A Neuromuscular Model of C. elegans with Directional Control

Michiyo Suzuki1, Toshio Tsuji1, Hisao Ohtake21 Department of Artificial Complex Systems Engineering, Hiroshima University, Higashi-Hiroshima, Japan

2 Department of Biotechnology, Osaka University, Suita, Japan

Abstract - This paper focuses on the nematode �� �������,which has a relatively simple structure, and is one of themost analyzed organisms among multicellular ones. We aim todevelop a computer model of this organism to analyze controlmechanisms with respect to its movements. First, a neuronalcircuit model for directional control and a kinematic model ofthe muscle body are proposed. Then, by integrating the twomodels, we construct the whole body model of �� �������.The effectiveness of the proposed model is verified through aseries of computer simulations.

Keywords - �� �������; directional control; steering circuitmodel; kinematic model; computer simulation

I. INTRODUCTION

Multicellular organisms have special and complex mech-anisms to adapt to various conditions of their external envi-ronment. However, despite of recent progress of the experi-mental techniques in biology, even the simple nematode hasnever been fully clarified. In recent years, a new approachfor analyzing functional mechanisms of living organisms, acomputer simulation of a mathematical model, has beendeveloped [1]. Using a computer model instead of theactual corresponding organism makes it possible to changeenvironmental conditions easily and to analyze behaviorrepeatedly under the same conditions. If the experimentalresults of an actual organism can be approximated withhigh precision by the computer model, these model couldbe some help for the biological experiments. Also, using themodel may make it possible to clarify some characteristicsthat cannot be measured in actual experiments.

Caenorhabditis elegans (a non-parasitic soil nematode)has a simple cylindrical body approximately 1.2 mm longthat includes such fundamental organs as a nervous system,muscles, a pharynx, a hypodermis, an alimentary canal andgenitals. The neuronal circuit of C. elegans processes vari-ous stimulations from the environment and controls forwardand backward movements, rest, and search movements [2]-[5].

Because all the neuronal cells (neurons) of C. eleganshave been identified and the connections have been ap-proximately clarified [3], several computer models of theneuronal circuit of C. elegans have been proposed in recentyears [6]-[8]. Much interest is centered on how C. elegansuses its muscles to perform and control various motions,and some body models of motor control have been proposed[9],[10]. The authors have also focused on the sinusoidalmovement that is peculiar to C. elegans, and developedboth kinematic and dynamic models of the muscle body[11],[12]. These models can express some patterns of two-dimensional movement. In addition, we modeled a neuronal

Muscle-body modelNeuronal circuit model

Stimuli Information processing Movements

Temperature

Good smell

Bad smell...Fig. 1. Schematic diagram of the proposed model.

circuit for touch response and integrated it to the kinematicmodel of the muscle body. The model was able to simulatethe motor control of responses to perceived touch stimula-tion [12].

In this paper, we propose a neuronal circuit model for di-rectional control that realized movements such as turning inresponse to environment stimuli. Also, a three-dimensionalkinematic model of the muscle body was constructed torealize the more complicated movements. As shown in Fig.1, the proposed model consists of a neuronal circuit model,which processes stimuli and controls the muscles, and amuscle-body model, which executes movement. This paperis organized as follows: In Section II, the overview ofC. elegans is explained. In Sections III and IV, a neuronalcircuit model for direction control and a muscle-bodymodel for movement are described in detail, respectively. InSection V, the behavior to environmental stimulation of theproposed model of C. elegans is verified through a seriesof computer simulations.

II. STRUCTURE AND MOVEMENT OF C. ELEGANS

The body is composed of 959 cells including the 302neurons [2]-[5]. The body-wall consists of four quadrantsof striated muscles enclosed by the hypodermis that executeforward, backward and turning movements. Each quadrantconsists of two closely apposed rows of muscles, as shownin Fig. 2. The pharyngeal muscles are located in the anteriorpart of the body and assume a leading role in searching theenvironment, turning the body and opening and closing themouth. The neurons are classified into three main groups byfunction: sensory neurons, interneurons and motoneurons[2]-[5]. The sensory neurons detect external stimuli, andthen the interneurons process information from the stimuli.Finally, the motoneurons control the muscles on the basisof signals from the interneurons.

