The relationship between leg stepping pattern and yaw torque oscillations in curve walking of two crayfish species

Arthropod locomotion has received much attention as a model for studying the generation and control of movement. While straight walking has been studied extensively in the past, especially in treadmill experiments (for reviews, see Clarac, 1985; Cruse, 1990), the study of arthropod manoeuvrability in more complex situations involving direction changes has recently received increased attention (e.g. Zollikofer, 1994; Cruse and Silva Saavedra, 1996; Domenici et al., 1998; Cruse et al., 1998; Jindrich and Full, 1999) because of its applicability in the field of robotics, in addition to its relevance to studies of the neuromuscular control of locomotion.

Arthropod manoeuvrability includes a whole spectrum of locomotor behaviour patterns that correspond to various combinations of body rotation and translation. Franklin et al. (1981) suggested that rotation during arthropod turns may be related to asymmetries between the inner legs (on the inside of the curve) and the outer legs resulting from (1) temporal asymmetries, e.g. the higher step frequencies of the outer legs, and (2) positional asymmetries, e.g. the larger step amplitudes of the outer legs. A recent study on the crayfish Astacus leptodactylus, based on treadmill locomotion, also concluded that curve walking is produced by changing the step amplitude and the step direction relative to straight walking (Cruse and Silva Saavedra, 1996).

More recently, Domenici et al. (1998) have shown that the inward rotation present in crayfish (Procambarus clarkii) walking freely along a curved path showed two characteristics: (1) the turn showed a discontinuous pattern linked to stepping, and (2) the power strokes of only three legs (outer leg 4 and inner legs 2 and 5) were in phase with the inward body rotation. However, the mechanisms underlying the temporal and/or geometrical patterns during curve walking remained unclear. The phenomenon described by Domenici et al. (1998) could be due to the fact that certain leg phase relationships (the turning tripod) are centrally pre-programmed and that the appropriate legs, when in stance, synchronously produce rotational torque by increased force production towards the inside of the curve. Alternatively, it is possible that only one principal leg produced the rotational torque, while the other two legs showed in-phase relationships (resulting in a tripod) due solely to certain coordinating trends among legs.

To test these alternative hypotheses, we investigate curve-walking behaviour in tethered individuals of the two species recently investigated, P. clarkii and A. leptodactylus. If curve walking in these species is accomplished through a mechanism implying a centrally pre-programmed tripod, one would expect also to observe such a tripod during tethered curve walking. If, however, there is only one principal leg producing the rotational torque and the phase relationship is not centrally pre-programmed, then, because of the diminished need for static stability, such a principal leg should show stronger coupling with the inward torque than all the other legs.

Here, we show that in both species, P. clarkii and A. leptodactylus, the yaw torque pattern observed during curve walking is temporally related to the movement pattern of outer leg 4, suggesting that the rotational force necessary in curve walking may be the result of the force produced by one principal leg, outer leg 4, and that the geometrical asymmetries as well as preferred phase relationships described in previous studies may be an ancillary phenomenon.

Adult crayfish of two species (Astacus leptodactylus Esdischoltz and Procambarus clarkii Girard) were obtained from local dealers. Individual animals were kept in separate tanks (40 cm×20 cm×20 cm) filled with fresh water at ambient temperature (12–15 °C) and fed with shrimps once a week. A. leptodactylus had a mean body length of 13.65±0.10 cm (mean ± S.E.M., N=7; mass 91.48±3.06 g), P. clarkii had a mean length of 11.18±0.61 cm (N=7) (mass 51.05±6.20 g).

For the experiments described here, we used the arrangement described by Cruse and Silva Saavedra (1996), which we adapted to measure the body yaw torque signal (Fig. 1). Individual animals were fixed dorsally by the carapace to a holder using dental glue (Protemp II, Espe, Germany) and were placed on a motor-driven belt in a water-filled tank. The glued end of the holder was positioned in the middle of the carapace and it was counterbalanced to allow the animal to determine its height above the belt. The holder consisted of a stainless-steel rod (3 mm in diameter) and it was guided by two ball bearings. The rod was connected to a torque transducer positioned at its upper end. The torque transducer consisted of a flexible brass foil (0.3 mm thick, 55 mm long, 20 mm wide) with strain gauges (Philips, type PR9833) attached on both sides. This arrangement allowed the measurement of the yaw torque in an isometric manner. The strain gauges complemented a Wheatstone half-bridge amplifier. The torque measurements were calibrated by adding a series of weights, exerting a left or right torque by means of a system of pulleys.

