Author Contributions
Conceptualization, Z.C., Y.Y. and X.X.; methodology, Z.C., Y.Y. and X.X.; software, Z.C.; validation, Z.C., Y.Y. and X.X.; formal analysis, Z.C.; investigation, Z.C.; resources, Y.Y. and X.X.; data curation, Z.C.; writing—original draft preparation, Z.C.; writing—review and editing, Z.C., Y.Y. and X.X.; visualization, Z.C.; supervision, Y.Y. and X.X.; project administration, Y.Y. and X.X.; funding acquisition, Y.Y. and X.X. All authors have read and agreed to the published version of the manuscript.
Figure 1.
The digital C. elegans body. (a) The digital C. elegans body is vermiform and positioned in a sinusoidal posture. It consists of 25 articulated body segments with the red tip indicating its head. (b) The body posture is determined by the angles of its joints between body segments, which are provided by the physics engine at each time step. (c) The digital C. elegans is actuated by 95 body wall muscles arranged in 4 (DL, DR, VL, and VR) quadrants. Each joint is actuated by 4 corresponding muscles, one in each quadrant, except for the last joint with only 3 muscles. The gray spheres are anchors for muscles, and the red rods connecting the anchors are muscle actuators.
Figure 1.
The digital C. elegans body. (a) The digital C. elegans body is vermiform and positioned in a sinusoidal posture. It consists of 25 articulated body segments with the red tip indicating its head. (b) The body posture is determined by the angles of its joints between body segments, which are provided by the physics engine at each time step. (c) The digital C. elegans is actuated by 95 body wall muscles arranged in 4 (DL, DR, VL, and VR) quadrants. Each joint is actuated by 4 corresponding muscles, one in each quadrant, except for the last joint with only 3 muscles. The gray spheres are anchors for muscles, and the red rods connecting the anchors are muscle actuators.
Figure 2.
The virtual chemotaxis environment with radial gradients of chemical attractant. (a) The rendered virtual chemotaxis environment, with the highest concentration of attractant indicated by the green cylinder at the center of the distribution. (b) The concentration of attractant in the virtual environment follows a radial Gaussian distribution ranging from 0 to 1.
Figure 2.
The virtual chemotaxis environment with radial gradients of chemical attractant. (a) The rendered virtual chemotaxis environment, with the highest concentration of attractant indicated by the green cylinder at the center of the distribution. (b) The concentration of attractant in the virtual environment follows a radial Gaussian distribution ranging from 0 to 1.
Figure 3.
The structure of the C. elegans neural network model with 469 nodes, 4869 chemical connections, and 1433 electrical connections. The model consists of 57 pharynx cells, 83 sensory neurons, 81 interneurons, 108 motor neurons, 21 end organs, and 24 sex-specific cells. The distribution of nodes reflects the biological positions of cells in the worm. The 95 body wall muscles, which are also part of the model, are not depicted in this graph.
Figure 3.
The structure of the C. elegans neural network model with 469 nodes, 4869 chemical connections, and 1433 electrical connections. The model consists of 57 pharynx cells, 83 sensory neurons, 81 interneurons, 108 motor neurons, 21 end organs, and 24 sex-specific cells. The distribution of nodes reflects the biological positions of cells in the worm. The 95 body wall muscles, which are also part of the model, are not depicted in this graph.
Figure 4.
The structure of proprioceptive feedback connections. (a) Each B-type and A-type motor neuron receives proprioceptive feedback from its anterior and posterior region, respectively, extending over 5 joints. (b) The connection matrix of proprioceptive feedback from joints to motor neurons, with 1 indicating a connection and 0 indicating no connection. Head motor neurons (HMNs) and sublateral motor neuron (SMNs) receive all proprioceptive feedback. B-type (DB, VB) and A-type (DA, VA) motor neurons receive feedback from their respective anterior and posterior regions.
Figure 4.
The structure of proprioceptive feedback connections. (a) Each B-type and A-type motor neuron receives proprioceptive feedback from its anterior and posterior region, respectively, extending over 5 joints. (b) The connection matrix of proprioceptive feedback from joints to motor neurons, with 1 indicating a connection and 0 indicating no connection. Head motor neurons (HMNs) and sublateral motor neuron (SMNs) receive all proprioceptive feedback. B-type (DB, VB) and A-type (DA, VA) motor neurons receive feedback from their respective anterior and posterior regions.
Figure 5.
