Effects of walking velocity on relative phase dynamics in the trunk in human walking

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EFFECTS OF WALKING VELOCITY ON RELATIVE PHASE DYNAMICS IN THE TRUNK IN HUMAN WALKING R. E. A. van Emmerik* and R. C. Wagenaar Department of Physical Therapy, Free University Hospital, Postbus 7057,1007 MB Amsterdam, The Netherlands Abstract-The nature of coordination changes and stability features in the relative phase dynamics of the trunk were examined in seven healthy subjects, while walking velocity on a treadmill was gradually increased and decreased. Predictions from Schoner et al. (J. Theor. BfoL 142,359-391, 1990) regarding transition mechanisms in quadrupedal walking generalized to pelvis-thorax phase relations in bipedal walking, in that more continuous transitions with and without loss of stability were observed when walking velocity was manipulated as a control parameter. Relative phase changed from more in-phase (about 2Y) at lower velocities to more out-of-phase (about 1loo) at higher velocities. Stability analysis of relative phase demonstrated the existence of more than one stable coordination pattern (‘multistability’). Total ranges of motion in pelvis, thorax, and trunk, as well as stride length were larger at the decreasing velocity range as compared to the increasing velocity range, showing dependence on direction of control parameter manipulation (‘hysteresis effect’). The nature of these transitions identifies phase relations in the trunk in human walking as lower symmetry dynamics, a finding consistent with the proposed dynamics of the quadrupedal walking mode. These results suggest the existence of different coordination patterns (multistability) in the human bipedal (walking and running) in human gait.

walking Copyright

mode and question traditional P 1996 Elsevier Science Ltd.

distinctions

in only

two

modes

Keywords: Locomotion; Trunk rotation; Walking velocity; Relative phase dynamics; Transition; Stability.

1FTRODUCTlON

The richness of different locomotory patterns and transitions between these patterns in animals have been described in great detail. Changes between different coordination patterns in quadrupedal locomotion like trot, pace, and gallop are oftentimes abrupt, but different patterns can emerge at similar speeds. These gait changes have been identified with different requirements or optimal rates in metabolic costs. Fowlers’ toad, for example,

can maintain speedsafarexceeding the maximum rate of oxygen consumption by switching gait from a walking to a hopping mode (Anderson et & 1991). In bipedal locomotion, coordination patterns like the walk and run have been identified as qualitatively different modes. It is often assumed that within these modes only linear scaling can occur: studies that were focused on the changes in spatio-temporal characteristics of the step patterns of the lower extremities reported systematic linear increases in stride frequency and stride length when walking velocity was increased (e.g. Andriacchi et ul., 1977; Larsson er ul., 1980). A number of studies, however, suggest a transition in the frequency and phase relations in bipedal walking within the velocity range 0.75-1.0 m s-l (e.g. Craik et al., 1976; Van Emmerik

and Wagenaar,

1992; Wagenaar

and

Beck, 1992; Wagenaar and Van Emmerik, 1994). Craik et a!. (1976) observed an ‘abrupt’ change in the frequency Received

in jinal form

12 June

1995.

Current address (and address for reprints): Department of Exercise Science, University of Massachusetts, Amherst, MA 01003, U.S.A. 1175

relations between upper and lower extremities as a function of walking velocity; below 0.75 Hz the frequency relation was 2: 1, above 0.75 Hz the frequency relation was 1: 1. It has been suggested that this switch from a 2: 1 to 1: 1 frequency coupling might arise within the upper extremity due to period doubling in the oscillation of the forearm. Wagenaar and Van Emmerik (1994) investigated the frequency relations in upper and lower extremities at a range of different velocities during treadmill locomotion. Using the model of the simple gravitational pendulum, two coordinative modes with preferred frequency were observed one at lower walking velocities around 0.5 m s- i in which the arms were dominantly locked onto the step frequency, and one at higher walking velocities around 1.2 m se ’ in which the arms were locked onto the stride frequency. Changes in coordination patterns as a function of walking velocity have also been observed for motions of the trunk and head during locomotion (e.g. Stokes et ~1.. 1989). Head, thorax and pelvis move in coordinated fashion in order to minimize mechanical energy variations during the walking cycle. Wagenaar and Beek (1992) observed in healthy subjects systematic changes in the phase relation between transverse pelvic and thoracic rotations by gradually scaling walking velocity. This phase relation changes from a more in-phase pattern (about 25’) at low speedsto a more out-of-phase pattern (about 120°) at higher walking speeds.Applying dimensionless analysis to trunk and pelvis rotations revealed that at velocities above 0.75 m se1 there appeared an optimal coupling in the coordination between thorax and pelvis, and the pelvic rotation started to contribute to lengthening the stride. On the basis of these

