## Abstract

To study intersegmental coordination in humans performing different locomotor tasks (backward, normal, fast walking, and running), we analyzed the spatiotemporal patterns of both elevation and joint angles bilaterally in the sagittal plane. In particular, we determined the origins of the planar covariation of foot, shank, and thigh elevation angles. This planar constraint is observable in the three-dimensional space defined by these three angles and corresponds to the plane described by the three time-varying elevation angle variables over each step cycle. Previous studies showed that this relation between elevation angles constrains lower limb coordination in various experimental situations. We demonstrate here that this planar covariation mainly arises from the strong correlation between foot and shank elevation angles, with thigh angle independently contributing to the pattern of intersegmental covariation. We conclude that the planar covariation of elevation angles does not reflect central constraints, as previously suggested. An alternative approach for analyzing the patterns of coordination of both elevation and joint (hip, knee, and ankle) angles is used, based on temporal cross-correlation and phase relationships between pairs of kinematic variables. We describe the changes in the pattern of intersegmental coordination that are associated with the changes of locomotor modes and locomotor speeds. We provide some evidence for a distinct control of thigh motion and discuss the respective contributions of passive mechanical factors and of active (arising from neural control) factors to the formation and the regulation of the locomotor pattern throughout the gait cycle.

## INTRODUCTION

Locomotor activity is characterized by rhythmic and well-coordinated movements of the lower limbs. Neuronal networks in the spinal cord, the so-called central pattern generators (CPGs), have been found to generate this basic motor activity (for a review see Grillner and Wallen 1985). Descending inputs, in particular from the brain stem, were found to trigger CPG activity (Shik et al. 1966). In humans, the gait cycle is initiated by the heel (and not the toe as in quadrupedal locomotion) strike; furthermore, reciprocal burst of flexor and extensor activity is not characteristic of lower limb coordination in humans. This change with respect to quadrupedal locomotion and other changes induced by the bipedal locomotion suggest that the locomotor control in humans needs to be understood as a special case (Capaday 2002); at a general level, the specific contribution of spinal structures to the generation and the regulation of the locomotor pattern in humans is still under debate (Capaday 2002; Dietz et al. 1994; Duysens 2002; Duysens and Van de Crommert 1998; Gurfinkel et al. 1998).

### The kinematic control of locomotion

Another general question relative to how the locomotor pattern is generated and controlled concerns the nature of the coordinate frames in which the CNS may encode such multijoint movements. Although many studies used the strong correlation between neuronal activity and a given movement parameter as evidence for an “encoding” process (e.g., the correlation between neurons’ firing patterns in the motor cortex and movement direction during reaching tasks in monkeys, in Georgopoulos et al. 1986), the nature of the coordinate system in which such encoding of movement is achieved remains an open question. Several potential candidates, like joint angular coordinates, body-centered coordinates (for instance, shoulder-centered in the case of arm-reaching movements), or task-related (external) Cartesian spatial coordinates (Ajemian et al. 2000) may be used by the CNS as control variables.

Although these questions were first and are still mainly considered for arm-reaching movements, they were also addressed for postural tasks in cats (Lacquaniti and Maioli 1994a; Lacquaniti et al. 1990). In particular, Lacquaniti and Maioli (1994b) tested the sensitivity of kinetic (tangential contact forces) and kinematic (joint absolute or elevation angles) parameters to static and dynamics perturbations (inclination of the support platform). Whereas major changes were observed in kinetic parameters for different inclinations, limb joint angles were found to covary linearly in a consistent manner under normal conditions. These authors also provided evidence that planar covariation of joint angles reflects some form of neural control of limb configuration in space and might represent a solution simplifying the problem of coordinate transformations in the control of cat posture.

### Planar covariation of elevation angles during human locomotion

This planar covariation of joint and elevation angles was tested in humans during straightforward walking (Borghese et al. 1996). In contrast with real joint angles of the lower limbs (hip and ankle anatomical flexion and extension), the elevation angles of the foot, shank, and thigh (i.e., the orientation of these segments with respect to the direction of gravity) were stereotyped across subjects and described over each step cycle as a regular loop lying on a plane. This planar covariation of elevation angles was consistently observed in other experimental situations. During backward walking, it was nearly identical to that of forward locomotion (Grasso et al. 1998) despite the different electromyographic (EMG) patterns observed in these conditions. This suggested that the planar covariation reflects a form of neural control rather than biomechanical constraints. The covariation plane (CP) parameters were found to predict mechanical energy expenditure (Bianchi et al. 1998) at different walking speeds. Interestingly, in a study performed in Parkinson patients, Grasso et al. (1999) showed that CP parameters were comparable to normal values only when electrical stimulation (by chronically implanted electrodes) was applied at the basal ganglia level (internal globi pallidi). A developmental study performed on children during their first unsupported steps (Cheron et al. 2001) provided some evidence for the progressive evolution of CP parameters toward adult values a few months after the first walking steps: this evolution was proposed to reflect “a continuous update of the neural command during this period.” Other evidence that the planar covariation of elevation angles mainly arises from neural factors rather than from purely biomechanical properties was provided by Ivanenko and colleagues (2002). These authors studied treadmill walking using body weight support (BWS) apparatus and observed that the planarity of elevation angles was consistently observed across different BWS conditions, whereas both EMG patterns of hip and ankle extensors (in particular) and ground reaction forces patterns drastically changed across BWS conditions. Recently, it was shown that planar covariation of elevation angles was consistently observed during curved walking, raising the possibility that both straight-ahead and curved walking are generated by a single motor pattern (Courtine and Schieppati 2004).

The covariation plane was proposed to be an “attractor plane common to both the stance and swing phases,” emerging from the coupling of neural oscillators with each other and with limb mechanical oscillators (Lacquaniti et al. 1999, 2002). Although the origins of this planar covariation law (PCL) are debatable, the existence of such a robust kinematic pattern suggests that this law may represent an independent control by the CNS of limb geometry. The existence of such a law is convenient for many reasons that have been reviewed elsewhere (Lacquaniti et al. 1999, 2002). Lacquaniti and colleagues (2002) proposed that CPG could control patterns of limb segment motion rather than patterns of muscular activity. Importantly, an implicit assumption of the planar covariation law is that the CNS controls lower limb motion (elevation angles of foot, shank, and thigh segments) with respect to the absolute vertical dictated by gravity. Thus both the nature of control variables (external space-referred motions of the legs) and the planar constraint observed at the level of elevation angles profile characterize PCL in humans.

