Human Movement Science 28 (2009) 371–386
Contents lists available at ScienceDirect
Human Movement Science
journal homepage: www.elsevier.com/locate/humov
Steady and transient coordination structures
of walking and running
C.J.C. Lamoth a,*, A. Daffertshofer a, R. Huys b, P.J. Beek a
a
Faculty of Human Movement Sciences, Research Institute MOVE, VU University Amsterdam, van der Boechorststraat 9,
1081 BT Amsterdam, The Netherlands
b
Theoretical Neuroscience Group, Institut des Sciences du Mouvement, CNR and Université de la Méditerranée, France
a r t i c l e
i n f o
Article history:
Available online 22 November 2008
PsycINFO classification:
2300
2330
2500
Keywords:
Gait transitions
Coordination patterns
Hysteresis
Critical fluctuations
Principal component analysis
a b s t r a c t
We studied multisegmental coordination and stride characteristics
in nine participants while walking and running on a treadmill. The
study’s main aim was to evaluate the coordination patterns of
walking and running and their variance as a function of locomotion
speed, with a specific focus on gait transitions and accompanying
features like hysteresis and critical fluctuations. Stride characteristics changed systematically with speed in a gait-dependent fashion, but exhibited no hysteresis. Multisegmental coordination of
walking and running was captured by four principal components,
the first two of which were present in both gaits. Locomotion speed
had subtle yet systematic differential effects on the relative phasing between the identified components in both walking and running and its variance, in particular in the immediate vicinity of
gait transitions. Unlike the stride characteristics, the identified
coordination patterns revealed clear evidence of both hysteresis
and critical fluctuations around transition points. Overall, the
results suggest that walking and running entail similar, albeit
speed- and gait-dependent, coordination structures, and that gait
transitions bear signatures of nonequilibrium phase transitions.
Application of multivariate analyses of whole-body recordings
appears crucial to detect these features in a reliable fashion.
Ó 2008 Elsevier B.V. All rights reserved.
* Corresponding author. Tel.: +31 20 5988522; fax: +31 20 5988529.
E-mail address: [email protected] (C.J.C. Lamoth).
0167-9457/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.humov.2008.10.001
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C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
1. Introduction
Human gait is characterized by two main modes of locomotion, walking and running that differ in
terms of both biomechanics and coordination. While the biomechanical differences have been studied
extensively over the years, differences in coordination have been examined only in piecemeal fashion,
as will become apparent shortly. An important biomechanical difference is that the body’s center of
mass (COM) gravitational potential and kinetic energy oscillate out-of-phase during walking and inphase during running (cf., Lee & Farley, 1998; Segers, Aerts, Lenoir, & De Clerq, 2007). However, the
usually considered defining difference is that walking entails a double (i.e., bipedal) support phase,
whereas running entails a flight phase. As a result, both gaits differ in the timing of key events in
the stride cycle: walking involves an alternating sequence of single and double support phases, while
running entails alternating sequences of support phases (during which one foot contacts the ground)
and nonsupport phases (during which both feet are off the ground). The fraction of the stride time that
a limb is in stance or with both feet off the ground is quantified by the duty factor, which is >.5 in
walking and <.5 in running (e.g., Farley & Ferris, 1998; Segers, Aerts, Lenoir, & De Clercq, 2006; Thorstensson & Roberthson, 1987). Importantly, most speed-dependent changes in body kinematics and
muscle activity scale with similar speed-dependent changes in stride length, stride duration, and
stride phase characteristics, and are dominated by amplitude effects (Cappelline, Ivanenko, Poppele,
& Lacquaniti 2006; Gazendam & Hof, 2007; Hreljac, 1995; Ivanenko, Cappellini, Dominici, Poppele,
& Lacquaniti, 2005; Milliron & Cavanagh, 1990; Thorstensson & Roberthson, 1987).
