Research Article  Open Access
Li Huang, Hongkyun Kim, Jacob Furst, Daniela Raicu, "A RunLength Encoding Approach for Path Analysis of C. elegans Search Behavior", Computational and Mathematical Methods in Medicine, vol. 2016, Article ID 3516089, 9 pages, 2016. https://doi.org/10.1155/2016/3516089
A RunLength Encoding Approach for Path Analysis of C. elegans Search Behavior
Abstract
The nematode Caenorhabditis elegans explores the environment using a combination of different movement patterns, which include straight movement, reversal, and turns. We propose to quantify C. elegans movement behavior using a computer vision approach based on runlength encoding of steplength data. In this approach, the path of C. elegans is encoded as a string of characters, where each character represents a path segment of a specific type of movement. With these encoded string data, we perform means cluster analysis to distinguish movement behaviors resulting from different genotypes and food availability. We found that shallow and sharp turns are the most critical factors in distinguishing the differences among the movement behaviors. To validate our approach, we examined the movement behavior of tph1 mutants that lack an enzyme responsible for serotonin biosynthesis. A means cluster analysis with the path stringencoded data showed that tph1 movement behavior on food is similar to that of wildtype animals off food. We suggest that this runlength encoding approach is applicable to trajectory data in animal or human mobility data.
1. Introduction
C. elegans is an important genetic model organism relevant to human biology and disease, as its genome is surprisingly similar to that of humans (40% homologous) [1]. Genes encoding tryptophan hydroxylase, which is the key enzyme for serotonin biosynthesis, are conserved in human and C. elegans [2]. The biogenic amine serotonin acts in C. elegans to modulate behaviors such as egglaying, pharyngeal pumping, locomotion, and learning in response to changing environmental cues [3]. Variants in human tryptophan hydroxylase are associated with a spectrum of neuropsychiatric disorders, including depression, bipolar disorders, and suicidality [4]. Therefore, by comparing the locomotory behaviors of C. elegans wildtype and tph1 animals under two different conditions (in the presence and absence of food), we expect to gain insights into understanding how certain genes influence human emotional and congenital disorders.
C. elegans locomotory behavior can be recorded and analyzed using computer vision and machine learning approaches. Several worm trackers have been developed to analyze worm behavior in different scenarios such as for recording multiple worms at the same time [5], imaging of specific neurons [6–8], and recording of single animals that are freshly removed from bacterial food, particularly over a long period of time [9]. Video data recorded with the trackers can be further analyzed by quantifying the differences in the worm body and movement characteristics. The differences in these characteristics allow comparing food search behavior across different types of worms and understanding what genetic mutation produces a certain defect in the worm movement. In this study, we quantify the path of a worm as a string of symbols where each symbol represents a segment of a certain type of movement (shallow or sharp turn) and learn the similarities and differences in C. elegans locomotory behaviors by comparing their stringencoded path data.
Previous C. elegans studies on the neurotransmitter serotonin focus on its role in the regulation of mating, egglaying, fat storage, reproductive lifespan longevity, and locomotory rate. For example, C. elegans males with reduced serotonin levels exhibited defects in tailcurling behavior during mating [10]. A similar observation was made in wildtype males in which serotonergic neurons were ablated. Serotonin signaling in C. elegans regulates the function of serotonergic motor neurons that stimulate egglaying behavior [11]. In particular, serotonin was hypothesized to control a switch between two distinct on/off states of egglaying behavior [12]. Further, serotonin modulates C. elegans locomotory rate in response to bacterial food differently for the wellfed and fooddeprived wildtype animals [13]. It is also known that wellfed tph1 mutants exhibit changes in behavioral and metabolic processes similar to those caused by starvation: slower rates of egglaying and pharyngeal pumping, dauer larval arrest, increased fat storage, and an extended reproductive lifespan [2]. Furthermore, a recent study showed that serotonin signaling regulates on food explorative behavior; lack of serotonin signaling increases roaming and decreases dwelling [14]. However, it is still not clear how wildtype and tph1 mutant animals compare in off food conditions, particularly with respect to their overall movement paths.