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R. E. A. van Emmerikand R. C. Wagenaar

dimensionless analyses differences in the disordered coordination patterns in the trunk of stroke patients could be identified. Coordination problems in trunk rotations have also been identified as major determinants of movement pathologies in parkinsonian patients (Van Emmerik et ul., 1993). The present study elaborates on these findings by investigating the nature of these coordination changes and the st&i&y of the patterns observed. Relevant to identifying the nature of coordination changes is the distinction in synergetics (Haken, 1977; Schoner and Kelso, 1989) between order and control parameters. Order parameters identify low-dimensional qualitative states of the system dynamics, in which changes between modes can be induced by manipulating an aspecific control parameter, such as frequency or velocity. Phase relations between body segments can be considered as order parameters because of their fundamental reflection of cooperativity between components in the system. The relative phase between component oscillators (fins, hands, legs, etc.) can identify different qualitative states of the system dynamics (i.e. in-phase, out-of-phase) on which basis changes in coordination patterns can be evaluated. Control parameter changes can also lead to hysteresis:where and when abrupt jumps in coordination patterns occur is dependent on the way the control parameter is changed. For example, the speed at which a transition takes place from walking to running can be different when increasing or decreasing walking velocity. The nature of transitions in relative phase dynamics can be identified by the changes in stability in relative phase. Discontinuous transitions are characterized by abrupt jumps between different coordinative modes; these abrupt jumps can occur for very small changes in the control parameter. Continuous phase transitions are more or less smooth and can occur over a large interval of control parameter values. Critical in distinguishing these two types of transitions is the stability of the order parameter; in abrupt transitions instability occurs before the transition point. This instability can be measured by means of fluctuations (standard deviation) in relative phase or the relaxation time after a transient perturbation (see Schoner and Kelso, 1989). Schoner et al. (1990) proposed a model of quadrupedal gaits and gait transitions based on the synergetics approach. They defined symmetry as the invariance of the phase vector representing the phases of the component oscillators. Patterns like the gallop, trot, and pace are symmetrical with respect to spatial symmetry operations of left-right limb and front-hind girdle exchanges, as well as temporal inversion, and can be considered stable independent attractors of the system dynamics. A pattern like the walk, however, appears multistable under these symmetry requirements, and two alternative modes, the ‘straight’ and the ‘reverse’walk, can exist as dynamically equivalent degeneracies. These can only become stable independent states when symmetry-breaking occurs through lowering the symmetry of the dynamics underlying the different coordinative modes (Schoner et cl., 1990). Under certain symmetry conditions and paramet-

ric changes both abrupt and continuous changes in the coordination between homologous and nonhomologous limbs are predicted. This is in contrast to more symmetric bimanual hand and finger movements in which abrupt changes have been observed (Kelso, 198% Kelso et al., 1986; Scholz et al., 1987).When biomechanical asymmetries arise and the number of degrees of freedom (e.g. number of independent oscillators) is increased, as in the case of human walking, both types of transition pattern might be observed, that can be related to different stabihties in the component oscillators. This paper focuses on the identification of different coordinative modes and the nature of transitions between these modes, and examines whether the changes described in the model of Schoner et al. (1990) on quadrupedal locomotion also occur in the human bipedal walking mode. More specifically, it examines the nature of coordination changes and corresponding stability aspects between different coordination patterns in trunk rotation when walking velocity is used as a control parameter. METHODS