The movements of the legs were recorded as described by Cruse and Müller (1986). However, we extended this method to the second dimension in the horizontal plane by applying two perpendicular electric fields of different carrier frequency. A pair of large-grid electrodes (37 cm×8 cm) was positioned under water to the front and the back of the treadmill to generate an electric field (1 V peak to peak, a.c., 20 kHz) parallel to the long axis of the animal (x-axis). Another pair of large-grid electrodes was positioned to the left and to the right of the treadmill to generate an electric field (1 V peak to peak, a.c., 4 kHz) perpendicular to the long axis of the animal (y-axis). Because of limitations in the number of channels available, we did not measure both x and y displacements for each leg. Measuring the x or y displacements allowed us to measure the beginning and the end of the stance phase of each leg on the basis of their longitudinal or latitudinal oscillations, while the actual real-world coordinates of the legs, which were not within the scope of this study, were not recorded.

The recording electrode on each leg was a thin copper wire (0.1 mm in diameter), isolated except at the tip, fixed by small pieces of adhesive tape to each leg. The tip of the wire was placed at the end of the propodite and served as the third electrode of a potentiometer that records a voltage proportional to the position of the recording electrode relative to the grid electrodes. The signal of each recording electrode was rectified and then fed into two parallel band-pass filters (with corner frequencies of 19 and 21 kHz, and of 3 and 5 kHz, respectively) to obtain the direct current signals corresponding to the leg’s x and y positions, respectively. The movement signal of each leg (leg signal) as well as the torque signal were A/D-converted (single-channel sampling rate 100 Hz) and stored on a computer for off-line evaluation.

Curve walking was elicited using the principles of classical optomotor stimulation (Hassenstein, 1958). Three video monitors (Sanyo, type 18112 CX, screen diagonal 25.4 cm) were arranged beside the aquarium, one in front of the animal and one on each side. The distance between the head of the animal and the screens was approximately 35 cm. The walls of the aquarium were covered with black plastic sheets except for the regions containing the monitors. The monitors were connected to a computer (Apple IIe) programmed to show vertical green and black stripes (height 14 cm, width 6 cm) moving in a horizontal direction. At the position of the animal, the width of a stripe subtended an angle of approximately 10 °. The speed of the moving stripes was fixed at 1.2 degrees s⁻¹, which is within the range of values used by Cruse and Silva Saavedra (1996).

The experimental procedure was similar to that used by Cruse and Silva Saavedra (1996). The animal was fixed to the holder, and the tips of the electrode wires were attached to the end of the propodites. The first pair of appendages, the chelipeds, are not used in underwater locomotion and were excluded from the analysis. Therefore, eight legs were considered, four inner legs (the legs on the inside of the turn, termed ‘i’, i.e. L2i–L5i) and four outer legs (the legs on the outside of the turn, termed ‘o’, i.e. L2o–L5o), respectively. The animal was then placed into the aquarium, and the holder was inserted into the ball bearings and clamped to the torque transducer. The treadmill belt was started and then run at a constant speed of 5 cm s⁻¹. After the animal had begun to walk steadily, the monitors and moving stripes were switched on. The rubber belt allowed the legs to move in both the longitudinal and latitudinal directions during power strokes. These semi-slipping conditions may have partially reduced the applicable torque, but did not confound our results because our main goal was to measure the time course of the torque and not its absolute magnitude.

The animals were filmed from above using a Canon Hi8 video camera at 50 frames s⁻¹. Video recordings were used to ensure that the minima and maxima in the x,y measurements corresponded to actual anterior extreme positions (AEPs) and posterior extreme positions (PEPs), respectively, of the power stroke. During the experimental phase, 14 animals (seven P. clarkii and seven A. leptodactylus) were used in 20 recording sessions. Each session lasted a maximum of 2 h, including pauses during which the belt and the video monitors were turned off. Of these, 18 sequences (10 right turns and eight left turns), representing a total of approximately 300 steps, from four A. leptodactylus and 15 sequences (eight right turns and seven left turns), representing a total of approximately 150 steps, from four P. clarkii were chosen. A minimum of three and a maximum of five stepping sequences were used for each individual crayfish. The selection criteria were that all crayfish legs were moving steadily and that the animals were showing curve-walking behaviour as indicated by asymmetries in antennal and claw positions and tail orientation. The torque values of such sequences were in the direction of the moving stripes.