Changing locomotion speed induces a switch from walking at lower speeds to running at higher
speeds (and vice versa), which usually occurs spontaneously at a certain critical but individually defined speed (Hreljac, 1993; Nilsson & Thorstensson, 1987). While empirical and theoretical studies on
animal locomotion abound (Alexander, 2002; Collins & Stewart, 1993; Golubitsky, Stewart, Buono, &
Collins, 1999; Haken, 1996; Pinto & Golubitsky, 2006; Schöner, Jiang, & Kelso, 1990), it remains largely
unclear how humans switch gaits, although many empirical studies have described speed-dependent
(local) kinematic changes in walking and running. For instance, the walk-to-run transition appears to
be preceded by several specific events in the lower limb kinematics and/or stride parameters (Raynor,
Yi, Abernethy, & Jong, 2002), including marked increases in stride duration variability (Brisswalter &
Mottet, 1996) and peak ground reaction force, changes in peak ankle angular acceleration, and decreases in step length (Hreljac, Imamura, Escamilla, & Edwards, 2007a, 2007b; Li & Hamill, 2002;
Segers, Lenoir, Aerts, & De Clercq, 2007; Segers et al., 2006). In addition, several authors have
compared the critical speed of the walk-to-run and the run-to-walk transition, as differences therein
are indicative of hysteresis, a hallmark feature of nonequilibrium phase transitions. However, while
some studies reported distinct hysteresis effects in that critical transition speeds and body kinematics
were found to depend on the direction of speed change (Diedrich & Warren, 1995; Hreljac et al.,
2007a,b; Li, 2000; Raynor et al., 2002; Turvey, Holt, LaFlandra, & Fonseca, 1999), such effects were
not observed in other studies (Segers et al., 2006).
The study of speed-induced switches between coordination patterns has thus far been restricted to
end-effector (i.e., intra- and inter-leg) coordination. For instance, Li and colleagues (Li, Haddad, &
Hamill, 2005; Li, van den Bogert, Caldwell, van Emmerik, & Hamill, 1999) showed marked similarities
in thigh–lower leg coordination between walking and running, and comparable thigh, lower leg, and
knee evolutions throughout the gait cycle. Moreover, findings of increased intersegmental variability
preceding the walk-to-run transition (Diedrich & Warren, 1995) – also known as critical fluctuations,
another profound characteristic of nonequilibrium phase transitions (cf. Gilmore, 1981; Kelso, Scholz,
& Schöner, 1986) – did not generalize to all lower extremity intra- and inter-segmental couplings (Kao,
Ringenbach, & Martin, 2003; Li et al., 2005; Seay, Haddad, van Emmerik, & Hamill, 2006; Segers et al.,
2006).
Thus, although several aspects of gait switches have been studied, few firm and incisive insights
have been obtained about the nature of gait coordination changes. The interpretation of gait switches
in terms of phase transitions appears troubled by contradictory evidence pertaining to a limited selection of gait kinematics, which may well explain this unsatisfactory conclusion. Moreover, due to the
interrelated nature of gait variables (e.g., step length and step frequency) it is to be expected that char-
C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
373
acteristics of gait transitions are not restricted to local changes in kinematics but pertain to more global patterns of multisegmental coordination. For instance, in a number of studies Jordan and colleagues (Jordan, Challis, & Newell, 2006; Jordan, Challis, & Newell, 2007) showed that during both
walking and running, changes in the structure of variability in stride time, do not occur in isolation
but are more generic and exist in a range of kinematic and kinetic gait variables.
The characterization of coordination patterns (or structures) in multisegmental data of walking and
running requires the recording of full-body kinematics and the subsequent application of a pattern
recognition technique like principal component analysis (PCA). Such an unbiased, statistical approach
can be considered more generic as it does not rely on an a priori, arbitrary selection of coordinative
features but instead addresses the entire kinematic gait pattern and enables the identification of gait
defining (e.g., walk-to-run transition and run-to-walk) coordinative features. Previous studies of multisegmental coordination during walking using PCA revealed that the first four components are sufficient to describe the walking pattern’s main features – two that account for the fundamental walking
frequency and two that represent its second harmonic (Daffertshofer, Lamoth, Meijer, & Beek, 2004;
Troje, 2002). To date, however, PCA has neither been used to investigate whole-body coordination differences between running and walking nor transient changes therein around gait transitions. Such an
analysis is not only crucial to reveal how gait coordination differs between walking and running, but
also to establish whether gait transitions constitute in fact nonequilibrium phase transitions. By combining PCA across a range of gait speeds with more conventional stride parameter analyses, the relation between gait coordination and stride characteristics can be elucidated.
Similarities in muscle activation patterns in walking and running in humans led to the suggestion
that the same set of neural oscillators (central pattern generators) underwrite the production of both
gaits. Apart from amplitude effects, five basic temporal activation components accounted for the timing of 32 limb and trunk muscle activations recorded at different locomotion speeds (Ivanenko, Poppele, & Lacquaniti, 2004). No differences in muscle activation patterns were found between walking and
running (Cappelline et al., 2006). Each of the five components was loaded on similar sets of leg muscles in both gaits but on different sets of upper trunk and shoulder muscles. Likewise, lower extremity
muscle activation patterns were found to be rather similar for walking and running, albeit in a speeddependent manner (Gazendam & Hof, 2007; Hof, Elzinga, Grimmius, & Halbertsma, 2002). Walking
and running appeared to be mainly differentiated by earlier activation of the calf muscles during running and changes in muscle activity amplitude. Thus, despite the clear differences between walking
and running in terms of amplitude and timing, rhythmic tasks like locomotion have been suggested
to share the same central pattern generators, while a reorganization of neural activity may give rise
to differences between both types of locomotion (Zehr et al., 2007).
