Biol. Cybern. 66, 345-355 (1992) Biological Cybernetics ~9 Springer-Verlag 1992

A model of leg coordination in the stick insect, Carausius morosus

IV. Comparisons of different forms of coordinating mechanisms

Jeffrey Dean

Abteilung fldr Biokybernetik und Theoretische Biologie, Universit~it Bielefeld, Postfach 8640, W-4800 Bielefeld !, Federal Republic of Germany

Received May 8, 1991/Accepted in revised form October 10, 1991

Abstract. The kinematic model presented in a separate report is used here to investigate several questions concerning the nature of the coordinating mechanisms. First, one or more mechanisms are inactivated in order to compare the relative etiiciencies of the different coor- dinating mechanisms in maintaining proper coordina- tion. Second, the most efficient mechanism, the position-dependent influence, is varied in order to illus- trate the consequences for coordination. Third, the strength of the contralateral coupling is varied in order to make predictions about how contralateral legs estab- lish alternation when started from symmetric positions. The consequences of adding reciprocal contralateral inhibition during swing is tested in the same context.

1 Introduction

One important use of a model is to explore alternative hypotheses when the experimental data are incomplete or permit several interpretations. This is the goal of the present study. Experimental findings demonstrate the existence of six mechanisms mediating interleg coordina- tion in the stick insect (review Cruse 1990), but they do not precisely specify all these mechanisms. Hence, sev- eral hypotheses are necessary in order to construct a model of leg coordination. Four of the six mechanisms are involved in normal step timing. (The other two correct errors or modify leg force.) The simulations presented in companion papers (Dean 1991b, 1992) show that a model incorporating these four mechanisms adequately simulates both normal step patterns and many responses to perturbations. The simulations re- ported here test the relative efficiency of the different mechanisms and then consider several alternative forms.

Abbreviations: AEP and iAEP, actual and intrinsic anterior extreme position; PEP and iPEP, actual and intrinsic posterior extreme posi- tion; VR, retraction velocity; VP, protraction velocity

The model assigns each leg a step pattern generator in the form of a relaxation oscillator with two states corresponding to swing and stance, a state variable corresponding to leg position, and position thresholds determining the end-points of swing and stance (the intrin- sic anterior and posterior extreme positions, iAEP and iPEP). During each state, a constant input corresponding to the velocity of protrac- tion (VP) or retraction (VR) is integrated until the state variable reaches the threshold for switching states. Interleg coordination de- pends upon the exchange of information between adjacent legs. The coordinating mechanisms modify the thresholds of the receiving leg's oscillator based upon the state, position, and velocity input of the sending leg's oscillator. Coordinating influences which shift the PEP- threshold caudally, delaying the beginning of swing, are referred to as inhibitory; influences which shift the PEP-threshold rostrally are referred to as excitatory. In accord with experimental data (Cruse et al. 1986; Cruse and Knauth 1989; Dean 1989), the contralateral coupling in the standard form of the model is strongest between front legs and weakest between middle legs.

Ipsilateral coordination depends upon four mecha- nisms acting between adjacent legs (review Cruse 1990). The beginning of a swing, and therefore the end-point of stance (PEP), is modulated by three mechanisms: 1) a rostrally directed inhibition during the swing of the next caudal leg, 2) a rostrally directed excitation when the next caudal leg begins active retraction (designated here as a time-dependent mechanism), and 3) a caudally directed influence depending upon the position of the next rostral leg. The beginning of stance (AEP) is modulated by a single, caudally directed mechanism depending upon the position of the next rostral leg (targeting behavior). With these four mechanisms, the standard form of the model produces the metachronal coordination seen in the adult stick insect; successive steps of ipsilateral rear, middle, and front legs are organized in waves progressing from back to front.

In the present simulations, the strength and form of several coordinating influences were varied in order to test the effect on leg coordination. The results show that, of the four ipsilateral mechanisms, the caudally directed position-dependent influence is the most efficient in maintaining proper coordination. Therefore, alternative forms of this mechanism were examined in several contexts. In the model, this influence is a func- tion of both the position and the retraction speed of the

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sending leg. Both parameters were varied. A third, unknown parameter is the value of the influence during the swing of the sender. Several treatments were tried; they will be described in the results section.

Results for a single pair of legs were described previously (Dean 1991a); results for interactions among all six legs are reported here. Variations in the strength and form of the two rostrally directed influences - the inhibition during swing and the excitation at the start of active retraction by the caudal leg - are considered elsewhere (Dean 1991b, 1992).

The two mechanisms mediating contralateral coor- dination are qualitatively similar to the ipsilateral time- dependent and position-dependent mechanisms. Unlike the ipsilateral influences, the contralateral influences are reciprocal in the sense that the right and left legs of a segment participate as both sender and receiver. One question addressed here concerns the form of these contralateral influences. A second question concerns the relative strength of the contralateral mechanisms in the different segments. A third question concerns the possi- ble role of contralateral inhibition during swing, an influence not yet conclusively demonstrated in behav- ioral experiments (e.g. Cruse and Knauth 1989; Dean and Wendler 1982; Foth and Graham 1983).

2 Relative efficiency of the coordinating mechanisms

Leg coordination in the standard form of the model, as in the insect, involves four ipsilateral and two contralat- eral mechanisms. One natural question concerns the relative importance of the different mechanisms in maintaining normal coordination. To answer this ques- tion, it is useful to study the individual mechanisms in isolation. This experiment is not possible in stick in- sects, but it can be performed with the model. In one set of simulations, one or more mechanisms were re- moved from the standard model. In a second set of simulations, each influence was tested on its own. Coor- dination of adjacent legs was evaluated on the basis of mean phase, concentration parameter and the number of overlapping swings. In order to see the effect of the mechanisms more clearly, the coordination was per- turbed by letting the retraction speed of the left legs slowly oscillate between lx and 1.88x that of the right legs.