The time of the beginning and the end of the power stroke (AEP and PEP, respectively) for each leg was calculated as the time at which minimum (AEP) and maximum (PEP) values along the longitudinal axis (the electrode signal along the x-axis) were reached in a leg stroke cycle. Sometimes, the most anterior legs (legs 2) moved diagonally (as observed by Cruse and Silva Saavedra, 1996) with a strong latitudinal component (the electrode signal along the y-axis). In these cases, the AEP and PEP for this leg were determined using the minimum and maximum values of the y-axis signal. The leg movements of all sequences were observed in the video recordings as well as by using the electrode x or y signals to ensure that the minima and maxima of either the x or y measurements corresponded to actual PEPs and AEPs. The leg signals recorded (see Fig. 3) therefore corresponded to arbitrary (relative) units in the x- or y-axis and not to absolute units of amplitude. The time elapsed between two successive AEPs defined the leg period. The relative duration of the stance phase over the period was expressed as the duty factor. All timing variables were analysed statistically using non-parametric tests (the Mann–Whitney U-test).

The phase relationship of each leg n relative to any leg n′ (Φ_{n in n′}) was defined as the occurrence of the AEP of the chosen leg_n (AEP_n) within the period of the given reference leg_n′ (P_n′). Phase relationships were calculated in degrees, with 0 ° representing an in-phase relationship and 180 ° representing alternation. Circular statistical methods (Batschelet, 1981) were used for the treatment of phase relationship data. By convention, the reference leg was always the anterior leg in ipsilateral and diagonal pairs and the inner leg in contralateral pairs.

Torque values were recorded throughout each walking sequence. The torque signal was calculated relative to the baseline for each sequence. The baseline was defined as the (flat) torque signal when the treadmill and the optical signals were off. Positive values indicated a torque in the direction of the optomotor stimulus. The time relationship between the torque signal and each leg signal was then computed using cross-correlation functions (CCFs), as used previously by Domenici et al. (1998) to describe the time relationship between body axis angular acceleration and the leg angles of freely walking crayfish. Third-power trend removal for leg signal was employed to remove low-frequency drifts of step ranges, which are not the subject of the present study. This trend removal does not affect the temporal position of the maxima and the minima (i.e. PEPs and AEPs) of the leg signal. For each sequence, the estimated CCF between each leg step cycle and the torque was computed, and the span of time lags or leads was analysed within a range from −1s to 1 s in increments of 40 ms. The CCFs for each sequence were then averaged using the procedure developed by Amblard et al. (1994). The correlation peaks of the averaged CCF were tested for a significant difference from zero using t-tests after z-transformation (see Amblard et al., 1994). The x-values of the resulting peaks provided an estimate of the time delay between the step cycle and the torque. A leg causing a peak torque is expected to show a period that is out of phase with that of the torque by a value between 0 ° and 180 ° (Fig. 2). Such a value implies that the peak torque is produced during the leg’s power stroke. Since the maximum force exerted by a leg is not likely to occur at the beginning (corresponding to a phase lag of 180 °, Fig. 2D) or at the end (corresponding to a phase lag of 0 °, Fig. 2B) of the power stroke (Klärner and Barnes, 1986), only intermediate values of phase lag suggest that the leg in question may be involved in producing the peak torque.

The analysis of the leg periods and the duty factors was aimed at testing the hypothesis that asymmetries in such variables between the inner and outer legs may be related to (although not necessarily the cause of) curve walking. For both species studied, the mean timing of the leg periods is shown in Table 1 for the inner and outer legs of all sequences combined.

In A. leptodactylus, the mean leg periods were in the range 0.74–0.98 s. There were some significant differences between contralateral legs, but these differences in the mean periods were small, ranging from 20 to 70 ms (2–9 % of the period), and did not show any consistent trend in the asymmetry between contralateral legs (i.e. the periods of the outer legs were not always longer than those of the inner legs).

The period of L4i was significantly longer than that of L4o, while the periods of L2i and L3i were shorter than those of L2o and L3o, respectively (Table 1). No differences were found between the periods of legs 5. Differences between periods are not likely to be due to the presence of double steps (two consecutive steps within the duration of one normal step) because the frequency distributions of the periods of each leg were unimodal.

Mean duty factors ranged from 0.48 to 0.61. In all cases, the duty factor of the outer leg was significantly higher than that of the corresponding inner leg (Table 1). Therefore, asymmetries in duty factor were consistent across legs and independent of asymmetries in leg period.

In P. clarkii, the leg periods were slightly longer than 1 s. The only difference between contralateral legs was in legs 5, with L5i period being significantly longer than L5o period (Table 1). No differences were found between the periods of the other legs.

The duty factors were all between 0.51 and 0.57, and contralateral pairs differed significantly only for legs 4, with L4i showing a higher duty factor than L4o (Table 1).