Based on those findings, we expected the whole-body kinematics during walking and running to be
characterized by rather similar coordination patterns and to find only subtle differences in the precise
definition and composition of components and the amount of variance accounted for by those components in the data. In view of the reported differences in stride parameters for the walk-to-run and
run-to-walk transition, we further expected to find differential effects within those modes and their
variability during and around gait transitions. To test these expectations, we examined the coordination patterns of walking and running across a range of speeds while eliminating all other known speed
effects like COM and amplitude changes.
2. Methods
2.1. Participants
Data were collected from nine healthy male participants who were recreational runners that engaged regularly in sport activities (e.g., distance running, squash, volleyball, and cycling). Participants
had no diseases that may have affected gait. Approval by the ethics committee of the Faculty of Human Movement Sciences, VU University Amsterdam, and written informed consent were obtained before the experiment was conducted. Participants had a mean age of 30.6 years (SD = 7.3), mean length
of 1.82 m (SD = 6.3 cm), and mean weight of 77 kg (SD = 6.2 kg).
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2.2. Procedure
The experiment was performed on a treadmill (Biostar Giant Biometrics, Almere, The Netherlands). After a brief familiarization and warm-up period, participants were asked to move at 10 incrementally increasing and subsequently decreasing speed levels (five walking speeds and five running
speeds), such that each speed level except the highest was performed twice. As each speed level lasted
60 s, total trial duration was 19 min (10 min of walking and 9 min of running without breaks). IndividðwalkÞ
ual comfortable walking speed mcomf was determined by increasing and decreasing the belt speed in
small increments with the participant indicating the most comfortable walking speed. Thereafter,
ðwalkÞ
treadmill speed was increased (0.1 km/h increments) and maximal walking speed mmax was defined
as the speed just before the participant switched from walking to running. Maximal running speed
mðrunÞ
max was defined as 80% of the fastest speed that a participant could sustain for 60 s. Finally, speed
levels for the experimental protocol were normalized with respect to the so-defined maximal walking
ðwalkÞ
ðwalkÞ
and running speeds: the initial walking speed was set as mstart ¼ 2mcomf mmax , and speed increments
ðwalkÞ
ðwalkÞ
ðrunÞ
ðwalkÞ
were DmðwalkÞ ¼ mmax mcomf =2 for walking and DmðrunÞ ¼ mmax mmax =5 for running. ParticiTM
pants were instructed not to resist switching gait pattern if they felt inclined to do so at a particular
speed.
2.3. Data recording
While walking or running on the treadmill, participants’ whole-body movements were recorded
with an active 3D movement registration system (Optotrak 3020, Northern Digital , Ontario, Canada).
Thirteen clusters of three markers were attached with neoprene bands to specific sites on the head,
trunk, pelvis, limbs, and feet. The data were analyzed in a xyz-coordinate system with the x-axis along
the line of gait progression, the y-axis pointing sideward, and the z-axis pointing vertically upwards.
To transform marker positions to anatomically defined coordinates, a stylus based on the Optotrak
Probing System was used to indicate the 3D spatial positions of the following anatomical landmarks:
left and right acromion, posterior superior iliac spine, lateral epicondyle of the humerus, the styloid
process of the radius, greater trochanter of the femur, lateral epicondyle of the tibia, calcaneus, and
the base of the fifth metatarsal. The 16 to-be-analyzed landmarks represented trunk, pelvis, upper
arms, lower arms, upper legs, lower legs, and feet segments, respectively (see Fig. 1 for an overview).
From the trajectories of the original 13 clusters we computed the locations of anatomical landmarks
around the joints and defined segment reference frames. To align the segment axes to a global reference frame of real-world Cartesian coordinates, we conducted a reference measurement in upright
stance to obtain the required rotation and translation parameters. The subsequently recorded signals
could thus be transformed to the reference frame of the body segments and expressed in real-world
coordinates (Söderkvist & Wedin, 1993). These data were sampled at 55 Hz and exported for off-line
analyses in MatlabÒ (version 7.1, Mathworks, Natic, MA, USA).
TM
2.4. Data analysis
To identify differences in coordination patterns between walking and running, PCA was applied to
48 time series representing the xyz-Cartesian coordinates of the 16 anatomical landmarks of the trunk
and limbs (Fig. 1a); see Daffertshofer et al. (2004), for an extended description. First, the data were
high-pass filtered at 0.5 Hz with a zero-lag 3rd-order Butterworth filter to remove slow frequency
drifts. To focus the analysis solely on the identification of co-varying patterns (i.e., coordination structures) and time-dependent changes of running and walking, amplitude and COM effects were eliminated. For this purpose, the COM trajectories were calculated and removed from the data (see
Appendix), and all time series were shifted to zero mean (dc-removal) and rescaled to unit variance.