The PID controller is used for closed-loop control of the digital C. elegans’ behavior. It continuously generates action signals of muscles to minimize the error of posture and reproduce the desired behavior. The controller comprises three terms: proportional (P), integral (I), and derivative (D), represented by the blocks P, I, and D, respectively.
Figure 5.
The PID controller is used for closed-loop control of the digital C. elegans’ behavior. It continuously generates action signals of muscles to minimize the error of posture and reproduce the desired behavior. The controller comprises three terms: proportional (P), integral (I), and derivative (D), represented by the blocks P, I, and D, respectively.
Figure 6.
An example trial of chemotaxis simulation with the digital twin C. elegans lasting 100 s: (a) The worm starts at mm and autonomously crawls towards the center; (b) The heatmap shows the joint angles exhibiting sinusoidal pattern with a cycle of ∼1.28 s; (c) The heatmap shows how the activation of 95 muscles changes overtime with a cycle of ∼1.28 s. The dorsal (DL, DR) and ventral (VL, VR) muscles activate in turns with ∼0.64 s of activation and ∼0.64 s of resting to generate dorsal–ventral sinusoidal movement; (d) The evolution of the worm’s posture during a turn, corresponding to the red section of the track; (e) The concentration detected at the worm’ head gradually increases along its track; (f) The chemical gradient detected at the worm’s head and the chemical gradient perpendicular to the traveling direction (normal gradient) detected at the worm’s center of mass along its track; (g) The activation of ASE neurons during chemotaxis.
Figure 6.
An example trial of chemotaxis simulation with the digital twin C. elegans lasting 100 s: (a) The worm starts at mm and autonomously crawls towards the center; (b) The heatmap shows the joint angles exhibiting sinusoidal pattern with a cycle of ∼1.28 s; (c) The heatmap shows how the activation of 95 muscles changes overtime with a cycle of ∼1.28 s. The dorsal (DL, DR) and ventral (VL, VR) muscles activate in turns with ∼0.64 s of activation and ∼0.64 s of resting to generate dorsal–ventral sinusoidal movement; (d) The evolution of the worm’s posture during a turn, corresponding to the red section of the track; (e) The concentration detected at the worm’ head gradually increases along its track; (f) The chemical gradient detected at the worm’s head and the chemical gradient perpendicular to the traveling direction (normal gradient) detected at the worm’s center of mass along its track; (g) The activation of ASE neurons during chemotaxis.
Figure 7.
Two-dimensional density plot of the digital C. elegans’ tracks over 100 trials (each 100 s) in different scenarios: (a) The worm starts at a random position in an environment with chemical gradients, resulting in convergence at the center. The average chemotaxis index is with an average initial concentration of ; (b) The worm starts at a random position in an environment without chemical gradients; (c) The worm starts at mm in an environment with chemical gradients, resulting in convergence at the center. The average chemotaxis index is with an initial concentration of 0.01; (d) The worm starts at mm in an environment without chemical gradients.
Figure 7.
Two-dimensional density plot of the digital C. elegans’ tracks over 100 trials (each 100 s) in different scenarios: (a) The worm starts at a random position in an environment with chemical gradients, resulting in convergence at the center. The average chemotaxis index is with an average initial concentration of ; (b) The worm starts at a random position in an environment without chemical gradients; (c) The worm starts at mm in an environment with chemical gradients, resulting in convergence at the center. The average chemotaxis index is with an initial concentration of 0.01; (d) The worm starts at mm in an environment without chemical gradients.
Figure 8.
Statistical results of chemotaxis simulation over 100 trials (each 100 s) with the C. elegans neural network model and the PID controller in environments with and without chemical gradients, as well as the results from biological experiments. The curving rate (degree/mm) and chemical gradient generated in the simulation are scaled to the same range as the results from biological experiments. (a,b) Relationship between curving rate and bearing in environments with and without chemical gradients. (c,d) Relationship between curving rate and chemical gradient in the normal direction in environments with and without chemical gradients.
Figure 8.
Statistical results of chemotaxis simulation over 100 trials (each 100 s) with the C. elegans neural network model and the PID controller in environments with and without chemical gradients, as well as the results from biological experiments. The curving rate (degree/mm) and chemical gradient generated in the simulation are scaled to the same range as the results from biological experiments. (a,b) Relationship between curving rate and bearing in environments with and without chemical gradients. (c,d) Relationship between curving rate and chemical gradient in the normal direction in environments with and without chemical gradients.