The study of C. elegans movement paths is focused on foraging speed, tail motion, crawling, and a specific posture in the movement of the worm. For example, Padmanabhan et al. studied bends during reversing [15], while Gray et al. focused on omega turns [16]. To date, there has been no research focusing on encoding and analyzing the whole path of C. elegans. In our paper, we propose to encode the path using runlength encoding, which is a data compression algorithm for sequences of data developed by Duda et al. [17] and extract a set of features that will quantify the characteristics of the path. Runlength encoding has been used in 2D and 3D image texture analysis to quantify texture information and improve image classification accuracy. For examples, Tang [18], Xu et al. [19], and Martinez [20] have also applied runlength encoding in data compression, such as the compression of images, sound, and program code.
Path analysis can be also used to find patterns in human mobility. For example, human flight length can be recorded and calculated from GPS services; Rhee et al. showed that the analysis of the distribution of the flight lengths of a walking path shows that human walk patterns contain statistically similar features observed in Lévy walks [21]. In our paper, by encoding the path with the clustering sequence on three feature variables derived from each video, we are able to generate features for each worm for further analysis. These results can be generalized to human locomotion, as well as technological advances such as selfdriving cars.
2. Materials and Methods
Our proposed methodology is based on both computer vision and machine learning components to quantify and compare C. elegans locomotory behavior. The raw data consists of image frames from the videos we record with our inhouse built worm tracker [9]. The path of each worm is generated from the worm body centroid location in each frame. We propose to encode each path as a string of movement patterns detected using clustering, (2) compress each path string using runlength encoding, (3) extract runlength descriptors that characterize the frequency and order of the detected movement patterns, and (4) use the runlength descriptors to understand the similarities in the locomotory behavior of wildtype N2 and tph1 worm data. Figure 1 illustrates the process of extracting the runlength encoding descriptors and Figure 2 illustrates the path similarity approach based on these descriptors.
2.1. C. elegans Data and Feature Extraction
Two types of C. elegans worm data, N2 and tph1, are collected using our inhouse built tracker, which records the movements of single animals at 30 frames per second [9], under three different conditions: on food (N2_f and tph1_f), off food (N2_nf and tph1_nf), and previously starved when placed on a plate without food (N2_nnf; the last condition was experimented for N2 wildtype worms only). Section 3 presents our findings based on 57 video recordings: 4 N2_f; 33 N2_nf, 2 N2_nnf; 13 tph1_f, and 5 tph1_nf.
For each video (see example in Figures 1(a) and 1(b)), the following features are extracted based on the worm body centroid position data (): steplength (mm), angle (degree) between two steps, and speed (mm/s) along the step. Rather than working with all the centroid points on the worm path, we divide the path into a series of line segments called “steps” and use the steplength data to quantify the path and infer patterns in search behavior. First, we sample the centroid coordinates of the worm () at a regular time interval sec forming a dataset of values (Figure 1(c)) where time and denotes the th point on the sampled path at a time rate of . The sample time interval can be adjusted to balance noise reduction and data fidelity. Second, a “turning event” (TE) is identified at a specific if the current movement heading deviates by more than some threshold angle Θ from the heading at the previous turning event. Studies show that the head angles of wildtype animals moving in a straight direction fluctuate between −30 and 30 degrees. We found experimentally that an angle Θ of 40 degrees is the best tradeoff between the correct sampling of the path and the predictive modeling power of the generated steplength data.
Therefore, a new subsample of centroids (Figure 1(d)) is generated from the sampled path, containing only the centroid locations for the turning events . Note that the first centroid location of the worm is considered to be the initial turning event . Once the turning events are identified, a step is defined as a line segment between two consecutive turning events and . The steplength is the displacement between the two turning events forming that step. The angle feature between two steps is defined as the degree between two movement steps joining three successive turning events, and the speed is calculated by taking the displacement between the two consecutive turning events forming the step and dividing by the interval of time taken to travel that step.
At the end of this feature extraction stage (Figures 1(a)–1(d)), each worm path is represented as a sequence of steps of different lengths that form different angles and are traveled at different speeds. A minmax normalization approach is further applied on the feature data to ensure that all variables have equal importance in determining the patterns along the worm path.
2.2. Worm Path Encoding as a String of Movement Patterns
To further quantify the worm path, we investigated the similarity among the steps in terms of their lengths, angles they form, and the speed at which they are traveled. We applied means clustering approach [17] with the Euclidian distance as the similarity measure and then analyzed the obtained clusters to find insights about any existing pattern characteristics of the worm paths. We varied , the number of clusters, from 2 to 5 and chose the one that generated the best classification results where each cluster was associated with a class of movement patterns. The classification and regression trees (CR&T) approach [22] was applied to validate the choice of using a holdout partitioning to divide the data into training and testing.