Subjects

Data were collected from seven healthy male subjects with a mean age of 25 years (range 22-29 yr). Apparatus

The experiment was performed on a treadmill (Woodway, Ergo esi-90 Special) with computer-controlled speed. The movements of pelvis and thorax in the transverse plane were recorded with an opto-electronic Selspot tracking system. Infrared light-emitting diodes (LEDs) were attached to two aluminum bars (‘Tmarkers’), one around the pelvis and one around the shoulder.* A special calibration frame consisting of eight LEDs was used to calibrate the two T-markers.? Spatiotemporal data of the lower extremities were recorded with uniaxial accelerometers (Coulbourn type T-45) attached to the distal parts of the fascies medialis tibiae.

*The T-markersweremade of light-weightharnesses, one for the shoulder and one for the pelvis.To each harnesswas attached a rigid rod in the shapeof a ‘T’ with two LEDs. The shoulderrod extended20 cm forwards at the level of the sternoid, and the pelvisrod 28 cm at the level of the spina iliaca

anterior-superior. For a detailed description, see Wagenaar and Beek (1992). The only differencewith the previousexperiment was that both pelvis and thorax markers were facing forwards so that the selpot-camera could see all 4 LEDs at the same time. tThe 3-D reconstruction error was within 7 mm, and instability of the LEDs, as measured by the standard deviation of stationary LEDs, was less than 0.5 mm. The calibration results yielded an angular reconstruction inaccuracy of fess than 0.8’. The vahre 0.8’ was obtained from the extremes of the field of view of the camera. Given that lens-nonlinearities affect these extremes more and the movements of the LED’s did not fill this field of view, the actual inaccuracy of the angular reconstructions would be less than 0.8’.

Effectsof walking velocityon relativephasedynamics The accelerometer signals were amplified through a transducer-coupler (Combourn A-s72-25). Procedures Before each measurement pelvic and thoracic LEDs were calibrated by means of a special frame that was adjustable along the longitudinal axis in the transverse plane. These calibration measurements allowed for later reconstruction of the real-world 3-D coordinates of the LEDs. Thoraoic and pelvic anguIar rotations were derived by means of a direct linear tran~o~ation method (Abdel-Aziz and Karara, 1971). All individuals were asked to walk barefooted on the treadmill, while the speed was gradually increased and decreased within the same trial. The velocity of the belt was increased from 0.3 to 1.3 m s r in steps of 0.1 m s- ‘, and subsequently decreased in similar steps to 0.3 m s- ’ to investigate possible hysteresis elIects in the gait pattern. Bach speed condition was sampled for 30 s at a frequency of 104 Hz. After a 15 min rest period the same procedure was repeated a second time.

Relative phase unalysis Continuous relative pIme. Stride cycles were determined from the accelerometer data based on the mument of heel contact. After no~~iza~ion the phase angles from the angular position/velocity diagrams (phase plane) for both pelvis and thorax were obtained. Angular velocities of pelvis and thorax were obtained after lowpass filtering of the angular rotation signals @utterworth 2nd order filter with a cut-off frequency of 5 Hz). Consequently, both pelvis and thorax were described by a pair of phase variables (x5,L?~ ): w

=

~.5@~~4I~s@~l,

Us(r)= r,Jt)sin[PJtJ],

Stride length, stride jkequency, and total runge of motion. After 3-D reconstruction the angular rotations

of thorax and pelvis were derived. Angular rotations of pelvis and thorax segments were obtained from the angles of these segments with respect to the horizontal in the transverse plane of motion. Stride length and stride duration were calculated from the accelerometer data of left and right legs. From the raw unfiltered time series of pelvis and thorax, total ranges of motion @IQ and Qt,,J were obtained from the distance from a maximum to a minimum within the same cycle. Trunk rotation, &, was obtained by subtracting the entire time series of the pelvis from the thorax, and then calculating the maximal difference between peaks and valleys of the resulting signal for each stride cycle. Dimensionless anulysis. Dimensionless analysis was applied to the movements of pelvis and thorax to investigate invariant strategies over a wide range of walking speeds in a similar fashion as in the earlier study by Wagenaar and Beek (1992). Dimensionless pelvis rotation was de&red as: @i,e,/arcsin(s/4h),