C. elegans exhibits various movements, such as goingforward and backward, and stopping and turn, that arecontrolled by the neuronal circuits [2],[8]. For example,motoneurons DB(1-7) and VB(1-11) control contractionand relaxation of the body-wall muscles, and overall control

167

Muscle

Dorsal side

Muscle

Muscle

Muscle

Ventral side

Hypodermis

Left

sideRight

side

Fig. 2. Cross-section of the muscles.

Temperature Good smell Good smell Bad smell

DR VLDL

RIA

L

AWA

R

AIZ

R

AIY

R

AFD

R

AWC

R

AWB

R

AWA

L

AFD

L

AWC

L

AWB

L

AIY

L

AIZ

L

RIA

R

VR

Synaptic connection

Gap junction

Fig. 3. A steering circuit model.

is performed by interneurons, AVB(L/R), in forward move-ment. DA(1-9), VA(1-12) and AVA(L/R) control backwardmovement [2]. Although the details of the neuronal circuitfor directional control have not been clarified, Kawamuraet al. identified the neurons that are likely concerned withdirectional control, and named the circuit that consists ofthese neurons “the steering circuit” [8].

C. elegans can respond to various kinds of environmentstimuli. For example, C. elegans moves forward when it per-ceives a comfortable temperature or good smell, and turnswhen it perceives a bad smell. Although C. elegans alsomoves forward when it receives no stimulation, sometimesit stops or turns spontaneously.

In this paper, we focus especially on the steering circuit,and propose a model that controls the movement at the timea stimulus is received.

III. MODELING OF STEERING CIRCUIT

A. Steering circuit model

Only the neurons that are supposed according to theliterature [8] to participate in the directional control wereextracted from the neuronal circuit map of C. elegans pub-lished by J.G. White et al. in 1986 [3]. Although C. eleganshas many sensory neurons, the neurons included in the

steering circuit are particularly those that sense temperatureor smell.

This model aims to realize control of movements, suchas forward movement and turning, stimulated by temper-ature and smell. The proposed steering circuit model ofC. elegans for directional control is shown in Fig. 3. In thefigure, eight sensory neurons are shown as rectangles, sixinterneurons as hexagons and four motoneurons as circles.AFD(L/R), which manage the sense of temperature, per-ceive comfortable temperatures. AWC(L/R) and AWA(L/R)perceive volatile attractants (good smells), although theyperceive different kinds of chemical materials. The stimuliperceived by the six sensory neurons cause a positiveresponse. To the contrary, AWB(L/R) perceives volatilerepellents (bad smells), and these stimuli cause a negativeresponse. The eight sensory neurons compose the amphid,which is the sense organ on the head tip, and play the mainrole in the perception of environment stimuli [4].

The 14 motoneurons of the pharynx, SMB(DL/DR/VL/VR), RMD(L/R), RMD(DL/DR/VL/VR) and SMD(DL/DR/VL/VR), are divided into four positional groups, DL,DR, VL and VR, for simplicity to confirm that the neu-ronal circuit model realizes directional control according tostimuli in this paper. On this occasion, multiple connectionsbetween two neurons are simply modeled as a single con-nection. These four motoneurons control the correspondingmuscles that exist in the four directions, respectively.

B. Characteristics of neurons

The model supposes that the outputs of sensory neuronsare not linear reactions to the strength of the stimulus, ac-cording to the general characteristics of neurons [15],[16],and they are expressed by the following nonlinear equationbased on a neuron model [16].

On =cn

1 + exp(−an(In − bn)) (1)

where an is an inclination with output function of sen-sory neuron n (n ∈ {AFD(L/R), AWC(L/R), AWA(L/R),AWB(L/R)}), bn is the value of the stimulatory input atwhich the output of the neuron takes a central value, andcn (0 ≤ cn ≤ 1) is the output gain and is equivalent tothe stimulation reception sensitivity. Therefore, On outputsthe continuation value of [0, 1], which is normalized bythe maximum output from the actual neuron. The stimu-lation input In to a sensory neuron, n (n ∈ {AFD(L/R),AWC(L/R), AWA(L/R), AWB(L/R)}), is a stepless inputof the range of [0, 1], which quantifies the strength of thestimulation.

The characteristics of interneurons and motoneurons arealso expressed by (1). The input In to the neuron, n (n ∈{AIY(L/R), AIZ(L/R), RIA(L/R), DL, DR, VL, VR}), isthe result of the multiplication of the connection weight bythe output of the connected neuron i or m, and is calculatedby the following equation.