Phase relationships were analysed for ipsilateral and contralateral pairs of legs. On the basis of previous studies (Clarac and Barnes, 1985; Domenici et al., 1998), we also analysed the phase relationships of leg pairs belonging to various sets of three legs (triplets) whose power strokes may be approximately synchronous and therefore form a tripod. The opposite triplets chosen for comparative purpose are 2i4o5i and 2o4i5o (2i4o5i was found to be in-phase in P. clarkii; Domenici et al., 1998) and 3i4o5i and 3o4i5o (which may form tripods moved in antiphase during straight walking; Clarac and Barnes, 1985). The phase relationships of the leg pairs from contralateral triplets (e.g. 2o4i5o and 2i4o5i; Tables 2, 3) were calculated and compared.

The mean values of the phase relationships are shown in Table 2. All pairs of ipsilateral legs on both sides showed a non-uniform distribution of phase relationships (Table 2), with mean phase increasing posteriorly from approximately 140 to 180 ° for inner and outer pairs of legs and therefore close to being in antiphase. Only pair 5o4o showed a distribution that did not differ from 180 ° (95 % confidence intervals test; Batschelet, 1981). Although they were very close in their mean phase values (the differences in the means of the distribution were in the range 3–13 °), the distributions of the phase values for corresponding pairs of ipsilateral legs (e.g. legs 3i2i and 3o2o) differed significantly (χ²-test, Table 2). This difference was related to a stronger coupling in the outer leg pairs, as shown by their higher mean vector lengths (Table 2).

Contralateral phase relationships resulted in non-uniform distributions in all pairs apart from the legs 2 pair (Rayleigh test; Table 2), and values for leg pairs 3 and 4 were not significantly different from an antiphase pattern (95 % confidence intervals test; Batschelet, 1981). Although non-uniform distributions suggest some inter-leg coordination, such distributions are quite broad and do not show any clear gaps, implying that this coordination may be relative and not absolute (absolute coordination implies strict inter-dependent rhythms with stable phase relationships, while relative coordination implies two weakly coupled rhythms with slightly different frequencies exerting an accompanying quantitative influence on one another; von Holst, 1973).

All leg pairs in the triplets considered showed distributions of phase relationships that were significantly different from uniform. For the opposite triplets 2i4o5i and 2o4i5o, the coupling between legs 4 or legs 5 with their corresponding legs 2i and 2o, respectively, was weak, as indicated by the small mean vector length, and had mean values between approximately 50 ° and 150 °. This implies that legs 2 were not on the ground in synchrony with the other legs of the triplet in question.

In contrast, the legs in each of the opposite triplets formed by legs 3i4o5i and 3o4i5o moved almost in phase with each other, as shown by the phase relationship values of each leg pair, which ranged between 320 ° and 3 °. In addition, leg pairs 4o3i and 5o4i showed significant in-phase patterns (95 % confidence intervals test; Batschelet, 1981). There was no difference in the distributions of the phase values between the two tripods, except for ipsilateral leg pairs 5i3i and 5o3o for which the outer pair showed a stronger coupling than the other ipsilateral pairs (Table 2). Therefore, the legs of each triplet (3i4o5i and 3o4i5o) were synchronized and formed two opposite tripods, which is compatible with observations on straight-walking crayfish (Clarac and Barnes, 1985).

In P. clarkii, all ipsilateral phase relationships resulted in non-uniform distributions (Rayleigh test; Table 3). The mean phase relationships increased posteriorly in the outer legs, and a significant antiphase pattern occurred only in the pair 5o4o (95 % confidence intervals test; Batschelet, 1981). Differences in the phase distribution on each side were found between pairs 4i3i and 4o3o only. L4i showed a significantly smaller phase lag (Φ) relative to L3i (Φ=55 °) than the corresponding contralateral pair L4o–L3o (Φ=133 °). This is in agreement with the larger phase lag in L5i–L4i (Φ=206 °) compared with L5o–L4o (Φ=174 °).

Contralateral coordinations were weak, as shown by their small mean vector lengths, and phase relationships resulted in non-uniform distributions only for leg pairs 3 (Φ=105±16 °) and 5 (Φ=23±17 °) (Rayleigh test; Table 3). For the latter, the phase distribution did not differ significantly from an in-phase pattern (95 % confidence intervals test; Batschelet, 1981). Although non-uniform distributions suggest some inter-leg coordination, asymmetries in the periods of legs 5 imply that this coordination may be relative and not absolute. The distribution of contralateral phases in P. clarkii differed strongly from that in A. leptodactylus. In P. clarkii, we did not find the alternate pattern of legs 4 that was present in A. leptodactylus.