Our results showed no difference in the classification accuracy when varying and, therefore, we hypothesize that the path discretized through steps can be simply further encoded as a string of two symbols, where each symbol represents a cluster of a certain movement pattern (Figure 1(e)). A test of significance was then applied to verify the differences between the two clusters with respect to the step data. The angle between two steps was found to be the most significant in differentiating the clusters (Table 1): cluster “1” was representative for angles smaller than 90 degrees and revealed the “sharp turn” movement pattern while cluster “2” was representative for angles larger than 90 degrees which corresponded to “shallow turn” movement patterns.

Identifying and quantifying these two types of turns are important in a variety of behaviors in the presence and absence of any external stimuli since the worm uses shallow turns to follow sensory information effectively [23–25]. However, only few studies have experimentally and theoretically characterized the shallow turns and quantified their occurrences [26].
2.3. RunLength Encoding Description of C. elegans Movement Path
The symbolbased encoded path is further quantified based on a runlength encoding approach [18] that quantifies the frequency of symbols as well as the order in which they appear. For example, for a sequence of symbols “,” the runlength encoding will be a sequence of pairs “.”
Based on the runlength encoding results for each video, a runlength encoding matrix is generated:The rows represent the symbols, columns represent the unique frequencies that symbol shows up in the sequence, and is the count of how many times symbol shows up for consecutive times in the sequence. Matrix can be generated in different ways depending on how the maximum value of , denoted by , is calculated: is the Maximum Frequency for Individual video data (Matrix IMF); Maximum Frequency across All videos (Matrix AMF); Binned and Maximum Frequency across All videos (Matrix BAMF), and Binned, Averaged, and Maximum Frequency across All videos (Matrix BAAMF). Table 2 shows an example of these four matrices for one of the worm paths.

The four runlength encoding matrices are Row 1: size(IMF) = 2 × 21 (21 is the length of the longest sequence of symbols for one worm path); Row 2: size(AMF) = 2 × 37 (37 is the length of the longest sequence of symbols across all worm paths); Row 3: size(BAMF) = 2 × 7 (7 is the number of bins in which the lengths were discretized: 1 = 1, 2 = 2, 3 = 3, 4 = 4⋯8, 5 = 9⋯16, 6 = 17⋯32, and 7 = 33⋯37); Row 4: size(BAMF) = size(BAAMF) rather than representing as the next available index; here it is represented by the average length within the corresponding bin. The benefit of having AMF versus IMF is that it takes into account the fact that videos have different lengths of time. Furthermore, by binning the lengths, we reduce the sparsity of the runlength encoding matrices as sequences with higher lengths will have less frequency values.
The runlength encoding matrix is further used to extract descriptors using an approach similar to [18] rather than encoding the texture information of an image we propose to encode the frequencies and orders of pattern movements (symbols) of a worm path. The definition of the proposed path descriptors (Table 3) involves four important concepts: short run, long run, low angle level, and high angle level.

Short run encoding (SRE) means that the same symbol is repeated shortly and quickly changes into another value; long run encoding (LRE) means the same symbol is repeated for a long time before changing into another value; low angle encoding (LAE) means the angle between two steps is less than 90 degrees (represented by cluster “1” and defined as a sharp turn); and high angle encoding (HAE) means the angle between two steps is more than 90 (represented by cluster “2” and defined as a shallow turn). Using these concepts, we extract eleven path descriptors quantifying the turns and reversals’ frequencies and their orders. Table 3 provides the formulas and descriptions where is the number of symbols found using clustering (Section 3.2), is the maximum run length, is the total number of distinct run lengths, and is the number of total steplengths.
2.4. Path Similarity Analysis
Once the path is decomposed into the basic locomotory gaits and transitions between them, the decomposition can be used to understand how animals respond to their environment through movement. In particular, since turning is a fundamental way by which C. elegans reorients itself in response to the presence or absence of external stimuli, we propose to use the RLE (runlength encoding) descriptors associated with the symbolencoded paths to understand how C. elegans wild type and tph1 behave in the presence and absence of food.