Here, we wanted to test the hypothesis that PCL reflects central constraints facilitating intersegmental coordination during human locomotion. To this purpose, we addressed the question of the origins of such planar covariation by testing several types of locomotor tasks (backward, normal, fast walking, and running). In particular, we wanted to understand how this planarity constraint could explain the way the pattern of intersegmental coordination of the lower limbs movements is generated and controlled. We also proposed an alternative approach for analyzing the patterns of intersegmental and interjoint coordination throughout the gait cycle and we described how these patterns were affected by changes in both locomotor mode and locomotor speed.

## METHODS

Ten healthy male subjects volunteered for participation in the experiments (no differences were observed between men and women in previous studies). Three of them repeated two, three, and four times the experimental sessions on different days and their results in these sessions were basically identical. Subjects gave their informed consent before the study. Experiments conformed to the Declaration of Helsinki and were approved by the local ethics committee. The age, height, and weight of the subjects were 25.4 ± 2.7 yr, 1.80 ± 0.07 m, and 77.0 ± 7.8 kg, respectively.

### Task

Subjects were asked to walk for about 10 m without any restriction straightforward at their self-selected “normal” (NORM) or “fast” (FAST) walking speed. Six of the subjects were also asked to walk backward (BACK) at their self-selected speed and to run (RUNN) at a “moderate” (not maximal) running speed (Fig. 1). They repeated each trial three times for each condition (4 conditions × 3 trials) so that for each subject we recorded 12 trials. No specific restriction was given in terms of arms posture, differently from most of the studies dealing with planar covariation in human locomotion where arms were folded on the chest (that was done mainly for technical reasons—to avoid masking of some markers by the arms—and no differences were reported in previous studies whether arm posture was restricted). A total of 407 and 395 steps were analyzed for the left (L) and right (R) segments, respectively: 183 (L) and 175 (R) steps, 139 and 138 steps, 51 and 51 steps, and 34 and 31 steps for the NORM, FAST, BACK, and RUNN conditions, respectively.

### Analysis

The data we recorded were analyzed rigorously using the same methodological procedure as described in the studies cited above. In particular, we replicated the analysis for deriving planar covariation parameters (Bianchi et al. 1998; Borghese et al. 1996; Grasso et al. 1998) for both elevation angles and joint angles.

### Definition of body segments

Three-dimensional (3D) positions of light-reflective markers were recorded using an optoelectronic Vicon V8 motion-capture system wired to 24 cameras at a 120-Hz sampling frequency. Markers were placed as in previous studies (see Bianchi et al. 1998). Briefly, markers were placed on the acromion (shoulder marker); the anterior superior iliac spine and the posterior iliac spine (pelvis markers); the external side knee joint; the lateral malleolus; at the top of the foot (subjects were allowed to wear shoes), between toes 2 and 3 (1 is for the big toe); and at the heel level (at the same height as toe markers). These markers were placed bilaterally and were used to define body segments. The TRUNK segment was defined as the line joining the shoulder and pelvis markers (the pelvis marker was calculated as the midpoint between the markers placed on the anterior and the posterior parts of the hip bone). The THIGH segment was defined using this pelvis marker and the knee marker. The SHANK segment was defined as the line joining the knee and ankle markers and the FOOT segment was defined as the ankle–toe markers segment. This was done for left and right sides of the body. It should here be noted that similar results were obtained using HEEL–TOE markers but only ankle–toe markers were used here, in agreement with the literature (Bianchi et al. 1998).

### Kinematic events

We used heel-strike and toe-off events for defining steps (Bianchi et al. 1998). These events were derived from the time course of heel and toe *Z*-position profiles (see Fig. 2). We considered one step as the interval separating two successive heel strikes (note that for backward walking, the step corresponded to the interval between two successive toe “strikes”). Results of both left and right steps are systematically presented.

### Calculation of elevation and joint angles

The projection of the trunk, thigh, shank, and foot segments onto the sagittal plane was performed in the following way. The sagittal plane was defined using the heading vector (provided by the velocity vector of the pelvis markers projected onto the ground) and gravity direction. Heading and gravity vectors X⃗_{t} and Z⃗_{t} defined the *X*- and *Z*-axes of a trajectory reference frame, respectively. The Yt⃗ vector corresponds to the *Y*-axis of this reference frame and was obtained by computing the cross-product of the gravity and heading vectors. Every projection of segment *S*_{i} onto the sagittal plane was performed using the formula where PS⃗_{i} represents the projected vector of the *i*th segment onto the sagittal plane and (S⃗_{i}, Y⃗_{t}) represents the scalar product.

Angles between each projected limb segment PS⃗_{i} and absolute vertical (provided by gravity) correspond to elevation angle of limb *i*. FOOT, SHANK, and THIGH elevation angles were calculated using this procedure. HIP, KNEE, and ANKLE joint angles in the sagittal plane were calculated as the angles between TRUNK and THIGH, THIGH and SHANK, and SHANK and FOOT segments, respectively.

### Covariation plane parameters

The general procedure for calculating the covariation plane parameters was the same as described in previous studies (e.g., Bianchi et al. 1998; Borghese et al. 1996; Grasso et al. 1998). Principal component analysis (PCA) was used to investigate the degree of interdependency of a set of three variables. Briefly, PCA was performed here on both elevation and joint angles (after subtraction of their mean value) for each gait cycle (step) independently. Three principal components with variabilities *V*_{1}, *V*_{2}, and *V*_{3} and eigenvectors *u*_{1}, *u*_{2}, and *u*_{3} were obtained for joint and elevation angles in the space of these angles.