These test conditions represent a quite strong but not totally unrealistic perturbation. The maximum difference in step ratios does not match that sometimes observed in rear legs when stick insects walk around tight curves, but it considerably exceeds that observed in front and middle legs (Jander 1985). There was no a priori reason for picking a particular difference in retraction speeds, so a slow oscillation was used as a way to rapidly test perturbations ranging from weak to strong. Inclusion of a strong perturbation was necessary to provide enough overlapping swings to reveal the effects of different coordinat- ing mechanisms. The difference in retraction speed was used here only as an experimental method; the intention was not to actually simulate turning, which also involves complex changes in tarsus trajectories and step end-points (Jander 1985). The method has the further advantage of reducing any influence of the starting configuration. Otherwise, the inertia of the model could give rise to significant mean phases even in the absence of coordinating influences.

2.1 Relative efficiency of the ipsilateral mechanisms

The effect of removing single ipsilateral mechanisms is shown in Table 1. The simplistic expectation was that removing an ipsilateral mechanism should affect the coordination of all ipsilateral leg pairs equally. How- ever, several interactions made the pattern of changes less uniform. In particular, the strong contralateral coupling between the front legs caused this pair to maintain a 1 : 1 step ratio. As a result, the right front leg stepped more frequently than the right middle and rear legs although the retraction speed of all three legs was constant and equal. When the ipsilateral mecha- nisms were strong, this faster rhythm was partially enforced upon the right middle leg, disturbing the coor- dination of right middle and right rear legs. Inactivating a caudally directed mechanism decreased this distur- bance, so the coordination of right middle and rear legs actually became better, not worse.

Despite these complications, several patterns were apparent. These were clearest in the incidence of over- lapping swings (Table lb). In the standard model, swings by adjacent ipsilateral legs did not overlap, despite the strong perturbations tested here (Table lb: none removed). Removing the rostrally directed inhibi- tion considerably increased this kind of error in adja- cent ipsilateral legs (Table lb: inhibition removed). The increase was primarily due to the rostral leg beginning a swing during the swing of the caudal leg: this is precisely the error which the inhibition helps prevent. Conversely, overlapping swings in which the caudal leg began a swing during the swing of the rostral leg increased in number when the caudally directed posi- tion-dependent mechanism was inactivated (Table lb: pos-dep removed). Removing either of the other two mechanisms alone did not increase the number of over- lapping swings by adjacent ipsilateral legs (Table lb: time-dep or targeting removed). Removing all four mechanisms greatly increased both kinds of overlapping swings (Table 2b: none present).

Changes in phase relationships formed a similar pattern (Table la). The ipsilateral changes were clearest in the left legs because these legs maintained 1 : 1 step- ping. Removing either the targeting mechanism or the ipsilateral time-dependent excitation had little effect on mean phases or concentration parameters. Removing the rostrally directed inhibition slightly decreased the concentration parameter of L1 in L2 (k = 0.84 versus 0.94 for the standard model). Removing the position- dependent mechanism elicited the largest decreases in coordination on the left side. The changes on the right side were similar, apart f rom the interaction noted above. Removing the position-dependent mechanism elicited the largest decrease in the coordination of right front and middle legs (k = 0.24). As a result, the right middle leg was subject to less perturbation via the front legs, so the right middle and rear legs remained well coordinated (R2 in R3: k = 0.78). Removing the target- ing influence produced little change. Removing either the rostrally directed inhibition or the time-dependent mechanism moderately decreased the coordination of

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Table 1. Comparison of leg coordination when different ipsilateral mechanisms were removed from the standard model. The mechanisms were compared in simulations in which the retraction speed of the right legs was constant and that of the left legs varied between lx and 1.88x that of the fight legs. Each simulation continued until one leg completed 100 steps. The mechanisms, their abbreviations, and the transition they affect are follows: caudally directed position-dependent mechanism (pos-dep, PEP), rostrally directed excitation (time-dep, PEP), rostrally directed inhibition (inhibition, PEP), and caudally directed control of the swing end-point (targeting, AEP). a) Mean phases (Ph) and concentration parameters (k) were calculated using circular statistics (Batschelet 1965); the significance (p) was determined using the Rayleigh test at both the 5% and 1% levels (ns, not significant). Most phase histograms were unimodal, so the mean phase and concentration parameter accurately represent the distribution. Values from distributions where a second peak was apparent are enclosed in parentheses, b) For each pair of adjacent legs, the number of overlapping swings was counted both for swings of leg 1 beginning in a swing of leg 2 and vice versa

Ipsilateral mechanism removed(-) from the standard model none

a) Phase relationships Ph k

contralateral leg pairs L1 in R1 114 0.75 L2 in R2 109 0.20 L3 in R3 123 0.22 ipsilateral leg pairs R2 in R3 99 0.75 R1 in R2 (61 0.36 L2 in L3 83 0.96 L1 in L2 83 0.94 b) Number of overlapping

contralateral leg pairs El/R1 L2/R2 L3/R3

ipsilateral leg pairs L2/L3 L1/L2 R2/R3 R1/R2

-pos-dep -time-dep -inhibition -targeting

p Ph k p Ph k p Ph k p Ph k p

1% 120 0.81 1% 112 0.82 1% 127 0.84 1% 117 0.7 1% 5% 115 0.17 ns (96 0.22 1%) 116 0.26 1% 115 0.26 1% 5% 110 0.22 1% 91 0.16 ns 120 0.21 5% 117 0.26 1%

1% 107 0.78 1% 122 0.58 1% 105 0.54 1% 110 0.77 1% 1%) (55 0.24 1%) (64 0.30 1%) (41 0.30 1%) 68 0.40 1% 1% 92 0.90 1% 87 0.95 1% 81 0.95 1% 88 0.96 1% 1% 81 0.83 1% 85 0.94 1% 80 0.84 1% 82 0.92 1%

swings (leg 1 in swing of leg 2/leg 2 in swing of leg 1)

0/0 2/0 1/0 0/0 1/2 7/3 7/11 12/0 3/1 4/0 9/11 7/7 10/13 13/8 12/4

0/0 0/0 0/0 15/0 0/0 0[0 0/4 0/0 8/1 0[0 0/0 0/0 0/0 11/0 0/0 0/0 0/9 0/0 22/0 0/0

the right f ront and middle legs and produced the largest decreases in the coord ina t ion o f right middle and rear legs.

