Towards a virtual C. elegans: A framework for simulation and ...

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In Silico Biology 11 (2011/2012) 137–147 DOI 10.3233/ISB-2012-0445 IOS Press

Towards a virtual C. elegans: A framework for simulation and visualization of the neuromuscular system in a 3D physical environment Andrey Palyanova,*, Sergey Khayrulina, Stephen D. Larsonb and Alexander Diberta,c a

A.P. Ershov Institute of Informatics Systems SB RAS, Lab. of Complex Systems Simulation, Acad. Lavrentjev pr., Russia b OpenWorm Project, http://openworm.org c Novosibirsk State University, Novosibirsk, Russia

Abstract. The nematode C. elegans is the only animal with a known neuronal wiring diagram, or “connectome”. During the last three decades, extensive studies of the C. elegans have provided wide-ranging data about it, but few systematic ways of integrating these data into a dynamic model have been put forward. Here we present a detailed demonstration of a virtual C. elegans aimed at integrating these data in the form of a 3D dynamic model operating in a simulated physical environment. Our current demonstration includes a realistic flexible worm body model, muscular system and a partially implemented ventral neural cord. Our virtual C. elegans demonstrates successful forward and backward locomotion when sending sinusoidal patterns of neuronal activity to groups of motor neurons. To account for the relatively slow propagation velocity and the attenuation of neuronal signals, we introduced “pseudo neurons” into our model to simulate simplified neuronal dynamics. The pseudo neurons also provide a good way of visualizing the nervous system’s structure and activity dynamics. Keywords: Virtual organism, C. elegans, simulation, neuro-cybernetics, muscular system, ventral neural cord

1. Introduction Artificial Intelligence (AI), the pursuit of computer systems that can rival human intelligence, has been sought after since the beginning of computers [1]. While investigation in this field has given rise to impressive progress in computational technologies and robotics, after over 50 years of work there are still no systems whose general intelligence across subject matters and mental faculties approaches that of human beings. In the absence of a complete reproduction of the human brain, previous approaches endeavored to build human-level artificial intelligence *Corresponding author: Andrey Palyanov, A.P. Ershov Institute of Informatics Systems SB RAS, Lab. of Complex Systems Simulation, Acad. Lavrentjev pr., 6, Novosibirsk, Russia 630090. Phone: +7(383) 330 7068; Fax: +7(383)332 3494; E-mail: [email protected].

without a mechanistic understanding of the original. In contrast to this approach, we believe that a functional virtual mechanistic reproduction of the simplest existing nervous system, simulated at the level of individual nervous cells and connections between them will enable a mechanistic understanding of nervous systems at the level necessary to build them. Citing P. Singh, “Most AI researchers are looking for simple explanations to inherently complex phenomena […] we should copy biology – there are doubtless hundreds of different kinds of mechanisms in the brain, specialized to different sorts of tasks, integrated in an equally complex management structure” [2]. While coming from a different direction – reproduction via simulation rather than investigation using experimentation – this endeavor shares the central goal of neuroscience – to understand how information is processed in neural

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circuits, and what physical principles underlie that information processing. Caenorhabditis elegans (C. elegans), one of the most well-studied species in the world, is an ideal candidate for this investigation. John Sulston, 2002 Nobel Prize laureate, who was awarded for study of this nematode, said about the C. elegans: “If we understand the worm, we understand life”. The C. elegans nervous system is invariant between animals of one sex (both male and hermaphrodite are possible). The hermaphrodite is composed of 959 cells; 302 are neurons and 95 are body wall muscles. It has about 7000 synaptic connections, over 2000 neuromuscular junctions [3,4]. The C. elegans connectome, in terms of synaptic partners of nerve cells and muscles, was determined about 25 years ago [3] and subsequently updated [5]. While C. elegans has a reasonable quantity of biological structures to understand, it is capable of rather complex behavioral patterns, as evidenced by its abilities for learning and memory [6]. Taken together, these facts imply that a rich spectrum of studies is possible to be carried out with C. elegans. In spite of continued interest to study C. elegans with experimental and computational methods, yielding a considerable amount of results in the field, there are still more questions than answers. Among the unresolved questions are the unknown neuronal mechanisms involved in the generation of sinusoidal patterns received by the body wall muscles during forward and backward locomotion [7,8]. Many species rely on a central pattern generator (CPG) for undulatory locomotion, but such a CPG has been elusive in the C. elegans wiring diagram [7]. A number of hypotheses explaining this mechanism have been proposed. Among them there are wave propagation via adjacent motor neurons of a given class that are connected by gap junctions [3] and the possibility that motor neurons could themselves act as stretch receptors, enabling wave propagation via contraction of body muscles. Additionally, the field has not settled on the principles of computation of interneurons, which seem to lack action potentials due to the high specificity of C. elegans neurons, and which are extremely small with high membrane resistance [9,10]. In order to begin to answer these questions, we believe that consolidating existing data sets into an integrated computational model is necessary. C. elegans, like all living organisms, is a physical system, albeit a complex one. Just as climate science has created integrated computational models for the weather, or astronomy has