In agreement with studies cited above, we calculated the “degree” of planarity or planarity index as the percentage of variance being accounted for by the first two principal components, defined as We calculated the covariation plane orientation as the angle between this plane and the thigh axis as previously performed by Bianchi et al. (1998). For discussion of the suitability of this planarity index as a measure of intervariables coupling, see appendix A.

### Relative error estimation of angle as predicted by planar covariation law

With PCL, we calculated the relative errors of the elevation angles estimation. The relative error *e*_{i} of the estimation of the *i*th angle (*i* = FOOT, SHANK, or THIGH) was defined as where ψ_{i} is for *i*th angle and ψ̃_{i} is for its estimation. See appendix b for details on how the estimation ψ̃_{i} was obtained.

### Correlation and timing pattern between pairs of angles

We calculated correlation coefficients between elevation angles of foot and shank, shank and thigh, and foot and thigh segments, respectively. The same calculation was performed for hip, knee, and ankle joint angles. A high value of correlation coefficient was observed only between foot and shank elevation angles so that we analyzed the angle between the foot–shank regression line and the covariation plane.

We analyzed the time shift between pairs of angular motion profiles as well as the similarity between these patterns. For this purpose, cross-correlation functions were calculated for each gait cycle: the maximum of correlation between a pair of time-varying variables (i.e., angles) as well as the time shift (between the two angular motion profiles) at this maximum were measured. It should be noted that the time shift can be correctly defined only for the variables with similar profiles (see Fig. 2). However, information gained using this method was preferred over the phase difference of first Fourier decomposition harmonics that was used by Bianchi et al. (1998); the latter is a suitable method when angular profiles are close to sinusoidal oscillations. Indeed, for joint angular motion (as depicted in Fig. 2), phase relationships between pairs of joint angles cannot be easily analyzed using such a method. Thus we investigated the mutual relationships between the patterns of joint angular displacements as follows (Kurz and Stergiou 2002; Schoner et al. 1990).

For every joint angle ψ_{i} we computed the phase plane by plotting the angle ψ_{i} and its time derivative ψ̇_{i} (angular velocity). The resulting curve (or trajectory) in the phase plane is used to calculate the phase variableφ_{i}, which corresponds to the angle between the positive direction along the *x*-axis and the line joining the baricenter of the trajectory to a particular point on the curve. Joint angular motion can be continuously described using the phase variable φ_{i}. For each pair of joint angles ψ_{i}, ψ_{j} with their absolute phases φ_{i}, φ_{j}, the relative phase φ_{i,j} is defined as φ_{i,j} = φ_{j} − φ_{i}.

### Magnitude of angular displacements

Angular displacement magnitude was measured as the root mean square value for both elevation and joint angles. We preferred this method rather than, for example, the difference between maximum and minimum values because it is more stable and less sensitive to artifacts.

### Statistical analysis

We performed repeated-measurements ANOVA and *t*-tests with the Statistica 5.1 software package (Statsoft) to compare results, for each calculated parameter, obtained in the different conditions, and for left and right legs, respectively. Intercondition tests were performed using the minima of number of steps recorded in one of the four conditions (usually the RUNN condition). A total of 31 left steps and 32 right steps for each of the four conditions (*n* = 244 steps) was included in the statistical comparison and correspond to steps recorded in the six subjects who performed every condition. Comparisons between left and right legs were performed on the entire number of analyzed steps (*n* = 802 steps), by independently comparing left and right limb parameters for each of the four tested conditions. For most of the comparisons, no statistically significant differences were observed when comparing left- and right-limb parameters. The few differences between left and right limbs are systematically mentioned when they exist. For each parameter, means and SDs across subjects and trials are presented.

## RESULTS

### Step parameters

Step parameters all significantly changed with locomotor task (see Fig. 1, *bottom*, for mean ± SD plots). Step speed increased by 0.29, 0.60, and 0.92 m/s respectively from BACK to NORM, NORM to FAST, and FAST to RUNN conditions [*F*(3,96) = 231.52; *P* < 0.001]. This increase in speed was realized through an increase in step length [by 0.27, 0.31, and 0.54 m, respectively, from BACK to NORM, NORM to FAST, and FAST to RUNN conditions, *F*(3,96) = 189.05; *P* < 0.001]. The step duration also changed with locomotor condition [*F*(3,96) = 96.04, *P* < 0.001] but in a different way: step duration was found not to significantly differ between NORM and BACK conditions (*P* > 0.01), although a significantly shorter duration was observed for BACK condition when *P* was fixed to 0.05 (*P* = 0.04). Thus the decrease in step speed during backward walking was mainly achieved through shortening of step length. For stance duration, statistically significant differences were observed when comparing the different conditions [*F*(3,96) = 201.66, *P* < 0.001] but here similar results for backward and normal walking were even more evident [*F*(1,32) = 0.72, *P* = 0.40]. Swing duration was found to significantly differ across conditions [*F*(3,96) = 55.436, *P* < 0.001] but there was no statistically significant difference between BACK, NORM, and FAST conditions (*P* > 0.01). When grouping these three conditions and comparing them (post hoc comparison) to the RUNN condition, we observed a significant difference between these two groups [*F*(3,96) = 70.64, *P* < 0.001]. Stance phase represented, respectively, 57.85 ± 4.99% (mean ± SD) (57.69 ± 6.25% for right limb) for BACK, 59.83 ± 4.55% (60.77 ± 4.36% for right limb) for NORM, 56.76 ± 4.55% (57.94 ± 4.10% for right limb) for FAST, and 30.30 ± 9.41% (29.79 ± 10.26% for right limb) for RUNN, with percentages referring to the whole step duration.

Taken together, all these results show that step parameters are not only determined by the locomotor speed, but they rather seem to change accordingly to the mode of locomotion.

### Elevation and joint angle profiles

Both elevation and joint angles followed a stereotyped profile across successive gait cycles and across subjects, as illustrated in Fig. 2.

