MetaWorm: An Integrative Data-Driven Model Simulating C.
elegans Brain, Body and Environment Interactions
Mengdi Zhao1†, Ning Wang1†, Xinrui Jiang1, Xiaoyang Ma1, Haixin Ma1, Gan He1,3, Kai Du1,4, Lei
Ma1,2,3*, Tiejun Huang1,3,4
1 Beijing Academy of Artificial Intelligence, Beijing 100084, China
2 National Biomedical Imaging Center, College of Future Technology, Peking University, Beijing
100871, China
3 National Key Laboratory for Multimedia Information Processing, School of Computer Science,
Peking University, Beijing, 100871, China
4 Institute for Artificial Intelligence, Peking University, Beijing 100871, China
*Corresponding author: Email: malei@baai.ac.cn (Lei Ma)
†These authors contributed equally to this work.
Abstract
The behavior of an organism is profoundly influenced by the complex interplay between its
brain, body, and environment. Existing data-driven models focusing on either the brain or the
body-environment separately. A model that integrates these two components is yet to be developed.
Here, we present MetaWorm, an integrative data-driven model of a widely studied organism, C.
elegans. This model consists of two sub-models: the brain model and the body & environment
model. The brain model was built by multi-compartment models with realistic morphology,
connectome, and neural population dynamics based on experimental data. Simultaneously, the
body & environment model employed a lifelike body and a 3D physical environment, facilitating
easy behavior quantification. Through the closed-loop interaction between two sub-models,
MetaWorm faithfully reproduced the realistic zigzag movement towards attractors observed in C.
elegans. Notably, MetaWorm is the first model to achieve seamless integration of detailed brain,
body, and environment simulations, enabling unprecedented insights into the intricate
relationships between neural structures, neural activities, and behaviors. Leveraging this model,
we investigated the impact of neural system structure on both neural activities and behaviors.
Consequently, MetaWorm can enhance our understanding of how the brain controls the body to
interact with its surrounding environment.
Introduction
The behaviors of an organism are not simply the product of brain activity but rather, emerge
from a dynamic interplay among the brain, body, and environment. To unravel the underlying
neural control mechanisms, it is crucial to develop an integrative simulation that incorporates
detailed models of the brain, body and environment, to both validate theories and make
predictions for biological experiments. Currently, there are specific data-driven models that
simulate either the brain, e.g., the mouse primary visual cortex1, the mouse striatum2, and the rat
somatosensory cortex3, or exclusively the body and environment to reproduce animal behaviors,
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such as neuromechanical models for Drosophila4 and rodent5. OpenWorm project6 aimed to
simulate C. elegans years ago, despite its detailed brain model and body model worked separately.
So far, no data-driven model exists that successfully integrates these elements, although such
integration is urgently needed to progress our understanding of neural control mechanisms7.
An integrative data-driven model must meet certain requirements. First, it should incorporate a
biophysically detailed brain model that has similar neural structures and neural activity patterns as
real organism. Secondly, it should consist of a realistic and high-performance body and
environment model that facilitates easy behavior quantification. Thirdly, and most crucially, the
brain model should not only control the body model interacting with the virtual environment, but
also receive the feedback from the body and environment model, establishing a closed-loop
interaction8,9.
C. elegans stands out as an exemplary model for developing such an integrative data-driven
model that bridges brain, body and environment. It has completely mapped morphologies of all
302 neurons10,11 (in adult hermaphrodite), along with the connectome and synapse-level
structure12-14. Recordings of single neural activity15-23, as well as brain-wide neural dynamics18,20,24
are available. Its body is well-reconstructed, consisting of only 95 muscle cells. It also exhibits
easily quantifiable behaviors, such as crawling, swimming and foraging25,26. Therefore, these
simple structures and abundant data provide solid foundations for creating an integrative data-
driven model.
In this study, we present an integrative data-driven model of C. elegans, MetaWorm (Fig. 1).
First, we developed a neural network model (brain model) of a C. elegans foraging neural circuit.
It was built using biophysically detailed compartmental neuronal models, with compartments
under 2 μm in length, encoding neural activity similar to live C. elegans. Second, we developed a
body & environment model of C. elegans. The body model contained 96 muscles and enabled
efficient real-time simulation at 30 frames per second and easy quantification of behaviors. Third,
we established a closed-loop interaction between the brain model and the body-environment
model to simulate C. elegans moving towards an attractor in a zigzag shape. Sensory neurons in
the neural network model were activated by attractor’s concentration in environment. Muscles in
the body model were controlled by motor neurons in the neural network model. Finally, we
performed synthetic perturbations on the neural network model and suggested that neurite or
synaptic/gap junction' absence disrupts global neural dynamics and hampers accurate forward
motion. This highlights the importance of network structure in shaping both network dynamics
and behavior patterns.
MetaWorm is open-source and modular as the brain, body, environment models and their
interactions can be independently modified, improved or extended by the scientific community.
MetaWorm will contribute to a deeper understanding of the intricate relationships between neural
structures, neural activities, and behaviors. This work also provides a valuable framework for
developing brain-body-environment model of other organisms. This innovative approach paves
the way for new avenues of systems biology and has the potential to revolutionize traditional
neural mechanism research.
Results
Constructing a data-driven biophysically detailed neural network model of C. elegans
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Our neural network model of C. elegans contained 136 neurons that participated in sensory and
locomotion functions, as indicated by published studies24,27-31. To construct this model, we first
collected the necessary data including neural morphology, ion channel models, electrophysiology
of single neurons, connectome, connection models and network activities (Fig. 2A). Next, we
constructed the individual neuron models and their connections (Fig. 2B). Subsequently, a
biophysically detailed model without functions was established (Fig. 2C). Finally, we optimized
the weights and polarities of the connections to obtain a model that reflected realistic dynamics
(Fig. 2D). An overview of the model construction is illustrated in Figure 2.
To achieve a high level of biophysical and morphological realism in our model, we employed
multi-compartment models to represent individual neurons. The morphologies of neuron models
were constructed based on published morphological data32,33. Soma and neurite sections were
further divided into several segments, where each segment was less than 2 μm in length. We
combined 14 known classes of ion channels (Supplementary Table S1-2)34 in neuron models, and
tuned the passive parameters and ion channel conductance densities by an optimization
algorithm35 to reproduced electrophysiological traces in the experiments36-39. Based on the few
available electrophysiological data, we digitally reconstructed models of five representative
neurons: AWC, AIY ,A V A,RIM and VD5. The I-V curves of our neuron models perfectly match
that in the experiments (Fig. 3). The optimized parameters of neuron models are listed in
Supplementary Table S3.
These five neurons are representative for five different functional group: sensory neuron,
interneuron, command neuron, head motor neuron and body motor neuron. They serve as the
initial pool of digital neuron models to reconstruct other neurons in the neural network model. For
neurons that are not representative neurons, their biophysical parameters were set the same as the
representative neuron models that belonged to the same functional group. The referenced neurons
of all neurons are listed in Supplementary Table S4. This approach allows us to achieve a high
level of biophysical and morphological realism in our C. elegans neural network model with
limited accessible experimental data.