In =i∑

wi,nOi +m∑

gm,nOm, (2)

where wi,n and gm,n are the connection weights of synap-tic connections (one-way) and gap junctions (interactive),respectively (wi,n �= wn,i, and gm,n = gn,m).

168

x l

y

z

x

(b) Side view

(a) Overhead view

Dorsal side

Left side

Ventral side

Right side

Pharyngeal

muscle

Body-wall

muscle

l i

i

1

1

Pharynx

Tail

Fig. 4. The 12-link body model of C. elegans.

Mjl

Mr

j

Mr

j-1Mj-1

l

Dorsal side

Ventral side

Left

sideRight

side

Fig. 5. Cross-section of the muscle model.

The muscle-body model of C. elegans, discussed in thenext section, receives the outputs of motoneurons ODL,ODR, OVL and OVR.

IV. MODELING OF MUSCLE BODY FOR MOVEMENT

A. Muscle-body model for movement

In this paper, the complete body of C. elegans is rep-resented by a rigid 12-link model, which consists of 11segments of body-wall muscles and one segment of pha-ryngeal muscle (Fig. 4). The relative angles, θi and φi, tothe (i + 1)-th link of the i-th link (i = 1, 2, · · · , 11) areangles of the dorso-ventral and left-right directions of eachjoint, respectively (−π ≤ θi, φi ≤ π [rad/sec]).

Four muscles shown in Fig. 2 are modeled at each jointas shown in Fig. 5. The internal states of the muscles onthe dorsal and ventral sides on the left are denoted as M ljand M lj−1, and the internal states of the muscles on theright are M rj and M

rj−1. Then, the i-th dorso-ventral joint

angle, θi, is expressed by using the internal states of themuscles as follows:

dθidt

=αi {(M lj + M rj )−(M lj−1 + M rj−1)}, (3)

where αi is a positive constant (−π ≤ αi ≤ π [rad/sec]),which is equivalent to the velocity gain for the i-th joint.The subscript, j and j−1, show the number of each muscle,the i-th muscle on the dorsal side is denoted by j = 2i(i = 1, 2, · · · , 11), and the one on the ventral side by j−1.θi is decided by the difference between the sum of thedorsal internal states and one of the ventral. In the same

way, the i-th left-right joint angle, φi, is

dφidt

= βi {(M rj + M rj−1) − (M lj + M lj−1)}, (4)where βi is a positive constant (−π ≤ βi ≤ π [rad/sec])that is equivalent to the velocity gain. φi is decided by thedifference between the sum of the right internal states andone of the left.

Because the internal state, M (M ∈ {M lj, M rj , M lj−1,M rj−1}), of the muscle is equivalent to the membranepotential, it is possible to express it as the sum of the directcurrent voltage DM corresponding to the bias componentand the oscillating voltage AM corresponding to the fre-quency component. For example, the internal state of thej-th muscle on the right of the dorsal side, M rj , can beexpressed as follows.

M rj =D M rj +

A M rj (5)

In this paper, the following assumptions are introduced.A.1 Two pairs of components, AM lj and

AM lj−1, andAM rj

and AM rj−1, produce the oscillations with the samephase. In this paper, two pairs of components on theleft and right sides oscillate identically at each joint.

AM lj =AM rj

AM lj−1 =AM rj−1

}(6)

A.2 The oscillation produced by each pair of components,AM , is expressed by using Matsuoka’s neuronal os-cillator model [13]. For example, the frequency com-ponent, AM rj , of the j-th muscle on the right of thedorsal side can be expressed as follows.

Trγ

d AM rjdt

+AM rj =∑n�=j

vrj,nVrn − brjf rj + γsrj , (7)

Taγ

df rjdt

+ f rj = Vrj , (8)

V rj ={

AM rj (AM rj ≥ 0)

0 (AM rj < 0),(9)

where Tr and Ta are time constants; vrj,n is theconnection weight from the j-th oscillator to the n-th oscillator; brj denotes a fatigue coefficient; f

rj is

the state of the oscillator; srj is a tonic input fromoscillators connecting with the j-th oscillator; and V rnis the output of the l-th oscillator. γ is a gain withrespect to the time constants [12], and γ = 1 inMatsuoka’s original model.