##### ELEVATION ANGLES.

Comparable temporal profiles were observed for foot, shank, and thigh elevation angles, although the THIGH profile was delayed relative to FOOT and SHANK profiles; the latter remained perfectly synchronized for the different tested conditions. We observed that peaks in the FOOT and SHANK elevation angle profile were systematically associated with transitions from stance-to-swing phases for forward walking (FAST and NORM), whereas peaks in the THIGH elevation angle profile corresponded to stance-to-swing transitions during running. For the BACK condition, it was more difficult to detect such relations between elevation angles profiles and transitions from stance-to-swing phases, although slight inflections were observable for each of the FOOT, SHANK, and THIGH elevation angles profiles near the stance-to-swing transition periods.

The amplitude of these angular oscillations is presented in Fig. 3 for the different conditions. Foot elevation angle amplitude significantly and gradually increased from BACK to RUNN conditions [by around 20°, *F*(3,96) = 247.29, *P* < 0.001] as did shank angle amplitude [but here by around 7°, *F*(3,96) = 47.237, *P* < 0.0001]. A similar pattern was observed for thigh elevation angle amplitude [*F*(3,96) = 162.12, *P* < 0.001], which increased by around 6° from BACK to FAST condition but no statistically significant change was observed here between FAST and RUNN conditions [*F*(1,32) = 6.48, *P* > 0.01]. Besides, statistically significant differences were observed when comparing the amplitude of angular oscillations across foot, shank, and elevation segments for all conditions, as revealed by a post hoc comparison [*F*(2,64) = 1,497.7, *P* < 0.001].

##### JOINT ANGLES.

Joint angle profiles were also characterized by periodic variations. However, only hip angle followed a profile comparable to that of the thigh elevation angle. The differences between hip and thigh angles can be explained by small oscillations of the trunk segment (which was used to calculate hip angle) about the absolute vertical. Ankle and knee angles were characterized by out-of-phase relationships with two periods over the gait cycle that correspond to the respective durations of the stance and swing phases. Interestingly, only one period was observable for joint angles in the BACK condition. We observed that peaks in the KNEE and HIP joint angle profiles were systematically associated with transitions from stance-to-swing phases for forward locomotion. For the BACK condition, peaks in ANKLE and KNEE joint angles were associated with transitions from stance-to-swing phases.

The amplitudes of joint angular oscillations are presented in Fig. 3. The ankle joint angle amplitude increased gradually from BACK to RUNN condition, by ≤8° [*F*(3,96) = 140.97, *P* < 0.001]. Statistically significant differences were also observed for knee joint angle amplitude (an increase of around 11° from BACK to RUNN) between conditions [*F*(3,96) = 141.18, *P* < 0.001]. However, knee angle amplitude for NORM condition was significantly higher than that of the FAST condition [*F*(1,32) = 7.84, *P* < 0.01; however, only 0.57 and 0.01° differences between the amplitude of the knee angle were obtained during FAST and NORM conditions, for left and right limbs, respectively]. Hip joint angle amplitude followed the same pattern as thigh elevation angle with increasing amplitude (by up to around 6°) from BACK to FAST [*F*(3,96) = 124.43, *P* < 0.001] and slightly higher amplitude for the FAST condition compared with the RUNN condition [*F*(1,32) = 15.02, *P* < 0.01]. However, only +0.76 and −0.22° differences between amplitudes of the hip angle were obtained during FAST and RUNN conditions, for left and right limbs, respectively. The comparison of angular displacements across joints always resulted in statistically significant differences [*F*(2,64) = 472.01, *P* < 0.001].

### Planar covariation of elevation and joint angles

The planarity index was found to be close to 100% for all conditions, for both elevation and joint angles (see Fig. 4). However, the mean values for joint angles were slightly lower than those obtained with elevation angles (mean values ranged between 98.26 and 99.50 for elevation angles vs. 94.92–97.23% for joint angles; but see appendix A for comments about the use of the planarity index). The variability was also slightly greater for joint angles (SD ranged between 0.32 and 1.21% for elevation angles vs. 1.26–2.93% for joint angles).

In agreement with previous findings, we also observed that plane orientation (elevation angles only) was related to locomotor speed (see Fig. 5). A statistically significant difference was observed when comparing the different conditions [*F*(3,96) = 13.98, *P* < 0.001]. However, the plane orientation was not found to significantly differ when comparing the FAST to the RUNN condition [*F*(1,32) = 0.12, *P* > 0.05]. For joint angles, such a relation between the locomotor speed and the plane orientation was less evident. In this case, the variability was about two- to threefold greater than the variability observed for elevation angles. The plane orientation was equal to 22.79 ± 9.58° (vs. 24.16 ± 5.49 ° for elevation angles) for BACK, 11.51 ± 5.80° (vs. 16.62 ± 2.02°) for NORM, 6.26 ± 4.66° (vs. 11.50 ± 2.43°) for FAST, and 34.66 ± 13.13° (vs. 11.56 ± 4.31°) for RUNN conditions.

##### CORRELATION BETWEEN PAIRS OF ELEVATION AND JOINT ANGLES.

In linear regression on pairs of elevation or joint angles, we noticed that the degree of correlation was not equivalent across pairs of segments (see Fig. 6). For elevation angles, only foot–shank elevation segments were highly correlated (*r* > 0.95 on average) in comparison with shank–thigh and foot–thigh angles (for which *r* was always significantly <0.7, except for the BACK condition where the shank–thigh correlation coefficient was close to 0.8). The correlation between pairs of joint angles (not shown) was not observed except for ankle–knee angles where a poor correlation was obtained (*r* ranged between 0.7 and 0.74 across conditions, but it was <0.3 for the RUNN condition).

It should be noted here that when performing PCA on ankle and knee joints only, we observed that only one (of two) principal components (PCs) accounted for 86.04 ± 6.48% of the total variance for BACK, 93.62 ± 2.67% for NORM, 94.68 ± 5.40% for FAST, and 79.84 ± 8.67% for RUNN conditions, respectively. This illustrates that the percentage of variance accounted for by PCs (previously used as a measure of interdependency between different variables) is a measure that is less revealing than linear regression for studying *correlation* between two variables (see appendix A for details). We subsequently addressed the origins of the planar covariation of elevation angles by comparing the changes in both covariation plane parameters and FOOT–SHANK linear regression values in the different tested conditions.