For connections, we first modeled gap junctions as simple ohmic resistances, and synapses as
continuously transmitting synapse, according to published models32,40-43. Then, we developed an
algorithm to determine the number and locations of connections. The number of connections
between two cells was from the cell adjacency matrix10,11 (Fig. 4A). For each connection, we
randomly assigned a distance following the neurite centroids distance distributions of synapse/gap
junction in experiment13 (Fig. 4B). Each connection linked two segments whose distance was
closest to the randomly assigned distance. This approach allowed us to generate connections that
were constrained by experimental connection numbers and distance distributions (Fig. 4C). Figure
4D-F show some examples of the connections between neurons in our neural network model.
Most synapses and gap junctions were located on neurites that were relatively close to each other.
Also, as shown in Fig. 4F, the constructed connections tended to cluster together, which followed
the clustered organization principle in published research13,14.
To determine the polarities of synapses (excitatory or inhibitory) and the connection weights,
we employed a gradient-descend-based algorithm to minimize the mean squared error (MSE)
between simulation results and the optimization target. In this study, our optimization target was a
Pearson correlation matrix of 65 identified neurons in a prior study24, where activity series were
recorded through whole-brain single-cell-resolution Ca2+ imaging. Figure 5A illustrates the
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correlation matrices generated from our simulation and the experimental data. These two maps
exhibited similar correlation strengths, whose mean squared error was only 0.076.
To further analyze the optimized neural network model, we conducted principal components
analysis (PCA) on the membrane potential of the 65 identified neurons, involving the calculation
of principal components (PCs) based on the covariance structure in the normalized data (Fig. 5B).
Our analysis unveiled that PC1 exhibits an opposite dynamic to that of PC2 (Fig. 5C). Upon
sorting based on PC1, most neurons controlling forward locomotion (e.g., A VB,VB, DB neurons)
and backward locomotion (e.g., A V A,V A,DA neurons) were distinctly classified into two groups
(Fig. 5B). This indicates that our neural network model accurately captures realistic neural
dynamical relationships, and the dynamics of neurons encode functions similar to those observed
in real C. elegans, to some extent.
Constructing a high-performance body & environment model of C. elegans
Next, we developed the body & environment model of the C. elegans. Our primary
contributions are: 1) modelling the soft body, muscles, and solving soft body deformation, 2)
creating the 3D simulation environment and solving soft body-fluid interaction, 3) proposed a
novel reference coordinate system for behavior analysis. The framework of this model is
illustrated in Figure 6A and B.
The body model was built with tetrahedral mesh, including 984 vertices, 3341 tetrahedrons and
1466 triangles on the body surface (Fig. 6C). Also, it had four muscle strings: ventral right (VR),
ventral left (VL), dorsal right (DR) and dorsal left (DL). Each string had 24 muscle cells arranged
from head to tail (Fig. 6D). Tetrahedrons were controlled by their near muscle strings. The
corresponding strings were visualized with different colors in Figure 6E. We used Finite Element
Method (FEM) solver to compute positions and velocities of all vertices on the deformable body
at any given time. In the computation, the solver used 4558 muscle constraints and 3341 corotated
elasticity constraints. Thus, our soft-body simulation engaged a sum of 7,899 mechanical
constraints. After the geometry construction, we need to solve the body deformation efficiently. In
our model, we selected a real-time, highly robust, and accurate approach named projective
dynamics as the FEM solver for calculation. The solved body state was visualized in Fig.6F.
We constructed a large 3D fluid simulation environment, where the body, foods were embedded.
This environment was essentially a large cube, with each edge equating to approximately 1200
times the length of a C. elegans. The mechanic of the body movement in fluid is essentially a soft
body-fluid coupling problem. To simplify this problem, we only calculated the thrust force (Fig.
6G) and drag force (Fig. 6H) on the body surface44. This simplification allowed for the simulation
of a much larger space and improved computational efficiency. To achieve forward locomotion
behavior, we used projective dynamics to calculate real-time bending waves that propagated from
head to tail. The interaction between these bending waves and the fluid generated impetus for
forward locomotion (Fig.6I).
C. elegans is an animal with soft body that exhibits special patterns during locomotion, wherein
every point on its body oscillates continuously. Therefore, it is difficult to quantify the trajectory
and body state during locomotion. Here, we proposed an effective and simple method to measure
C. elegans locomotion. The main idea of this method is to establish a numerically stable
coordinate system relative to the body itself, named Target Body Reference Coordinate System
(TBRCS) (Fig. 7A). First, we defined the target body, standard body and Standard Body
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Coordinate System (SBCS). Then, we computed transformation M between the target body and
the standard body. Finally, we used M to perform three-dimensional transformation on the SBCS
to obtain the TBRCS. Details of this method are described in Methods. Using this method, we can
stably quantify not only the body’s locomotion trajectory (Fig. 7A), but also the body states
including the relative position and velocities of any points on the C. elegans body (Fig. 7B and Fig.
7C) and steering angle (Fig. 7D). For example, figure 7E displays the relative positions of 17
sampled 3D tracking points, while Figure 7F illustrates their corresponding velocities. In Figure
7E, the swing direction of body was Z direction of the TBRCS (dorsal-ventral direction). Also,
there was a spring movement in X direction of the TBRCS (head-tail direction) during forward
locomotion. Locomotion in Y direction of the TBRCS (left-right direction) was nearly flat since
muscle activations were approximately symmetric in this direction. Figure 7F shows that, in this
case, the primary propulsion of forward locomotion originated from tail (pink line), since speed
represented power. Moreover, our method enables the quantification of the body's steering. Figure
7D shows an example of a steering angle θ, which was defined as the angle between TBRCS x-
axis and the direction of velocity. The steering angle θ was also numerically stable.
In summary, our body model is simple yet preserving anatomical authenticity. Our environment
model provides a large-scale 3D scene for simulations. Our simulation is real-time and allows
simulations of C. elegans populations. Importantly, our model facilitates easy quantification of
behaviors.
Replicating locomotion behavior of C. elegans
In our model, we successfully replicated the locomotion behavior of C. elegans by a closed-
loop interaction between the neural network model and the body & environment model.
Specifically, the neural network model sensed the dynamic food concentration in the environment
by sensory neurons. This sensory information was used to calculate the membrane potentials of
motor neurons. The membrane potentials of motor neurons were subsequently translated into
muscle activation signals, which controlled the movement of C. elegans' body to navigate towards
the food in the environment. An overview of this closed-loop simulation is illustrated in Figure 8A.
Notably, we observed that the trajectory of the C. elegans movement in our model closely
resembled the trajectory recorded in the experiments45, with both exhibiting a zigzag pattern (Fig.
8B). Additionally, C. elegans in both the simulations (Supplementary Movie S1 and
Supplementary Movie S2) and experiments exhibited a dorsoventral fluctuation during 3D
movement46,47.
Regarding sensation, we assumed that the food concentration in the environment follows a
linearly distribution. To incorporate this sensory information into our model, we used the
derivative of the food concentration as the amplitude of an external current injected into the soma
of all sensory neurons. Regarding neuromuscular coupling, we employed a linear transformation,
treating the neural network model as a reservoir to generate muscle activities48 (Methods 3.6).
During locomotion, the input of sensory neurons exhibited fluctuations due to the zigzag
movement of the body’s head (Fig. 8C). The membrane potentials of each individual motor neuron
also oscillated in response to the sensory input, especially the head motor neurons (Fig. 8D). The
activation of muscle cells revealed traveling waves from the head to the tail, accompanied by
alternating contractions and relaxations of dorsal and ventral muscle cells (Fig. 8E). These
findings closely resemble observations in biological experiments49. To facilitate visualization, we
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have incorporated the locomotion of C. elegans, the activities of its neural network model, and
muscle activities into our graphical user interface (GUI) (Fig. 9).