Under the above assumptions, (3) and (4) are respectivelyreduced to

dθidt

=αi{4AM lj +(DM lj +DM rj )−(DM lj−1+DM rj−1)},(10)dφidt

= βi{(DM rj +DM rj−1) − (DM lj +DM lj−1)}, (11)

where AM lj = −AM lj−1.For example, if the following equality is true in (10),

movement to the dorso-ventral direction is determined only

169

Body-wall muscle

DR VLDL VR

Pharyngeal muscle

Fig. 6. Motor control by a steering circuit model.

by the frequency components, AM , of the internal state ofthe muscle.

DM lj +DM rj =

D M lj−1 +DM rj−1 (12)

Furthermore, from (11), the left-right joint angle, φ1,with respect to the pharyngeal movement is derived only bythe bias components ,DM , of the internal states of muscles.If the right-hand side of (11) is positive, the pharynx liftsto the left. In this case, the dorso-ventral contraction andrelaxation of the muscles are realized only by A M basedon the neuronal oscillator model, and left-right contractionand relaxation only by DM .

B. Integration of neuronal circuit model and muscle-bodymodel

The steering circuit model described in Section III isconnected to the muscle-body model in Section IV.

In this study, directional control of the muscle-bodymodel in forward movement is realized by giving outputsof the steering circuit model to the muscle-body model asthe bias components, DMd1 and

DMd2 (d ∈ {l, r}). Therelationship between outputs of motoneurons of the steeringcircuit model and the bias components of the muscle-bodymodel is as follows:

DM l1 = OVLDM r1 = OVRDM l2 = ODLDM r2 = ODR

(13)

The motoneurons of the pharynx and body-wall musclesare partially connected in the actual organism. As shownin Fig. 6, outputs of motoneurons of the pharynx aretransmitted through the pharynx to body-wall muscles.Therefore, outputs of steering circuit model are given tothe corresponding muscles as the bias component, DMdjand DMdj−1 (d ∈ {l, r}; j = 2i = 4, · · · , 22), as wellas those in the first joint. As mentioned in Section III, asinusoidal wave of forward movement is realized in thismodel by adjusting the frequency component, AM , basedon Matsuoka’s neuronal oscillator model [13].

Using the above system, directional control of themuscle-body model based on the outputs of the steeringcircuit model can be realized. In the next section, a com-puter simulation, which represents a series of stimulationresponses by the integration model of C. elegans, is carriedout.

TABLE I

DESIRED OUTPUTS OF MOTONEURONS FOR STIMULATION INPUTS TO

SENSORY NEURONS.

u = 1 u = 2 u = 3 u = 4 u = 5

IAFD(L/R)(u) 1 0 0 0 0IAWC(L/R)(u) 0 1 0 0 0IAWA(L/R)(u) 0 0 1 0 0IAWB(L/R)(u) 0 0 0 1 0

O∗DL(u) 0.1 0.1 0.1 0 0O∗DR(u) 0.1 0.1 0.1 0 0O∗VL(u) 0.1 0.1 0.1 0.1 0O∗VR(u) 0.1 0.1 0.1 0.1 0

movement FD FD FD TR FD

V. COMPUTER SIMULATION

A. Simulation Settings

In this simulation, the coefficients for sensory neuronsincluded in (1) are set as an = 15, bn = 0.6 andcn = 1, based on references giving data on the neuronalcharacteristics of higher organisms [14],[15]. Also, thosefor the interneurons and motoneurons are set as the samevalues except for cn =0.1 in motoneurons.

In the muscle-body model, the link lengths, li, are setto l1 = 0.2, l2 = l3 = · · · = l12 = 0.1 [mm] toapproximate the actual body length, which is about 1.2[mm]. In the neuronal oscillators, the initial values ofAMdk are all set to 1, b

dk = 18, f

dk = 1 and s

dk = 5

(d ∈ {l, r}; k = 1, 2, · · · , 22), and the time constants areset to Tr = 0.12 [sec] and Ta = 0.24 [sec] by trial anderror based on the literature such as [18]. Under the abovesetting, the differential equations included in (6), (7) and (8)are calculated every 1.0 × 10−3 [sec] by using the fourth-order Runge-Kutta method.