We performed three PCAs, one covering all walking speeds (PCA(w)), one covering all running
speeds (PCA(r)), and one covering all speeds (PCA(wr)). In each PCA, all time series of all participants
were included. Concatenation of all 19 speeds and nine participants provided a data set of 48 time series of 9 19 60 s 55 Hz = 564,300 samples each, including consecutive stride cycles. Note that the
C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
375
Fig. 1. (a) Illustration of PCA and the contribution of the eigenvector coefficients. PCA was applied to the time series resulting in
eigenvalues that can be interpreted as the corresponding modes’ contribution to the overall variance. The coefficients of the
eigenvectors corresponding to the eigenvalues quantify the (signed) contribution of each segment within a mode. (b) Example
of the first principal mode or component displaying a main contribution of the x-coordinate across segments except the pelvis.
Length and width of the arrows indicate the contribution to the component. (c) The fourth principal component shows the
pelvis motion in the y- and z-direction. (d) All eigenvector coefficients; significant ones are indicated with an asterisk (see text).
(e) The cumulative significant contributions for the first four modes.
separate PCA for walking and running were deemed necessary to test if it was justified to concatenate
all data. Each PCA was examined further in terms of its composition (e.g., the relative contributions of
individual time series to relevant components) and their time-course as described below in detail.
2.5. Number of PCA eigenvalues
The data’s dimension was estimated as the number of relevant eigenvalues after applying the broken-stick test for each of the three aforementioned PCAs (Jackson, 1993; Peres-Neto, Jackson, & Somers, 2003). In brief, if the total variance in a multivariate data set is randomly spread across all
components, then the distribution of the eigenvalues kk follows a so-called broken-stick distribution,
reflecting uniformly distributed chunks. The observed eigenvalues kk are considered significant if they
exceed values generated by the corresponding broken-stick test: the proportion of total variance assoP
ciated with kk is bk ¼ 1=n nk¼1 1=k, where n is the number of encountered time series or observables
and bk is the expected proportion of variance represented by kk; if kk > bk, the component in question
is retained.
To verify the aptness of the selected components, we further plotted the eigenvalues on a log-log scale
to identify steps in eigenvalue spectra of all three PCA’s and evaluated the time evolutions, i.e., projecðwÞ
ðwÞ
ðrÞ
mðwÞ
¼ ðmk;1 ; ::::; mk;48 Þ, ~
mðrÞ
tions nk ðtÞ, along with the corresponding eigenvector’s coefficients ~
k
k ¼ ðmk;1 ; ::::;
ðrÞ
ðwrÞ
ðwrÞ
ðwrÞ
~
mk;48 Þ, and mk ¼ ðmk;1 ; ::::; mk;48 Þ for PCA(w), PCA(r), and PCA(wr), respectively. A negative sign of an
eigenvector’s coefficient of a component implies that the movement of the corresponding segment is
opposite to that of a segment associated with a positive sign (e.g., 180° phase difference as observed
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C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
for movements of the legs. Note that 90° phase differences between segment movements were reflected
in two principal components since cosine and sine time series are orthogonal in that they form a circle in
state space, whose description requires two dimensions.
2.6. PCA eigenvector coefficients
To test the significance of eigenvector coefficients we used a bootstrapped broken-stick test (PeresNeto, Jackson, & Somers, 2005). The underlying idea is to randomize the original data (here by means
of re-sampling with replacement) before determining principal components. This procedure was remðwÞ
,~
mðrÞ
, and ~
mðwrÞ
, which we subsequently analyzed using
peated 1000 times yielding distributions for ~
k
k
k
conventional statistics. Furthermore, a correspondence index (CI) was defined as the number of coefðwÞ
ðrÞ
ficients for which jmk;j j ¼ jmk;j j held, divided by the total number of eigenvector coefficients for the x-,
y- and z-coordinates separately (1 = exact correspondence between eigenvector coefficients for walk
and run).