Simulations with only one ipsilateral mechanism active produced similar results (Table 2). Overlapping swings by adjacent ipsilateral legs were frequent in the absence o f ipsilateral coordinat ing mechanisms (Table 2b: none present); they did no t occur when all four mechanisms were active (Table lb: none removed). Nei ther the caudally directed targeting mechanism nor the rostrally directed t ime-dependent mechanism alone reduced the number o f overlapping swings. The ros- trally directed inhibition and the caudally directed posi- t ion-dependent influence did reduce the number o f overlapping swings. They again acted in a complemen- tary fashion: the inhibition allowed overlapping swings only when the rostral leg began its swing first and the posi t ion-dependent mechanism only when the caudal leg began its swing first.

Changes in phase relationships were analogous. Both the targeting mechanism and the t ime-dependent mechanism enforced significant but weak coord ina t ion for all ipsilateral pairs except the right f ront and middle legs. O f the two, the targeting mechanism was some- what more effective. The rostrally directed inhibition produced still better coord ina t ion on the side subjected to var ia t ion in retract ion speed (the left side), but had

very little effect on the side with constant retract ion speed. In contrast , the posi t ion-dependent influence p roduced coordina t ion nearly equal to that o f the stan- dard model (Table 2a: pos-dep versus Table la: none removed).

2.2 Relative efficiency of the contralateral mechanisms

Contralateral mechanisms were compared in the same manner (Table 3). Fo r the f ront legs, the s tandard model produced good coord ina t ion and few overlap- ping swings despite the s trong perturbat ion. The s trong posi t ion-dependent coupl ing made the right f ront leg maintain 1 :1 stepping for quite large differences in retract ion speeds. Fo r middle and rear legs, overlapping swings were more frequent and concentra t ion parame- ters were small but significant. In the absence o f bo th contralateral mechanisms, phase distributions were ran- d o m and overlapping swings by contralateral leg pairs were frequent. When the posi t ion-dependent mecha- nism was removed, leaving only the t ime-dependent mechanism active, coord ina t ion was no better than in the absence o f bo th mechanisms. W h e n the t ime-depen- dent mechanism was removed, leaving only the posi- t ion-dependent mechanism active, coord ina t ion was similar to that o f the s tandard model; in fact, it was slightly improved. Adding contralateral inhibition

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Table 2. Comparison of the ability of single ipsilateral coordinating mechanisms to maintain normal coordination. The mechanisms were compared in simulations in which the retraction speed of the right legs was constant and that of the left legs varied between lx and 1.88x that o f the right legs. Each simulation continued until one leg completed 100 steps. Mechanisms are abbreviated as in Table 1. a) Mean phases (Ph) and concentration parameters (k) were calculated using circular statistics (Batschelet 1965); the significance was determined using the Rayleigh test at both the 5% and 1% levels (ns, not significant), b) For each pair of adjacent legs, the number of overlapping swings was counted both for swings of leg 1 beginning in a swing of leg 2 and vice versa

lpsilateral mechanism present

none

a) Phase relationships

Ph k

contralateral leg pairs

L1 in R1 126 0.88 L2 in R2 131 0.11 L3 in R3 136 0.49

ipsilateral leg pairs

R2 in R3 15 0.09 RI in R2 220 0.08 L2 in L3 281 0.46 L1 in L2 339 0.13

b) Number of overlapping

contralateral leg pairs

L1/RI 0/0 L2/R2 9/11 L3/R3 4/7

ipsilateral leg pairs

L2/L3 8/32 L1/L2 24/23 R2/R3 10/14 RI /R2 12/7

targeting time-dep inhibition pos-dep

p Ph k p Ph k p Ph k p Ph k p

1% 125 0.86 1% 124 0.85 1% 122 0.84 1% 126 0.87 1% ns 90 0.06 ns 114 0.15 ns l l2 0.14 ns 133 0.24 1% 1% 182 0.21 5% 139 0.37 1% 128 0.45 1% 125 0.25 1%

ns 124 0.74 1% 87 0.57 1% 192 0.20 5% 119 0.72 1% ns 27 0.12 ns 19 0.16 ns 63 0.16 ns 64 0.25 1% 1% 110 0.70 1% 31 0.74 1% 117 0.80 1% 87 0.95 1% ns 54 0.55 1% 31 0.21 5% 92 0.77 1% 84 0.95 1%

swings (leg 1 in swing of leg 2/leg 2 in swing of leg 1)

0/0 0/0 0/1 0/0 8/10 10/10 6/13 5/1 3/8 5/9 6/5 5/9

6/1 49/11 0/0 11/0 43/5 18/25 0/3 7/0 4/0 9/1 0/12 3/0 22/7 21/8 0/5 22/0

between front and rear legs had little effect on the mean contralateral phase values. It did slightly increase con- centration parameters for the front legs and reduce the incidence of overlapping swings by the rear legs.

3 Simulations with different forms of the coordinating mechanisms

One goal of the study was to compare several alterna- tive forms of coordinating mechanisms which are im- precisely specified by the experimental data. Since the position-dependent mechanism was most efficient in maintaining both ipsilateral and contralateral coordina- tion, the comparison focused on four aspects of this mechanism.

