Figures and data in A neural network model that generates salt concentration memory-dependent chemotaxis in Caenorhabditis elegans

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Videos
Tables
Additional files

9 figures, 3 videos, 4 tables and 1 additional file

Figures

Figure 1

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A neuroanatomical minimal network circuit for salt klinotaxis in *C. elegans*.

The white circles represent chemosensory neurons, the gray circles represent interneurons, and the black circles denote motor neurons. The black and green connections between neurons represent the chemical synaptic connections and electrical gap junctions, respectively. The minimal circuit was derived from the *C. elegans* connectome, with two constraints applied as described in the text (Izquierdo and Beer, 2013).

Figure 2

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The trajectories of the worm’s locomotion.

The highest-performing network models, with and without the AIY–AIZ connections constrained to be inhibitory, were placed at the initial position, 4.5 cm away from the salt gradient peak, at 10 different angles of worm orientation, and allowed to move freely for 250 s. The salt concentrations were represented by a Gaussian distribution. The color of the trace represents the passage of time. (a) The highest-CI network model that evolved without any constraints. The CI is 0.855, with a standard deviation of 0.006. (b) The highest-CI network model that evolved with the constraint that the AIY–AIZ connection be inhibitory. The CI is 0.877, with a standard deviation of 0.002. The insets provide an enlarged view of the sinusoidal locomotion and turning processes.

Figure 3 with 3 supplements

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The most optimized neural network circuits, with and without constraining the AIY–AIZ connections to be inhibitory, and the resulting network responses to step changes in salt concentration.

The most optimized neural network circuits without (a) and with (b) the constraints. The blue arrow and the red blunt arrow indicate an excitatory and an inhibitory synaptic connection, respectively. The green connections represent electrical gap junctions. The color intensity of these connections indicates the strength of the synaptic connections. (c) The neurotransmitter release, $z_{i}$ , from each neuron in the most optimized network without the constraints is illustrated as a response to step changes in the salt concentration. The responses to positive and negative step changes in the salt concentration are represented by the colors blue and red, respectively. (d) The illustration is the same as (c), but the outcome was obtained from the most optimized network with the constraints. In both (c) and (d), the black horizontal line represents the level of the bias term, $θ_{i}$ , in the interneurons and motor neurons. In both (c) and (d), the turning angle $φ$ , as defined by Equation A13a (see Figure 8b), is illustrated in the second panel from the bottom. In order to identify the direction of the sweep of the head sensory neurons upon introducing step changes in salt concentration, it is necessary to confirm the ideal sinusoidal trajectory of the model worm in the absence of sensory input. This is illustrated in the lowest panel from the bottom in (**c, d**).

Figure 3—figure supplement 1

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Signal transmission through chemical synaptic connections and electrical gap junctions.

The most optimized neural circuits developed without (a) and with (b) the constraints mentioned in the text. For purposes of reference, the same as in Figure 3a, b are shown again. (c, d) The neurotransmitter release, $z_{i}$ , from each neuron in the network (a, b) in response to step changes in salt concentration of positive (blue) and negative (red) is shown in (c, d), respectively, along with the signaling through the chemical synaptic connection and the electrical gap junction. The $z_{i}$ s shown herein are the same as those shown in Figure 3c, d. (e, f) The membrane potential $y_{i}$ that was observed in the network (a, b) is shown in (e, f), respectively. In (c–f), the blue (red) arrow and the blue (red) line with an ending in a filled circle show the excitatory and inhibitory synaptic transmission, respectively, in response to positive (negative) step changes in salt concentration. The blue (red) bidirectional arrow indicates gap junction signal transmission, in response to positive (negative) step changes in salt concentration.

Figure 3—figure supplement 2

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Blocking electrical gap junctions significantly has a marked effect on neurotransmitter release $z_{i}$ and the resulting turning angle $φ$ .

The most optimized neural circuits without (a) and with (b) the constraints. With the exception of the gap junctions that have been blocked, all other parameters remain consistent with those illustrated in Figure 3.

Figure 3—figure supplement 3

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The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the time half a cycle later than those shown in Figure 3.