2.7. PCA projections
If significant, differences between the components of PCA(w) and PCA(r) were analyzed in more detail by PCA(wr) through calculation of the projections’ effective values and analysis of the mean and
variance of the relative phase between those components. The contribution of each speed level to
the relevant principal components was determined by calculating the effective value over the recording time interval 0 6 t 6 T for each participant and speed level as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z t0 þDT
1
Effk ¼
n2k ðtÞdt
DT t 0
where nk ðtÞ represents the aforementioned projection of principal component k – note that time t0 is
the time of the beginning of a speed level of a given participant and time interval DT the duration of
the speed level (recall that time series of all speed levels and participants were concatenated). To compute the time-dependent relative phase between pairs of principal components the projections nk ðtÞ
were filtered with a 3rd-order Butterworth bandpass filter with cut-off values of 0.7 and 1.5 Hz, prior
to PCA(wr). By choosing this filter band we included step frequencies of the highest speed plateaus but
excluded second harmonics of the lowest speed levels, ensuring that the time series of identified pairs
only oscillated at the (fundamental) frequencies at all speed levels. For each nk ðtÞ phase and amplitude
ðHÞ
ðHÞ
were determined via the analytical signal nk ðtÞ þ ink ðtÞ ¼ Ak ðtÞ expfi/k ðtÞg, in which nk ðtÞ denotes
the Hilbert transform of nk ðtÞ (see, e.g., Rosenblum, Pikovsky, Schafer, Tass, and Kurths (2001)). The
relative phase between nk(t) and nl(t) was defined as D/kl ¼ ½/k ðtÞ /l ðtÞmod2p; here we used
{k,l} = {principal components 1–4}.
The continuous relative phases were calculated for each participant and speed level and the resulting distributions of continuous relative phases were evaluated statistically (see below). If the relative
phase distribution differed significantly between speed levels, the mean relative phase and the distribution’s uniformity (variance) of the corresponding speed levels were calculated.
To zoom in on the walk-to-run and run-to-walk transitions we selected a region of twenty strides
before transition and twenty strides after transition and subsequently computed the mean and variance of the relative phases. This was done in a time-resolved way by defining a small, sliding window
over which mean and variance were computed (window size was one stride cycle, given by the foot
contact data; windows were shifted by 20% of the stride cycle).
2.8. Stride parameters
Heel strikes and toe-offs were derived from the vertical velocities and the position profiles of the
heel and toe markers. The distance between subsequent heel strikes of the same leg defined a stride
cycle. Stride length (SL) was calculated by multiplying the stride time intervals by the treadmill speed
while correcting for the spatial separation of consecutive ipsilateral heel strikes on the treadmill.
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C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
Stride frequency (SF) was defined as the inverse of mean left and right stride time intervals. The stride
parameters’ variability was quantified by the coefficient of variation (CV). The relative change of stride
length and frequency from one speed to the next was expressed as ((speedm+1/speedm)1)100%.
In addition, we identified the first stride at which the duration of the stance phase was reduced to
50% or the first stride with an aerial phase (i.e., when both feet were off the ground). However, the
usual definition of walking and running by the absence or presence of an aerial phase and the instant
at which double support is first observed is not entirely correct (Hreljac, 1995; Hreljac et al., 2007a);
therefore, 20 strides before and after this ‘‘transition stride” were taken and stride parameters were
calculated for both the walk-to-run and run-to-walk region.
2.9. Statistical analysis
Outcome measures were means and CV’s of the stride parameters SL and SF, and PCA outputs:
eigenvalues kk, eigenvector coefficients mk;1 ; . . . ; mk;48 , and effective values Effk of PCA(w), PCA(r), and
PCA(wr). As explained, the statistical significance of the kk was determined via the broken-stick test
and that of the mk;j via the bootstrapped broken-stick test.
Main effects of speed were examined with a repeated measures analysis of variance (ANOVA) with
speed (nine levels) and hysteresis (increasing and decreasing speed (two levels) as within-participant
factors. Significant main effects were followed-up by post-hoc paired t-tests with Bonferroni adjustments for multiple comparisons. Paired t-tests were used to test for differences in stride parameters
before and after the transition step and to compare walk-to-run and run-to-walk transitions. All significant levels were set at p < .05.
The relative phase distributions at the different speed levels were evaluated with Kuiper’s twosample test (Upton & Fingleton, 1989), which estimates the discrepancy between the cumulative distribution function of two empirical distributions (see Figs. 4A–C). Bin-width for evaluation of relative
phase distributions (resolution) was set to 5°. To test for differences in mean relative phase between
pairs of projections, a non-parametric test for the equality of circular means was applied (Fisher,
1993). For the circular statistics, ps < .001 were considered significant. Notice that uniformity and variance of a phase distribution are non-circular variables that can hence be submitted to conventional,
linear statistics.
3. Results
On average, comfortable walking speed was 4.95 km/h (SD = 0.59 km/h), maximal walking speed
was 6.95 km/h (SD = 0.49 km/h), and maximum running speed was 14.3 km/h (SD = 1.25 km/h).