In summary, our model successfully reproduced locomotion behavior in this closed-loop
simulation and facilitated the visualization and analysis of neuron dynamics, muscle activities, and
behavioral patterns.
Exploring the impact of neural network structure on dynamics and behaviors
We are interested in exploring the extent to which network structure influences the neural
dynamics and behaviors, thus we conducted synthetic perturbations on the structure of our neural
network model. Synthetic perturbations of the neural system often exceed individual tolerance
level in animals. However, our model enables researchers to carry out experiments without
concerns about experimental technique limitations or animal tolerance. Based on MetaWorm, we
performed experiments involving synthetic perturbations to quantitatively examine the impact of
network structure on network dynamics and behaviors.
In the absence of neurites, the body exhibits a higher degree of twisting during locomotion, as
indicated by the larger relative locations of tracking points on the head and tail compared to the
control group. This suggests that neurites play a crucial role in the computations associated with
network information processing (Fig. 10B). Moreover, shuffling connection locations within the
model leads to faster oscillations of the head and tail but slower forward locomotion (Fig. 10C).
This suggests that the specific locations of connections along the neurites are important,
potentially facilitating localized computations within neurons. Additionally, despite the neural
network model contains fewer gap junctions than synapses, the role of gap junctions in C. elegans
neural network appears to be more important than that of synapses (Fig. 10D-G), because
perturbations on gap junctions result in more obvious changes in the Pearson Correlation matrix
compared to perturbations on synapses. Perturbations on gap junctions accelerate the twisting of
the body compared to the control group, whereas perturbations on synapses result in slower
twisting. These results emphasize the importance of network structure in shaping network
dynamics and behavior patterns. By utilizing our model, researchers can gain valuable insights
into the contributions of network structure to neural network.
Discussion
In this study, we present MetaWorm, a data-driven model of C. elegans that integrates models
of neural network, body and environment. The neural network model of MetaWorm is highly
detailed and data-driven, capturing neural population activity patterns and encoding functions
observed in real C. elegans. The body & environment model of MetaWorm is realistic, efficient,
and allows for the quantification of behaviors. Based on the interaction of these two sub-models,
we achieved locomotion behavior in C. elegans. This represents the first time that an integrative
data-driven model replays a behavior with motor control and sensory feedback.
The neural network model of MetaWorm integrated experimental data including ion channel
dynamics, neural morphologies, electrophysiology, connectome, synaptic organization rules on
neurites, as well as neural activities. Although we collected and integrated as much data as
possible up until 2023, there remained a substantial amount of detail that could not be fully
implemented due to lack of data. Also, the physical parameters in body & environment model did
not correspond precisely to those of actual C. elegans, because they were difficult to measure
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accurately. However, with the influx of new experimental data, these models can be enhanced and
refined.
For the neuromechanical connections from motor neuron to muscle, the neural network model
served as a reservoir that processed data in a complex and nonlinear way. The input was mapped
onto an output linear layer to produce muscle activation. With only one linear readout layer, the
neural network model successfully controlled the body’s locomotion. This demonstrates the great
information processing ability of our neural network model. In the future, more realistic models of
the neuromuscular junction can be used in our neural network model, reflecting the real synaptic
properties from motor neuron to muscle.
When the input of the neural network model was a periodic standard signal, i.e. without sensory
feedback from the environment, the membrane potentials of motor neurons and the activations of
muscle cells were periodic (Supplementary Figure S1). This resulted in the movement trajectory
of the C. elegans body following a standard Z-shape, which was extremely similar to the
trajectory in the experiment45,50 (Supplementary Figure S1B and Supplementary Movie S3).
Conversely, When the input was a sensory signal dependent on the body’s location in the
environment (Fig. 8), the input was not periodic, leading to non-periodic dynamics in both
neurons and muscle cells. Therefore, the trajectory of the body was not as regular as that observed
in the experiment (Fig. 8B). Future studies will focus on designing more reasonable Central
Pattern Generators (CPGs) or motor control systems to achieve more realistic movement.
Training biophysically detailed neural network model is challenging and consumes much time
and GPU memory resources. In the future, we will improve the optimization algorithm to
effectively train larger detailed neural network that contains all 302 neurons, with more constrains
such as single neuron voltages, and reproduce more behaviors beyond zigzag locomotion.
MetaWorm serves as a white box model of C. elegans. Its two sub-models and their interaction
can be independently modified or extended. We provide researchers with user-friendly interfaces
to design their own model based on their experimental data. Moreover, our model serves as a
valuable reference for designing integrative data-driven models for other animals. In this era of
exponential growth in experimental data, these integrative data-driven models will enable us to
gain a deeper understanding of the neural computation that underlies specific behaviors.
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Figures
Fig 1. Overview of MetaWorm.
MetaWorm comprises two sub-models: the neural network model and the body & environment
model. The neural network model is a biophysically detailed. Its neuron models are multi-
compartmental, exhibiting electrophysiological features similar to real C. elegans neurons. These
neuron models are connected by synapses and gap junctions on neurites. An optimization tool is
employed to tune the weights and polarities of these connections. The body & environment model
consists of a biomechanical body, a 3D fluid environment and a real-time simulation engine. The
soft body is constructed using 3341 tetrahedrons, with 96 muscles arranged from head to tail as
actuators. The large-scale 3D environment allows for the integration of various agents and
elements including C. elegans, E. coli (food), and fluid. And it provides a range of auxiliary tools
for dynamic lighting, visualization, tracking, and behavior analysis. Real-time simulation is
achieved through a Finite Element Method (FEM) solver and simplified hydrodynamics. To
achieve foraging behavior in C. elegans, the neural network senses attractor’s concentration in the
environment, performs neural computations to control muscles’ contraction and relaxation, and
ultimately deforms the body to propel it towards the attractor.
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Fig 2. Construction of biophysically-detailed neural network model of C. elegans.
A. Experimental data collection to constrain models. These data include neural morphologies, ion
channel models, electrophysiological recordings of single neurons, connectome, connection
models, and neural network activities. B. Construction of multi-compartmental neuron models and
connection models (synapses and gap junctions). C. The biophysically detailed C. elegans neural
network model without functional neural activities. D. Optimization of the biophysically detailed
C. elegans neural network model to achieve realistic network dynamics. Neurons are color-coded
to represent membrane potential.
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Fig 3. Models of five representative neurons fitted to electrophysiological recordings in
experiments.
AWC(L), AIY(L), A V A(L),RIM(L), VD5 were selected as the representative neurons for each of
the five functional groups, respectively. Left, 3D morphology of each neuron32,33. Middle,
membrane voltage/current dynamics induced by current or voltage clamp steps. The simulated
responses (blue lines) are overlayed with biological experimental data (gray lines). The
corresponding current or voltage clamp protocol are shown in left. AWC(L): voltage clamp
protocol, in spans of 0.1 s, starting from -10 mV and increasing to 110 mV by 20 mV increments.