The first concerned the form of the shift in the PEP-threshold imposed on the receiver during the stance of the sender. The dependence of the PEP- threshold on both the retraction speed and on the position of the sender were varied. In the model, the speed-dependence is expressed by the position (P0) of the sender for which the PEP-threshold function inter- sects the abscissa, i.e. does not change the iPEP of the receiver. For fast walking, the speed-dependence fol- lowed the minimum required to avoid overlapping swings in an isolated pair of legs (Dean 1991a). This minimum requirement is that P0 move rostrally with increasing retraction speed such that the time for the

sender to retract from Po to its iPEP equals the time for the receiver to make a full-length swing. When this mechanism modulates the swing of the receiver, the subsequent swing of the sender begins immediately after the swing of the receiver is completed. For slow walk- ing, three forms were tested: a slight modification of this minimum requirement, a strong form which elicits symmetric alternate stepping (phases of near 180 deg), and an intermediate form.

For ipsilateral coupling, the minimum required speed-dependence was modified slightly to allow a delay between successive swings by adjacent ipsilateral legs to develop during slow walking. This modified speed-de- pendence still produced the metachronal coordination of the adult stick insect rather than tripod coordination. Hence, phase distributions were unimodal and very asymmetric: typical mean phases for one leg in the step cycle of the next caudal leg were about 75 deg.

When used for reciprocal contralateral coupling, this same speed-dependence led to equally asymmetric distri- butions of contralateral phases. The two possible asym- metries produced peaks in the phase distributions at positions corresponding to the single peak in ipsilateral phase histograms, or its mirror image, or both. The mean phase in the example shown (cross-hatched histogram in Fig. 1), where only one asymmetry occurred, was 97 deg. With this speed-dependence, the mechanism was unable to maintain 1 : 1 stepping when the retraction speeds of left and right sides differed (Fig. 2b; see also Sect. 5).

349

Table 3. Comparison of the effectiveness of various contralateral coordinating mechanisms. The riaechanisms were compared in simulations in which the retraction speed of the right legs was constant and that of the left legs varied between Ix and 1.88x that of the right legs. Each simulation continued until one leg completed I00 steps. Mean phases (Ph) and concentration parameters (k) in part a were calculated using circular statistics (Batschelet 1965); the significance was determined using the Rayleigh test at both the 5% and 1% levels (ns, not significant). The number of overlapping swings was counted both for swings of the left leg beginning in a swing of the right leg and vice versa. The strength of the position-dependent mechanism (pos-dep), expressed as a fraction of the full-strength influence, was 1.0, 0.25, and 0.5 for front, middle, and rear legs, respectively. The contralateral time-dependent (time-dep) influence was equally strong in all three segments. The contralateral inhibition (inhibition) was introduced only between front and between rear legs; its strength, expressed as the shift in the PEP-threshold, was 50 units or about 60% of the intrinsic step amplitude

Contralateral coordinating mechanisms

removed(-) -pos-dep -time-dep

a) Phase relationships

Ph k

leg pairs

L1 in R1 327 0.04 L2 in R2 324 0.05 L3 in R3 299 0.05

standard added(+) -pos-dep (pos-dep + inhibition

-time-dep time-dep)

p Ph k p Ph k p Ph k p Ph k p

ns 278 0.06 ns 128 0.79 1% 114 0.75 1% 114 0.84 1% ns 233 0.04 ns 124 0.28 1% 109 0.20 5% 107 0.20 5% ns 166 0.05 ns 117 0.24 1% 123 0.22 1% 129 0.23 1%

b) Number of overlapping swings (L-leg in swing of R-leg/R-leg in swing of L-leg)

L1/R1 11/15 11/8 0/1 0/0 0/0 L2/R2 9/17 12[ 11 8[0 7/3 8[2 L3/R3 10/16 13/15 10/9 9/11 0/0

40

30

Z 20

10

0 0 gO 180 270 360

phase (degrees) Fig. 1. Histograms of contralateral phase relationships between the front legs for different forms of the position-dependent mechanism. Results from three different simulations are superimposed. The con- tralateral position-dependent mechanism used either a speed-depen- dence similar to that used for ipsilateral coordination (cross-hatched histogram) or one favoring symmetric alternation (open and dotted histograms). Both forms were tested in simulations in which the retraction speed of all six legs varied together between 18% and 33% of protraction speed (cross-hatched and open histograms). The form favoring symmetric alternation was also tested in a simulation in which retraction speed of the right legs remained at 18% of protrac- tion speed and only that of the left legs varied (dotted histogram)

When the speed-dependence for eo followed a straight line from the iAEP for equal retraction and protraction speeds to the mid-point between iAEP and iPEP for very slow retraction speeds, distributions of contralateral phases were unimodal and symmetric (open histogram in Fig. 1). This speed-dependence pro- vided stronger coupling; it was better able to maintain 1 : 1 stepping when retraction speeds differed on the two sides (Fig. 2c).

As a compromise, the speed-dependence used for contralateral coupling in the standard model was one derived from the lag measured between steps of ipsilat- eral rear and front legs. For slow to moderate speeds, the step alternation of these legs is less asymmetric than

that of adjacent ipsilateral legs. With the speed-depen- dence derived from this relationship, the coupling is weaker than that required for strict alternation but stronger than the minimum required to prevent over- lapping swings. As a result, phase distributions were typically bimodal with peaks at 120 to 170 and 200 to 240 deg.

Regardless of which speed-dependence was chosen, when the same form was used for both ipsilateral and contralateral coupling, each front leg exerted similar influences on the contralateral front leg and on the ipsilateral middle leg, encouraging these legs to step in-phase. Hence, either the left front and the right middle or the right front and the left middle legs stepped nearly in-phase. Middle and rear legs adopted the same diagonal pairing.