3.1. Coordination patterns of walking and running: PCA eigenvalues and eigenvectors
For all PCA’s (PCA(w), PCA(r), and PCA(wr)) the first four components covered 65% and 70% of the data’s spread in walking and running, respectively (see Table 1). Comparing components of PCA(w), PCA(r),
and PCA(wr) revealed that the first two principal components captured dynamical features that were
common to both walking and running.
Table 1
Percentage of variance covered by the eigenvalues kk of the first four principal components (k = 1,. . .,4) for the walking speeds,
PCA(w), the running speeds, PCA(r), and all speeds, PCA(wr).
k
PCA(w) (%)
PCA(r) (%)
PCA(wr) (%)
1
2
3
4
35.33
12.22
9.01
8.06
36.07
23.10
5.49
4.93
33.76
16.31
7.60
6.22
64.62
69.59
63.89
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C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
ðwÞ
ðrÞ
Fig. 2. Significant eigenvector coefficients ~
mk , (left panel) and ~
mk (right panel) of the first four principal components (stacked).
Table 2
~ðrÞ
mÞ for walking ð~
mðwÞ
Summary of the significant eigenvectors ð~
k Þ and running ðmk Þ and the correspondence index (CI).
k
x-coordination
y-coordination
m
# Significant ~
1
2
3
4
CI
~
mðwÞ
k
~
mkðrÞ
14
0
4
5
13
4
3
3
.94
.75
.94
.88
z-coordination
# Significant ~
v
CI
~
mðwÞ
k
~
mðrÞ
k
6
14
7
8
5
13
3
7
.75
.82
.44
.56
# Significant ~
v
CI
~
mðwÞ
k
~
mðrÞ
k
8
8
11
12
3
7
6
6
.44
.56
.44
.12
Fig. 2 shows the significant eigenvector coefficients (components 1–4) in terms of their absolute
values and cumulative components. Table 2 displays the corresponding number of significant eigenvector coefficients of each component and the amount of correspondence (CI) between the coefficients
for PCA(w) and PCA(r). All x-coordinate time series except that of the pelvis contributed similarly to the
first component, with minor contributions of the y-coordinates. Recall that the original time series
were rescaled to unit variance, implying that this differential effect was not caused by differences
in the segment amplitudes. By and large the coefficients played the same role in both gaits, regardless
of differences in speed. The second component, in contrast, predominantly reflected contributions of
the y-coordinate time series of all body segments. For both components only small differences were
found for the z-coordinates. Components 1 and 2 represented movements of the body segments that
oscillated at the stride frequency, i.e., they largely reflected three-dimensional pendulum-like oscillations with a 90° phase shift between the time evolutions of components 1 and 2 (see also Daffertshofer
et al., 2004). However, the sign of the eigenvector coefficients, representing the (significant) contribumðwÞ , was opposite to that of ~
mðrÞ and ~
mðwrÞ , implying a 180° difference
tions of the z- and y-coordinates of ~
in phase coupling within component 2, which is attributable to the speed levels.
While the first two components were common to walking and running, components 3 and 4 differed markedly between both gaits, particularly with respect to the y- and z- coordinate time series
as evidenced by CI < .6 (see Fig. 2 and Table 2). The PCA(wr) allowed for a comparison of the time evolutions of the identified global pattern, as summarized by the first four components, across speed levels. This analysis confirmed the correspondence of the first component for walking and running: Eff1
was roughly constant across all speeds (Fig. 3). In contrast, significant effects of speed were found for
Eff2, F(8, 64) = 24.34, p < .001, Eff3, F(8, 64) = 12.83, p < .001, and Eff4, F(8, 64) = 19.72, p < .001. Post-hoc
pair wise analysis confirmed that Eff2 was significantly larger (p < .002) for running than for walking
without significant differences between speeds within these gait patterns. Eff3 was significantly smal-
C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
379
Fig. 3. Means of the effective value (Eff1,...,4) of the time evolutions (n1,...,4) Error bars indicate standard errors. Each bar
represents a speed level, e.g., w1 = slowest walking speed, r5 = fastest running speed.
ler for all running speeds compared to all walking speeds (p < .005). In addition, Eff3 for the speed just
before (after) the first (last) running speed always differed significantly from that for the lower walking speeds (p < .01), but not from that for the running speeds. For Eff4 not only significant differences
between walking and running were observed, but also between speed levels within both gaits
(p < .01). For none of the four components projections significant speed by hysteresis interactions
were found.
