field diffraction signatures. A single C. elegans is suspended in a water column inside an optical cuvette. A 632 nm continuous wave HeNe laser is directed through the cuvette using front surface mirrors. A significant distance of at least 20-30 cm traveled after the light passes through the cuvette ensures a useful far-field (Fraunhofer) diffraction pattern. The diffraction pattern changes in real time as the nematode swims within the laser beam. The photodiode is placed off-center in the diffraction pattern. The voltage signal from the photodiode is observed in real time and recorded using a digital oscilloscope. This process is repeated for 139 wild type and 108 "roller" C. elegans. Wild type worms exhibit a rapid oscillation pattern in solution. The "roller" worms have a mutation in a key component of the cuticle that interferes with smooth locomotion. Time intervals that are not free of saturation and inactivity are discarded. It is practical to divide each average by its maximum to compare relative intensities. The signal for each worm is Fourier transformed so that the frequency pattern for each worm emerges. The signal for each type of worm is averaged. The averaged Fourier spectra for the wild type and the "roller" C. elegans are distinctly different and reveal that the dynamic worm shapes of the two different worm strains can be distinguished using Fourier analysis. The Fourier spectra of each worm strain match an approximate model using two different binary worm shapes that correspond to locomotory moments. The envelope of the averaged frequency distribution for actual and modeled worms confirms the model matches the data. This method can serve as a baseline for Fourier analysis for many microscopic species, as every microorganism will have its unique Fourier spectrum.

Single wavelength shadow imaging of Caenorhabditis elegans locomotion including force estimates.

This study demonstrates an inexpensive and straightforward technique that allows the measurement of physical properties such as position, velocity, acceleration and forces involved in the locomotory behavior of nematodes suspended in a column of water in response to single wavelengths of light. We demonstrate how to evaluate the locomotion of a microscopic organism using Single Wavelength Shadow Imaging (SWSI) using two different examples. The first example is a systematic and statistically viable study of the average descent of C. elegans in a column of water. For this study, we used living and dead wildtype C. elegans. When we compared the velocity and direction of nematode active movement with the passive descent of dead worms within the gravitational field, this study showed no difference in descent-times. The average descent was 1.5 mm/sec ± 0.1 mm/sec for both the live and dead worms using 633 nm coherent light. The second example is a case study of select individual C. elegans changing direction during the descent in a vertical water column. Acceleration and force are analyzed in this example. This case study demonstrates the scope of other physical properties that can be evaluated using SWSI while evaluating the behavior using single wavelengths in an environment that is not accessible with traditional microscopes. Using this analysis we estimated an individual nematode is capable of thrusting with a force in excess of 28 nN. Our findings indicate that living nematodes exert 28 nN when turning, or moving against the gravitational field. The findings further suggest that nematodes passively descend in a column of water, but can actively resist the force of gravity primarily by turning direction.

Quantitative locomotion study of freely swimming micro-organisms using laser diffraction.

Soil and aquatic microscopic organisms live and behave in a complex three-dimensional environment. Most studies of microscopic organism behavior, in contrast, have been conducted using microscope-based approaches, which limit the movement and behavior to a narrow, nearly two-dimensional focal field.(1) We present a novel analytical approach that provides real-time analysis of freely swimming C. elegans in a cuvette without dependence on microscope-based equipment. This approach consists of tracking the temporal periodicity of diffraction patterns generated by directing laser light through the cuvette. We measure oscillation frequencies for freely swimming nematodes. Analysis of the far-field diffraction patterns reveals clues about the waveforms of the nematodes. Diffraction is the process of light bending around an object. In this case light is diffracted by the organisms. The light waves interfere and can form a diffraction pattern. A far-field, or Fraunhofer, diffraction pattern is formed if the screen-to-object distance is much larger than the diffracting object. In this case, the diffraction pattern can be calculated (modeled) using a Fourier transform.(2) C. elegans are free-living soil-dwelling nematodes that navigate in three dimensions. They move both on a solid matrix like soil or agar in a sinusoidal locomotory pattern called crawling and in liquid in a different pattern called swimming.(3) The roles played by sensory information provided by mechanosensory, chemosensory, and thermosensory cells that govern plastic changes in locomotory patterns and switches in patterns are only beginning to be elucidated.(4) We describe an optical approach to measuring nematode locomotion in three dimensions that does not require a microscope and will enable us to begin to explore the complexities of nematode locomotion under different conditions.

Optomechanical shape analysis using group theory.

We describe an optomechanical technique using a knife-edge, which is scanned spatially across a beam of light to identify shape-based irradiance. Symmetry groups are identified through linear and rotational scanning signatures of illuminated shapes. The scanning signature is used to classify the shape into a symmetry group. To demonstrate the shape analysis technique, we have classified basic geometric shapes, which belong to the orthogonal and dihedral symmetry groups O(2), D(2), D(3), and D(6).

Optomechanical integration method for finite integrals.

We present, for the first time to our knowledge, an optomechanical integration method for finite functions. This technique allows for the integration of any finite function by combining optical and mechanical principals. The integrated function can then be determined using curve fitting methods. Furthermore, the original function can be reproduced through numerical or analytical integration.