This simulation considers only whether each sensoryneuron receives stimulation or not for simplicity, althoughthe sensory neurons in this model can sense differencesin stimulus strength. In = 1 when stimulation is given,and 0 when no stimulation is given. Also, since neuronsAFDL and AFDR, AWCL and AWCR, AWAL and AWAR,and AWBL and AWBR are in the same class, they sensethe same stimulation input at same time, respectively. Inaddition, some kinds of stimulation are not given at sametime. Therefore, responses to five patterns of stimulation,i.e., four kinds of stimuli and non-stimulation are dealt within this simulation. The desired outputs of four motoneurons,O∗n(u) (n ∈ {ODL, ODR, OVL, OVR}), for five patterns ofstimulation inputs to eight sensory neurons, In(u) (n ∈{AFD(L/R), AWC(L/R), AWA(L/R), AWB(L/R)}; u =1, · · · , 5), are shown in Table I. In the table, FD meansforward movement, and TR turning.

As shown in the table, it is assumed that the body movesforward even when no stimulation is given (u = 5). Also,the body moves forward in response to stimulation when asuitable temperature or good smell is sensed (u = 1, 2, 3),and it turns when a bad smell is sensed (u = 4).

Although turns to the ventral and dorsal sides are possi-ble, for simplicity it is assumed that a turn occurs alwaysto the ventral side in this simulation. At each joint, theset value of bias components of the muscles for a turn isunderspecified. It is assumed that motoneurons, DL and

170

1.5

50 1000 150 200 250

Generation

Eli

te f

itn

ess,

Fsu

p

1

0.5

0

Fig. 7. Evolution of the elitist fitness value.

DR, which correspond to the dorsal muscles, output 0, andVL and VR output given positive values. Therefore, oneach joint, the bias components of ventral muscles, DMdj−1(d ∈ {l, r}; j = 2, 4, . . . , 22), are larger than those ofdorsal muscless, and thus, the muscle-body model turns ina ventral direction. On the other hand, four motoneuronsoutput the same value when a positive stimulation is given,and 0 when no stimulation is received.

B. Optimization of the neuronal circuit model by a real-coded GA

In the steering circuit model based on the actual structuredescribed in Section III, the connection weights of thechemical synaptic connections and the gap junctions mustbe appropriately set to realize the desired output accordingto the stimulation inputs. However, at the present moment,it is almost impossible to measure these values by biologicalexperiments with actual organisms. Therefore, in this paper,we employed a real-coded genetic algorithm (GA) [17]in order to adjust these connection weights, 59 chemicalsynaptic connections and nine gap junctions.

All connection weights are adjusted to obtain the ap-propriate outputs to the U patterns of stimulation inputsto sensory neurons. In this paper, a method of real-codedGA [17] is employed, where each parameter is includedas the r-th component (r = 1, 2, · · · , 68) in a GA stringqp, and the three steps of (i) selection, (ii) crossover and(iii) mutation are carried out in each generation G, wherep (p = 1, 2, · · · , P ) is the serial number of the individuals[12].

The fitness function F (p), which is the evaluation stan-dard of the solutions, is defined as (14) in order to reducethe difference between the desired output signal of themotoneurons, O∗n (n ∈ {DL, DR, VL, VR}), and theactual output signal, pOn(u). The steering circuit modelis optimized to minimize the value of F (p).

F (p) :=m∑

{ 1U

U∑u=1

|O∗n(u) −pOn(u)|(p = 1, 2, · · · , P )

(14)

where the U = 5 patterns with P = 50 individuals and G =250 generations. The evolution of the elite fitness Fsup(g)of the elite individual, qsup(g), in each generation, g (g =1, 2, · · · , 250), is shown in Fig. 7. The figure confirms thatthe value of the fitness function, F (p), almost convergedat the 50th generation. The optimal set of 68 parametersat the 250th generation of the above GA was used as theconnection weights of the steering circuit model.

a

b

c1mm

Fig. 8. Motion trajectory of he proposed neuromuscular model.

Time [sec]

Inte

rnal

sta

te

0 1 2 3 4

Non-stimulation Bad smell

5 6 7 8 9

Good smell

M1

M2

0

0.5

1

1.5

2

Fig. 9. Internal states of the muscles, M1 and M2

C. Responses to stimuli

To confirm that the proposed model can realize motorcontrol according to stimulation inputs to the steeringcircuit model, a series of computer simulations were carriedout.