The second parameter investigated was the position- dependence of the change in PEP-threshold. Several variants were tried for ipsilateral coupling: 1) replacing the standard sigmoid function by the saw-tooth func- tion used in the contralateral mechanism, or 2) using only the positive part of the sigmoid curve with the maximum threshold shift either left unchanged or dou- bled. None of these substitutions had much effect on mean phases or concentration parameters. The maxi- mum and minimum values differed by less than 0.05 for the latter and by less than 8 deg for the former. None of these variants provided inhibition during the swing of the sender. As a result, the frequency of overlapping swings between ipsilateral front and middle legs in- creased from virtually none to as many as 1 in 20 steps.

The third parameter considered was the value of the position-dependent influence during the swing of the sender. The standard sigmoid function used ipsilaterally imposes different PEP-shifts on the receiver at the be- ginning and end of the sender's stance, so there are

350

o) VR 0.3(R): 0.35(L) " / / " i " : : : '% : "

R2 . ~ : ' ~ -

�9 ",, . . . . x ; R3

, ,

time --~

b) 360

~. 240

._c

"~ 180

0

~, 90

0 0

Fig. 2 a - c . Comparison of two forms of contralateral coupling when retraction speed differs on the two sides of the body. Legs are numbered from front to back and designated as right (R) or left (L). The retraction speeds (VR), expressed as fractions of the protraction speed, were 0.3(R) and 0.35(L). a The six traces represent the rostrocaudal movement of the legs with forward movement corre- sponding to upward change in the trace. Front legs were more likely to make extra steps, disrupting the metachronal waves of successive rear, middle and front leg steps. After such a step by R1 (arrow), the

lb

" , . , �9 ' , . � 9

2'0 sb step number

c) 360

~, 240 .c_

-~ 180

0 r

Q. 90 . " � 9 1 4 9 , . �9 " . .

~ .~

step number

strong caudally directed position-dependent mechanism then delays the ensuing steps of R2 and R3. Parts b and e illustrate the phase of L1 relative to R1 for successive steps, b The minimum position-de- pendent excitation required to prevent overlapping swings for equal retraction speeds produced only relative coordination when retraction speeds differed. The faster, left side continually overtakes the right side. a, e Changing the speed-dependence so as to produce alternation for equal retraction speeds elicited stronger coordination with only intermittent gliding

several possibilities for extending this function through the swing�9 First, the position-dependence can simply continue as in stance, providing a continuous curve over the whole step cycle. Second, the influence can be inactive during the swing�9 Third, the maximum inhibi- tion can be imposed throughout the swing of the sender�9 This last form acts like a weaker version of the rostrally directed inhibition; it was used for most simu- lations discussed above�9

When the step coordination was unperturbed, the value of the position-dependent influence during the swing of the sender made no difference�9 Therefore, the alternative forms were compared in simulations where the retraction speed of the three left legs oscillated about that of the right legs, as described above�9 Even under these conditions, removing the constant inhibi- tion during the sender's swing caused little change�9 Mean ipsilateral phases differed by less than 5 deg; concentration parameters were virtually unchanged or, for contralateral leg pairs, slightly larger�9 Using the same sigmoid position-dependence during swing and stance did not increase the frequency of overlapping swings�9 Making the influence neutral throughout the sender's swing did increase the frequency with which a middle leg began a swing during the swing of the ipsilateral front leg: such overlapping swings occurred as often as one step in twenty versus only one in 160 with inhibition during swing. Overlapping swings ini- tiated by the front leg or between middle and rear legs did not occur�9

A final aspect considered was the relative strengths of the different mechanisms�9 Ipsilateral mechanisms were not tested in detail, but the model did not appear particularly sensitive to the choice of parameters�9 For example, reducing the strength of the rostrally directed

inhibition by 40% and doubling the maximal ipsilateral position-dependent influence produced virtually no change in mean ipsilateral phases or in concentration parameters of ipsilateral or contralateral leg pairs�9

The relative strengths of the two contralateral mechanisms did have one curious effect. In an early version of the model, the amplitude of the time-depen- dent excitation did not change with retraction speed�9 For fast walking, this model produced good metachronal coordination. As walking speed decreased, the strength of the position-dependent influence de- creased relative to that of the time-dependent influence and the asymmetry of the contralateral phases permit- ted by the former increased�9 As a result, the time-de- pendent excitation from one front leg occasionally excited the other front leg to step out of turn in the ipsilateral metachronal sequence�9 Making the amplitude of the time-dependent influence decrease with retraction speed led to regular metachronal sequences at all speeds�9

4 Effect of coupling strength in different segments on contralateral coordination

Behavioral experiments indicate that contralateral cou- pling is not equally strong in all segments�9 In accord with these results, the strength of the position-depen- dent influence in the standard model is strongest be- tween front legs and weakest between middle legs. The coupling strength was adjusted by multiplying the PEP- threshold function by a constant between 0 and 1 while leaving the speed-dependence unchanged�9 For equal retraction speeds of left and right legs, the strong coupling between front legs ensured that these legs

351

Table 4. Recovery from in-phase stepping in simulations beginning with contralateral legs in bilaterally symmetric positions. The table compares the speed of recovery for different relative strengths of the contralateral position-dependent influences and for two levels of contralateral inhibition. The strength of the position-dependent influence (pos-dep) is indicated as a decimal fraction of the standard form; the inhibition strength is the step-change added to the PEP-threshold during the swing of the contralateral leg, where the intrinsic step amplitude was 80 units. Ten simulations of the first 8 steps beginning from the same leg configuration were performed for each set of parameters. The speed of recovery is indicated by the median and range for the number of overlapping swings (N steps) by contralateral front (FL), middle (ML) and rear (RL) legs. The table also gives the number of simulations (N sim) in which the leg pair at some point adopted the asymmetry opposite to that finally established (op sym)