The results obtained from the most optimized neural circuits without and with the constraints are shown in (a, b), respectively. With the exception of the timing of the step changes in salt concentration, all other conditions are identical to those depicted in Figure 3.

Figure 4

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The analysis of klinotaxis in the most optimized model with the constraints.

(a) The curving rate as a function of the bearing. (b) The definitions of the curving rate and the normal direction of translational movement, which are utilized in the klinotaxis analysis, are illustrated. (c) The curving rate as a function of the normal gradient of salt concentration. (d) The positive (red) and negative (blue) components of the curving rate as a function of the translational gradient of salt concentration. The black dots represent the mean value of the two components. In the analysis, the salt concentration profile was modeled with a Gaussian distribution. All the error bars represent the standard deviation.

Figure 5 with 5 supplements

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The reversal of salt concentration memory-dependent preference behavior in klinotaxis is attributed to the alteration from inhibitory to excitatory connections between ASER and AIY.

In the most optimized neural circuit with the constraints, it was postulated that the synaptic connections between ASER and AIY would be altered from (a) inhibitory connections to (b) excitatory connections when the cultivated salt concentration was replaced from (a) a higher to (b) a lower concentration than the current environment. The figure shown in (a) is identical to Figure 3b and #0 of Figure 5—figure supplement 1. (c) The curving rates obtained from the networks illustrated in (a) and (b) (corresponding to #0 and #15 in Figure 5—figure supplements 1 and 2, respectively), are presented as a function of the normal gradient of salt concentration. In addition, the curving rate obtained from the neural circuit with an intermediate nature in the neural circuit between those shown in (a, b) (corresponding to #9 in Figure 5—figure supplements 1 and 2) is also presented. (d) The curving rates as a function of the normal gradient of salt concentration, which were experimentally determined when the cultivated salt concentration was higher (black) and lower (red) than the current environment (Kunitomo et al., 2013). The case in which the cultivated salt concentration was close to the current environmental concentration (50 mM) is also shown in green. (e) The analysis presented here is identical to Figure 3d, except that the ASER-AIY inhibitory connections in the most optimized model with the constraints have been replaced with excitatory connections as illustrated in b.

Figure 5—figure supplement 1

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The weight of the ASER–AIY synaptic connection, $w_{j i}$ , in the most optimized network with the constraints is increased from a negative value (inhibitory connection) to a positive value (excitatory connection) by introducing an increment of 1.5 to $w_{j i}$ at each step.

Figure 5—figure supplement 2

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As the weight of the ASER–AIY synaptic connection $w_{j i}$ is increased from negative (inhibitory connection) to positive (excitatory connection) in the most optimized network with the constraints by introducing an increment of 1.5 to $w_{j i}$ at each step, the curving rate is varied from an increasing function with an increasing normal gradient of salt concentration to a function that shows a decreasing trend.

The curving rate shown in the #0, #9, and #15 is also shown as that for the inhibitory, intermediate, and excitatory connections, respectively, in Figure 5c.

Figure 5—figure supplement 3

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The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the timing half a cycle later than those shown in Figure 5e.

The changes in $φ$ in response to step increases in $z_{ON}$ and $z_{OFF}$ at the timing of half a cycle later ensured that the regulation mechanism of $φ$ discussed in the text (Figure 5e) was maintained.

Figure 5—figure supplement 4

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The trajectories of the worm’s locomotion simulated by the network shown in Figure 5a, b.

(a) For purposes of comparison, the same trajectories depicted in Figure 2b are presented, again. (b) The trajectories given by the network depicted in Figure 5b, wherein the inhibitory connections between the ASER and AIY in the most optimized model with the constraints have been substituted with excitatory connections. The majority of the trajectories exhibited a directionality that was opposite to the peak of the salt concentration gradient. However, a subset showed a trajectory where the worm proceeded toward the peak of the gradient and passed it without any discernible response.

Figure 5—figure supplement 5

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Signal transmission through chemical synaptic connections and electrical gap junctions.