##### LINEAR COVARIATION OF FOOT AND SHANK ELEVATION ANGLES.

Linear regression performed on foot and shank elevation angles did not just reveal that these two variables (only) were highly correlated; indeed, as depicted in Fig. 6 (*top*), it was found that the slope of the foot–shank (FS) regression line (which may be considered as a measure of the orientation of the regression line, in the plane defined by foot–shank axis in the joint space) differed significantly as a function of locomotor speed, following a comparable profile with that of the plane orientation [*F*(3,96) = 337.20, *P* < 0.001]. However, a decrease in the slope of the FS regression line was consistently observed with increasing speed (*P* < 0.001 for every comparison between pairs of conditions), differently from what was observed for plane orientation (where this parameter was found not to differ significantly between FAST and RUNN conditions). Amplitude of angular oscillations of FOOT and SHANK as well as ankle joint (Fig. 3) also followed a common profile with the slope of the regression line (Fig. 5). Amplitude of THIGH as well as knee and hip joint angles followed a different profile, which is more closely related to the plane orientation profile. The observation of a strong correlation only between foot and shank elevation angles might be explained by the lower amplitude of the ankle angular displacement (compared with knee and hip angles; see Fig. 3).

##### CONTRIBUTION OF THIGH ELEVATION ANGLE TO PLANAR COVARIATION.

In calculating the angle between the FS regression line and the covariation plane, we found that this angle was nearly constant [*F*(3,96) = 1.2055, *P* = 0.31] across conditions and close to 1.2 ° (see Fig. 5), meaning that the covariation plane nearly passes through the regression line. Given that plane orientation is calculated with respect to the thigh axis, an angle of 90° would mean that the thigh elevation angle is the only contributor to PCL, whereas an angle of 0° would mean that the thigh elevation angle does not contribute to PCL at all. However, we noted that the plane orientation parameter is always small (about 10 to 25° for left leg and 10 to 20° for right leg) and the magnitude of THIGH angular motion is always significantly less than that of foot and shank angles. Taken together, these observations show that thigh angle is not essential for planar covariation (and that the planar covariation law arises mainly from strong foot–shank correlation). The following analyses, based on “PCL-based angle reestimation,” will provide additional evidence for the independent contribution of THIGH segment to the pattern of intersegmental covariation.

### Estimating THIGH using the “planar covariation law”

As illustrated in appendix B, if PCL holds (i.e., if it is considered as an essential constraint on the pattern of intersegmental coordination), it is possible to estimate one elevation angle if the other two are known. Indeed, PCL stipulates elevation angles are constrained to the covariation plane: thus patterns of two elevation angles determine that of the third elevation angle (of course with some error because PCL does not hold to 100%).

Such estimations were quantified for every angle and were found to be <30% for FOOT and SHANK parameters. Relative errors for these parameters were equal to 23.25 ± 9.18% for the FOOT segment (18.38 ± 5.29% for SHANK) for BACK, 16.71 ± 4.08% (14.84 ± 3.50%) for NORM, 16.75 ± 5.40% (15.15 ± 4.73%) for FAST, and 28.03 ± 9.40% (29.84 ± 10.69%) for RUNN conditions, respectively.

In contrast, it was not possible to estimate THIGH angle with acceptable error, both in terms of error magnitude and variability of the estimation: error for THIGH, using the same procedure, was equal to 217.11 ± 375.90% for BACK, 67.33 ± 22.50% for NORM, 126.32 ± 162.30% for FAST, and 287.30 ± 294.51% for RUNN conditions, respectively (comparable results were obtained for contralateral side). As illustrated in Fig. 7, even for the NORM condition where THIGH estimation error is the lowest (with respect to other conditions), reestimated angle profiles differ distinctly from the actual ones. At the same time, even for the greatest FOOT and SHANK estimation errors the estimated angles profiles are comparable to the real ones. Large THIGH reestimation errors coupled with a good estimation of FOOT and SHANK angles proves that the THIGH contribution to the pattern of covariation is inconsequential, with PCL mainly arising from high FOOT–SHANK correlation.

### Replacing THIGH with arbitrary sinusoidal function

For clarity, the thigh elevation angle is replaced with a cosine function in Fig. 7 (for all steps and all subjects of the condition NORM): planar covariation of FOOT, SHANK, and this cosine function still remain. The choice of the cosine function is not determinant because comparable plots can be obtained using sine function or even white noise or any arbitrary function (under the condition that the magnitude of the function must match that of the recorded variable). The same procedure repeated with either foot or shank results in completely different observations (no planar covariation plane, not shown), demonstrating that planar covariation of elevation angles mainly arises from the strong linear covariation between FOOT and SHANK angles.

As noticed above, the elevation angles profiles are not similarly synchronized, raising the possibility that phase shifts between the set of elevation and joint angles explain the unequal correlation between pairs of adjacent segments. Thus results from cross-correlation analysis are now presented to further examine the spatiotemporal pattern of elevation angles.

### Pattern of elevation and joint angles

##### TIMING PATTERN OF ELEVATION ANGLES.

Correlation coefficients between time-shifted elevation angles were close to 1.0 for all conditions and elevation angles (see Fig. 6). Time lags between FOOT and SHANK were nearly equal to 0, confirming the high synchronicity between these variables (Fig. 6). Time lags between elevation of foot and shank segments on one hand, and thigh elevation on the other hand, were found to significantly differ [*F*(1,32) = 1,009.08; *P* < 0.001], revealing that the thigh segment is first “elevated” and is followed (100 to 160 ms later) by synchronous foot and shank elevations. No statistically significant difference was observed for shank–thigh and foot–thigh time lags, respectively [*F*(1,32) = 5.64; *P* > 0.01]. Thus foot–shank covarying angles were found to be strongly synchronized, whereas thigh segment motion systematically occurs before motion of foot and shank segments. Interestingly, despite significant changes in locomotor speed, the time lags were of comparable magnitudes for BACK, FAST, and RUNN conditions (although statistical significance was observed when comparing the three former conditions), and close to 110 ms on average and around 60 ms less than the NORM condition. It should be noted here that for the BACK condition, signs of time delays are reversed to facilitate comparison with other conditions. Surprisingly, the shortest time lags were observed for FAST conditions but not RUNN conditions, time lags for the BACK condition also being shorter than those obtained for RUNN conditions. However, testing higher running speeds could likely result in shorter time lags between foot–shank limbs elevations on one hand and that of the thigh segment on the other hand. Nevertheless, the locomotor speeds tested here for RUNN are about 0.95 m/s higher than those recorded for the FAST walking, on average.