Strength of contralateral mechanisms:

1) Pos-dep i) 1.0, 0.5, 0.25 ii) 0.25, 0.5, 1.0 iii) 1.0, 0.25, 0.5 2) Inhibition 0 0 0

recovery op sym recovery op sym recovery op sym N steps N sim N step N sim N steps N sim median range median range median range

FL 2.5 2-3 0 5 3 >8 1 2 2-3 0 ML 3 3-4 0 6 4 - > 8 3 3.5 3-5 0 RL 4 4-5 2 5.5 4 >8 4 4 4-5 5

Strength of contralateral mechanisms:

Pos-dep iv) 1.0, 0.25, 0.5 v) 1.0, 0.25, 0.5 Inhibition 50, 0, 50 25, 0, 25

recovery op sym recovery op sym N steps N sim N steps N sim median range median range

FL 0 0 0 0.5 0-1 0 ML 1 0 2 0 2 1 2 4 RL 0 0 2 0 2 2-3 4

stepped alternately. On its own, the weaker coupling used between middle legs and between rear legs did not exclude overlapping swings. However, the ipsilateral mechanisms enforced tight metachronal rhythms and imposed the alternation of the front legs on the other leg pairs.

The effect of contralateral coupling was more appar- ent when the normal coordination was perturbed by either starting the simulation from unusual configura- tions or varying the retraction speed of one or more legs. Symmetric positions of the left and right legs of each segment is one unnatural configuration. When a stick insect begins a walk from such a configuration, segmen- tal leg pairs often make several in-phase steps with overlapping swings before switching to alternation. The transition from in-phase to alternate stepping is not well-studied. Here the model can generate predictions.

For example, in an isolated leg pair, the strong form of the position-dependent mechanism prevents overlap- ping swings and converts in-phase configurations to alternation within the first step (Dean 1991a). In the six-leg model, recovery was slower, but alternation was achieved within a few steps (Table 4iii). When con- tralateral coupling was strongest between front legs, alternation was first established rostrally and then propagated to the rear (Fig. 3a). The recovery was slower due to the stability of the ipsilateral metachronal rhythms facilitated by the ipsilateral configuration cho- sen. In particular, the rostrally directed inhibitory and excitatory influences prevented the contralateral influ- ence between the front legs from effecting a correction within one step.

The strong form of the position-dependent mecha- nism did not permit sustained in-phase stepping. To obtain longer gallops, the strength of the contralateral coupling had to be decreased. When the position-de- pendence of the PEP-threshold function was reduced by a small additive constant or varied from step to step, then the front legs sometimes switched from in-phase to alternate stepping in an abrupt manner qualitatively more like that seen in stick insects.

When the contralateral coupling was strongest be- tween rear legs, then the pattern of recovery was the same but alternation was established more slowly (Fig. 3b, Table 4ii: pos-dep strength 0.25, 0.5, 1). Because the ipsilateral mechanisms enforced a tight ipsilateral coor- dination, the symmetry ultimately adopted was still determined by the front legs. In some simulations, middle and rear legs initially shifted towards the asym- metry opposite to that of the front legs (e.g. rear legs in Fig. 3b; Table 4). Recovery was slower because the coupling between the front legs was weaker. The strong coupling between the rear legs sometimes brought this pair into alternation at the same time as the front legs and before the middle legs.

The strength of the contralateral coupling between the middle legs was less important when the strong position-dependent influence was used between the front legs (e.g. Table 4: iii versus i). Using a weak influence between the middle legs helps balance the number of influences on the PEP-thresholds in the different segments: if the weak position-dependent influ- ence is not counted, then the PEP-threshold of the middle legs is subject to 3 ipsilateral influences and 1

352

controlotecol COUl~lng

o} FL >ML>RL

J . l : l . : : - : / - " \ , \ , \ -w<

b) FL<ML<RL

. " , \ ; 'N / \ i \ i \ . , , \ i \ i \ i "<. R2 ~ \ i \ i ~

.I. I i.

c) FL>RL>ML and swing inhibition

�9 3

tlme ~ l r swings overtop

L1 l l l . :N~

time

Fig. 3a-e . Simulations of recoveries from in-phase stepping. The walks began with bilaterally symmetrical leg positions, which lead to in-phase stepping ( g a l l o p s ) . The strength of the contralateral coupling for front (FL), middle (ML) and rear (RL) legs, expressed as a fraction of the minimum required to prevent overlapping swings, was varied as follows: a 1.0, 0.5, and 0.25; b 0.25, 0.5, and 1.0; c 1.0, 0.25,

tlme

and 0.5 with inhibition during the swing of the contralateral leg. The arrows mark swings which overlap with that of the adjacent con- tralateral leg and indicate the time at which the overlap begins, a Contralateral legs switched more rapidly from in-phase stepping ( a r r o w s ) to alternation when the contralateral position-dependent influence was strongest between front legs. c Recovery was faster still when contralateral inhibition was added

contralateral influence, that of the front legs is subject to 1 ipsilateral influence and 2 contralateral influences, and that of the rear legs is subject to 1 ipsilateral influence and 2 contralateral influences. Reducing the number of influences acting on the middle legs reduces the chance of unwanted interactions in the additive algorithm used for the threshold calculation.

The effect of varying contralateral coupling strength was also measured by letting the legs of the two sides step with different frequencies. The result was qualita- tively the same. When contralateral coupling was strongest at the front, the front legs made fewer over- lapping swings than did middle and rear legs and vice versa.