AIY(L): current clamp protocol, in spans of 5 s, starting from -15 pA and increasing to 35 pA by 5
pA increments. A V A(L):current clamp protocol, in spans of 5 s, starting from -20 pA and
increasing to 20 pA by 5 pA increments. RIM(L): current clamp protocol, in spans of 5 s, starting
from -15 pA and increasing to 25 pA by 5 pA increments. VD5: voltage clamp protocol, in spans
of 1.2 s, starting from -10 mV and increasing to 40 mV by 10 mV increments. Right, comparison
between experimental (red dot) and simulated (black star) I-V curves, including I-V curve at
steady-state (left) and I-V curve at the time when the rates of voltages/currents change reach a
certain threshold (right). Experimental data are from published articles36-39.
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Fig 4. Connecting multi-compartmental neuron models in the neural network model.
A. Synapse and gap junction adjacency matrices of the neurons from published data10,11. Matrix
elements represent the total number of electron microscopy series capturing synapses (or gap
junctions) between two neurons. The color indicates the series number. The order of cells in rows
and columns corresponds to Supplementary Table S4. B. Distribution of distances between
centroids coordinates of neurites at connected synapse (upper) and gap junction (lower) locations.
Experimental data (red/blue bars) from a published paper13 is fitted by two inverse Gaussian
distributions (black/red lines). Random data (yellow/green bars) is produced following the fitted
inverse Gaussian distributions. C. Similar to B, with red and blue histograms indicating
distributions in our neural network model constructed in our model. D. 3D visualization of a
synapse (cyan) between RIBR (pre-synaptic, yellow) and AIBL (post-synaptic, blue). E. 3D
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visualization of a gap junction (yellow) between RIVR (red) and SMDVR (blue). F. 3D
visualization of clusters of synapses (cyan) and gap junctions (yellow) between PQR (blue) and
A V AR(red).
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Fig 5. Neural network model of C. elegans fitted to functional map in biological experiment.
A. Pearson correlation matrix of neurons in simulation and experiment24. The simulation matrix
was derived from our optimized neural network model. Red and cyan color indicate positive and
negative correlation coefficients, which were calculated by membrane potentials of two neurons.
B. Membrane potentials of 65 neurons, each represented in a separate row. Principal component
(PC0, PC1, PC2) weights of each neuron are shown by the bar plots on the right. Neurons were
sorted by PC1 weights. Color of the bars indicates the reported role of the neuron in locomotion
from published research24 (blue: forward neurons, red: backward neurons, green: turning motor
neurons, black: no clear role).
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Fig 6. A biomechanical body model of C. elegans.
A. Schematic diagram of C. elegans body model. The body was composed of tetrahedrons. 96
muscle cells along 4 muscle strings were serving as actuators for soft body deformation. Different
colors on muscles indicate distinct muscle cells. The body was situated in a large 3D fluid
environment and can move in the environment. Projective dynamics functions were used as the
soft body solver, and simplified hydrodynamics calculated thrust force and drag force on the body
surface. B. Internal view of the body model, consisting of 4 muscle strings: ventral right (VR),
ventral left (VL), dorsal right (DR), dorsal left (DL). C. Triangulated tetrahedron mesh of the body.
D. Four muscle strings: VR (green), VL (yellow), DR (red) and DL (blue). Color intensity reflects
the strength of muscle contraction, with high intensity indicating contraction and transparent
intensity indicating relaxation. E. Tetrahedrons driven by four muscle strings: VR (green), VL
(yellow), DR (red) and DL (blue). F. Deformation of the soft body. Blue tetrahedrons are increase
their volume, while red ones are decreasing. G-H. Thrust force (G) and drag force (H) on body
surface. Red indicates strong force and blue indicates weak force. I. Positions and velocities of the
body at four moments. The body was moving from left to right. Color indicates velocities, with
red representing fast, blue representing slow.
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Fig 7. Quantification of body’s behavior.
A. Schematic diagram of Target Body Reference Coordinate System (TBRCS) to quantify
behavior stably. The standard body and SBCS are depicted on the left. The target body and
TBRCS are illustrated on the right. Matching points (magenta dots) are positioned on both the
standard and target body. Magenta lines links the corresponding points on two body. The
transformation matrix M is obtained through singular value decomposition, enabling the
conversion from SBCS to TBRCS. The trajectory of the target body center (orange line) exhibits a
zigzag pattern, whereas the trajectory of TBRCS (green line) is smooth. B. Tracking body
movement with TBRCS. 17 tracking points were defined on the surface of both the standard body
and the target body. Magenta lines represent relative positions, while blue arrows indicate relative
velocities. Coordinate systems of the head on standard body and target body are drawn out to
display translation and rotation of the head on target body. C. Trajectories of 17 tracking points on
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the body. D. Steering angle θ of the body. The velocity vector is depicted as the yellow arrow, and
the XYZ axes of TBRCS are represented by red-green-blue arrows. θ is defined as the angle
between the red arrow and the yellow arrow and remains stable over time. E-F. Relative positions
(E) and velocities (F) of 17 tracking points on the body during the movement.
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Fig 8. The closed-loop 3D simulation of C. elegans locomotion behavior.
A. The information flow diagram of the closed-loop interaction between neural network model
and body & environment model. B. Comparison of C. elegans trajectory during forward
locomotion in simulation (left) and experiment (right, from a published article45). C. Input current
injected to sensory neurons during locomotion. D. Left, neural network model of C. elegans. Right,
membrane potentials of all motor neurons in the model during locomotion. E. Left, the body
model with 96 muscle cells. Right, the activation signals of all muscle cells during locomotion.
The dashed red line indicates that the activation signal shifting from head to tail.
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Fig 9. Graphical User Interface (GUI) of MetaWorm.
Graphical User Interface (GUI) of MetaWorm, including: the neural network of C. elegans where
the color indicates membrane potentials, a C. elegans body actively moving towards a food source
(E. coli population) in the 3D fluid environment, the locomotion trajectory, activations of four
muscle strings extending from head to tail.
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Fig 10. Synthetic perturbation of on the neural network model in MetaWorm.
A. Column 1: Schematic diagram illustrates connections between two neurons in the neural
network model. Ball and stick denote soma and neurite, respectively. Rectangles with solid
and dashed lines indicate synapses and gap junctions. Rectangles with different colors
represent different connections. The width of rectangles signifies connection weights. The
depicted connections between this neuron pair represent all connections in the neural network
model. Column 2: Pearson correlation matrix of neurons during simulation of the neural
network model. Column 3: Locomotion behavior of the C. elegans body controlled by the
neural network model in Column 1. Column 4: Relative position of 3D tracking points on
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head (blue), center (orange) and tail (green) of the C. elegans body. Column 5: Relative
velocity of a 3D tracking point on head of C. elegans body. B-G. Diagrams represented as in
A, but with different perturbation types: (B) Removal of neurites from all neurons in the
neural network model, relocating connections onto the soma, (C) shuffling the location of
connections on neurites while fixing source and target neurons, (D) shuffling the weights of
synapses while fixing their locations, (E) shuffling the weights of gap junctions while fixing
their locations, (F) removal of all synapses in the neural network model, (G) removal of all
gap junctions in the neural network model. These perturbations are sequenced based on their
distance to the control group in terms of the Pearson Correlation matrix (B: closest, G:
farthest).