##### PHASE RELATIONSHIPS BETWEEN JOINT ANGLES.

Because locomotor activity is a cyclic process, we verified that all trajectories (for all conditions) in the phase space converged to a single limit cycle across consecutive steps. The relative phase curves, calculated from these trajectories, showed a very reproducible profile across steps for all conditions (although more variability was observed in the BACK condition; see Fig. 8). This reproducibility of the relative phase profile was systematically observed for all steps of the different subjects, although more intersubject variability was observed for the BACK condition (but as illustrated in Fig. 8, reproducibility across steps was observed for all subjects).

Although the relative phase was found to be maintained around a constant value for different walking speeds and shifted to another value for running (Diedrich and Warren 1995), we found, for most of the analyzed joint angles, that relative phase significantly varied during the different phases of the step cycle for a single locomotor speed. More interesting was the observation that phase profiles were very reproducible across steps and were marked by progressive 180° shifts that corresponded to an alternation between periods of coextension (or coflexion) and periods of antagonistic activity of two joints (extension of one joint and flexion of the other joint), rather than to a transition between two stable patterns (Diedrich and Warren 1995). Thus from these phase plots it is possible to differentiate the distribution of synergistic/antagonistic periods during the locomotor cycle and to point out some differences in the way interjoint coupling is modulated in the different conditions.

The main difference in terms of relative phase between two joints between forward and backward locomotion was observed at the level of ankle–knee angular motions, where the relative phase profile was reversed for backward walking. In-phase relationships between knee and hip joint angles were observed during the stance phase (coextension of knee and hip joints) of NORM and FAST walking and out-of-phase relationships were observed in most of the swing phase (while the thigh segment is elevated—hip flexion—the knee is extending until the next heel strike). This pattern of hip–knee coordination was reversed for BACK walking but here, rather than hip-knee out-of-phase relationships during the stance phase, the knee joint was stabilized at a fixed value while the hip is flexed, propelling the body in the backward direction. Interestingly, the hip–knee coordination pattern was reversed in the RUNN condition (out-of-phase relationships during the stance phase and in-phase relationships during most of the swing phase).

Ankle–hip relative phase progressively shifted from out-of-phase relationships (around −180°) during the stance phase to in-phase relationships during the swing phase. During the stance phase, the hip is extending while the ankle is flexing, although at the end of stance, the ankle is extending while the hip is flexing. This was true for forward locomotion and revealed a synergistic activity where shank and foot segments are elevated while hip flexion moved the thigh segment forward; this pattern was reversed for backward locomotion but no specific ankle movement was observed at the stance-to-swing transition, where hip flexion associated with slight knee extension, propelled the body backward. During the swing phase, hip flexion is associated with ankle flexion, but here the ankle is maintained around a fixed value (a reversed pattern is observed for the BACK condition). However, in the second part of the swing phase during RUNN, the ankle and hip shifted back to out-of-phase relationships mainly through a hip extension period that was not observed for NORM and FAST walking.

In general, as mentioned above, the relative phase between pairs of joint angular displacements within the gait cycle considerably varied, in particular at the swing-to-stance and stance-to-swing transitions. Even for a single gait period (stance or swing), shifts between two coordination modes can be observed (see, for instance, ankle–hip relative phase in the second part of the RUNN swing phase). However, as was observed for elevation angles, highly reproducible patterns of interjoint coordination patterns were observed across steps. Phase relationships for specific joints were modulated with changes in locomotor mode (phase patterns obtained for all pairs of joints were very similar for NORM and FAST walking conditions). These results are reminiscent of the changes in the coupling of elevation angles presented earlier, where only a few parameters (like the time lag between thigh and foot–shank elevation angles and amplitude of angular motion) were modified in the different conditions.

## DISCUSSION

Intersegmental coordination was investigated during different locomotor tasks in humans (backward, forward walking, at normal and fast speeds, and running). We first focused on the origins of planar covariation of elevation angles before analyzing the effects of different modes of locomotion on the pattern of intersegmental coordination.

### Planar covariation of elevation angles

We tested the planar covariation law of both elevation and joint angles for these different conditions and obtained results similar to those of previous reports (e.g., (Bianchi et al. 1998; Borghese et al. 1996; Grasso et al. 1998). Elevation (but also joint angles, with a greater variability) angles were found to invariantly covary within a plane (Fig. 4) and the plane orientation significantly changed for different walking speeds (e.g., in Fig. 5, for running, the plane orientation was comparable with that of fast walking). This robust observation might well have represented a convenient solution for intersegmental coordination, which would have reduced the dimension of the motor system by constraining three angles to evolve along a plane throughout the gait cycle. However, a detailed analysis of the elevation segments motion showed that planar covariation of elevation angles mainly arises from a strong coupling between foot and shank segments, with thigh angle independently contributing to the pattern of intersegmental covariation. The planarity of the set of these time-varying angles as well as the plane orientation was explained by the properties of the linear regression fit between foot and shank segments (Fig. 6). Thus it is unlikely that planar covariation of elevation angles represents a specific law of intersegmental coordination reflecting some form of central constraints (Grasso et al. 1998). This conclusion appears to contradict conclusions of earlier studies (see introduction). However, our study does not challenge the existence of planar covariation of elevation angles, which was systematically observed for all of the conditions we studied. Rather, our results simply demonstrate that planar covariation of elevation angles arises from purely methodological factors: performing PCA on a set of three variables with two of them being strongly correlated will always lead to planarity of the set of the three analyzed variables.