(1)

where the denominator stands for the leg swing, ‘s’ is the stride length and ‘M the distance from trochantor major of the femur to lateral maleolus of the foot. Dimensionless trunk rotation was defined as: arc fdl@~e~ + @& - %&V

QpeL @dl/4,

(2)

where 4 is the actually measured phase difference (for a full derivation, see Wagenaar and Beek, 1992). When the dimensionless trunk rotation approaches a value of one, there is an optimal coupling between pelvic and thoracic rotations.

(4

where x8is the position and usthe velocity variable of the specified segment s (pelvis or thorax), rs the amplitude? Ps the original signal, and ‘t’ time (e.g. time frame). From these two phase variables the phase angle was determined for thorax and pelvis. This phase angle was calculated as follows: Pa&) = arctan tvJ#x,JN.

Daru aealysis

(1%)

(51

where Pa stands for phase angle. Both thorax and pelvis phase angles were calculated in the range O--180’. To allow for a comparison across cycles,two normalization procedures on the phase angles were performed. In the first procedure the maxima and minima in the phase plane were normalized to 1 and - 1. and the other data accordingly, to eliminate amplitude effects [r$, see equations (3) and (4)] on the observed phase angles. The second normalization procedure was performed to obtain a measure of the standard deviation of relative phase across stride cycleswithin one velocity condition. Earlier investigations (e.g. Keiso et al., 1986) have used the standard deviation around the mean relative phase over constant control parameter values, assuming stationarity of relative phase. During the gait cycIe, however%relative phase might change within a cycle. In this case the standard deviation around the mean relative phase would not be a good index of stability and contaminated by these systematic changes in relative phase within a cycle.To obtain a stability measure of relative p&se we therefore calculated the standard deviation between cycles.To superimpose different cycleswithin a velocity level a procedure had to be performed in which al1 cycles were normalized to the shortest stride cycle at that speed, keeping the start and end points (foot-contact times) intact. Variability in relative phase was calculated from the standard deviation between corresponding points in the different stride cycles. The continuous relative phase measure, 4c0n, is the difference between these two norma?ized phase angles. As stability measure was taken the standard deviation of relative phase over the stride cyclesat one specific walking velocity. In addition to the relative phase over the entire stride cycle, the relative phase over the nrst and second halves of the stride cycle was calculated (approximating &,,= and &,.i, respectively).

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R. E. A. van Emmerikand R. C. Wagenaar

Discrete relative phase. Whereas continuous relative phase is a descriptor of the coordination between pelvis and thorax during the entire stride, discrete relative phase provides information about this coordination at specificpoints in the cycle.We calculated discrete relative phase at the extreme positions of the thorax and evaluated the coordination of pelvic and thoracic at those points. Discrete relative phase, ~~~~~ was defined as follows: 4dis

=

(@th(i)

-

@pe(i))/(@th(i)

-

%h(j

+ l)),

01

where @*h(i)and @pecijare the times of maximum (right rotation) or minimum (left rotation) thorax (th) and pelvis (pe) angles during a stride cycle ‘2. Statistical analysis. Statistical analyses were carried out by means of Repeated Measures Analysis of Variance with the factors Hysteresis (increasing, decreasing velocity) and Velocity (11 levels).

0.3

0.5

0.7

0.Q

I.1

1.3

I.1

0.Q 0.7

0.5

0.3

walking velocity (m/s} Fig. 1. Changesin total range of motion in trunk rotation as a function of increasingand decreasingwalking velocity.