created integrated computational models for planetary motion, we believe that biology must also move towards simulation-based research, and that an integrated computational model of C. elegans represents an important first step. A high degree of interest in the idea of the creation of a virtual organism based on the C. elegans has resulted in several important efforts. Experimental data about the C. elegans nervous system were updated several times and at the end of 2009. An interactive C. elegans neural network connectivity browser [11] provided a straightforward way to work with it. More recently the Virtual Worm project, a project that aims to create a cell-by-cell reconstruction of the adult hermaphrodite so as to generate a virtual, 3-dimensional, interactive atlas of the anatomy of C. elegans, (http:// caltech.wormbase.org/virtualworm/) accumulated even more detailed information. They have presented the worm body and all its inner structure, including neurons, muscle cells, sensory cells and connections between them as a Blender (http://blender.org) file available for download, containing all these data in the form of 3D objects. This has opened some new options in building a biologically reasonable detailed 3D model of C. elegans, and has helped to catalyze an international project named OpenWorm (http://openworm.org, http:// openworm.googlecode.com) aimed at developing a fully digital lifeform – a virtual nematode – in a completely open source manner. Here we present a proof of concept prototype of a virtual 3D C. elegans. The prototype is an extensible interactive simulation with a 3D graphical interface and its purpose is to accumulate existing and future data describing the worm’s biological systems in the form of a functional integrated model. This article describes the current state of the project and reports on our preliminary results.

2. Materials and methods Our prototype is a 3D dynamic model which consists of partial nervous and muscular systems interacting with each other in a simulated physical environment. We chose this set of systems to simulate because together they enclose the feedback loop between the worm’s nervous system, the movement of muscles caused by the nervous system, and the changes of the local environment of the worm, which in turn provides updated sensory data back to the nervous system.

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In earlier modeling approaches investigating the C. elegans locomotion mechanism and its neural control, often only two-dimensional models were used. [12,13,7]. In contrast, here we have built a dynamic, interactive 3D simulation into which we have consolidated, simulated and visually represented data about the C. elegans anatomy. The C. elegans muscular system is, of course, three-dimensional. It consists of 4 muscle quadrants running longitudinally along the inside of the body cavity [3] that two-dimensional models are unable to fully represent. As a result, two-dimensional models introduce approximations at the level of connections between motor neurons and muscles which, in turn, require compromises and approximations in the model. Additionally, visualization of the nervous system attached to the worm body (see below, Neurons and Connections) enables a user to observe the relationship between simulated nervous system activity and outward behavior. In addition, in order to properly collect sensory information from the surface of the C. elegans body once the model is supplemented with a sensory system, it will be necessary to place sensory receptors at their correct locations on the worm body. For this purpose a full three-dimensional model that corresponds to the body plan of the animal is preferred. To build the model, we have developed a physics simulation engine that facilitates the construction of complex dynamic objects. Objects can be composed of the following primitives: mass points, springs (which connect a pair of mass points), muscles (active springs, which contract proportionally to the intensity of a signal from a connected motor neuron), neurons, and lastly, connections between two neurons or between a neuron and a muscle. Any combination of objects composed of primitives can themselves be combined into more complex objects in a virtual physical environment. Between objects, relations can be defined that enable physical forces to act upon them. One type of relation is a physical connector, which enables mass points to be influenced by force from one another due to their connection to a spring or muscle. Another one is a physical repulsion, such as the interaction of a mass point with a plane or surface, such as a virtual ‘Petri dish’, that prevents penetration through the object. These relations, along with forces derived from gravity and friction, sum together to determine the total force acting on each mass point at any moment of time t. Based on the total force, the movement of each point