Acknowledgements
Funding: this research was funded by National Key R&D Program of China (2020AAA0105200)
Author Information
Contributions
M.D.Z. developed the neural network model module, designed and conducted experiments, and
interpreted results. N.W. developed the body & environment module. H.X.M. and L.M. developed
the 3D neural visualization software. X.R.J. and M.D.Z. collected experimental data for
constructing the neural network model. X.Y .M.and M.D.Z. optimized the connection algorithm
and rewrote the code of the neural network model module for open source. G.H. developed the
optimization algorithm for the multi-compartmental neural network model. M.D.Z., N.W., and
X.R.J. wrote the article. K.D. reviewed the research and provided advice. L.M. supervised the
research, coordinated and conceptualized the study, and contributed to writing the article. All
authors participated in the paper revision process.
Corresponding authors
Correspondence to Lei Ma (malei@baai.ac.cn)
Ethics declarations
The authors declare no competing interests.
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Methods 
1 Details of Neural Network Model 
1.1 Neuron models 
Neurons were modeled by morphologically derived multi-compartmental models with 
somatic Hodgkin-Huxley dynamics1 and passive neurites. These neuron models captured both 
the biophysical properties and spatial complexity of neurons. The models integrated three 
kinds of neuron data: ion channel, morphology, and electrophysiology.  
Ion channel models were determined according to published studies2. There were 14 ion 
channels divided into 4 categories: voltage-gated potassium channels (SHK-1, EGL-36, SHL-
1, KVS-1, KQT-3, EGL-2, IRK1/3), calcium-regulated potassium channels (SLO1, SLO2, 
KCNL), voltage- gated calcium channels (EGL19, UNC2, CCA1), and passive sodium 
channel (NCA). Model equations and parameters for ion channels are given in Supplementary 
Table S1 and Supplementary Table S2. 
As for morphologies of neuron models, the spatial structures were adopted from the C. 
elegans neuronal anatomy models created by the Virtual Worm Project3 and the OpenWorm 
Project4. Each neuron model had a soma section and several neurite sections. Neurite sections 
were further divided into several compartments, each of which was shorter than 2 μm. 
Only soma compartment had active i on currents. Soma compartments and neurite 
compartments shared the same passive properties. To identify the passive and active 
parameters of neuron models, a deep learning tool5, was used to fine-tune these parameters to 
reproduce experimental data of voltage /current clamp. The value of the parameters can be 
found in Supplement Table S3. 
Less than 20% of the C. elegans  neurons have been recorded for the desired 
electrophysiology data. To estimate model parameters for those neurons that were not 
recorded, a method was implemented to generalize parameter values. First, considering the 
available data and hierarchy of information flow in C. elegans nervous system6,7, the neurons 
were divided into 5 functional groups (sensory neuron, interneuron, command interneuron, 
head motor neuron, and body motor neuron, see Supplement Table S4). Second, 5 neurons of 
each functional group (AWA8, AIY8, AVA9, RIM8, and VD510) were selected and their model 
parameters were optimized as described above. These parameter values of optimized neuron 
models were then generalized to other neuron models according to their functional categories. 
In other words, neuron models in the same functional group share identical electrical 
parameters. Each neuron models have their own morphologies yet similar electrical properties 
to those neurons in the same functional group. 
1.2 Graded synapse and gap junction models 
The neuron models were connected by graded synapses8,11 and gap junctions12 in C. elegans 
neural network. Graded synapses, include excitatory and inhibitory synapses, were modeled 
according to published C. elegans  models 4,13,14. The synaptic conductance is continuously 
changed with presynaptic membrane potential. Postsynaptic current is given by: 
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⎩
⎪
⎨
⎪
⎧ 𝐼𝐼 = −𝑤𝑤𝑤𝑤(𝐸𝐸 − 𝑉𝑉 )
𝑑𝑑𝑤𝑤
𝑑𝑑
𝑑𝑑 = 𝑤𝑤∞ − 𝑤𝑤
𝜏𝜏
𝑤𝑤∞ = 1
1 + 𝑒𝑒
𝑉𝑉𝑡𝑡ℎ−𝑉𝑉𝑝𝑝𝑝𝑝𝑝𝑝
𝛿𝛿
, 𝜏𝜏 = 1 − 𝑤𝑤∞
𝑘𝑘
 
where 𝑤𝑤 is maximal postsynaptic membrane conductance for the synapse, 𝑤𝑤 is a voltage-
sensitive parameter controlling the conductance, 𝐸𝐸 is the reversal potential, 𝜏𝜏 is the time 
constant of s, 𝑉𝑉𝑡𝑡ℎ  is the presynaptic equilibrium potential at the middle of the voltage 
sensitive range, 𝑉𝑉𝑝𝑝𝑝𝑝𝑝𝑝 is the pre-synaptic membrane potential. 
Ga
p junctions were modeled as simple resistances, where current flowing from presynaptic 
cell to postsynaptic cell is given by: 
𝐼𝐼 = −𝑤𝑤�𝑉𝑉pre − 𝑉𝑉� 
where 𝑤𝑤 is the conductance of gap junction. 
The parameter values of these synapses were set according to published models12: the 
conductance of the synapses and gap junctions had the same order of magnitude of 1e-4 μS. 
The specific weight of each synaptic connection in a specific neural network can be set 
randomly or be optimized according to a target, as described in Method 1.4. 
1.3 Connection locations 
Detailed neuron connections in the neural network model were located on the neuronal 
neurite (axons or dendrites) of the neurons, which were based on the neuron connectivity 
matrices and a neurite connectivity algorithm. 
The neuron connectivity matrices were acquired from published connectome data6,15. The 
original data in the matrices was the total number of electron microscopy (EM) series of all 
synapses (or gap junctions) between two neurons. In our work, the number of EM series was 
transformed into the number of synapses (or gap junctions) by: 
𝑁𝑁 = 𝑎𝑎 ∙ tanh(𝑏𝑏 ∙ 𝐶𝐶) 
where 𝐶𝐶  is the number of EM series, 𝑎𝑎  and 𝑏𝑏  are constants. For synapses number 
transformation, 𝑎𝑎 = 23.91,𝑏𝑏 = 0.02285, while for gap junction, 𝑎𝑎 = 20.49, 𝑏𝑏 = 0.02184. 
These parameters were inferred from partial-animal connectome data6. 
After the number of synapses (or gap junctions) was determined, the locations of synapses 
(or gap junction) on neural neurites were identified by a connection algorithm. The algorithm 
was based on the rule that distances between the centroid coordinates of possible presynaptic 
and postsynaptic neurites follow a certain statistical distribution16. We fitted synapse and gap 
junction distribution by two Inverse Gaussian distributions: 
𝑓𝑓(𝑥𝑥, 𝜇𝜇) = 1
𝑘𝑘 ∙ √2𝜋𝜋𝑥𝑥3 exp �− (𝑥𝑥 − 𝜇𝜇)2
2𝑥𝑥𝜇𝜇2 � 
where 𝜇𝜇 = 0.44, 𝑘𝑘  = 0.63 for synapse distribution, 𝜇𝜇 = 0.70, 𝑘𝑘 = 0.40 for gap junction 
distribution. The algorithm of generating connection locations was designed: 
Algorithm 1. Generate connection locations. 
Input: 
𝐴𝐴�𝑎𝑎0, 𝑎𝑎1, … ,𝑎𝑎𝑁𝑁𝑎𝑎−1�: set of all compartments of neuron A. 