In the simulation, no stimulation is given for the first3 [sec], and then a bad smell is given for 3 [sec] and agood smell for the final 3 [sec]. First, the motion of themodel controlled by the stimulation that is received by thesensory neurons is plotted every 1 [sec] in Fig. 8. In thefigure, ● is the head of C. elegans, and a, b and c are thehead positions at 3, 6 and 9 [sec], respectively. The modelmoved forward in a sinusoidal wave for the first 3 [sec],then turned for the next 3 [sec] and finally moved forwardagain. The result confirmed that the stimulation inputs areprocessed appropriately by the steering circuit model, andthat motor control appropriate to stimulation can be wellrealized by the proposed muscle-body model.

Next, the sums of the internal states of the muscles on thedorsal side, M1=M l1+M

r1 , and ventral sides, M2=M

l2+M

r2 ,

are shown (Fig. 9). The result demonstrates that the muscleson the dorsal and ventral sides repeated contraction andrelaxation alternately and that an approximately oppositephase could be successfully expressed. It demonstrates thatthe sinusoidal movement can be realized by the behavior ofthe neuronal oscillators. Also, when a bad stimulus is given,

171

0.2 0.40.6

0.81.0

1.2

00.2

0.40.6

0

0.1

x [mm]y [mm]

z [m

m] Pharynx

Tail

0

Fig. 10. A resulted form of the model in pharyngeal movement.

only the internal states of the muscles on the ventral sidehave large values. All internal states in the other joints showthe same behavior. This means that the inputs to the muscleswere changed on the basis of the outputs of the steeringcircuit model and the muscle-body model was induced toturn toward the ventral side. It is can be seen that forwardmovement and turning, which correspond to stimulus input,can be realized by using the proposed model of the steeringcircuit.

Although turning is realized by adjusting the values ofeach bias component in this model, the set of values ofbias components on each joint for turns are underspecified.Therefore, it will be necessary to consider the way toset these values. Also, actual responses, including thefrequency and angle of turns, must be realized in moredetail by considering various patterns of stimulation thanjust the existence of stimuli.

D. Pharyngeal movement of muscle-body modelIn Subsection 5.3, it was confirmed that turning and

forward movement as a sinusoidal wave were realized bythe proposed model. Then, in this simulation, the left-right movement of the muscle-body model, such as liftingthe pharynx, was confirmed. Left-right movement of themuscle-body model was realized by adjusting the values ofeach bias component, as in the case of the dorso-ventral di-rection. In this simulation, the lifting of the pharynx is con-sidered. The bias components of muscles of the first jointwere set as DM l1 =D M l2 = 0, DM r1 =D M r2 = 0.5, andones of the other joints were DM lj−1=

DM lj=DM rj−1=

DM rj =0 (j = 4, 6, . . . , 22).

A resulted form of the pharyngeal lifting of the model isshown in Fig. 10. From the figure, it can be confirmed thatthe whole body forms a sinusoidal wave and the pharynxis lifted. By adjusting the bias components on the otherjoints, a form, such as lifting the whole body, may also berealized. If the neuronal circuit for control of the left-rightmovement is identified, it is expected that the movementwill be realized based on outputs of the neuronal circuitmodel.

VI. CONCLUSIONThis study focused on C. elegans and proposed its neu-

romuscular model. This model consisted of a neuronalcircuit model for directional control and a kinematic modelof the muscle body for movement. It was possible togenerate the sinusoidal movement of C. elegans by usingthe proposed rigid model of a muscle body with 11 jointsthat incorporates the neuronal oscillators. Responses corre-sponding to stimulation inputs were realized by connectingthe steering circuit model with the muscle-body model.Also, the expression of three-dimensional movements, suchas lifting the pharynx, in addition to the planar movementof crawling on the ground became possible.

In future research, a series of behaviors of actualC. elegans, such as a series of forward movement, stimulusreception, backward movement, stopping, forward move-ment, stimulus reception and turn, will be realized byintegrating some neuronal circuit models for various kindsof stimulation into the muscle-body model.

ACKNOWLEDGMENTThe authors would like to thank Dr. Kiyoshi Kawamura

of the Cybernetic Caenorhabditis Elegans Program, KeioUniversity, for information about C. elegans.

A part of this study was supported by a Grant-in-Aidfor Scientific Research (No. 1605189) by the ResearchFellowships of the Japan Society for the Promotion ofScience for Young Scientists.

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