5 Possible role of contralaterai inhibition

Some behavioral experiments suggest that a leg is inhib- ited from beginning a swing during the swing of the contralateral segmental leg, but the evidence is not conclusive. When this inhibition was added to the model, the contralateral coordination was stronger and the frequency of overlapping swings was reduced (Table 3). Contralateral inhibition as strong as that used ipsilaterally virtually prevented in-phase stepping. This change was particularly evident for the rear legs, where the position-dependent mechanism was weaker in the standard model. In-phase steps by the strongly coupled front legs were already quite infrequent. Inhibi- tion between the front legs did shorten gallops when the simulation was started with bilaterally symmetric leg positions (Fig. 3c, Table 4iv, v). However, strong con- tralateral coordination sometimes disturbed ipsilateral coordination by inducing adjacent ipsilateral legs to make overlapping swings.

Foth and Graham (1983) postulated a more com- plex form of contralateral inhibition lasting longer than the swing of the sender. This hypothesis was based on an analysis of phase-response curves obtained when the resistance to leg retraction differed on the two sides of the animal. When the difference in load was small, 1 : 1 stepping was maintained, but the natural asymmetry in contralateral phases was biased such that the side which faced more resistance led and the amount of asymmetry increased. When the difference in load was larger, 1:2 and 1:3 step rhythms occurred. Transitions were abrupt and gliding coordination was rare. Foth and Graham interpreted these results as evidence for con- tralateral inhibition which imposed the rhythm of the slower, more heavily loaded side on the faster side. For 1:2 and 1:3 rhythms, this hypothesized inhibition would explain the characteristic lengthening of the step which included the swing of the slower side.

This situation was simulated by specifying different retraction speeds for the two sides. As in the animal, the result was to bias the asymmetry in contralateral phase relationships. The side with the slower retraction speed usually led. However, unlike the animal, the model showed relative or gliding coordination even for small differences in retraction speed (Fig. 2b). More- over, for larger differences and unequal step ratios, the excitatory influences used in the present model tended to shorten rather than lengthen the step of the faster side which included the swing of the slower side.

Making the contralateral position-dependent mech- anism stronger removed some discrepancies. In the standard model, the strength of the contralateral cou- pling was the minimum required to prevent overlapping swings in an isolated leg pair when both legs step at the same speed. Furthermore, the change in the PEP- threshold depends on the speed of the sender, not that

of the receiver. As a result, when one leg retracted slower than the other, its influence on the faster side was too weak. It could only slow the phase advance of the faster side and produce relative coordination (Fig. 2b). Changing the speed-dependence to reduce the amount of contralateral asymmetry also reduced the tendency to glide (Fig. 2c). Glide would also be reduced if the contralateral coupling becomes stronger when a leg experiences increased resistance. This is probably true of the insect. In contrast, coupling becomes weaker in the present kinematic model when increased load is simulated by reducing retraction speed. Nevertheless, one fundamental difference between excitatory and in- hibitory mechanisms remains. The latter cause the slower side to control the faster side and induce caudal shifts in the mean PEP. The former elicit rostral shifts.

6 Discussion of the form and relative strength of the position-dependent mechanisms

To ensure stable leg configurations, the coordinating mechanisms need to prevent overlapping swings by adjacent legs. This applies to both ipsilateral and con- tralateral leg pairs. The neural circuits for the position- dependent mechanism would be simplified if the signals a leg sends to adjacent ipsilateral and contralateral legs have a common source. The results presented here suggest that this simple hypothesis does not entirely apply. Whenever the same form is used for ipsilateral and contralateral coordination, in-phase stepping of diagonal leg pairs is favored, a coordination which is sometimes but not invariably seen in stick insects. Us- ing the minimal form of the speed-dependence for both ipsilateral and contralateral mechanisms does lead to good alternation for fast walking, but it permits unreal- istically asymmetric step alternation during slow walk- ing (Dean 1991b). By varying the speed-dependence, as shown in Sect. 3, varying amounts of asymmetry can be obtained. Behavioral measurements show that mean contralateral phases lie between perfect alternation and the very asymmetric phase of adjacent ipsilateral legs (Graham 1972). Using the alternation of ipsilateral front and rear legs as a model for the speed-dependence in the contralateral mechanism provides a more realis- tic, moderate asymmetry. However, the coordination of these legs is generally believed to arise indirectly by way of the middle leg (see, however, Wendler 1968), so it does not represent a physiological source which could be shared with the contralateral influence.

The form of the PEP-threshold imposed by the contralateral position-dependent mechanism was cho- sen on the basis of considerations discussed elsewhere (Dean 1991a). At full strength, this influence follows the ideal form for preventing overlapping swings in adjacent, reciprocally coupled legs. With increasing re- traction speed, the influence becomes active at more rostral positions of the sender and the maximum change in the PEP-threshold of the receiver increases. These two changes may appear arbitrary, but both could reflect phasic neural response characteristics.

353

They could also be related to increases in general excitation associated with faster walking. The same considerations can be advanced to justify the speed-de- pendence incorporated in the time-dependent mecha- nism, the rostrally directed excitation accompanying the start of leg retraction.

The contralateral position-dependent mechanism, like the time-dependent excitation, effects non-overlap- ping swings indirectly. Both favor leg configurations in adjacent legs such that only one leg nears its PEP- threshold at any time. The position-dependence used contralaterally follows the minimum required for a single pair of legs, but this is no guarantee that it will be adequate for systems of three or more legs. Because the influence acts indirectly by exciting the receiver to swing early and move out of stance configurations leading to overlapping swings, it requires that the course of the sender's stance and the receiver's swing be predictable. Unlike an inhibition during the sender's swing, the basic position-dependent mechanism cannot directly suppress an incipient overlapping swing if random vari- ations, external perturbations, or actions of other legs bring about inappropriate leg configurations.

Hence, swings by the left and right legs of a segment can overlap for two reasons. First, the minimal form of the position-dependent influence is calculated as if the sender will always continue to its iPEP after exciting the receiver to a swing. If ipsilateral influences excite the sender to an earlier swing, this swing will overlap with that of its segmental partner. Second, overlapping swings can occur if the actual contralateral coupling is less than this minimum requirement. Therefore, either centrally mediated, reciprocal inhibition of swings or local, intraleg influences sensitive to leg loading are necessary to totally exclude overlapping swings by seg- mental legs.