𝐵𝐵�𝑏𝑏0, 𝑏𝑏1, … ,𝑏𝑏𝑁𝑁𝑏𝑏−1�: set of all compartments of another neuron B. 
𝑁𝑁𝑠𝑠 �𝑁𝑁𝑔𝑔�: desired synapses (gap junctions) number between these two neurons. 
𝐷𝐷𝑠𝑠 �𝐷𝐷𝑔𝑔� : Inverse Gaussian distribution of synapses (gap junctions) fitted 
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above. 
𝑑𝑑𝑡𝑡ℎ: distance threshold of one synapse (gap junction).  
𝜌𝜌: arrangement priority factor between synapse and gap junction.  
Output: 
𝐶𝐶𝑠𝑠��𝑎𝑎𝑖𝑖1
𝑠𝑠 , 𝑏𝑏𝑗𝑗1
𝑠𝑠 �, �𝑎𝑎𝑖𝑖2
𝑠𝑠 , 𝑏𝑏𝑗𝑗2
𝑠𝑠 �, … ,�𝑎𝑎𝑖𝑖𝑁𝑁
𝑠𝑠 , 𝑏𝑏𝑗𝑗𝑁𝑁
𝑠𝑠 ��  ( 𝐶𝐶𝑔𝑔 ): set of synapse (gap junction) 
compartments that best approximates 𝐷𝐷𝑠𝑠 (𝐷𝐷𝑔𝑔) distribution. 
A
lgorithm: 
r
andomly sample synapses (gap junction) distances 𝑇𝑇𝑠𝑠�𝑑𝑑0
𝑠𝑠, 𝑑𝑑1
𝑠𝑠, … ,𝑑𝑑𝑁𝑁𝑠𝑠−1
𝑠𝑠 � ( 𝑇𝑇𝑔𝑔 ) 
according to 𝐷𝐷𝑠𝑠(𝐷𝐷𝑔𝑔). 
in
itialize distance matrix 𝑀𝑀 
for 𝑖𝑖 = 0 … ( 𝑁𝑁𝑎𝑎 − 1) do 
for 𝑗𝑗 = 0  …  (𝑁𝑁𝑏𝑏 − 1) do 
 𝑀𝑀𝑖𝑖,𝑗𝑗 ≔ Euclidean_Distance�𝑎𝑎𝑖𝑖, 𝑏𝑏𝑗𝑗� 
        end for  
end for 
i
nitial index pointer 𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠(𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔) of synapses (gap junction) allocation status 
𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠  ≔  0  
𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔  ≔  0  
w
hile (𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠 < 𝑁𝑁𝑠𝑠) 𝑜𝑜 𝑟𝑟 �𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔 < 𝑁𝑁𝑔𝑔� do 
r
andomly sampled a uniform[0,1] variable 𝑟𝑟 
if �(𝑟𝑟 < ρ) 𝑜𝑜
𝑟𝑟 �𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔 == 𝑁𝑁𝑔𝑔��  𝑎𝑎𝑎𝑎𝑑𝑑 (𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠 < 𝑁𝑁𝑠𝑠) then 
f
ind connection �𝑎𝑎𝑝𝑝, 𝑏𝑏𝑞𝑞� in distance matrix 𝑀𝑀 closest to 𝑑𝑑𝑝𝑝𝑡𝑡𝑝𝑝𝑠𝑠
𝑠𝑠  
if �𝑀𝑀𝑝𝑝,𝑞𝑞 < 𝑑𝑑𝑡𝑡ℎ � then 
Append �𝑎𝑎𝑝𝑝, 𝑏𝑏𝑞𝑞� to 𝐶𝐶𝑠𝑠 
 
𝑀𝑀𝑝𝑝,∗ ≔ 𝑖𝑖 𝑎𝑎𝑓𝑓  
 
𝑀𝑀𝑞𝑞,∗ ≔ 𝑖𝑖 𝑎𝑎𝑓𝑓  
e
nd if 
           𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠 ≔ 𝑝𝑝𝑑𝑑𝑟𝑟𝑠𝑠 + 1; 
el
se 
f
ind connection �𝑎𝑎𝑝𝑝, 𝑏𝑏𝑞𝑞� in distance matrix 𝑀𝑀 closest to 𝑑𝑑𝑝𝑝𝑡𝑡𝑝𝑝𝑔𝑔
𝑔𝑔  
if �𝑀𝑀𝑝𝑝,𝑞𝑞 < 𝑑𝑑𝑡𝑡ℎ � then 
Append �𝑎𝑎𝑝𝑝, 𝑏𝑏𝑞𝑞� to 𝐶𝐶𝑔𝑔 
 
𝑀𝑀𝑝𝑝,∗ ≔ 𝑖𝑖 𝑎𝑎𝑓𝑓 
 
𝑀𝑀𝑞𝑞,∗ ≔ 𝑖𝑖 𝑎𝑎𝑓𝑓 
e
nd if 
           𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔 ≔ 𝑝𝑝𝑑𝑑𝑟𝑟𝑔𝑔 + 1; 
e
nd if 
end while 
return 𝐶𝐶𝑠𝑠, 𝐶𝐶𝑔𝑔 
1.4 C. elegans neural network model 
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Our neural network model consists of 136 neurons, including 15 sensory neurons for inputs 
and 80 motor neurons for outputs (Supplement Table S5). Here are the reasons why we chose 
these neurons. First, 65 head ganglia neurons, whose neural dynamics were recorded in the 
published whole- brain Ca2+ imaging experiment 17 should be selected to enable the 
comparison between experiment and simulation. Second, other 71 neurons, most of which 
were motor neurons in the locomotion circuit of C. elegans7,18, were added to the network so 
that the model could be complete in a sensory-motor functional structure. In summary, this 
model contained neurons for chemosensory, decision, and locomotion. 
1.5 Connection parameters optimization 
The weights of synapses and gap junctions in the neural network was initialized randomly. 
The polarities of synapses were initialized randomly according to a given 
excitatory/inhibitory ratio (2.33). These parameters were optimized using an unpublished 
optimization algorithm for multi-compartment neural network. The algorithm employed a 
mathematical approximation approach to calculate gradients for Back Propagation Through 
Time (BPTT) to update parameters. Except for the weights (of synapses and gap junctions) 
and the polarities (of synapses), the inputs (of some sensory neurons) also can be optimized 
by this optimization algorithm. 
1.6 PCA of neural activities 
Principal Component Analysis (PCA) was applied to the time series data of soma membrane 
potential of the 65 head neurons. Before conducting the analysis, the time series data was 
preprocessed by subtracting the mean value of the trace. 
1.7 Correlation analysis  
The correlation between two neurons was quantified by the Pearson correlation coefficient 
of their neural activities, which ranged from -1 to 1. A coefficient of -1 or 1 indicates a strong 
negative or positive correlation of neural activity, respectively. A coefficient of 0 indicates 
that the neural activities of the two neurons are unrelated. In this analysis, the neural activity 
was represented by the membrane potential of soma. 