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at t + dt is calculated by means of numerical integration (currently our time step dt = 8 ∙ 10−3 s). 2.1. Some technical issues The C. elegans neuromuscular simulation and visualization system described here is written in C++, STL and OpenGL. It is fast enough to run on a personal computer with modern processor resources, in our case a Core2Duo 3 GHz. A video illustrating the main aspects of the simulation is available here: http:// www.youtube.com/watch?v=3uV3yTmUlgo. In addition, the latest version, compiled for Microsoft Windows, is available for download here: “http://open worm.googlecode.com/files/c. elegans 2.087+[upd. 2012] full.rar”. Instructions are available by pressing the “F1” key. The file “readme.txt” in the archive also provides a list of available commands as well as instructions on how to build the source code included in the archive. 2.2. Flexible body model C. elegans, which lacks any form of rigid skeleton, has a spindle-shaped body with a diameter of ~60 μm across its midsection and a length of ~1 mm in its adult phase [14]. An elastic cuticle filled with pressurized fluid maintains its body shape, which remains flexible [15]. Our physical model of the worm’s body, in many aspects inspired by [16] is composed of mass points and springs (Fig. 1). Springs running along the body wall serve to imitate the cuticle, while an internal spring structure imitates the fluid pressure. The springs used include a damping factor to prevent infinite oscillations, similar to those used by Boyle [17]. To reproduce the geometrical relations of the worm in the model, we have adhered to the width and length profile described by Altun and Hall [18, Fig. 7]. This construction enables the model to have enough flexibility to reproduce different body configurations produced by C. elegans (sinusoidal shape, omegaturn, coil). 2.3. Muscular system C. elegans muscles are attached to the inside of the cuticle and are grouped into 4 longitudinal tracts. To our best ability, we have approximated the muscular system according to Altun and Hall [18, Fig. 15]. In our

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Fig. 1. Scheme of muscle layout used in the model and connections from forward-movement motorneurons to muscles. For each group (VB and DB) only one side is shown for comfort perception, opposite side is connected symmetrically except one muscle in VR quadrant, which has no symmetrical pair in VL (shown with shading).

Fig. 2. Screenshot of C. elegans simulation representing its general view.

model, the worm has 26 intermediate segments with identical spring structures as well as a head and a tail segment on either end with a modified structure. Muscles are attached to the mass points of the body wireframe (Fig. 1). Each muscle is implemented as a combination of an ordinary spring object that contracts with a force proportional to an incoming activating signal. For better flexibility, the muscle is split into two consecutive segments connecting three corresponding mass points of the worm body wireframe (Fig. 2 and Fig. 4). We have included connections between ventral nerve cord motor neurons and the body wall muscles. However, muscles MVR01 through MVR06, MDR01 through MDR06, MDL01 through MDL06 and MVL01

through MVL06 are innervated instead with nerve ring motor neurons. This difference in innervation source causes that head and neck muscles allow the real worm to bend in 3D while the remaining body wall muscles only in a 2D plane. At this time we have omitted these connections as we are unable to simulate this part of the neural network.

2.4. Neurons and connections The activity of the neurons is defined by the continuous version of the McCulloch-Pitts model [19] using a linear transfer function on the interval (−1,1).

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We used the following data to construct the prototype: neuron names and properties ([5]; http://www.worm atlas.org/images/NeuronType.xls), connections between neurons (http://www.wormatlas.org/neuronalwiring.html; http://www.wormatlas.org/images/NeuronConnect.xls) and connections to body muscles / sensory organs from (http://www.wormatlas.org/images/NeuronFixedPoints .xls). The 3-dimensional positions of all 302 neuron cell bodies were extracted from open access data provided by VirtualWorm (http://caltech.wormbase.org/ virtualworm/).

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where dt is the time step, α(0,1] specifies the decay rate of i-th neuron potential with the course of time and wj defines transfer rate from j-th (pre-synaptic) neuron to i-th (post-synaptic). The activity level of pseudo neuron units is calculated in the same way as those for neurons. Propagation delays along pseudo neurons are handled by passing activity along one segment each time the simulation clock ticks. In case of branching the signal is passed to neighbor neurons or pseudo neurons, and each receives the same value as in case of only one branch.

