Elsevier

Neurocomputing

Volume 131, 5 May 2014, Pages 248-264
Neurocomputing

A 3D undulatory locomotion model inspired by C. elegans through DNN approach

https://doi.org/10.1016/j.neucom.2013.10.019Get rights and content

Abstract

In this work, a 3D undulatory locomotion model inspired by Caenorhabditis elegans is constructed. Following the anatomical structure of C. elegans, the body of the model is represented as a multi-joint rigid link system with 12 links. The angle between two consecutive links is determined by the muscle lengths in four quadrants that are controlled by the nervous system. The nervous system of this locomotion model is represented by a dynamic neural network (DNN) that involves three parts: head DNN, central pattern generator (CPG), and body DNN. The head DNN decides turning or not, and CPG produces the sinusoid waves that are transmitted through the body DNN to control the lengths of muscles. The 3D locomotion behavior is achieved by using the DNN to control the muscle lengths, and then using the muscle lengths to control the angles between two consecutive links on both horizontal plane and vertical plane. In this work, the relations between the outputs of DNN and muscle lengths, as well as the muscle lengths and the angles between two consecutive links, are determined. Furthermore, due to the learning capability of DNN, a set of nonlinear functions that are designed to represent the chemotaxis behaviors of C. elegans are learned by the head DNN using Differential Evolution Algorithm. The testing results show good 3D performance of this locomotion model in both forward and backward locomotion, as well as slight turn and Ω turn. Furthermore, this locomotion model performs the chemotaxis behaviors of finding food and avoiding toxin successfully. Finally, quantitative analyses by comparing with the experiment results are provided to verify the realness and effectiveness of this locomotion model, which could serve as a prototype for the worm-like robot.

Introduction

Undulatory locomotion is one of the fundamental behaviors of the footless animals, such as larva, worm, snake, and even some mammals. Among these animals, the snake has been widely studied to disclose the mechanism of undulatory motion. However, due to the huge amount of neurons and muscle-bone structures of the snake, it is difficult to study the motion mechanism from cellular level. Fortunately, Caenorhabditis elegans offers us an idea model to study the mechanism of undulatory locomotion behavior. The nervous system of C. elegans contains 302 neurons and 95 muscles, and all the neuronal connections are visibly known [1]. The undulatory locomotion of C. elegans is similar to other limbless animals, such as snake, and the neural circuit of C. elegans is much simpler than that of snake. So the clearly described nervous and muscular systems of C. elegans provide us a good opportunity to investigate the essence of undulatory locomotion from cellular level. Furthermore, if we realize these mechanisms on the computer, it may be possible to incorporate these biological or biomimetic methods into the undulatory robots, which can achieve at least four tasks: (1) rescuing survivors in complex areas where human cannot enter [2]; (2) checking the inner side of the industrial equipment pipes [3]; (3) crawling on the ground, under water, or inside the pipes for military utilities [4]; (4) checking the stomach, blood vessels, or intestine for clinical use [5].

The study of undulatory locomotion behavior of C. elegans begins in recent years. From the biological aspect, Lockery et al. investigated the locomotion of C. elegans for the chemotaxis behaviors [6], [7], [8]. Wakabayashi et al. investigated the mechanism of neurons to regulate the duration of forward locomotion of C. elegans [9], and de Bono et al. studied the neuronal substrates of complex behaviors in C. elegans [10]. From the engineering aspect, Suzuki et al. explored the locomotion behaviors of C. elegans in forward movement, backward movement, and turning [11], [12], [13], [14]. Boyle et al. verified that the body wall muscles of C. elegans were simple actuators [15] and the different patterns for swimming and scrawling represented the modulation of a single gait that was generated by a single neural circuit [16], [17], [18]. They also investigated the sensory feedback mechanism of C. elegans, which produced the undulatory wave instead of CPG [19]. Lastly, inspired by C. elegans they designed and constructed a micro-robot with polymeric actuators [20] and a giant robot based on the sensory feedback mechanism [21]. In work [22], a locomotion model was constructed based on the experimental results. This computational model can predict the mechanisms underlying various behaviors of mutant C. elegans. In [23], [24], two robots based on Shape Memory Alloy (SMA) are constructed to mimic the undulatory behavior of C. elegans. The above models of C. elegans crawl in 2D, namely, performing the undulatory behavior on the horizontal plane. The model in [14] can lift up its head during moving, but the rest of its body crawls in 2D on the ground. The 3D locomotion behaviors of C. elegans was first verified by [25], but no mathematical locomotion model was provided. Till now the only 3D locomotion model of C. elegans was constructed in [26]. The body of this 3D model was assembled by 25 boxes and the muscles were represented by the springs. Without preserving the biological anatomical structure, the locomotion behavior of this 3D model was displayed mainly in the game engine, and no precise mathematical description was given.