Synchrony within triplets was investigated for P. clarkii as for A. leptodactylus (i.e. for the opposite triplets 3i4o5i–3o4i5o and 2i4o5i–2o4i5o). The strength of coupling between the various pairs of legs was weak, as indicated by the small mean vector lengths (Table 3). Some leg pairs showed a uniform distribution, and only pair 5o2o showed a significant in-phase pattern (95 % confidence intervals test; Batschelet, 1981). The uniform distributions observed in one or more leg pairs (e.g. 5i4o and 5o4i) of the triplets in question are inconsistent with the hypothesis that these triplets form tripods in our experimental situation.

The yaw torque signal showed an oscillating pattern over time. The example in Fig. 3 shows that such a discontinuous pattern may be related to the stepping pattern of one particular leg, L4o, because the rise and fall in the torque correspond to L4o the AEP and PEP, respectively of L4o.

To investigate the relationship between yaw torque and stepping pattern, we calculated the mean torque values at regular 10 % intervals of the power (stance) and return (swing) strokes, and we defined this as the average torque over the standardised period (ATSP). This resulted in mean torque profiles over standardised power and return strokes (Fig. 4). The assumption is that any leg with a role in producing a body rotation would show a high ATSP during its power stroke. In both A. leptodactylus (Fig. 4A) and P. clarkii (Fig. 4B), the highest peak in ATSP occurred during the power stroke of L4o. In A. leptodactylus, the other legs showed either a lower peak in ATSP during stance (L2o, L3i and L5i) or a maximum ATSP during swing, in agreement with an antiphase coordination with L4o (L3o, L5o, L2i and L4i). In P. clarkii, a clear single peak ATSP during stance occurred only for L4o, while the curves of other legs did not show any clear in-phase pattern with ATSP.

In addition, L4o showed the largest ATSP amplitude in both P. clarkii and A. leptodactylus. ATSP amplitude is related to the level of synchronisation of each leg with the torque signal. If a leg signal is completely unrelated to the torque oscillations, the torque level during the standardised period of such a leg would be relatively flat. However, if the torque signal is synchronised with the leg signal, standardization of the leg period should reveal the torque cycle.

The results shown in Fig. 4 suggest that L4o may be the most important leg in producing a rotation during curve walking. However, other legs must contribute to the yaw torque because the averaged torque signal is positive, albeit at its lowest value, during the swing phase of L4o in both P. clarkii and A. leptodactylus.

To study the relationship between the stepping pattern and the torque signal in more detail, we performed cross-correlation function (CCF) analysis. For both species, L4o showed the profile closest to that implying a peak in torque during the leg’s power stroke (see Fig. 2). In A. leptodactylus, L4o, L3i and L5i showed phase lag values between 0 ° and 180 ° (Fig. 5A). Of these, only L4o and L5i showed correlation peaks of the averaged CCF that were significant. In addition, the correlation coefficient of L4o was approximately twice those of L3i and L5i, suggesting that L4o was the main contributor to the peaks in yaw torque.

In P. clarkii, intermediate values of phase lags were found in L4o only (Fig. 5B). In addition, as for the ATSP curve (Fig. 4B), the curve for L4o stands out from the curves of all other legs, having the highest correlation coefficient and being the only one that is significantly different from zero.

The aim of this study was to test whether the mechanism underlying curve walking is the synchronous action of a centrally pre-programmed leg tripod or whether it is the action of one principal leg that produces the main body yaw torque. The hypothesis tested here rests on the results obtained in a recent study on curve walking in P. clarkii (Domenici et al., 1998) in which it was shown that the power strokes of only three legs (outer leg 4 and inner legs 2 and 5) were in phase with the inward body rotation. However, the mechanisms underlying the temporal and/or geometrical patterns during curve walking remained unclear.

Domenici et al. (1998) suggested that it is possible that only one of the three legs of the tripod contributed actively to curve walking, while the other legs may simply be approximately in phase. Our experiments here on two species of crayfish in tethered conditions show that the yaw torque oscillations produced by the animals during curve walking are related to the movement pattern of outer leg 4, suggesting that the rotational force necessary during curve walking may be the result of the force produced by one principal leg.

Our results suggest that most of the rotational force necessary during curve walking may be produced by one principal leg, outer leg 4. This is based on the observation that L4o showed the expected synchrony for producing an inward torque and that, considering all the legs, the highest ATSP occurred during the stance of L4o. Although our results are based on temporal analyses of total yaw torque rather than force measurements of single legs, the ATSP observed during the stance of L4o in A. leptodactylus (160 mN cm, see Fig. 4A) was within the range of maximum torques found by Klärner and Barnes (1986) to be applied by leg 4 of crayfish of the same species and size. Klärner and Barnes (1986) found that the maximum forces exerted by legs 4 were in the range 15–20 mN for both laterally directed and posteriorly directed forces. The maximum force vector (obliquely directed) was probably higher (20–25 mN) since it corresponded to the vectorial sum of laterally and horizontally directed forces. For an 8 cm lever arm, corresponding to the approximate distance between the dactyl position on the ground and the leg insertion on the body of a 90 g animal, as in Klärner and Barnes (1986) and in our present experiment, the maximum torque applied by L4o may therefore be in the range 160–200 mN cm, which is near the maximum ATSP observed during the present study.