2.5. Pseudo neuron paradigm 2.6. Forward/backward locomotion The neurons of C. elegans from the ventral cord have long neuronal processes that run almost through the whole length of the animal. According to [10], signal transmission via action potentials in C. elegans continues to be unobserved, and moreover it is highly unlikely due to absence of any identified genes encoding voltage gated sodium channels in the worm’s genome [20], and because of specific properties of its neurons. Propagation of graded potentials produced by the neurons is relatively slow and therefore can be a factor affecting the dynamics of locomotion. Because of this, we needed to include a mechanism in our model for representing this relatively slow propagation. Point neurons are insufficient for this because the activity of the neuron is restricted to a single location in space – it does not propagate through space. Therefore we chained units together as segments of a larger neuron model, enabling activity to propagate from one unit to the next. Inspired by the multi-compartmental approach for the neuronal modeling, described by Gerstner & Kistler [21] (chapter 2.6) we followed the main principles but implemented a significantly simpler variant due to consideration of graded potentials only. We call our approach to this the “pseudo neuron paradigm”. A pseudo neuron works as “retransmitting station”, receiving signal from a previous neuron or pseudo neuron and then sending it to the next one (or more, if the current pseudo neuron represents a branching point) taking into account attenuation of signal at its length and providing a corresponding time delay (illustrated by example at Fig. 3), which can be written in the following way:   ui ðt þ dtÞ ¼ min ui ðtÞ ⋅ α þ ∑ wj ⋅ uj ðtÞ; 1 j

When moving the worm lies on its side, with the ventral and dorsal muscles at any longitudinal level contracting in anti-phase [15]. The first simple test consisted of sending sinusoidal signals directly to the dorsal muscles and complementary anti-phase signals to the ventral muscles. This caused sinusoidal forward or backward (depending on signal parameters) movement similar to a real worm’s movement. The next test included the portion of the neuronal system that was built in compliance with the pseudo neuron paradigm. This contains all connections within the group of neurons comprising AVBL, AVBR, PVCR, PVCL, DVA, VB01...VB11 and DB01...DB07 (see Fig. 5) which form the circuit that drives forward locomotion [22]. Here we sent a sinusoidal signal to VB and a complementary anti-phase signal to DB motor neurons. In this case we also observed forward locomotion, but at reduced speed and not perfectly sinusoidal because head and neck muscles (VR01 through VR06, DR01 through DR06, DL01 through DL06 and VL01 through VL06), as it is seen at Fig. 1, are not innervated by VB and DB motor neurons, so in this simulation they remained immobile (see related video material). They are innervated with head interneurons which are part of a group of neurons not considered in this work. Fig. 6 shows the screenshot of our C. elegans simulation, which contains muscles, neurons and connections displayed in Fig. 1 and Fig. 5.

2.7. Visualization and interface Every object is visualized within the 3D scene; neurons, pseudo neurons (described below) and muscles

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Fig. 3. Illustration of signal propagation along axon composed of a number of consequently connected pseudoneurons. Connections between adjacent neurons or pseudoneurons are represented as a narrow cone; the sharp end of it indicates the direction of signal propagation.

Fig. 4. Simple fragment of neural network illustrating the conception of pseudoneurons. It includes DB03 neuron, which has post-synaptic connections to DB02 neuron and a group of dorsal body muscles which are innervated with DB03.

and are drawn with a color corresponding to their current activity. Additionally, neurons and pseudo neurons can be selected with a mouse and moved using the “neural network editor” mode or to artificially inject current into selected objects (pressing

the ‘A’ key when one or more of them are selected will cause their synchronous activation). In order to ensure that neurons were in the right locations, were functioning and connected appropriately to the muscles and each other, we augmented

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Fig. 5. Scheme of connections within the group of neurons including AVBL, AVBR, PVCR, PVCL, DVA, VB01...VB11 and DB01...DB07. Distances between neurons do not reflect the reality, but its sequence order is reproduced.