We analyze the 3D locomotion behavior of C. elegans by observing the video record, as shown in Fig. 1. The arrows in Fig. 1 indicate the lift parts when C. elegans is in high-speed crawling. This 3D locomotion phenomenon is similar to the result of [25], and it is also analogous to the locomotion of snake that lifts up the most bent parts during full speed movement [27]. According to this phenomenon, in this work we construct a bio-inspired locomotion model based on the biological structure of C. elegans and use the DNN approach to produce the 3D undulatory locomotion pattern. The bio-inspired locomotion model can overcome the limitation of the biological locomotion behaviors of C. elegans and serve as the prototype for the worm-like robot. The novelty of this work involves six aspects.

First, without losing the reality, the bio-inspired locomotion model is based on the anatomical muscle structure of C. elegans provided by [28]. The whole body is divided into 11 muscle segments, and in each muscle segment there are four pieces of muscles located in four quadrants. These 11 muscle segments are represented as a multi-joint rigid link model with 12 links and 13 joints. The first joint and the last joint denote the head and the tail tips, and other eleven joints stand for the center of each muscle segment.

Second, DNN is adopted to represent the nervous system of C. elegans in this model. DNN has the function of mimicking the nervous system and realizing some important functions of real animals, such as dynamic locomotion [29], associate memory [30], etc. Furthermore, the strong learning capability of DNN can make the locomotion system adapt different environment quickly, and it also can generate different locomotion patterns according to different environments. In this model, DNN involves three parts: head DNN, CPG, and body DNN. The head DNN contains two sensory neurons, three interneurons and one output neuron. The function of the head DNN is to smell the outside concentration and make the decision of turning or not. CPG produces the sinusoid waves for undulatory locomotion. The body DNN receives the sinusoid waves generated by the head DNN and CPG, and passes the waves through each segment. In each segment, the structure of the body DNN is inspired by or similar to the real connectivity, and the function of the body DNN is to produce the signals to the muscles in four quadrants. The phase lag of the sinusoid wave is produced by incorporating a time delay when the wave is transferred through DNN in consecutive segments.

Third, the relation between 3D body shape and muscle lengths is determined. Biologically, C. elegans has a compliant body. For the bio-inspired model, its 3D shape is determined by the joint angles projected onto the sagittal plane (dividing the worm into left and right parts, namely, xy plane) and the coronal plane (dividing the worm into dorsal and ventral parts, namely, xz plane). Comparing with other existing models, our model uses the lengths of muscles to control the joint angles directly. In our work, the relations between these angles and the lengthes of four quadrant muscles are determined. In other words, once we know the lengths of muscles, the 3D shape is fixed.

Fourth, the muscle model is designed to associate with the outputs of DNN. When receiving the sinusoid wave from the DNN, the length of muscle will change according to the magnitude of signal, and the joint angle will also change accordingly. In this way, we can use the outputs of DNN to control the shape of the model.

Fifth, with the strong learning capability of DNN, our model can implement the chemotaxis behaviors of finding food and avoiding toxin concurrently after the head DNN learns a set of nonlinear functions. These nonlinear functions are called switching logic functions (SLFs) in our work and are designed to approximate the logic of C. elegans for chemotaxis behavior. Once SLFs are well learned, our model can perform the chemotaxis behaviors in different simulated environments.

Lastly, image records of actual C. elegans are analyzed to verify the effectiveness of our model. The mechanisms to generate the slight turn and sharp turn are also investigated. Furthermore, quantitative analyses of the trajectories of our model are carried out by comparing with the experiment results. Finally, we simplify the well optimized head DNN to smaller ones according to the method of [31] and find out two patterns that are similar to the result in [31].

This paper is organized as following. Section 2 provides the biological background of C. elegans, which includes of muscle structure and neuronal structure. Section 3 discusses the locomotion system of the model, which includes the head DNN, CPG, body DNN, and muscle system. The structure of DNN and muscles, as well as the mathematical models are provided in details. In Section 4, we first investigate the 3D body shape during locomotion, and then explore the relation between the muscle lengths and the joint angles on the sagittal plane (xy plane) and the coronal plane (xz plane). Lastly the relation between muscle lengths and motor neuron outputs is determined. Thus, once the outputs of DNN are known, the 3D shape of the locomotion model is determined. In Section 5, the methods to optimize the head DNN and the body DNN are provided. Section 6 gives the testing results. In Section 7, the analyses of the testing results are provided. Section 8 concludes this paper and discusses the future works.