For purposes of reference, those shown in Figure 5a, b are presented again in (a) and (b), respectively. (c) The neurotransmitter release $z_{i}$ in response to step changes in the salt concentration of positive (blue) and negative (red) for the most optimized circuit with the constraints shown in (a), is shown together with the signal transmissions through the chemical synaptic connections and electrical gap junctions. (d) The same as (c) is shown, except that the inhibitory connections between the ASER and AIY are replaced with the excitatory connections, as shown in (b). The $z_{i}$ shown in (c) and (d) are the same as those shown in Figures 3d and 5e, respectively. (e, f) The membrane potentials $y_{i}$ resulting in $z_{i}$ , as illustrated in (c) and (d), are shown in (e) and (d), respectively. In (**c-f**), the blue (red) arrow and the blue (red) line with an ending in a filled circle indicate the excitatory and inhibitory synaptic connections, respectively, in response to step changes in the salt concentration of positive (negative). The blue (red) bidirectional arrow indicates the transmission via gap junctions, in response to positive (negative) step changes in salt concentration.

Figure 6 with 1 supplement

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Inhibition of SMB activity suppresses the salt concentration memory-dependent preference behavior in klinotaxis, as observed experimentally.

(a) The curving rates as a function of the normal gradient of salt concentration that were experimentally observed in starved individuals cultivated at a salt concentration higher, comparable, and lower than the current environment (Kunitomo et al., 2013). (b) The curving rates as a function of the normal gradient of salt concentration that were obtained from the neural circuits that yielded the results in Figure 5c, except that here the synaptic connections between AIZ and SMB and the self-connections of SMB were multiplied by 0.9 to inhibit the SMB activity. (c) The trajectories of the model worm obtained by inhibiting the SMB activity in the most optimized network, where the ASER–AIY connections remained inhibitory, as shown in Figure 5a (or #0 of Figure 5—figure supplement 1). (d) The trajectories of the model worm obtained by inhibiting the SMB activity in the most optimized network, where the ASER–AIY connections were altered to be excitatory, as shown in Figure 5b (or #15 of Figure 5—figure supplement 1).

Figure 6—figure supplement 1

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The inhibition of SMB activity by reducing the bias term $θ_{SMB}$ has the effect of suppressing the salt memory-dependent preference behavior observed in klinotaxis, similar to that illustrated in Figure 6b.

(a) For purposes of comparison, the same experimental data on the curving rate as presented in Figure 6a is provided here again. The figures shown in (**b–d**) are the same as shown in Figure 6b–d, respectively, except that in the most optimized models depicted in Figure 5c, the bias terms for the SMB motor neurons were reduced, rather than the synaptic connections between AIZ and SMB being weakened as was done in Figure 6.

Figure 7

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Modeling of the synaptic transmission from the ASEL and ASER sensory neurons in response to changes in NaCl concentration.

Figure 8

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Worm locomotion model.

(a) The body of the worm, consisting only of the idealized head and neck regions of *C. elegans*. The worm model was represented as a point (*r_x*, *r_y*), located at the center of the boundary between the head and neck regions of the model. The μ represents the angle between the velocity vector v and the positive x-axis. In this context, a counterclockwise angle is considered positive. The dorsal (gray) and ventral (black) motor neuron pairs receive an out-of-phase constitutive oscillatory input from the motor systems, respectively. (b) Changes in the direction of locomotion. In the interval between time steps i −1 and i, the orientation of the velocity vector undergoes a change of the turning angle *φ_i*. The gray arc represents the path of the worm.

Figure 9

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Terminology used in the analysis of the worm’s locomotion.

Orientation vectors used in the analysis of sinusoidal locomotion. Undulations occur in the *x–y* plane. The white circles represent the start and end points of n-cycles of locomotion, where n was set to three throughout the analysis of the worm’s locomotion characteristics.

Videos

Video 1

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posterframe for video — The video of the worm’s locomotion simulated by the most optimized network with the constraints.

Video 2

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Video 3

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Tables

Appendix 1—table 1

The parameters that were optimized by the genetic algorithm in the worm’s chemotaxis simulation.