2 Details of Body & Environment Model 
2.1 The body model 
Tetrahedron is the simplest ordinary convex polyhedra in three-dimensional space. It has 
four vertices, six edges, and is bounded by four triangular faces. We used tetrahedron as the 
basic element to construct the body model. Surface shape of the body was extracted as 
triangle list from Sibernetic’ membrane data19. Then we triangulated the body with a robust 
and high-quality algorithm20. The triangulated tetrahedron mesh was refined for latter 
mechanics solver. The final body mesh was in a straight, natural resting pose that we defined 
as the standard body (Fig.6A). The triangulated tetrahedron body was not strictly symmetric 
in dorsal-ventral direction and left-right direction. Since triangulation algorithm had 
additional constraints on quality of tetrahedrons. This non-strict symmetry is probably more 
consistent with real C. elegans. 
2.2 Muscles 
96
 muscles were arranged from head to tail according to C. elegans  anatomy. Each 24 
muscles were grouped into longitudinal string, marked as ventral right (VR), ventral left 
(VL), dorsal right (DR), dorsal left (DL) in four quadrants (Fig.6B)19. We imported body 
membrane mesh into blender21 and modeled VR, VL, DR, DL by four Bezier curves. Then 
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muscle geometry was exported as wavefront obj file. We used Finite Element Method (FEM) 
to simulate the soft body, and used finite element constraints to simulate elastic muscle cells 
of body. Each muscle cell (Fig.6A) corresponds to one or more FEM constraints according to 
the discretized tetrahedron body (Fig.6C). Only nearby tetrahedrons can be driven by 
corresponding muscle cells.  
2.3 Soft body solver  
We used projective dynamics22 as the solver of the soft body. The purpose of mechanics 
solver is calculating positions and velocities of all 3D vertices of the deformed body over 
time. Key idea of projective dynamics is to compute a local step and a global step iteratively. 
At local step, vertex positions of the body are fixed and projection variables are updated. At 
global step, projections are fixed and vertex positions of the body are updated. The local step 
is well suited for parallelism, thus can be optimized for performance. The global step is a 
linear system and can be precomputed if the body geometry and constraints are not changed. 
After a few local-global iterations the vertex positions of deformed body are converged. Then 
velocity of deformed body can be computed by Euler integration.  
2.4 Interaction with environment 
A simplified model of hydrodynamics was used to simulate the interaction between the 
body and the fluid, similar to the published study 23. For simplicity, we supposed external 
force is only formed at the surface of body. Two kinds of external forces were considered 
here: thrust force offered by body movement, and drag force offered by fluid. With this 
simplification, computation in the whole 3D fluid environment is avoided. Thus, more agents, 
larger environments, more complex tasks are achievable through this simplification. 
2.5 Numerical stable coordinate system for body movement 
C. elegans  is a soft body animal which has special patterns during movement. Every 
position of the body is oscillating during movement. Moreover, there is no apparent important 
marks on the body during locomotion, therefore it’s hard to find an apparent stable coordinate 
system for C. elegans movement. Simply using head or body center as a coordinate system 
can’t analysis the body’s movement quantitatively and stably. Here we proposed an effective 
and simple method to measure C. elegans locomotion. The core of our method is to find a 
numerically stable coordinate system relative to the body itself. The main steps of our method 
are as follows:  
1) Define the target body, standard body and standard body coordinate system (SBCS).  
2) Obtain transformation between the target body and the standard body. 
3) Use the transformation in 2) to perform three-dimensional transformation on the SBCS to 
obtain the target body reference coordinate system (TBRCS). 
Details of the three steps are described below: 
Step 1) The target body is the research subjects. It can change its shape over time and 
exhibits various behaviors. The standard body is a special case of the target body. Here we 
define the standard body in a natural state where the head and tail are in a straight line, 
without stretching and bending. There is no internal force in the standard body soft body, and 
no external force acting on the standard body surface. The S BCS is a three-dimensional 
orthogonal coordinate system used to reflect the overall structure and function of the standard 
body. In this case, we define the SBCS with origin at the standard body center. X-axis of the 
SBCS is towards the head of the standard body. Z- axis of the S BCS is towards ventral to 
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dorsal direction of the standard body. Three axes of the SBCS form a right-handed coordinate 
system. Schematic diagram of the target body, the standard body and the SBCS can been seen 
in Figure. 7A. 
Step 2) In this step, we first define the corresponding matching points on the target body 
and the standard body. For example, simply use all the vertices of the body. The 
transformation between the matching points on the target body and the standard body is 
calculated as the transformation between the target body and the standard body. Calculation 
formulas of the transformation are as follows: 
𝑝𝑝0��� = 1
𝑚𝑚 � 𝑝𝑝𝑖𝑖
0
𝑚𝑚
𝑖𝑖=1
 
𝑝𝑝̅ = 1
𝑚𝑚 � 𝑝𝑝𝑖𝑖
𝑚𝑚
𝑖𝑖=1
 
𝑞𝑞𝑖𝑖
0 = 𝑝𝑝𝑖𝑖
0 − 𝑝𝑝0��� 
𝑞𝑞𝑖𝑖 = 𝑝𝑝𝑖𝑖 − 𝑝𝑝̅ 
𝐻𝐻 = 𝑞𝑞𝑖𝑖
0𝑞𝑞𝑖𝑖𝑇𝑇 
𝑅𝑅 = 𝑉𝑉𝑈𝑈𝑇𝑇 
𝑇𝑇 = 𝑝𝑝̅ − 𝑅𝑅𝑝𝑝 0��� 
𝑀𝑀 = �𝑅𝑅 𝑇𝑇
0 1 � 
The matching points are represented by column vectors. 𝑝𝑝𝑖𝑖
0  are matching points on the 
standard body, and 𝑝𝑝𝑖𝑖 are matching points on the target body . m is the number of matching 
points. 𝑝𝑝0��� is the center point of matching points on the standard body , and 𝑝𝑝̅ is the center 
point of matching points on the standard body. 𝑞𝑞𝑖𝑖
0 refers to the standard body matching points 
with the center offset subtracted, and 𝑞𝑞𝑖𝑖 denotes the standard body matching points with the 
center offset subtracted. 𝐻𝐻  is a matrix computed by 𝑞𝑞𝑖𝑖
0𝑞𝑞𝑖𝑖𝑇𝑇 . 𝑞𝑞𝑖𝑖𝑇𝑇  is transpose of 𝑞𝑞𝑖𝑖 , 𝑞𝑞𝑖𝑖 =
[𝑞𝑞1, 𝑞𝑞2 … 𝑞𝑞𝑚𝑚], 𝑞𝑞𝑖𝑖
0 = [𝑞𝑞1
0, 𝑞𝑞2
0 … 𝑞𝑞𝑚𝑚0 ]. 𝑈𝑈 and 𝑉𝑉  are obtained by the singular value decomposition 
𝐻𝐻 = 𝑈𝑈Λ𝑉𝑉 of the matrix  𝐻𝐻. 𝑈𝑈𝑇𝑇 is transpose of 𝑈𝑈. 𝑀𝑀 is the transformation matrix between the 
target body and the standard body, which will be used in next step. 𝑅𝑅 is rotation part of 𝑀𝑀, 𝑇𝑇 
is translation part of 𝑀𝑀. A schematic diagram of matching points and transformation 𝑀𝑀 are 
shown in Fig. 7A. 