Section snippets

Muscle and body structure

C. elegans with a simply cylindrical body is about 1 mm in length. The body wall muscles can be classified into 4 quadrants on the transverse plane, as shown in Fig. 2(a). These four quadrants are dorsal-left (DL), ventral-left (VL), ventral-right (VR), and dorsal-right (DR). Its 95 body wall muscle cells are arranged as pairs located in four quadrants along the body, as shown in Fig. 2(b). Each of DL, VR, and DR quadrants contains 24 muscle cells, except for the VL quadrant, which contains 23

Locomotion model

The locomotion of our model requires the undulatory signals being sent from the nervous system to the motor neurons, and then to the muscles. Muscles receive the undulatory signals to perform the corresponding behaviors. In our work, we adopt DNN to represent the nervous system of C. elegans, which is classified into three parts: head DNN, CPG, and body DNN. Muscle is modeled by investigating the relation between the muscle length and the neuronal inputs.

Motion modality

Undulatory movement is the primary way for C. elegans to move. According to [22], the whole body of C. elegans shapes as a sinusoid wave about 1.5–2 periods during locomotion on agar gel. In our work we consider that the wave length of C. elegans is 1.5 sinusoid wave periods. Thus, the form of our model during locomotion is shown in Fig. 7.

Fig. 7(a) shows the shape of the locomotion model on the horizontal (xy) plane by overlooking, and (b) shows its shape on the vertical (xz) plane. During

Optimization

For our model, the parameters of the head DNN and body DNN should be optimized to implement the undulatory locomotion behavior. In this section, the optimization procedures are discussed in detail.

Testing results

In this section, we test (1) the lengths of muscles in four quadrants, which vary periodically; (2) behaviors of forward and backward locomotions; (3) shape of C. elegans during locomotion in 3D; (4) finding food; (5) avoiding toxin; (6) finding food and avoiding toxin simultaneously. Except the head DNN is optimized by DE, other parameter values are obtained by trial and error. Parameter settings are listed in Table 1. All the tests are conducted in Matlab 2011(b), which is set up in Win7

Validation by analyzing the video of C. elegans

The video of actual C. elegans is provided by Yong Loo Lin School of Medicine, National University of Singapore, which can be accessed at website [54]. C. elegans is enlarged 50 times in the video. A software, Nandub, is used to transfer the video to image files in JPG format with 30 frames per second. These images are imported into the image processing software, Scion Image, for analyzing the velocity, joint angle, and body shape.

The velocity of the head of C. elegans is computed by using the

Conclusion

In this work, a 3D undulatory locomotion model inspired by C. elegans is constructed to perform the forward and backward locomotion, as well as the chemotaxis behaviors of finding food and avoiding toxin. The locomotion model is represented as 12 multi-joint rigid links. Its nervous system is divided into three parts: the head DNN, CPG, and the body DNN. The head DNN has the learning ability to learn the SLFs, which represents the chemotaxis behaviors of C. elegans. CPG generates the sinusoid

Xin Deng received the Bachelors degree from Department of Computer Science and Technology, Jilin University, Changchun, China, in 2004, and Masters degree from Department of Computer Science, Chongqing University, Chongqing, China, in 2007. He was awarded the Ph.D. degree in computer engineering from National University of Singapore, Singapore, in 2013. He is now an assistant professor in College of Computer Science and Technology, Chongqing University of Posts and Telecommunications in China.

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    Xin Deng received the Bachelors degree from Department of Computer Science and Technology, Jilin University, Changchun, China, in 2004, and Masters degree from Department of Computer Science, Chongqing University, Chongqing, China, in 2007. He was awarded the Ph.D. degree in computer engineering from National University of Singapore, Singapore, in 2013. He is now an assistant professor in College of Computer Science and Technology, Chongqing University of Posts and Telecommunications in China.

    Jian-Xin Xu received the Bachelors degree in electrical engineering from Zhejiang University, Hangzhou, China, in 1982, and the Masters and Ph.D. degrees in electrical engineering from the University of Tokyo, Tokyo, Japan, in 1986 and 1989, respectively.

    He spent one year at the Hitachi Research Laboratory, Ibaraki, Japan, more than one year in Ohio State University, Columbus, as a Visiting Scholar, and 6 months in Yale university, New Haven, CT, as a Visiting Research Fellow. He joined the National University of Singapore, Singapore, in 1991, and is currently a Professor at the Department of Electrical Engineering. He is entitled as the IEEE Fellow since January 2012. His current research interests include learning theory, intelligent control, nonlinear and robust control, robotics, and precision motion control.

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