The observation of a strong correlation only between foot and shank elevation angles might be explained by the lower amplitude of the ankle angular displacement (compared with knee and hip angles; see Fig. 3). In effect the displacements of the foot and shank segments in the sagittal plane are constrained to evolve together, thus minimizing the variations of the ankle angle, particularly in the last part of the swing phase just before heel contact. This could be part of a strategy that helps control the vertical displacement of the foot so that the toe (whose elevation reaches as little as 1 cm just before heel contact) does not hit the ground. An independent reanalysis of foot and shank movements on data previously published would help test this assertion.

### Joint versus elevation angles: nature of the control variables

As mentioned earlier (see methods), although calculating variables that relate limb orientation to gravity is an interesting approach, the term “elevation” is equivocal: indeed, changes in “elevation” angles are only partly induced by the elevation of a limb with respect to ground level; changes in this angle also correspond to forward and backward oscillations of the limbs around the absolute vertical throughout the gait cycle. Thus given that elevation angles account for upward as well as forward–backward movements of the limbs it is logical to observe that the covariation pattern of elevation angles is related to the pendulum-like mechanical exchange between potential and kinetic energy characteristic of human walking (Bianchi et al. 1998). It was found (Borghese et al. 1996) that elevation angles present a much more stereotyped pattern than joint angles for different locomotor conditions, suggesting that locomotor control is realized “in a gravity-based body-centered frame of reference” (Grasso et al. 1998). Our approach allowed us to consistently (across subjects and conditions) describe highly reproducible coordination patterns for both joint and elevation angles. Although our results concerning angular motion of segments (elevation of foot, thigh, and shank segments) were less variable, it is difficult to emphasize the proposed role of elevation angles (to the detriment of joint angles) that are considered as the principal kinematic variables under CPG control. Further examination and studies are needed to test the hypothesis that intersegmental coordination is controlled using such a gravity-based body-centered reference frame. Nevertheless, the present study indicates that the planar covariation of elevation angles does not provide any convincing evidence to confirm this hypothesis.

### Intersegmental and interjoint coordination during human locomotion

##### ELEVATION AND JOINT ANGULAR MOTION PATTERNS: DIFFERENTIATED EFFECTS OF LOCOMOTOR MODE AND LOCOMOTOR SPEED.

At a general level, our results describe changes in the locomotor pattern that are associated with either locomotor speed (from NORM to FAST walking) or locomotor mode (BACK, NORM, and RUNN). These changes are related both to amplitude and timing (or phase) pattern of both elevation and joint angles.

Amplitude of angular displacements (both elevation and joint angles) significantly and systematically increased with increasing locomotor speed (FOOT and SHANK elevation angles and ANKLE joint angle). This observation did not hold for HIP and THIGH angle (comparable results between FAST and RUNN) and KNEE angle (comparable amplitude between NORM and FAST walking). These last parameters are all related to displacement of the thigh segment.

Further evidence for a distinct control of thigh motion comes from both timing of elevation angles and phase relationships between joint angles. Indeed it was found that the thigh is first (elevated) moved, followed by foot and shank segments. Besides, changes in the time lag between thigh elevation profile on one hand, and foot and shank movements on the other hand, were found to be associated with changes in the locomotor mode (statistically significant differences between conditions) and for forward walking, the time lag was considerably reduced from the NORM to the FAST condition by about 60 ms. The functional significance of these distinct results for thigh motion pattern is not fully understood. However, some hypotheses have been put forward stating, for instance, that proprioceptive flow from the thigh segment mainly determines (or is associated with) the timing of step cycle (Capaday 2002). We observed that in conditions other than BACK, peaks in the THIGH, KNEE, and HIP profiles were indeed associated with stance-to-swing phases transitions, whereas slight inflections of all elevation angles curves as well as peaks in ANKLE and KNEE angles profiles were associated with stance-to-swing transitions. We also observed a particular (specific) role of hip flexion in the transitions from stance to swing during backward walking and in the preparation of the swing-to-stance transition during running. Recently, McVea and colleagues (2005) showed in decerebrate cats that assisting the flexion of the hip joint during swing advanced the onset of activity in ankle extensor muscles and reduced the duration of hip flexor muscle activity. They found that the hip angle at the time of onset of the flexion to extension transition was similar during assisted and unassisted steps. These findings provide evidence for a distinct role of hip position in the regulation of the temporal pattern of gait cycle, in general, and in the swing-to-stance transitions in particular.

Diedrich and Warren (1995) showed that ankle–knee and ankle–hip relative phases vary around a single value (90 and 45°, respectively) for different walking speeds, and an abrupt phase shift was observed at the transition between walking and running. This was taken as evidence for a bifurcation between two attractors, whereas phase variability was associated with instable coordination patterns. Both of these interpretations have recently been challenged (Kao et al. 2003; Li et al. 2005).

Rather than using the relative phase as an order parameter indicating the stability of the coordination pattern, we used this parameter as a coordination variable, i.e., a variable continuously describing the strength and the nature of the coordination between a pair of variables along the whole step cycle or during specific periods of the step. A first observation was that relative phase profiles were highly repetitive across steps and subjects (for BACK condition, profiles were more variable across subjects). A second observation was that the relative phase is not fixed at a specific value for a single walking speed condition (as found in Diederich and Warren 1995). Two reasons might explain this difference: in their study, these authors presented ankle–hip and ankle–knee relative phases using ankle and thigh elevation angles rather than actual ankle and hip angles (see Fig. 3 of their paper); more important, they used a point estimate of relative phase rather than continuous relative phase.

After analyzing continuous variations of relative phase, we found that phase coupling between two specific joints considerably varies during the gait cycle, either because of stance-to-swing or swing-to-stance transitions or because of the different locomotor modes. Although the inverted pendulum adequately models human walking patterns, elastic storage mechanisms characteristic of running (for a recent review see Saibene and Minetti 2003) suppose a different temporal or phasic organization of the joint angular motion pattern). Phase analysis allowed us to describe a reversal of the relative phase variations between anatomic joint angles from forward to backward walking: such a reversal was previously observed only for hip angle (Grasso et al. 1998), but not for the ankle and knee. Rather than studying absolute joint phase shifts, we used relative phases between pairs of joints; this might explain the differences with the latter study. For each locomotor condition, we described alternation between periods of “synergistic” or antagonistic activities for specific pairs of joint angles that draw a first (yet basic) picture of the phase relationships between joint angles along the gait cycle. Testing different walking cadences (e.g., during backward walking and running) would provide more insights into the reorganization of the relative phase pattern.