RESUL’IS

Stride length, stride frequency,

and total range of motion

Averaged across subjects, stride length increased from about 0.6 m at a walking velocity of 0.3 ms-r to about 1.4 m at 1.3 m s-l, and this velocity effect was significant, F(lO,60) = 5.73, p x 0.0001. Stride length was systematically larger in the decreasing velocity range than in the increasing range, as shown by an almost significant hysteresis effect, F(l,6) = 5.73, p < .06, and a significant interaction of hysteresis by velocity, F(lO,60) = 4.34, p < 0.0001. Stride frequency changed, averaged across subjects, from 0.5 Hz at 0.3 ms-r to 1.0 Hz at 1.3 ms-r and this velocity effect was significant, F(lO,60) = 205.49, p K 0.0001, An almost significant main effect of hysteresis, F(l,6) = 0.59, and a significant interaction effect of hysteresis by velocity, F(iO,60) = 4.41, p < OBOE, showed lower stride frequencies at the decreasing velocity range. The total range of motion (TRM) of pelvis rotation first decreases in the interval 0.3-0.7 rns- ’ and then increasesin the interval 0.7-1.3 m s-l, and this effect was significant, F&6) = 15.37, p < 0.01. Pelvis rotation was also larger when decreasing walking velocity as compared to increasing walking velocity, and this was supported by an almost significant hysteresis effect, F(l, 6) = 6.72, p = 0.0539, and a significant interaction effect of hysteresisby velocity, F( 10,60) = 2.97, p < 0.01. The TRM in thorax rotation, in contrast, first increasesand then decreasesas a function of walking velocity, and this effect is signi~cant, F&6) = 14.83, p < 0.01. A significant hysteresis effect, F(l,6) = 11.86, p < 0.01, showed larger ranges in thorax rotation in the decreasing as compared to the increasing velocity interval. There also was a significant increase in TRM of trunk rotation, F(lO,60) = 35.93, p < 0.0001. TRM in trunk rotation was also significantly larger in the decreasing as compared to the increasing veIocity range, as shown by

a significant hysteresis effect, F(l,6) = 32.86, JJ< 0.001 (seeFig. 1). Dimensionless

analysis

Dimensionless pelvis and trunk rotations are depicted in Fig. 2. Dimensionle~ pelvis rotation [Fig. 2(a)] decreaseswith increasing walking velocity and converges to a mean value of about 0.4 at 0.8 ms-‘. From 1.0 rns-l there is a subsequent increase to a dimensionless value of about 0.5. This increase indicates the contribution of the pelvic step in lengthening the stride. Dimensionless trunk rotation [see Fig. 2(b)] converges onto a mean value of 1.0 from 0.8 m s- r onwards. Continuous

relative phase

The continuo~ relative phase between thorax and pelvis increased significantly from a phase difference of about 25’ at a walking speed of 0.3 ms-’ to a phase difference of about 1lp at a walking speed of 1.3 m s-r [F(lO, 60) = 28.18, p < 0.0001, see Fig. 31. Decreasing walking velocity resulted in a similar decrease in the relative phase (no significant hysteresis effect, p > 0.05). Co~esponding changes in stability of relative phase are also shown in Fig. 3. Averaged across subjects there appeared a significant velocity effect [F(lO, 130) = 2.68, p < 0.011 and no significant hysteresis effect @ > 0.05). Further analysis demonstrated a signiiicant first order velocity effect, F(l, 13) = 5,69, p < 0.05, which indicated a linear increase in standard deviation of relative phase with walking velocity. This averaged pattern, however, obscures differences in individual stability patterns, as shown in Fig. 4. Two different patterns of stability change can be discerned as a function of walking velocity, namely: (1) no systematic changes (subjects 1, 2, and 6); and (2) smaller standard deviations at lower (0.3-0.5 m s-r) and higher velocities (1.2-1.3 m s-r), with increased standard deviations (decreased stability) at intermediate velocities (subjects 3,4,

E&cts of walking velocity on reIativc phase dynamics

V

*

s5

El

s&

s7

-~-~~-----

i

+

?