Fig. 6. Screenshot of C. elegans simulation. Body structure (springs and mass points) are not drawn to show muscles, neurons and connections between them.

the 3D visual representation of the worm to include dynamic visual elements that conveyed extra information. Initially, the neurons were visually represented as simple spheres attached to the body model in the locations of the neuronal cell bodies. To display the connections between neurons, again as a visual mechanism to inspect the correctness of the underlying model, we found that a simple initial method of displaying all of them simultaneously using straight lines connecting the neuron spheres together made interpretation difficult. To simplify the visually complex scene created by hundreds of lines crossing

over each other, we implemented a user-driven method for filtering the number of connections displayed at a single time. To facilitate this, we enabled user input for selecting a neuron via mouse click on a neuron sphere, implemented with a “picking ray” method [23]. Selected neurons are made visually distinct from unselected neurons by being surrounded with a wireframe sphere. A user may select any neuron or group of neurons. The simple built-in editor allows a user to modify the structure of the neural network in visual 3D form by adjusting the positions of neurons and pseudo neurons

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in all dimensions with a possibility to save updated positions into the file “celegans302positions.new.txt” (see 2.1, the program archive is available for download). The latter and another file “celegans302connections. txt”, containing information about connectivity between neurons and pseudo neurons in the current version, should be modified manually in case of adding new pseudo neurons and connections between them. Further it is planned that this labor-consuming process will be performed automatically. Rather than being drawn as a straight line, connections between neurons are drawn as a thin cone that points in the direction of signal propagation. This means that the visualization allows a user to see the paths of the neurons irrespective of the bending of the body model and along with the surrounding of other neurons and muscles. To represent the electrical activity of neurons and muscles, each group of neurons is associated with its own color, which is modulated with current neuron activity represented as brightness, with maximal brightness for the maximal level of activity and minimal brightness for the absence of activity.

3. Discussion The work we have described here outlines a first step in a long road towards an integrated simulation of the C. elegans. While many biological details have been left out of the present prototype, its value lies in a proof of concept that a wide range of physical phenomena, from forces acting on a body model, to activity patterns in neurons, to the movement of muscles, can be integrated into a single software system. While more detailed models have been built for the locomotor system [17], and better representations of neurons exist (e.g. NeuroML [24] for multi-compartmental modeling), we believe that the value of our system lies in its integration.

3.1. Comparison with Boyle et al., 2012 The work done by Boyle et al. [17], published recently after our initial manuscript was submitted describes a model of C. elegans forward locomotion that includes a neuromuscular control system operating on a two-dimensional body model. Its control

system relies on a sensory feedback mechanism to generate undulations and is integrated with a physical model of the body and environment. The physical model and worm body architecture in their work has much in common with ours. Both models use point masses and passive springs with dampers to construct a flexible wireframe which reproduces the worm body geometry. The Boyle worm body contains 48 segments in the longitudinal direction, which indicates that four sequential body segments together have a length corresponding to one real muscle cell (approximately). In our model we have half the number of longitudinal segments, one muscle cell for two body segments only. Muscles are also connected to point masses of body wireframe, and muscle forces are modeled qualitatively following Hill’s relations [25]. Also our model has much in common with the Boyle approach to the calculation of forces responsible for locomotion. Obvious Boyle’s advantages are taking into account mechano-sensory feedback, the neural circuit responsible for this, and a number of locomotion tests performed with various environment parameters. In comparison, our model’s special features are 3-dimensionality and the ability to observe all system components and their condition in visual form within a single simulation.

3.2. Going beyond point neurons for C. elegans modeling The neurons of the C. elegans are small (2–3 μm) [26], have an unusually high unit membrane resistance, estimated as ~150 kΩ ∙ cm2 [28] (for ordinary neurons 1–50 kΩ ∙ cm2 depending on number of ion channels [27]), have axons with a small diameter (0.2–1 μm by Roehrig [28]; 0.1–0.2 μm by Lockery & Goodman [29]), and have a small number of ion channels per neuron, from tens to hundreds, ~50 [29]. However, the absence of action potentials still leaves the possibility of graded (electrotonical) potentials. C. elegans electrophysiological parameters – length constant λ ≈ 1 mm and time constant τ = Rm ∙ Cm ≈ 0.15 s for AVB neuron (Rm = 11 GΩ, Cm = 14 pF) [28,30] make this mechanism potentially suitable for a 1 mm long worm. This, in turn, suggests a relatively slow propagation velocity and a near-exponential attenuation of the signal with time and distance. This possibility requires taking into account the real length

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of connections between each pair of neurons for a better understanding of its nervous system. The real lengths of the connections between the neurons in the C. elegans were not included in the published connectivity matrix. This was problematic for our modeling effort, because a straight connection between a neuron and its post-synaptic partner often is significantly shorter than it is in the real worm due to the 3D structure of the body plan. In the case of graded potentials this difference in lengths can significantly affect the character of signal propagation, because signal amplitudes and time synchronization for the receiving neuron can vary widely. Pseudo neurons have been placed along real or approximate path of connection between two neurons, which provides a discretization of the neurite both for visualization and calculation of signal propagation. This allows for an approximate solution for signal propagation and generates a useful visualization both for the geometrical structure of the neural network as well as the dynamic representation of neuronal activity not only at cell bodies, but also along connections between neurons. This can be illustrated with a simple example shown in Fig. 4 and in related video material.