To find the similarities and differences among different search behaviors, we apply means clustering approach on the C. elegans data encoded using the set of eleven RLE descriptors. Given the five different types of experiments (N2_f, N2_nf, N2_nnf, tph1_f, and tph1_nf), we began means clustering with and then gradually reduced the value of . The analysis of each cluster will then show which C. elegans paths are similar as a function of turn type as well as turn type’s frequency and order.
Table 4 shows examples of paths for either low or high values of these descriptors to exemplify how the differences in the descriptor values can help differentiate among different worm paths. Since some of these descriptors are correlated with each other, a data redundancy step is applied on the descriptor data to diminish the effects of this redundancy on the next stage of finding path similarities.

3. Results and Discussion
3.1. Encoding the Path as a Sequence of Symbols Using Clustering of Step Data
The path was decomposed into steps as described in Section 2.1, resulting in 30,125 steps across all 57 videos for the two types of worms under different food conditions. means algorithm was applied on the step data (step length, turning angle, and speed along the step) and the number of clusters was varied from 2 to 5 (as described in Section 3.2). Table 5 shows the characteristics of the clusters for each value of . While the values of the steplength data overlap for length and speed across the clusters for each value of , the angle clearly differentiates between the types of clusters. To validate the clustering results and the selection of , we performed a decision tree classification approach and recorded the accuracy of each of the four classifiers (one classifier for each value of ) in predicting the classes associated with the clusters. In all four scenarios, the classification accuracy was 99% meaning that, based on the angle feature, the step data can be clearly divided into 2, 3, 4, or 5 groups of turning angles.

In this study, we decided to choose a coarse level of granularity in quantifying the path movement patterns based on the steplength data by putting the data into two categories () and show that the two identified turning patterns are important in differentiating behavior in the presence and absence of food. We noticed that when (Table 5), the two clusters represent two movement patterns, shallow and sharp turns, characterized by angles less than 90 degrees and angles larger than 90 degrees, respectively. In other words, cluster symbol “1” will denote a sharp turn defined as an angle between two consecutive steps being less than 90 degrees and cluster symbol “2” will denote a shallow turn defined as the angle between two consecutive steps being larger than 90 degrees. Within each of the two clusters, although there is a significant overlap between the ranges for speed and length, we noticed that cluster “1” had larger values for its outliers with respect to speed and cluster “2” had larger lengths for its step data (Figure 3). In other words, there are some sharp turns that are executed at higher speeds and some shallow turns that correspond to larger steps.
(a)
(b)
(c)
(d)
3.2. RunLength Encoding Descriptors
The symbolencoded paths of 1s and 2s were further encoded using the runlength encoding approach presented in Section 3.3. Using (1) we generated four runlength encoding matrices (IMF, AMF, BAMF, and BAAMF) and two additional matrices that were the logarithm transformation of the matrices IMF and AMF. For each RLE matrix, we extracted the set of eleven runlength encoding descriptors (using the formulas from Table 3) and analyzed the power of these descriptors in differentiating between N2 and tph1 on food and off food. As some of the 11 features are highly correlated, to reduce the redundancy of the data, we performed a correlation analysis for each set of descriptors and kept only the descriptors with correlations smaller than 0.9 for further analysis. Table 6 shows that only five RLE descriptors can be used further to learn the similarities among the different types of worms and food conditions.

3.3. Clustering of C. elegans Using RLE Descriptors
The C. elegans data was grouped into five clusters using all eleven features and the selected set of features for each one of the six RLE matrices. The analysis of the clustering showed the same groupings for the 57 videos regardless of the calculation of the RLE matrix and the use of all or reduced set of RLE descriptors. Therefore, we present in Table 7 the results for one of the combinations, BAMF matrix and its two selected RLE descriptors, RP and ALN.

The results show that, using the RLE descriptors, we are able to separate 75% of the N2 on food data (cluster 5) from all the other C. elegans data. In addition, N2 off food data are relatively well clustered together with tph1 off food data in cluster 1 and cluster 3. Most importantly, 100% of the tph1 on food data is grouped with 64% of N2 off food data (clusters 3 and 4). Our observation of tph1 mutants, which fail to produce serotonin, suggests that tph1 mutants continue food search behavior even on food. 100% of the tph1 off food data were also grouped with 91% of the N2 off food data across three clusters (clusters 1, 2, and 3). These observations show that tph1 mutants either on food or off food behave like wildtype off food animals. Interestingly, through our approach we were also able to distinguish between the two differently collected N2nnf data, one in which the worm was starved for a longer period of time.