The assumption that the contralateral position-de- pendent influence reaches a maximum and then declines as the sender retracts may at first appear artificial. However, the symmetry of the contralateral coupling usually causes changes in the PEP-threshold larger than the distance of the sender from its iPEP to be masked by the reciprocal influence. Thus, the monotonic, sig- moid position-dependence used in the ipsilateral mecha- nism can be used in the contralateral mechanism without significant change as long as the legs alternate. This substitution does reduce the possibility of in-phase stepping and causes the PEPs of such gallops to be shifted forward. Qualitative observations of gallops in which both legs proceed to near-normal PEPs suggest that, in the animal, the change in the PEP-threshold imposed on the receiver does not exceed the distance of the sender from its iPEP. Physiologically, such a reduc- tion when segmental legs reach caudal positions at the same time could reflect reciprocal inhibition.

The function of the ipsilateral position-dependent mechanism resembles that of the contralateral mecha- nism: when the sender is trailing, it induces the leading leg to swing early in order to avoid overlapping swings. However, four differences are worth noting. First, the mechanism acts only in one direction. Second, the

354

ipsilateral mechanism includes an inhibitory component for rostral positions of the sender: this component extends its influence over additional leg configurations in which the sender is trailing. Third, the excitation remains high for caudal positions of the sender: this extends its influence to leg configurations in which the sender is leading. This part of the curve is not masked because the ipsilateral influence is unidirectional. Fourth, the excitatory part of the sigmoid PEP- threshold function used here is less than the minimum required to prevent overlapping swings. Thus, adjacent legs can enter stance configurations in which the caudal leg will begin its stance too late and must depend on the rostrally directed inhibition to delay the swing of the rostral leg - this interaction can create the illusion that inhibition from the caudal leg is controlling the timing. Physiologically, the graded, monotonic increase in the ipsilateral position-dependent influence is intuitively ap- pealing because it might simply reflect steadily increas- ing activity in proprioceptors excited during leg retraction.

Inhibition and excitation, as used here, merely de- note the direction of a shift with respect to the iPEP. If the iPEP is redefined as the PEP expressed when the sender is far forward, then the ipsilateral position-de- pendent mechanism can be reformulated as a purely excitatory influence. The true intrinsic thresholds are unknown, but some evidence suggests that the caudally directed mechanism does contain both excitatory and inhibitory components. Shifts in the mean PEP after severing ipsilateral connectives indicate that the iPEP for the middle leg lies near the mean PEP for normal walking whereas that of the rear leg may lie more caudally (Dean 1989). Foth and B~issler (1985) postu- lated a caudally directed inhibition depending on phase rather than on constant time delays; this could reflect the inhibitory component of the position-dependence discussed here. Finally, changes in mean PEPs caused by restraint of the adjacent rostral leg also point to an inhibitory component in the caudally directed influence (Dean and Wendler 1983).

Letting the position-dependent mechanism con- tribute inhibition during the swing of the sender intro- duces a reciprocal inhibition between adjacent ipsilateral legs. This has not been unequivocally demonstrated in behavioral experiments although electrophysiological evidence from cockroaches does indicate a reciprocal inhibition for some alternating leg movements (Pearson and Iles 1973). Overlapping swings might be expected to increase in frequency if the position-dependent mecha- nism is neutral or maintains its position-dependent influence during the swing of the sender. In the latter case, the mechanism would actively promote an overlap- ping swing until the sender protracts past the neutral position, P0. The simulation results show that such a component would not have a large effect on coordina- tion. In free-walking insects, local intraleg influences may provide a final check to prevent overlapping swings: when a leg makes a swing, the increased load experi- enced by adjacent legs may prevent them from beginning swings (Bfissler 1977; Cruse 1985).

In the complete model, several mechanisms combine to influence the PEP, the critical transition of a leg from support to swing. Their relative strengths determine how the model assumes a stable coordination. The position-dependent mechanism generally provides the most effective coordination because it exerts a graded modulation on many leg configurations. In contrast, the inhibition during swing is less effective although it causes a larger change in the PEP-threshold. When all the legs have the same natural frequency, the rostrally directed inhibition does not specify a particular ipsilat- eral phase relationship because it modifies only some leg configurations. This mechanism is effective in pre- venting overlapping swings. With a suitable hierarchy of natural frequencies, an inhibition coupled to swing duration can organize the steps of ipsilateral legs into metachronal waves with a constant intersegmental lag. To obtain natural lags, the inhibition must last longer than the swing and depend upon walking speed (e.g. Graham 1977, 1978). Both the rostrally directed, de- layed excitation, which is similar to a mechanism pro- posed by Wilson (1968), and the targeting mechanism are less effective in maintaining overall coordination. Like the inhibition, the time-dependent mechanism is active during only part of the stance. Moreover, its maximum influence is small, so it can only modulate step timing in conjunction with the other influences. This excitation does help promote a swing at a time when the neighboring legs are particularly well placed to assume extra load. Thus, each mechanism con- tributes to the model's stability, although some mecha- nisms appear more important than others. The results reported here illustrate how important assumptions of the model affect its behavior in various situations and thus suggest experimental tests of these assumptions.

Acknowledgements. I thank Professor H. Cruse for helpful comments on the manuscript and Ms. A. Exter for help with the figures. The research was supported by a grant from the Deutsche Forschungs- gemeinschaft (Cr 58/8-1).

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Dr. Jeffrey Dean Universit~it Bielefeld Abteilung 4/Fakult~it f/ir Biologie Postfach 8640 W-4800 Bielefeld 1 Federal Republic of Germany FAX (0)521-106 5844