Parameter	The explanation of parameter	The range of parameter
$N$	The duration of the second interval over which the salt concentration is averaged in Equation A1.	[0.1, 4.2] (Izquierdo and Beer, 2013)
$M$	The duration of the first interval over which the salt concentration is averaged in Equation A1.	[0.1, 4.2] (Izquierdo and Beer, 2013)
$θ$	The bias term in all the neurons in Equations A5, A11a, A11b, A13b, and A13c.	[−15.0, 15.0] (Izquierdo and Beer, 2013)
$w_{i}^{O N}$	The strength of synaptic connection between interneuron i and the ASEL sensory neurons in Equation A7.	[−15.0, 15.0] (Izquierdo and Beer, 2013)
$w_{i}^{O F F}$	The strength of synaptic connection between interneuron i and the ASER sensory neurons in Equation A7.	[−15.0, 15.0] (Izquierdo and Beer, 2013)
$w_{j i}$	The strength of synaptic connection between interneurons i and j in Equations A4 and A8.	[−15.0, 15.0] (Izquierdo and Beer, 2013)
$w_{i}^{s e l f}$	The strength of self-connection of interneurons i in Equations A12a and A12b.	[−15.0, 15.0] (Izquierdo and Beer, 2013)
$g_{k i}$	The strength of gap junction between interneurons k and I in Equation A4.	[0.0, 2.5] (Olivares et al., 2018)
$w_{O S C}$	The strength of the connection to the oscillatory component in Equation A9.	[0.0, 15.0] (Izquierdo and Beer, 2013)
$w_{N M J}$	The strength of the connection from motor neurons to muscles in Equation A13a.	[1.0, 3.0] (Izquierdo and Beer, 2013)

Appendix 1—table 2

The parameters used in the simulations of worm’s chemotaxis.

Parameter	The explanation of parameter	The range of parameter
$α$	The range of randomly chosen steepness of the salt concentration gradient in Equation A15.	[−0.38,–0.01] (Izquierdo and Beer, 2013)
$x_{p e a k}$	The x coordinate of the peak position of salt concentration in Equation A15.	4.5 cm (Izquierdo and Beer, 2013)
$y_{p e a k}$	The y coordinate of the peak position of salt concentration in Equation A15.	0.0 cm (Izquierdo and Beer, 2013)
$Δ t$	The time step used in Euler integration of Equations. A4, A8. A13a, and A14.	0.01 s was mostly used, while 0.001 s was only used to display the trajectories and determine the CI values for the most optimized networks.
$T$	The duration of a one cycle of sinusoidal locomotion in Equation A10a and A10b.	4.2 s (Ferrée and Lockery, 1999; Izquierdo and Beer, 2013)
$v$	The velocity of locomotion in Equation A14.	0.022 cm/s (Ferrée and Lockery, 1999; Izquierdo and Beer, 2013)
$T_{s i m}$	The total simulated assay time for the calculation of CI by Equation A17.	500 s for the evolutionary algorithm (Izquierdo and Beer, 2013). 200 s for the analysis of klinotaxis behaviors while 1000 s for the CI calculation of obtained individuals.
$τ$	The time constant in Equations A4 and A8.	0.1 s (Izquierdo and Beer, 2013)
$C_{0}$	The parameter in the Gaussian distribution of salt concentration by Equation A16	1.0 mM (Ferrée and Lockery, 1999; Ward, 1973)
$λ$	The parameter in the Gaussian distribution of salt concentration by Equation A16	1.61 cm (Ferrée and Lockery, 1999; Ward, 1973)

Appendix 1—table 3

The parameters that control the evolutionary algorithm.