4. Conclusions
In this study we aimed to quantify the movement behaviors of wildtype N2 and tph1 mutant animals, so that we could identify if tph1 mutants show defects in foraging and food search behaviors. By encoding worms’ path, extracting features for each worm, and performing clustering analysis, we conclude that the locomotory behavior of tph1 on food resembles the wildtype N2 animal off food. In addition, we found that five of the eleven RLE descriptors can be used to efficiently represent the path characteristics for C. elegans data. Our unbiased, unsupervised analysis provides evidence that shallow and sharp turns are the most critical factor that distinguishes C. elegans movement behaviors in the presence and absence of food. This finding is surprising in that we expected that based on previous studies speed is one of the important factors that determine on food and off food behaviors [13, 16].
As future work, we will investigate the relationships among the C. elegans data when encoding the path with more than just two movement patterns. We will look into the cluster patterns formed for larger values of and identify subcluster patterns within the two identified patterns. For the latter, we expect that, for example, if duration of a traveling a step will be important, we can further refine the turns into either short or long turns. We will be also looking into other similarity metrics for the clustering approaches. While we normalized the features individually to the range 0 to 1 using the minmax normalization and then used the Euclidean distance, we have not accounted for the fact that one of the features is an angular component. We will address this limitation by designing a combined similarity metric that will include a cosine similarity metric for the angular component similarity comparisons.
While the RLE descriptors provide important information about the distribution of different path characteristics, the timecomponent is not integrated in the analysis. To address this limitation, our future work will include the investigation of dynamic time warping (DTW) to compare C. elegans paths encoded through sequences of pattern movements. Furthermore, we will be looking into the generalizability of the proposed approach to other domains that make use of trajectory data such as for animal or human mobility studies based on GPS data.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by the DePaulRosalind Franklin University Grant Program.
References
 D. D. Shaye and I. Greenwald, “OrthoList: a compendium of C. elegans genes with human orthologs,” PLoS ONE, vol. 6, no. 5, Article ID e20085, 2011. View at: Publisher Site  Google Scholar
 J. Y. Sze, M. Victor, C. Loer, Y. Shi, and G. Ruvkun, “Food and metabolic signalling defects in a Caenorhabditis elegans serotoninsynthesis mutant,” Nature, vol. 403, no. 6769, pp. 560–564, 2000. View at: Publisher Site  Google Scholar
 D. L. Chase and M. R. Koelle, “Biogenic amine neurotransmitters in C. elegans,” WormBook, vol. 20, pp. 1–15, 2007. View at: Google Scholar
 K. Andre, O. Kampman, M. Viikki et al., “TPH1 A218C polymorphism and temperament in major depression,” BMC Psychiatry, vol. 13, article 118, 2013. View at: Publisher Site  Google Scholar
 D. Ramot, B. E. Johnson, T. L. Berry Jr., L. Carnell, and M. B. Goodman, “The parallel worm tracker: a platform for measuring average speed and druginduced paralysis in nematodes,” PLoS ONE, vol. 3, no. 5, Article ID e2208, 2008. View at: Publisher Site  Google Scholar
 A. M. Leifer, C. FangYen, M. Gershow, M. J. Alkema, and A. D. T. Samuel, “Optogenetic manipulation of neural activity in freely moving Caenorhabditis elegans,” Nature Methods, vol. 8, no. 2, pp. 147–152, 2011. View at: Publisher Site  Google Scholar
 S. Faumont, G. Rondeau, T. R. Thiele et al., “An imagefree optomechanical system for creating virtual environments and imaging neuronal activity in freely moving caenorhabditis elegans,” PLoS ONE, vol. 6, no. 9, Article ID e24666, 2011. View at: Publisher Site  Google Scholar