The relative phase between components (e.g., component 1 vs. 2 = Du12) showed a consistent pattern of differential effects for both walking and running. Kuiper’s two-sample test revealed that the
relative phase distribution for all walking speeds differed significantly from that for all running speeds
(Figs. 4A–C). Furthermore, the relative phase distribution for walking and running speeds within the
walk-to-run and run-to-walk region differed significantly from that for the other speeds.1
Mean Du12 was about 150° at lower than preferred and preferred walking speeds, as opposed to
about 90° across all running speeds. This analysis thus confirmed that components 1 and 2 oscillated
at a phase shift of 90° due to the pendulum-like oscillation of the body segments (except the pelvis)
(Fig. 4A, left panel). For running the variance (phase uniformity) dropped significantly to near zero
(Fig. 4A, middle panel).
Analysis of the walk-to-run and run-to-walk region in isolation revealed ‘‘local” changes both preceding and following the transition step in accordance with the observed speed-dependent changes in
mean and variance of the relative phase. First, the phase difference preceding the transition step was
significantly greater than afterwards (Table 3) for both increasing (p = .040) and decreasing speeds
(p = .026). Second, relative phase variability was significantly higher preceding than following the
transition step (walk-to-run: t = 101.90, p < .001; run-to-walk: t = 63.90, p < .001). Third, variability
was significantly higher for speed increments towards the walk-to-run transition than for the speed
decrements towards the run-to-walk transition, indicating hysteresis (t = 5.59, p < .001). In particular,
1
For all reported significant differences in relative phase distribution, p<.001 and 2.5 > K < 10.96.
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Fig. 4A. Relative phase distributions (smoothed) of components 1 and 2 (Du12; left panel). Each gray scale represents the
relative phase distribution for a given speed level, with increasing darkness reflecting increasing speed (black represents the
maximal running speed). The distribution is centered between 185° and 180° for the slowest two walking speeds (white, light
gray) and switches to 90° at higher speeds. With the switch in phase, the speed level phase variability (uniformity) decreases
drastically to near zero (middle panel). Prior to the transition relative phase variance is amplified (right panel), especially with
increasing speed (solid line) compared to decreasing speed (dotted line), indicating hysteresis. Walk-to-run and run-to-walk
transitions are indicated by the vertical gray lines.
Fig. 4B. Relative phase distributions between components 1–3 (Du13) and 2 and 3 (Du23; left panels). Du13 shows a shift in
relative phase with increasing walking speed from 90° to 150° followed by a transition to the 300°–330° region during running.
Relative phase distribution of Du23 is centered around 60° for walking and switches to 210–270° region when running (left
panels). Both switches are associated with an increase in phase variability at the speed level before and a decrease after the
switch (middle panels). Variance at the transition step increases in the presence of a marked hysteresis effect for Du23 for
increasing (black line) and decreasing speed levels (dotted line).
within the walk region variability was significantly higher for speed increments than for speed decrements (t = 5.39, p < .001; compare solid and dotted lines in Fig. 4A, right panel, left of transition).
Relative phase distributions of Du13 revealed differences between walking and running and among
walking speeds, but not among running speeds. Mean Du13 increased during walking from about 90°
to the range of 120°–150° and shifted to the range of 300°–330° for running (Fig. 4B, left upper panel).
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Fig. 4C. Relative phase distributions of components 2–4 ((Du24) were almost uniform (left panel), as is also evident from the
large variability (phase uniformity) at all speed levels (middle panel). No significant differences were observed in variability
before and after the transition step (right panel) or between increasing (black line) and decreasing speeds (dotted line).
Table 3
Changes in mean relative phase (Du) for the phase couplings between principal components 20 stride before and 20 strides after
the transition stride for the walk-to-run transition and run-to-walk transition.
Walk-to-run – accelerating speed
Du12
Du13
Du23
Run-to-walk – decelerating speed
Before
Transition
After
Before
Transition
After
(159±40)°
(129±12)°
(92±36)°
(116±37)°
(90±25)°
(8±68)°
(82±33)°
(35±30)°
(210±45)°
(86±11)°
(4±35)°
(261±31)°
(72±24)°
(20±25)°
(32±68)°
(120±40)°
(86±23)°
(103±49)°
Speed significantly affected the variance of Du13 (F8,64 = 3.23, p = .004); variance was smaller at higher
than lower walking speeds and increased just before running to then decrease again while running
faster (Fig. 4B, right upper panel). Mean Du23 changed from 20° to 240° while its variance dropped
significantly during running (p < .001). The variance was largest at the highest walking velocity. All
segments contributed considerably to this pattern, albeit with a significant contribution of the pelvis
lateral movement (y-coordinate) and shoulder lateral and vertical movement (y- and z-coordinates)
for Du23 alone.