### Functional organization of gait cycle: active and passive contributions to the shaping of the locomotor pattern

Interestingly, although we described *continuous* changes (in phase or out of phase, or progressive phase shifts along the step cycle) at the level of joint angular oscillations (see Fig. 8), patterns of muscular activation that are associated with such joint angular motion are characterized by bursts of activity at specific periods of the step cycle only. For example, under normal walking condition, flexors and extensors of both hip and knee joints are activated during the first half of the stance phase and during the last half of swing phase where knee flexors are first activated (for a review see Capaday 2002; Hicheur et al., unpublished observations). Between these periods, near the end of the stance phase until half of the swing phase, only the hip flexors are activated. Although it is evident that biarticular muscles also partly contribute to the formation of the angular motion pattern at a given joint (such as knee joint), joint angular motion patterns are determined by both a *noncontinuous* muscular activation pattern and passive mechanical factors like the gravitational moment of the falling shank during the second part of the swing phase (see also Winter and Eng 1995). Thus the CNS, rather than continuously activating the muscles at a particular joint, is assisted by such passive mechanical factors: for example, during the first half of swing, Winter (for a review see Winter and Eng 1995) assessed to around 20% the contribution of muscular activity to the angular acceleration of the leg, passive gravitational and coupling moments contributing to around 80%. As stated by these authors, “… the important message here is not to interpret joint movements as being caused only by muscles crossing these joints. Here the CNS has learnt about these gravitational and coupling contributions and intervenes with only a 20% contribution to achieve the desired trajectory of the leg and foot.”

It would be interesting to distinguish between respective contributions of purely passive mechanical factors (for instance, see work by Ruina’s group about legged passive robots; Collins et al. 2005) and active (arising from neural control) factors to the shaping of the locomotor pattern. Discriminating between active and passive elements may help in better understanding the specific contribution of neural tuning to the modulation of intersegmental coordination patterns in different locomotor tasks such as those tested in the present study.

## APPENDIX A: THE PLANARITY INDEX IN INTERSEGMENTAL COORDINATION ANALYSIS

Although PCA provides a powerful tool for analyzing the coordination of three and more variables, there are a number of possible misinterpretations that can arise if planarity index is used as the only measure of three variables interdependency.

1)Coupling of two variables is interpreted as interdependency of three variables. Having R

_{PCL}close to 100% can lead to the conclusion that the three variables are interdependent, even when it is the case for only two of them. Consider three variables x, y, z with variances V_{x}, V_{y}, V_{z}, respectively. Let V_{y}= 1, V_{z}= 1, and x = y + ε, where ε is a random variable with variance V_{ε}≪ 1. Consider y, z, and ε independent. Principal components are approximately θ_{1}= y, θ_{2}= z, θ_{3}= ε, and the planarity index R_{PCL}= (2V_{y}+ V_{z})/(V_{x}+ V_{y}+ V_{z}) × 100% = 2/(2 + V_{ε}) × 100%. Thus it approaches 100% when V_{ε}approaches 0, whereas only two of three variables, x and y, are interdependent and the third one, z, is arbitrary.2)One variable having small variance is interpreted as interdependency of three variables. Consider x, y, z are independent having V

_{x}= 1, V_{y}= 1, and V_{z}≪ 1. In this case principal components coincide with x, y, and z and thus R_{PCL}= (V_{x}+ V_{y})/(V_{x}+ V_{y}+ V_{z}) × 100% = 2/(2 + V_{z}) × 100%, which is close to 100% when V_{z}is close to 0.3)Variance of the third principal component may not be negligible. For example, having R

_{PCL}≈ 95% could signify that the variables are constrained to the plane with only minor deviations orthogonal to the plane (roughly 5% of the greatest variable magnitude). Consider x, y, z, with V_{x}= 1, V_{y}= 1, V_{z}= 1 such that R_{PCL}= 95%. Then the variance of the third principal component θ_{3}is V_{θ3}= 0.15 and the magnitude of deviations orthogonal to the plane is ≈ 0.4, e.g., 40% of the magnitude of x, y, z and thus not negligible.

## APPENDIX B: ESTIMATION OF PCL-BASED ANGLES

If *R*_{PCL} is close to zero and if PCL holds, the relative contribution of each variable to the whole coordination pattern can be measured. Such a test can help to avoid possible misinterpretations of the result.

Assuming that three variables are strongly coordinated and that each one of them is determined by two others by virtue of the coordination pattern, this permits a comparison of the real values of variables with those estimated from PCL.

Consider the third principal component direction vector *u*_{3} = {*A*, *B*, *C*}^{T}, where *A*^{2} + *B*^{2} + *C*^{2} = 1. A perfect PCL implies that the variation along *u*_{3} is zero: *A*ψ_{1} + *B*ψ_{2} + *C*ψ_{3} = 0, where ψ_{1}, ψ_{2}, and ψ_{3} are the angle variables. From this simple formula follows where ψ̃_{1}, ψ̃_{2}, and ψ̃_{3} are the estimated values of the corresponding angles.

For each angle variable the relative estimation error *e*_{i} (*i* = 1, 2, 3) is defined as The relative estimation error can be used as a measure of the contribution of a particular variable to the whole covariation pattern. If for all variables the relative estimation error is relatively small, then it can be stated that these variables are mutually dependent. However, if estimation errors are not of the same magnitude for all variables—for instance, if the error of one of them is of significantly greater magnitude—then it can be concluded that this variable is not linearly dependent on the others.

## GRANTS

The present study was partly supported by a Human Frontiers Science Program HFSP n° RGP0054/2004 and a CNES grant. A. V. Terekhov was partly supported by Russian Foundation for Basic Research Grant 05-01-00418.

## Acknowledgments

We thank Dr. Sidney Wiener for suggestions in the text and S. Dalbera (Atopos, Paris) for technical assistance during the experiments.

## Footnotes

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- Copyright © 2006 by the American Physiological Society