f J 0.3

0,5

0.7

,

L

0.9

1.1

walking

0.3 @I

-i-

sl

+

s2

v

s5

l

s6 ---.-

0.5

0.7

0.9

,

, i .3

A

walking

f

1.3

velocity

i

1.1

velocity E -----..-

1.1

3

i

l"-.L-L-L--J

0.9

0,7

0.5

R-3

(m/s) A

s3

s4

$37. ---

1.1

0.9

0.7

0.5

-y

0,3

(m/s)

Fig. 2. Changes in d~~e~s~onless pelvic (a) and tm~k (b) rotation as a function of increasing and decreasiag walking velocity for all individual subjects.

walking velocity (m/s) Fig. 3. Changes in continuous relative phase between pelvis and thorax (man, in degrees) and stability in relative phase (SD, in degrees) as a function of increasing and decreasing walking velocity averaged across suejects.

1180

R. E. A. van Emmerik and R. C, Wagenaar

continuous

contmuous

rp

sl D

ln.a"

=

rp

S4 Inam

sd

m

d

s6

Fig. 4. Individual

patterns of continuous relative phase (hathched bars, mean) and stability of relative phase (dark bars, SD) for six of the seven subjects.

and 7).* In Fig. 5 these two patterns of stability change are presented in the form of superimposed relative phase time series over the entire stride cycle. The beginning and end of each cycle mark consecutive heel contacts in the right leg. *Subject 5 (not displayed here) showed very little change in relative phase as a function of increasing walking velocity, and also showed an increase in standard deviation (decrease in stability) at higher walking velocities (1.1-1.3 ms-r) as compared to lower and intermediate velocities.

No significant differences were observed in both the mean and standard deviation of relative phase between the first and second half of the stride cycle (ps > 0.05). As can be seen in Fig. 5, however, variation in the relative phase over the stride cycle could occur and this was observed for all subjects. Discrete relative phase A significant increase in discrete relative phase with walking velocity was observed both for the right side

Effectsof walking velocityon relativephasedynamics Finally, the observed stability changes in discrete relative phase [Fig. 7(a)] differ from the peak to peak variability in pelvic and thoracic rotation separately [Fig. 7(b) and (c)], which show a systematic decrease in variability with increasing walking velocity.

DISCUSSION

Fig. 5. Examplesof suprimposed normalized relative phase valuesfor all stride cyclesat three representativevelocities (upper panel: low velocity,middle panel intermediatevelocity, and bottom panel high velocity).(a) subject1 (b) subject7. rotation, F(lO,60) = 22.83, p -C0.0001 (see Fig. 6) and the left side rotation, F(lO,60) = 18.47, p < 0.0001. No significant hysteresis effects were observed (ps > 0.05). The stability of discrete relative phase (see Fig. 6) showed similar patterns as the stability in continuous relative phase measure, but some differences appeared. In subjects 1 and 2 the variability in discrete relative phase decreasesat higher walking velocities. In subjects 3,4,6, and 7 the standard deviation of the relative phase is larger at intermediate velocities as compared to lower and higher velocities. This latter pattern determines the overall shape in the mean variability plot in Fig. 7(a). In the averaged stability plot in Fig. 7(a), there is a significant increase in variability from low to intermediate velocities, followed by a decrease [second order velocity effect, F(l, 13) = 5.11, p C O.OSJ.