The prominent role of legs 4 in turning applies to the two different species of crayfish studied here. The characteristics of leg 4 are in accord with its major role in turning. Legs 4 produce the highest force and have the largest step amplitude during walking, and their force includes both a pulling and a pushing component, which can facilitate the control of turning (Klärner and Barnes, 1986; Jamon and Clarac, 1995). Our results are consistent with data for free-walking crayfish, which showed that L4o was the leg most actively involved in the turn (1) because the value of correlation between L4o step cycle and the angular acceleration of the body axis was the highest among all legs, and (2) because the angular acceleration of the body axis was twice as high for L4o as for the other legs of the tripod suggested to produce the rotation necessary for curve walking (Domenici et al., 1998). We suggest that other arthropods may also show functional differences among legs in such a way that torque is maximized during the power stroke of one particular leg.

Recent work on turning in untethered cockroaches (Blaberus discoidalis) showed that the ground reaction forces produced by different legs differed in magnitude and direction, and that one leg produced a majority of the total force impulse responsible for changing the heading of the animal (Jindrich and Full, 1999).

In freely walking P. clarkii, the peak angular acceleration of the body axis was produced during the power strokes of legs 2i, 4o and 5i, which tended to be in phase (Domenici et al., 1998). L4o therefore operated in synchrony with two inner legs (2i and 5i), forming a tripod. In the present study, such a tripod is not as evident in P. clarkii, as shown by the observation that the phase relationships between each leg pair of the tripod in question are not always significant. The coupling between the legs of each pair was weak and differed slightly from those observed during curve walking of freely moving P. clarkii (Domenici et al., 1998). Ipsilateral phase relationships tended to increase from front to back, as in freely moving animals, although with much higher variability. Contralateral coupling was weak and significant only for legs 3 and 5. This differs from the results obtained in freely moving curve-walking crayfish (Domenici et al., 1998), in which significant contralateral relationships were found in all legs pairs.

The low values of the mean vector found in the phase relationships among legs suggest that the yaw torque produced by P. clarkii in tethered conditions is relatively independent of a particular contralateral coupling between legs. By using a protocol that requires the animals to be tethered, we have removed the need for static stability. This would allow the central mechanism, which determines turning, to be active regardless of the leg phase relationships. Thus, our protocol allows a de-coupling between the torque-generating mechanism and the inter-leg coordination. We found weak leg coordination during turning and, in spite of this, we found a clear correlation between torque pattern and the movement of one leg (L4o). This reveals that L4o is the main leg used for the rotational component of curve walking.

It is possible that, whereas during untethered curve walking, L4o maximizes its contribution to body rotation when in phase with the other legs of the tripod, during tethered curve walking its contribution may be relatively independent of its phase relationships with the other legs. This could be because, during tethered walking, the animals are fixed by their backs, thus allowing L4o to exert a force that causes a torque with the rod as the fulcrum. Such a force can be exerted independently of the phase relationship with the other legs. Regardless of these considerations on methodological differences between experiments on tethered and untethered anmials, our results emphasise the prominent role of outer leg 4 in providing the rotational component of curve walking.

Unfortunately, there are no data on freely moving A. leptodactylus engaged in curve walking to be compared with the present results. We can, however, make inter-specific comparisons with P. clarkii. The results of the torque versus standardised period plot (ATSP; Fig. 4) and of the cross-correlation function (CCF; Fig. 5) are similar in P. clarkii and in A. leptodactylus, suggesting that L4o is the leg causing the main torque oscillations in both species. The main inter-specific differences seem to be in the ‘uniqueness’ of L4o. In particular, when considering CCF analysis, L4o in P. clarkii is the only leg whose lag corresponds to a phase relationship with the torque of between 0 and 180 °, i.e. with the peak torque occurring during the power stroke of the leg. In contrast, in A. leptodactylus, there are at least three legs with such a phase lag (L4o, L3i and L5i). However, both the amplitude of the ATSP and the correlation coefficients are much smaller for L3i and L5i than for L4o. This suggests that the synchrony of the phase lags between L5i, L3i and L4o is not due to all the legs modulating the torque signal, but to L4o modulating the torque and the other legs being almost in phase with it. Legs 3i, 4o and 5i are close to an in-phase pattern in A. leptodactylus. Therefore, it is not surprising that they show a similar phase relationship with a given oscillating signal (the torque signal).