Figure 9.
The average crawling distance results of node ablation experiments: (a) The average crawling distance of all nodes grouped by node types. The control value is the average crawling distance without ablation (9.19 mm). The average result of individual node ablation for all nodes is 5.81 mm. Node types with result lower than the overall average (5.81 mm) are predicted to be significant for the behavioral modulation of sinusoidal crawling and are colored blue. Node types that are experimentally proven to be involved in sinusoidal crawling are marked with asterisks; (b) The average crawling distance of different types of ventral cord motor neurons, including B-type, A-type, D-type, and AS motor neurons; (c) The subcircuit responsible for sinusoidal crawling consists of 119 neurons with ablation result lower than the overall average (5.81 mm), including 69 motor neurons, 22 interneurons, and 28 sensory neurons. The distribution of nodes reflects the biological positions of cells in the worm.
Figure 9.
The average crawling distance results of node ablation experiments: (a) The average crawling distance of all nodes grouped by node types. The control value is the average crawling distance without ablation (9.19 mm). The average result of individual node ablation for all nodes is 5.81 mm. Node types with result lower than the overall average (5.81 mm) are predicted to be significant for the behavioral modulation of sinusoidal crawling and are colored blue. Node types that are experimentally proven to be involved in sinusoidal crawling are marked with asterisks; (b) The average crawling distance of different types of ventral cord motor neurons, including B-type, A-type, D-type, and AS motor neurons; (c) The subcircuit responsible for sinusoidal crawling consists of 119 neurons with ablation result lower than the overall average (5.81 mm), including 69 motor neurons, 22 interneurons, and 28 sensory neurons. The distribution of nodes reflects the biological positions of cells in the worm.
Figure 10.
The average chemotaxis index results of node ablation experiments: (a) Nodes with average chemotaxis index after ablation lower than the overall average (0.43). The control value is the average chemotaxis index without ablation (0.65). The concentration of the worm’s initial position is 0.14. Nodes previously identified as responsible for sinusoidal crawling are excluded; (b) The subcircuit responsible for chemotaxis navigation consists of 40 neurons with ablation results lower than the overall average (0.43), including 10 sensory neurons, 15 interneurons, and 11 motor neurons. The distribution of nodes reflects the biological positions of cells in the worm.
Figure 10.
The average chemotaxis index results of node ablation experiments: (a) Nodes with average chemotaxis index after ablation lower than the overall average (0.43). The control value is the average chemotaxis index without ablation (0.65). The concentration of the worm’s initial position is 0.14. Nodes previously identified as responsible for sinusoidal crawling are excluded; (b) The subcircuit responsible for chemotaxis navigation consists of 40 neurons with ablation results lower than the overall average (0.43), including 10 sensory neurons, 15 interneurons, and 11 motor neurons. The distribution of nodes reflects the biological positions of cells in the worm.
Table 1.
Trainable parameters of the C. elegans neural network model.
Table 1.
Trainable parameters of the C. elegans neural network model.
Parameter | Unit | Initialization | Constraint | Number |
---|
| second (s) | | 1 | 469 |
| millivolt (mV) | | - | 469 |
| millivolt (mV) | | 1 | 3324 2 |
| - | | 3 | 4869 |
| - | | 3 | 1433 |
| - | | - | 1223 |
4 | - | 1 | 3 | 3 |
Table 2.
Behavioral performance comparison between the C. elegans neural network model and the PID controller over 100 trials (each 100 s) with different initial positions. The crawling distance and chemotaxis index represent the performance of sinusoidal crawling and chemotaxis navigation, respectively. Results are presented as mean ± standard deviation (s.d.).
Table 2.
Behavioral performance comparison between the C. elegans neural network model and the PID controller over 100 trials (each 100 s) with different initial positions. The crawling distance and chemotaxis index represent the performance of sinusoidal crawling and chemotaxis navigation, respectively. Results are presented as mean ± standard deviation (s.d.).
Controller | PID Controller | C. elegans Neural Network |
---|
Initial Position (mm) | Crawling Distance (mm) | Chemotaxis Index | Initial Concentration | Crawling Distance (mm) | Chemotaxis Index | Initial Concentration |
Random | | | | | | |
| | | 0.61 | | | 0.61 |
| | | 0.14 | | | 0.14 |
| | | 0.01 | | | 0.01 |