This paper focused on the effects of walking velocity on coordination patterns in the trunk during walking. Its main aim was to identify different patterns and the nature of change between these patterns. The results support earlier fin&gs from Wagenaar and Beck (1992), who found a systematic change between pelvis and thorax from a more in-phase pattern to a more out-of-phase pattern when increasing walking velocity. As Fig. 3 demonstrates, the nature of the changes in the relative phase between pelvis and thorax appears more or lessgradual, and does not show the abrupt changes that were observed earlier in the bimanual transition experiments (Sch6ner and Kelso, 1989). This more gradual change is consistent with the model put forward by Schtiner er LZ[. (1990) as it relates to lower symmetry groups like the walk in quadrupedal locomotion. These patterns are predicted to exhibit multistability, in which different coordination modes can easily change from one to the other. The observed changes in maximal pelvic, thoracic, and trunk rotations as a function of walking velocity are generally consistent with previous findings in the literature (e.g. Stokes et al., t989; Wagenaar and Beck, 1992). Pelvic rotation increases considerably from about 1.0 m s- ’ onwards. Total range in thoracic rotation did not change as much with walking velocity. The3e data add to the observations from Stokes er LL (1989) in which, averaged across subjects. no changes were observed in the velocity range l&2.4 m s. The present data show a decrease in the total range of thoracic rotation with increasing speeds and this is consistent with the earlier report by Wagenaar and Beek ( 1992).At the lower and intermediate speeds there were some differences, in that Wagenaar and Beek (1992) observed a systematic decrease from 0.25 to lSms-‘, while in the present study there appeared to be a slight increase in thoracic rotation up to 0.7ms-‘, followed by a systematic decrease at higher velociGes.These discrepancies could be due to differences in measuring equipment (video in the Wagenaar and Beek study vs selspot in the present study) or the different increments in walking velocity in the two studies (0.25 m s--’ vs 0,l m s .-‘). Indeed, it has been suggestedfrom synergetics (e.g. Haken, 1977) that different scaling processesof the control parameter can lead to different coordination dynamics. This difference in scaling could be in the stepsize with which the speed is increased or decreased, but could also be the duration over which a particular level of the control parameter (e.g. movement speed) is maintained. Another important finding was that stricle length, stride frequency, and TRM in pelvic. thoracic. and trunk

R. E. A. van Emmerik and R. C, Wagenaar

1182

discrete

2 % z

discrete

rp

105

rp

loa

i =i

l-2

+!

35

0

72

36

0.3

0.5

0,7

0.0

1.1

1.3

1.1

0.0

0.7

0.5

0.3

0 0.3

0.5

0.7

0.0

t 1

s2

1.3

I ,

0.9

0.7

0.5

0 3

s6 ,*0

144

1*5

72

30

0 0.3

0.6

0.7

0.0

1 ,

13

I,1

03

0.7

s 5 9.3

0.3

0.5

0,7

0,s

1.1

t.3

+ ,

O,$

9 7

0 5

0 3

Fig. 6. Individual patterns of discrete reIative phase (hathched bars, mean) and stability of relative phase (dark bars, SD) for six of the seven subjects.

rotation were not fixed at certain levels of walking velocity. In the decreasing velocity range stride length and all three rotation variables were larger as compared to similar speeds in the increasing velocity range (stride frequency was lower). These results indicate that these gait variables are not fixed at specific speed leveh, and that walking velocity is an important control parameter in the coordination of gait, These so-called ‘hysteresis’ effects

could have ~plications for rehabiIitation, in that at lower speedslarger rotations can be established (e.g. Van Emmerik et & 1993; Wagenaar and Beek, 1992). Results from dimensionless pelvis and thorax rotations were also very similar to those reported earlier by Wagenaar and Beek (1992). These analysesdemonstrated the emergence of invariant features and changes in pelvis and trunk rotations; dimensionless pelvis rotations

Effects of walking velocity on relative phase dynamics

walking

0.3

0.5

0.7

0.9

walking

velocity

I.1

I.3

1.1

velocity

(m/s)

0.9

0.7

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(m/s)

60

0,3 (C)

0.5

0.7

0.9

walking

1.1

1.3

1.1

velocity

0.9

0.7

0.5

0.3

(m/s)

Fig. 7. Average variability across subjects in (a) discrete relative phase, (b) peak to peak intervals in pelvis, and (c) peak to peak intervals in thorax as a function of walking velocity.

converged to a value between 0.4 and 0.5, indicating the contribution of the pelvic rotation to lengthening of the stride. The convergence of the dimensionless trunk rotation to a value of 1.0 confirms the model of Wagenaar and Beek (1992), and shows that optimal coupling in the trunk arises at higher walking velocities.