Unlike freely moving P. clarkii engaged in curve walking (Domenici et al., 1998), in which only the triplet 2i4o5i moved as a tripod, in A. leptodactylus the legs of triplet 3i4o5i and triplet 3o4i5o were synchronised. Synchrony in these two triplets is consistent with observations on straight-walking A. leptodactylus (Clarac and Barnes, 1985). Therefore, curve walking in A. leptodactylus is not related to patterns of leg coordination that differ from those for straight walking.

Phase relationship values were more stable in A. leptodactylus than in P. clarkii, as indicated by their higher mean vector lengths. In A. leptodactylus, ipsilateral phase relationships values were all between approximately 140 ° and 180 °, i.e. close to an antiphase pattern and consistent with previous observations on straight walking in this species (Clarac and Barnes, 1985). Phase relationships between contralateral pairs were weaker than those between ipsilateral pairs, and the contralateral pairs were close to an antiphase pattern (except for legs 2, whose phase relationship value was not statistically significant). This observation is in line with previous studies on straight-walking crustaceans on a treadmill (Müller and Cruse, 1991; Clarac, 1982). Clarac and Barnes (1985) have found that, although for legs 4 and 5 there is contralateral coordination showing an approximately alternating pattern, the other leg pairs show a much more variable distribution of phase relationship. These results are compatible with our observations of significant contralateral phase relationships (all close to 180 °) in contralateral leg 3, 4 and 5, with legs 3 showing the weakest coupling. Alternation of contralateral pairs and ipsilateral pairs gives rise to the approximate in-phase pattern between diagonal pairs constituting the tripods 3i4o5i and 3o4i5o. This pattern was found to be stable for A. leptodactylus loaded with a weight increase of 25–50 % (Clarac and Barnes, 1985). These authors suggest that such a pattern reflects a situation of maximum stability equivalent to the alternating tripod gait of insects.

The main difference compared with straight walking was related to the higher variability of coordination in the inner legs. This result should be related to the coordination pattern observed in split treadmill experiments in which the leading side is more stable than the trailing side (Müller and Cruse, 1991), and may indicate that the outer legs drive inter-leg coordination during the turn.

In both P. clarkii and A. leptodactylus, it should be noted that the ATSP observed during the swing of L4o is positive (i.e. in the direction of the ‘turn’), although at its lowest, and it corresponds to only half the ATSP produced during the stance of L4o. This means that other legs are also producing an inward torque. Probably most, if not all, of the crayfish legs can produce a torque in the inward direction, similar to the rotatory response observed in Drosophila melanogaster, which was accomplished by the fore, middle or hind legs when some of the other legs were amputated (Götz and Wenking, 1973). This situation is somewhat different from free turns in which the angular acceleration of the body axis (comparable with the torque of the present experiment) was negative (i.e. in the opposite direction to the turn) during the swing phase of L4o (Domenici et al., 1998). Freely moving crayfish may produce a torque in the inward direction only periodically and during the power stroke of L4o. The observation that not all legs may produce an inward rotation in freely moving conditions may be explained from a behavioural point of view. When crayfish are walking freely along a curved trajectory, they are in a closed-loop condition and may be able to accomplish the task by adding a transitory rotational component only periodically to the translatory locomotion. Such an inward rotational component is present during the in-phase power strokes of L4o, L5i and L2i, which may represent an optimal configuration for turning. However, when tethered and presented with a constant optomotor stimulus, crayfish are in an open-loop condition. Since the animals cannot compensate for the stimulus, they may continually produce a torque, even with those legs that are not preferentially used for rotation in freely moving conditions.

Any temporal (or spatial) asymmetry found in curve walking may imply that such a phenomenon is related to turning, but such a finding does not necessarily mean that turning is caused by the asymmetry in question. However, a lack of asymmetry or of any consistent trend would reveal that asymmetry cannot be involved in turning.

The torque produced by legs other than L4o may be due to asymmetries in step amplitudes (which we did not measure but were observed by Cruse and Silva Saavedra, 1996, using a similar experimental arrangement) and/or in the forces applied by each single leg, but are not caused by temporal asymmetries, since these do not show any consistent trend. Our results showed no clear-cut asymmetry in the step frequency of contralateral legs in either species. In particular, A. leptodactylus showed differences in three out of four leg pairs, but these differences were not consistent across leg pairs, because the period of the outer leg was longer in legs 2 and 3, but shorter in leg 4. The only significant difference between the periods of each pair of contralateral legs in P. clarkii was that the period of L5o was shorter than that of L5i. This is the opposite of what was found in freely moving P. clarkii (Domenici et al., 1998), in which the shorter L5i period was due to the double steps needed to maintain certain phase relationships.