3.3. Locomotion and friction Once the muscular structure and motor neurons were created and connected, we tested and observed the dynamics of the system. The frictional forces that act on C. elegans facilitating locomotion were studied in detail by Berri et al. [8]. They measured the ratio of effective drag coefficients, K = C⊥/C||, (C|| – tangential, C⊥ – normal to the local body surface) and concluded that C. elegans locomotion in gelatin and agar environments is determined and adequately described by a single-parameter (K) viscoelastic model of the environment. This approach was also used in a recently published work [17] using estimated C||,agar = 3.2∙10−3 kg ∙ s−1 [12] and C⊥≈ (30 − 40) ∙ C||,agar [8]. In our model, the drag coefficients have been used to calculate drag force on the underside of the worm body in contact with the floor of the environment. The same value of K worked fine in our case producing visually realistic forward locomotion, which was achieved by propagating sinusoidal undulations along the body from head to tail.

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3.4. Future plans Future results in this field can give new knowledge about neuronal mechanisms and patterns, such as the one, still unknown, which is responsible for the generation of sinusoidal patterns for locomotion. These findings can be used in the field of artificial neural networks design to broaden its possibilities. The study and reproduction of the C. elegans nervous system in detail will be only the beginning in context of more complex nervous systems investigation. Although it is often said that the C. elegans nervous system is almost completely known, this implies only the topology of connections between neurons (wiring diagram), but at deeper levels corresponding to mechanisms of signal transduction and types of neuromodulators involved, much is still unknown. In 2007, a technology was developed at MIT that allows tracing of the fine wiring of the brain (from a set of thin slices of nerve tissue) more accurately than ever before [31]. It can be applied to any nerve tissue to reconstruct the precise 3D structure of neurons and connections between them. This makes it possible to build virtual copies of more complex organisms in future. In an ideal case it should be an animal with “ordinary” neurons but still quite simple nervous system, and possibly with a basic visual system. For example, the sea slug Aplysia californica has about 18000–20000 nervous cells and its simple eye contains approximately 7000 receptors [32]. Depending on the native habitat and musculoskeletal system of an organism, it may be more optimal to implement the model body within the simulation or simulate only the nervous system and connect it to a robotic animal operating in the real world. But in any case we believe that the conception of a 3D dynamic physical model and a neural network supplemented with pseudo neurons for visualization, simulation, observation of dynamics and “debugging” of virtual nervous system will be a powerful tool. We believe that it will provide the easiest way to keep positions of branching points and nerve fibers pathways instead of significantly less informative connections between pairs of neurons. Our work in this field was recently noticed by the OpenWorm community and our research group was invited to join. Our future plans are now closely coupled with this collaboration that will allow us, on the one hand, to focus on improvement and enhancement of the physical engine aspect of the model, and on the other hand, connect with specialists in adjacent

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fields of knowledge, especially neuroscience and anatomy of C. elegans. We therefore plan to form close collaborations with experimental neurobiologists investigating the C. elegans nervous system to implement all necessary refinements to continually make our simulation more accurate. For example we could replace the current muscle model with that of Huxley [33]. The spring mechanism model of the C. elegans body could be enhanced by taking into account surface tension and liquid viscosity. Additionally we seek to ensure that the system is maximally convenient and helps with actual tasks relating to the investigation and simulation of the C. elegans nervous system. We also plan to implement a web-based version to make it even easier to access. Here we have presented a design and implementation of a tool for future research of the C. elegans nervous system. We believe that it is now possible with further investigation and additional rigorous testing that a model of the C. elegans ventral neural cord based on graded potentials and therefore relatively slow and spatially attenuated signal propagation speed can help explain the mechanisms behind C. elegans locomotion. Acknowledgements

[9]

[10]

[11] [12] [13]

[14]

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[16]

[17]

[18] [19]

The work of A.P., S.S. and A.D. was supported by SB RAS Integration project #136.

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