Step 3) In this step, we use matrix 𝑀𝑀 to perform three -dimensional transformation on the 
SBCS to obtain the TBRCS . The matrix of the TBRCS can be formulated as: 𝑀𝑀� = 𝑀𝑀𝑀𝑀0 =
�𝑅𝑅𝑅𝑅0 𝑅𝑅𝑇𝑇0 + 𝑇𝑇
0 1 � . 𝑀𝑀 = �𝑅𝑅 𝑇𝑇
0 1 � ,  𝑅𝑅  and 𝑇𝑇  have been described in last section. 𝑀𝑀0 =
�𝑅𝑅0 𝑇𝑇0
0 1 � is matrix used to represent the S BCS. 𝑇𝑇0  is the origin of the S BCS, and 𝑅𝑅0 
indicates orientation of the S BCS. The column vector of 𝑅𝑅0  represents the orthogonal 
coordinate axis. Finally, we can make the TBRCS through 𝑀𝑀�: 𝑅𝑅𝑇𝑇0 + 𝑇𝑇 as the origin and 𝑅𝑅𝑅𝑅0 
as the orthogonal coordinate axis.  
2.6 Numerically stable measurements of body movement 
1) Measurement of body trajectory 
During behavior analysis, C. elegans  can be seen as a deformable biology composed of 
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multiple points. When studying the locomotion of a deformable body, generally one or more 
points of biology body is defined as tracking object, and locomotion of deformable body is 
studied through tracking these defined points. For deformable biology like C. elegans , 
position and velocity of local tracking points are hard to reflect overall state of body 
locomotion (Fig. 7C). As a result, values such as body  trajectory or steering angle obtained 
through these tracking points are usually unstable over time. Using the TBRCS, we can better 
describe the overall trend of body movement. Comparison of trajectories are shown in Figure. 
7A. If the center point of body is used as trajectory, the trajectory is a zigzag shape. However, 
if the TBRCS is used as trajectory of body movement, the trajectory is a smooth curve which 
means stable values. It can be seen the TBRCS  is more suitable as a measurement of overall 
path and direction of body locomotion. 
2) Tracking body movement 
H
ere, we demonstrate how to track body movement with the TBRCS. If the standard body 
is drawn in the TBRCS , then it can be used as a reference for the target body . The standard 
body and the target body will be center alignment, translate and rotate together. We define 𝑤𝑤𝑘𝑘
0 
as 3D tracking points of the standard body , and 𝑤𝑤𝑘𝑘 as 3D tracking points of the target body. 
k = 1, 2 … n. n is the number of tracking point. Then relative movement of body can be 
defined as 𝑤𝑤𝑘𝑘 − 𝑤𝑤𝑘𝑘
0. It’s a measurement of body locomotion relative to itself (the standard 
body), which is more numerically stable than relative to world. 𝑤𝑤𝑘𝑘 − 𝑤𝑤𝑘𝑘
0 is also better than 
relative to body center, since both translation part and rotation part of whole deformation 
body are considered in the TBRCS. Moreover, sampling bias of tracking points is eliminated 
through subtract operation, since bias minus bias equals zero. 
In
 Figure 7B, we sampled 17 tracking points on the body uniformly. Offset of the tracking 
points are indicated by pink lines between tracking points. Relative velocity of 3D tracking 
points is drawn in blue arrows. As a common method, trajectories of sampled 17 tracking 
points that relate to world are also shown in Fig ure 7C. We can see that compared to all 17 
tracking points trajectories, trajectory of the TBRCS  is the smoothest. Relative position and 
relative velocity of 3D tracking points over time are plotted in Figure 7 E and Fig ure 7 F 
respectively. In Figure 7E, Positions of 3D tracking points are tiled from bottom to top for 
clear seeing. In this case, swing direction (Z direction of the TBRCS ) of swimming body can 
be clearly reflected with the TBRCS . We can also find that there is a spring movement in X 
direction of the TBRCS (tail to head) during forward swimming. Locomotion in Y direction 
of the TBRCS is nearly flat since in this case muscle activations are approximately 
symmetrical in this direction. From Fig ure 7 F we can further deduced that the biggest 
propulsion of forward swimming is come from tail (purple line), since speed means power. 
3) Steering angle of body movement 
If 𝑂𝑂𝑡𝑡𝑖𝑖   is the trajectory of TBRCS origin at time 𝑑𝑑𝑖𝑖 . 𝑑𝑑𝑖𝑖 =  𝑑𝑑1, 𝑑𝑑2 … 𝑑𝑑𝑚𝑚 . Then we can get 
velocity of TBRCS  by 𝑣𝑣𝑖𝑖 =
𝑂𝑂𝑡𝑡𝑖𝑖−𝑂𝑂𝑡𝑡𝑖𝑖−1
𝑡𝑡𝑖𝑖−𝑡𝑡𝑖𝑖−1
. In this way, velocity magnitude and velocity direction 
over time are numerically stable. We used angle θ which is between TBRCS x-axis and 𝑣𝑣𝑖𝑖 as 
a measurement of steering, as shown in Fig ure 7D. Both 𝑣𝑣𝑖𝑖 and the TBRCS  are numerically 
stable over time, thus θ is also numerically stable. 
3 Locomotion Behavior Simulation of C. elegans  
3.1 Simulation configuration 
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The neural network simulation was coupled with the physical body & environment 
simulation to achieve closed-loop interaction. Food was at the center of the simulated 
physical environment, acting as an attractor to the C. elegans . The food concentration was 
linearly distributed in the environment. The neural system received currents regarded as 
sensory signals. The amplitude of the currents was proportional to the gradient of food 
concentration. The outputs of the neural network controlled the movements of the body in the 
environment through a reservoir readout weight matrix (see Method 3.2). As the body moved 
in the physical environment, the distance between the body and the food changed, causing the 
food concentration gradient updated and transformed to currents injected into the soma of 
sensory neurons. 
During the interaction of the two sub-models, the simulation time step of the neural network 
was 5/3 ms and that of physical environment simulation was 100 ms. The two systems 
synchronized every 100 ms to update the sensory signals and muscle forces. 
3.2 Reservoir configuration 
To establish the neuron- muscle connection between the neural system and physical body 
muscles, a reservoir computing framework was used. The neural network model acted as the 
non-linear reservoir. The membrane potentials of 80 motor neurons (Supplement Table S5) 
were linearly combined and mapped to 96 muscle forces via an 80 by 96 matrix.  
The r eservoir  was trained using data extracted from a 10-second movement generated in the 
simulated physical environment. The pairs of food concentration and muscle force during the 
movements were used as inputs to the neural network and target outputs of the readout layer, 
respectively. Only the reservoir readout matrix was optimized, while the parameters in the 
neural network model was fixed. 
4 User-friendly Interfaces 
The neural network model consists of ion channel, neuron, and network connection 
modules. The parameter and algorithm settings of each module can be modified 
independently. Parameters were saved in JSON files that are easy to read and modify.  
Code Availability 
The C. elegans neural network simulation was performed with the code written in Python 
3.7 and employing NEURON 8.0.0 24 as a simulation engine. The custom-developed code 
used in this work will be made publicly available in an online repository latest upon journal 
publication. If you wish to obtain the code before then, please contact us. To facilitate better 
evaluation of this article by reviewers, we also provide the code that can be run (we will 
continue to optimize our code details): https://www.dropbox.com/s/ep2yv3rmlp67eex/worm-
simulation-share.zip?dl=0, Please contact us if you would like to have access to the latest 
code before our article is published. 
 
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