Parameter	The explanation of parameter	The range of parameter
average	The number of simulations that were performed to average the CI values of an individual.	50 (Izquierdo and Beer, 2013)
gen_size	The length of gene that was equal to the number of parameters that were searched.	22 This work
ga_count	The number of iterations for which the evolutionary algorithm was run.	100 (Izquierdo and Beer, 2013)
n_gen	The number of generations for which the populations evolved.	300 (Izquierdo and Beer, 2013)
pop_size	The number of individuals that were included in the populations	60 (Izquierdo and Beer, 2013)
sel_top	The number of top individuals that were selected.	20 This work
mat_pb	The probability by which a pair of parents was selected for a two-point crossover.	0.6 This work
mut_pb	The probability by which a mutant was selected.	0.5 This work
mut_in_pb	The probability with which a mutation occurs to the 22 elements in the vector of a selected mutant.	0.4 This work

Appendix 1—table 4

The parameters of the most optimized network model with the constraints, as obtained from the evolutionary algorithm with the assumption that the AIY–AIZ connections are inhibitory.

Parameter	The optimized parameter
N (s)	N = 0.4907
M (s)	M = 0.7618
$θ$	$θ_{A I Y L} = 0.8839$ $θ_{A I Y R} = - 7.3416$ $θ_{A I Z L} = 2.3906$ $θ_{A I Z R} = 5.3649$ $θ_{S M B D L} = θ_{S M B V L} = - 8.4964$ $θ_{S M B D R} = θ_{S M B V R} = - 11.7800$
$w_{i}^{O N}$	$w_{A I Y L}^{O N} = 9.8280$ $w_{A I Y R}^{O N} = - 9.7935$
$w_{i}^{O F F}$	$w_{A I Y L}^{O F F} = 8.2233$ $w_{A I Y R}^{O F F} = - 14.3481$
	$w_{A I Y L, A I Z L} = - 15.0000$ $w_{A I Y R, A I Z R} = - 11.0792$ $w_{A I Z L, S M B D L} = w_{A I Z L, S M B V L} = 0.3112$ $w_{A I Z R, S M B D R} = w_{A I Z R, S M B V R} = 10.7255$
$w_{i}^{s e l f}$	$w_{S M B D L}^{s e l f} = w_{S M B V L}^{s e l f} = - 13.8653$ $w_{S M B D R}^{s e l f} = w_{S M B V R}^{s e l f} = 2.0301$
$g_{k i}$	$g_{A I Y L, A I Y R} = g_{A I Y R, A I Y L} = 2.4368$ $g_{A I Z L, A I Z R} = g_{A I Z R, A I Z L} = 2.2160$
$w_{O S C}$	$w_{O S E} = 2.9655$
$w_{N M J}$	$w_{N M J} = 2.7969$

Additional files

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Masakatsu Hironaka
Tomonari Sumi

(2025)

A neural network model that generates salt concentration memory-dependent chemotaxis in Caenorhabditis elegans

eLife 14:RP104456.

https://doi.org/10.7554/eLife.104456.3

Figures

A neuroanatomical minimal network circuit for salt klinotaxis in C. elegans.

The trajectories of the worm’s locomotion.

The most optimized neural network circuits, with and without constraining the AIY–AIZ connections to be inhibitory, and the resulting network responses to step changes in salt concentration.

Signal transmission through chemical synaptic connections and electrical gap junctions.

Blocking electrical gap junctions significantly has a marked effect on neurotransmitter release $z_{i}$ and the resulting turning angle $φ$ .

The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the time half a cycle later than those shown in Figure 3.

The analysis of klinotaxis in the most optimized model with the constraints.

The reversal of salt concentration memory-dependent preference behavior in klinotaxis is attributed to the alteration from inhibitory to excitatory connections between ASER and AIY.

The weight of the ASER–AIY synaptic connection, $w_{j i}$ , in the most optimized network with the constraints is increased from a negative value (inhibitory connection) to a positive value (excitatory connection) by introducing an increment of 1.5 to $w_{j i}$ at each step.

The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the timing half a cycle later than those shown in Figure 5e.

The trajectories of the worm’s locomotion simulated by the network shown in Figure 5a, b.

Signal transmission through chemical synaptic connections and electrical gap junctions.

Inhibition of SMB activity suppresses the salt concentration memory-dependent preference behavior in klinotaxis, as observed experimentally.

The inhibition of SMB activity by reducing the bias term $θ_{SMB}$ has the effect of suppressing the salt memory-dependent preference behavior observed in klinotaxis, similar to that illustrated in Figure 6b.