 J. Ben Arous, Y. Tanizawa, I. Rabinowitch, D. Chatenay, and W. R. Schafer, “Automated imaging of neuronal activity in freely behaving Caenorhabditis elegans,” Journal of Neuroscience Methods, vol. 187, no. 2, pp. 229–234, 2010. View at: Publisher Site  Google Scholar
 K. Moy, W. Li, H. P. Tran et al., “Computational methods for tracking, quantitative assessment, and visualization of C. elegans locomotory behavior,” PLoS ONE, vol. 10, no. 12, Article ID e0145870, 2015. View at: Publisher Site  Google Scholar
 C. M. Loer and C. J. Kenyon, “Serotonindeficient mutants and male mating behavior in the nematode Caenorhabditis elegans,” The Journal of Neuroscience, vol. 13, no. 12, pp. 5407–5417, 1993. View at: Google Scholar
 J. E. Tanis, J. J. Moresco, R. A. Lindquist, and M. R. Koelle, “Regulation of serotonin biosynthesis by the G proteins $\text{G}{\alpha}_{\text{o}}$ and $\text{G}{\alpha}_{\text{q}}$ controls serotonin signaling in Caenorhabditis elegans,” Genetics, vol. 178, no. 1, pp. 157–169, 2008. View at: Publisher Site  Google Scholar
 L. E. Waggoner, G. T. Zhou, R. W. Schafer, and W. R. Schafer, “Control of alternative behavioral states by serotonin in Caenorhabditis elegans,” Neuron, vol. 21, no. 1, pp. 203–214, 1998. View at: Publisher Site  Google Scholar
 E. R. Sawin, R. Ranganathan, and H. R. Horvitz, “C. elegans locomotory rate is modulated by the environment through a dopaminergic pathway and by experience through a serotonergic pathway,” Neuron, vol. 26, no. 3, pp. 619–631, 2000. View at: Publisher Site  Google Scholar
 S. W. Flavell, N. Pokala, E. Z. Macosko, D. R. Albrecht, J. Larsch, and C. I. Bargmann, “Serotonin and the neuropeptide PDF initiate and extend opposing behavioral states in C. elegans,” Cell, vol. 154, no. 5, pp. 1023–1035, 2013. View at: Publisher Site  Google Scholar
 V. Padmanabhan, Z. S. Khan, D. E. Solomon et al., “Locomotion of C. elegans: a piecewiseharmonic curvature representation of nematode behavior,” PLoS ONE, vol. 7, no. 7, Article ID e40121, 2012. View at: Publisher Site  Google Scholar
 J. M. Gray, J. J. Hill, and C. I. Bargmann, “A circuit for navigation in Caenorhabditis elegans,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 9, pp. 3184–3191, 2005. View at: Publisher Site  Google Scholar
 R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, John Wiley & Sons, New York, NY, USA, 2000.
 X. Tang, “Texture information in runlength matrices,” IEEE Transactions on Image Processing, vol. 7, no. 11, pp. 1602–1609, 1998. View at: Publisher Site  Google Scholar
 D.H. Xu, A. S. Kurani, J. D. Furst, and D. S. Raicu, “Runlength encoding for volumetric,” in Proceedings of the 4th IASTED International Conference on Visualization, Imaging, and Image Processing, 2004. View at: Google Scholar
 I. Martinez, Using the Run Length Encoding Features on the MPC5645S, Freescale Semiconductor, 2011.
 I. Rhee, M. Shin, S. Hong, K. Lee, S. J. Kim, and S. Chong, “On the Levywalk nature of human mobility,” IEEE/ACM Transactions on Networking, vol. 19, no. 3, pp. 630–643, 2011. View at: Publisher Site  Google Scholar
 L. Breiman, “Random forests,” Machine Learning, vol. 45, no. 1, pp. 5–32, 2001. View at: Publisher Site  Google Scholar
 L. Luo, D. A. Clark, D. Biron, L. Mahadevan, and A. D. T. Samuel, “Sensorimotor control during isothermal tracking in Caenorhabditis elegans,” Journal of Experimental Biology, vol. 209, no. 23, pp. 4652–4662, 2006. View at: Publisher Site  Google Scholar
 Y. Ino and K. Yoshida, “Parallel use of two behavioral mechanisms for chemotaxis in Caenorhabditis elegans,” The Journal of Neuroscience, vol. 29, no. 17, pp. 5370–5380, 2009. View at: Publisher Site  Google Scholar
 G. J. Stephens, B. JohnsonKerner, W. Bialek, and W. S. Ryu, “From modes to movement in the behavior of Caenorhabditis elegans,” PLoS ONE, vol. 5, no. 11, Article ID e13914, 2010. View at: Publisher Site  Google Scholar
 D. Kim, S. Park, L. Mahadevan, and J. H. Shin, “The shallow turn of a worm,” Journal of Experimental Biology, vol. 214, no. 9, pp. 1554–1559, 2011. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2016 Li Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.