1in3

The relative phase analysis performed here added additional insights into the different coordination patterns and stability features. Based on both the discrete and continuous relative phase, two general classes of coordination change were observed, namely with and without loss of stability. In five of seven subjects?this loss of stability seems to spread out over difIerent values of control parameter walking velocity, indicating that there is an unstable region in the coordination patterns between 0.5 and l.Oms-r. Even in the subject in which no changes in relative phase were observed as a function of walking velocity, increasing walking velocity beyond 1.3 m se1 might have induced a coordination change. Another important finding from the present paper is that stability changes were picked up more sensitively by the discrete as compared to the continuous relative phase analysis. Although the relative phase changes appeared very similar (cf. Figs 4 and 6), only for the discrete relative phase was the loss of stability at the intermediate velocities significant over the group. The observed instability in relative phase seems to spread out over longer time scalesas compared to phase transitions in finger and hand movements !Kelso, 1984; Scholz et uL, 1987). Of course, the degree of stability of the in-phase and out-of-phase patterns at high and low velocities would have to be determined in more detail, for example through perturbation experiments. The pattern in Fig. 5(a) suggests multiple coordination patterns. as the 20’, 90 , and 12P patterns exhibit similar stability properties. The pattern in Fig. 5(b), in contrast, suggests ‘bistabihty’ with more stable coordination dynamics at low and high walking velocities. In addition, the analysis of discrete relative phase demonstrated that variability in coordinative measures (i.e. relative phase) can add valuable information regarding the changes in stability as a function of walking velocity. Whereas stability in the individual segments is continuously increasing with walking velocity (sd decreasing), analysisof their relative phase dynamics revealed a much richer pattern of stability change in the velocity range 0.3-1.3 m se ’ and the possible existence of ‘multistabiiity’ in the human walking mode. In the model of Schoner et UL (1990),a lower symmetry pattern like the walk in quadrupedal locomotion is characterized by fluctuating phase relations, whereds higher symmetry patterns (like the trot or gallop) show more fixed phase relations. These different forms of symmetry in coordination

might be examined

through

a more for-

mal examination of the coupling between two or more oscillators and the degree of mode locking that can be achieved. A model that is being used in the area of nonlinear dynamics to investigate these mod? locking features is the so-called circle map (e.g. Bak e( (I[.. 1984: Jensenera/.. 1984 Glass and Mackey, 1988). This mode)

can provide insight into the degree of frequency or phase coupling between oscillators, in which stability (periodic locking) and adaptability (phase or frequency huctuations) can be directly investigated (Beck* 1989: Van Holst, 1969 1973

R. E. A. van Emmerik and R. C. Wagenaar

1184

In this context, coordination dynamics of lower symmetry during locomotion can be made stable by introducing ‘functional asymmetries’ (seeSchoner et ul., 1990). In various movement disorders, there is possibly a decreasein the potential of imposing these functional asymmetries. This decreasecan have an effect on the stability of the different movement coordination patterns and the transitions between them. Pathological asymmetries, like in hemiplegia after stroke or Parkinsonian tremors and rigidity, could interfere with this ability to manipulate the symmetry in the system. These results demonstrate the richness of coordination patterns in human walking, and important similarities between quadrupedal and bipedal adaptations in the coordination dynamics with movement speed. Both transition mechanisms as predicted by the Schoner et tsl. model (1990) on quadrupedal locomotion exist in the human walking mode: (1) a more or lessgradual change in the relative phase between pelvis and thorax with a loss of stability at intermediate velocities; and (2) more gradual changes with multistability and without loss of pattern stability. These different transition mechanisms are consistent with earlier findings from Wagenaar and Beek (1992); in this study a variety of strategies were observed in dimensionless trunk rotation when walking velocity was increased. In another paper, Wagenaar and Van Emmerik (1994) demonstrate the existenceof similar qualitative modes based on frequency patterns in arm-leg coordination during walking. ,4cknowledgemenfs-The preparation of this paper was supported by a fellowship from the Royal Dutch Academy of sciences, awarded to R.E.A. van Emmerik, and by a grant from the Dutch Organization for Scientific Research (NWO), no. 900-565-031. We would like to th+ik A.F. Housheer and M.J. Melchers for their help in data collection and analysis and two anonymous reviewers for their helpful comments.

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