The mean leg period of L4o was significantly shorter than that of L4i in A. leptodactylus and was also slightly shorter in P. clarkii, although not significantly. Leg 4o also showed a shorter period than L4i in freely moving P. clarkii walking along a curved trajectory (Domenici et al., 1998). It is interesting to note that L4o, which is possibly the leg contributing the highest power to turning, has a shorter period than L4i. We did not measure step velocity or step amplitude, and we cannot therefore know whether such a phenomenon may be related to higher step velocity, as found by Domenici et al. (1998). A higher step velocity of the outer legs could be related to turning, as suggested by Cruse and Silva Saavedra (1996).

Although no obvious trend in the asymmetries of the leg periods was found, another source of temporal asymmetry could derive from differences in the relative duration of the power strokes. This would result in asymmetric duty factors. However, our results on two species of crayfish do not show any consistent trend in duty factors. In A. leptodactylus, the duty factors are greater in the outer than in the inner legs, as hypothesized by Cruse and Silva Saavedra (1996). However, in P. clarkii, the duty factor of L4o was smaller than that of L4i, and no differences were found between the other inner and outer legs. Domenici et al. (1998) found no differences in the duty factors of the inner and outer legs of freely walking P. clarkii engaged in curve walking.

It should be noted that the relationship between asymmetries (spatial or temporal) and the torque observed in a treadmill situation are ambiguous and are to be treated with caution. In a tethered animal, turning could result from the action of one leg without any temporal or positional difference, because of the constraint of the treadmill. This is illustrated by L4o, which generates a large torque with reduced step periods.

The observation that A. leptodactylus and P. clarkii showed some differences in both step timing and inter-leg coordination may be related to postural changes resulting from the larger mass and/or the higher step frequency of A. leptodactylus compared with P. clarkii.

Some previous studies on torque fluctuations or on other kinematic measures of rotation (such as angular acceleration) in tethered arthropods suggested that the fluctuation in the turning tendency is influenced by the type or the frequency of the visual stimulation (e.g. Poggio and Reichardt, 1973; Heisenberg and Wolf, 1979; Lönnedonker, 1991). Horn and Mittag (1980) suggest that the body movements of walking flies (Calliphora erythrocephala) approaching a stationary object were of two types: one type was exploratory and visually controlled, while the other was possibly caused by the alternate and coordinated movements of the legs, although such a relationship was not experimentally established and was not therefore specified. Therefore, fluctuations in turning tendency may be related either to a sensory input or to a given frequency of the motor system. This relationship can be relatively complex, so different torque oscillation frequencies may be due to different properties of the sensory or motor systems, and possible interactions (e.g. superposition of two or more torque frequencies) cannot be excluded. Therefore, although the torque fluctuation we observed was related to an intrinsic property of the motor system (the stepping pattern of L4o), we cannot exclude the possibility that manipulation of the sensory environment may alter such a relationship.

A recent study by Kanzaki (1998) suggests that the characteristic side-to-side zigzagging path exhibited by male silkworm moths (Bombyx mori) walking while attracted by pheromones is not dependent on sensory feedback but is synchronous with the pattern of wing kinematics and, in particular, with the timing of alternation in wing retraction pattern. To explain these results, Kanzaki (1998) suggests that zigzagging locomotion in male silkworm moths depends on centrally generated instructions that are distributed to both the wings and legs in such a way as to initiate and coordinate these behaviours.

It is possible that, like the male silkworm moths, crayfish and other arthropods exhibiting characteristic zigzagging locomotion or fluctuating torque in tethered conditions may show synchronization of locomotor pattern and rotational behaviour. In crayfish, although it is apparent that L4o is not the only leg that can produce an inward torque, this leg is the main contributor to the torque fluctuations in two species of crayfish under tethered conditions, and it is also one of the main legs implicated in the inward rotation of freely moving P. clarkii engaged in curve walking (Domenici et al., 1998). At a neural control level, our findings imply that there may exist a specific, centrally generated control of outer leg 4 during curve walking in crayfish. The pattern of leg coordination found in freely moving animals (Domenici et al., 1998) suggests that, in those conditions, the output of such a control may be modulated by the phase relationships of L4o with the other legs.

We wish to thank two anonymous reviewers for useful comments on this paper. Financial support was provided by a Human Frontier Science Program long-term fellowship to P.D. and by Procope grants to M.J. and H. Cruse (PROCOPE 312/pro-gg).