Modeling of the synaptic transmission from the ASEL and ASER sensory neurons in response to changes in NaCl concentration.

Worm locomotion model.

Terminology used in the analysis of the worm’s locomotion.

Videos

The video of the worm’s locomotion simulated by the most optimized network with the constraints.

The video of the worm’s locomotion simulated by the network shown in Figure 5b.

The videos of the worm’s locomotion displayed in Figure 6c and d are shown on the upper and lower, respectively.

Tables

The parameters that were optimized by the genetic algorithm in the worm’s chemotaxis simulation.

The parameters used in the simulations of worm’s chemotaxis.

The parameters that control the evolutionary algorithm.

The parameters of the most optimized network model with the constraints, as obtained from the evolutionary algorithm with the assumption that the AIY–AIZ connections are inhibitory.

Additional files

MDAR checklist

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A neuroanatomical minimal network circuit for salt klinotaxis in C. elegans.

The trajectories of the worm’s locomotion.

The most optimized neural network circuits, with and without constraining the AIY–AIZ connections to be inhibitory, and the resulting network responses to step changes in salt concentration.

Signal transmission through chemical synaptic connections and electrical gap junctions.

Blocking electrical gap junctions significantly has a marked effect on neurotransmitter release zi and the resulting turning angle φ.

The changes in the turning angles φ in response to step changes in the salt concentration of positive and negative at the time half a cycle later than those shown in Figure 3.

The analysis of klinotaxis in the most optimized model with the constraints.

The reversal of salt concentration memory-dependent preference behavior in klinotaxis is attributed to the alteration from inhibitory to excitatory connections between ASER and AIY.

The weight of the ASER–AIY synaptic connection, wji, in the most optimized network with the constraints is increased from a negative value (inhibitory connection) to a positive value (excitatory connection) by introducing an increment of 1.5 to wji at each step.

The changes in the turning angles φ in response to step changes in the salt concentration of positive and negative at the timing half a cycle later than those shown in Figure 5e.

The trajectories of the worm’s locomotion simulated by the network shown in Figure 5a, b.

Signal transmission through chemical synaptic connections and electrical gap junctions.

Inhibition of SMB activity suppresses the salt concentration memory-dependent preference behavior in klinotaxis, as observed experimentally.

The inhibition of SMB activity by reducing the bias term θSMB has the effect of suppressing the salt memory-dependent preference behavior observed in klinotaxis, similar to that illustrated in Figure 6b.

Modeling of the synaptic transmission from the ASEL and ASER sensory neurons in response to changes in NaCl concentration.

Worm locomotion model.

Terminology used in the analysis of the worm’s locomotion.

The video of the worm’s locomotion simulated by the most optimized network with the constraints.

The video of the worm’s locomotion simulated by the network shown in Figure 5b.

The videos of the worm’s locomotion displayed in Figure 6c and d are shown on the upper and lower, respectively.

The parameters that were optimized by the genetic algorithm in the worm’s chemotaxis simulation.

The parameters used in the simulations of worm’s chemotaxis.

The parameters that control the evolutionary algorithm.

The parameters of the most optimized network model with the constraints, as obtained from the evolutionary algorithm with the assumption that the AIY–AIZ connections are inhibitory.

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Downloads (link to download the article as PDF)

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Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Blocking electrical gap junctions significantly has a marked effect on neurotransmitter release $z_{i}$ and the resulting turning angle $φ$ .

The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the time half a cycle later than those shown in Figure 3.

The weight of the ASER–AIY synaptic connection, $w_{j i}$ , in the most optimized network with the constraints is increased from a negative value (inhibitory connection) to a positive value (excitatory connection) by introducing an increment of 1.5 to $w_{j i}$ at each step.

The changes in the turning angles $φ$ in response to step changes in the salt concentration of positive and negative at the timing half a cycle later than those shown in Figure 5e.

The inhibition of SMB activity by reducing the bias term $θ_{SMB}$ has the effect of suppressing the salt memory-dependent preference behavior observed in klinotaxis, similar to that illustrated in Figure 6b.