Detailed examination of the walk-to-run and run-to-walk regions confirmed the speed-dependent
changes in mean and variance of both phase couplings. After the transition step, mean relative phase
Du13 increased significantly in the walk-to-run region and decreased significantly in the run-to-walk
region (p < .001 for both), whereas mean relative phase Du23 decreased significantly after the transition step, again both when accelerating and decelerating (p < .01 for both). For both phase couplings
the transition step itself differed significantly from both preceding and subsequent steps (Table 3).
Relative phase variability changed in association with changes in mean relative phase. For Du23 the
variance was higher before the transition step than after (t = 32.78, p < .001), while variance was higher after the transition step from running to walking (t = 66.49, p < .001). Moreover, for both gaits the
variance increased significantly at the transition step (p < .001). As for Du12, a significant hysteresis
effect was observed for Du23 (walk-to-run vs. run-to-walk; t = 4.91, p < .001), due to a significantly
higher variance for increasing walking speeds than decreasing walking speeds (t = 4.03, p < .001).
No overall significant hysteresis effect was found for Du13, as can be appreciated from Fig. 4B right
panel.
For the relative phase distributions of Du14, Du24, and Du34 no significant overall speed effects
were observed due to large variation (see Fig. 4C for an example).
3.2. Stride parameters
Both stride parameters changed significantly with speed (see Table 4). On average, mean SL
increased from 1.24 m (SD = 0.25) at the slowest walking speed to 2.94 m (SD = 0.3) at the highest
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Table 4
Effects of speed and increasing versus decreasing speeds (i.e., hysteresis) on the stride parameters (repeated measures ANOVAs).
Variable
Speed
F(8, 64)
p
Hysteresis
F(1, 8)
p
Interaction
F(8, 128)
p
SL
Mean
CV
259.00
4.70
<.001
<.001
4.62
1.62
.06
ns
1.32
2.32
ns
.031
ST
Mean
CV
27.66
4.7
<.001
<.001
5.18
1.26
.052
ns
2.26
0.93
ns
ns
SF
Mean
CV
149.56
4.68
<.001
<.001
4.82
2.25
.06
ns
1.47
0.98
ns
ns
ns = not significant.
running speed. Similarly, SF increased from 0.7 Hz (SD = 0.13) to 1.23 Hz (SD = 0.05). Changes in walking speed, however, were realized by an increase in SL and SF, whereas faster running was accomplished predominantly by an increase in SL, and relatively constant SF. SL and SF were higher at
deceleration than acceleration, but this difference was not significant (Table 4). No significant speed
by hysteresis interactions were found.
CVs of SL and SF changed significantly with speed in similar fashion. CVs were higher for deceleration than acceleration and during the walk-to-run transition than during the run-to-walk transition.
SL decreased significantly at the highest walking speed just before running, whereas SF increased
with respect to the previous speed level (see Fig. 5). At descending speeds these shifts began one speed
level earlier and covered the running to walking speed level and the next speed level.
Detailed analysis of the walk-to-run and run-to-walk regions revealed that mean SF was significantly (p < .001) lower just before the transition step in the walk-to-run region (mean = 1.03 strides/s) than immediately afterwards (mean = 1.26 strides/s), as well as directly after the
transition step in the run-to-walk region (mean = 1.06 strides/s) than just before (mean = 1.27 strides/s). No significant differences were found between these regions for mean SL.
Fig. 5. Percentage of change in stride length and stride frequency with respect to the previous speed, e.g., w21 represents the
difference between the slowest walking speed (1) and the next to slowest walking speed (2). Note the dramatic increase in
stride frequency and decrease in stride length at the fastest walking speed (w54) prior to the first running speed (r1w5). Error
bars indicate standard errors.
C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
383
SL and SF were significantly more variable before than after the transition step in both the walk-torun and run-to-walk region (p < .05 and p < .001, respectively). For the walk-to-run region the differences in variability in SF and SL were 1.3% and 1.1%, respectively, whereas for the run-to-walk region
those differences were markedly larger, namely, 5.6% and 4.7%, respectively. Indeed, SL and SF variability of walking after running (i.e., in the run-to-walk region) was higher (p < .01) than of walking
in the walk-to-run region.
4. Discussion
In this study, we examined whole-body gait kinematics over a wide range of treadmill speeds using
PCA to determine and compare the coordination patterns of walking and running. We further analyzed
transient changes in the composition and variance of the identified coordination patterns during
walk-to-run and run-to-walk transitions to investigate if those changes exhibit hysteresis and critical
fluctuations, and thus constitute nonequilibrium phase transitions. We focused on the coordination
patterns by rescaling the signals’ amplitudes and removing COM contributions to gait kinematics
and also investigated conventional stride parameters like stride frequency and length. We sought to
elucidate the relation between gait coordination and stride parameter adjustments to speed variations
by taking individually preferred walking and running speeds and individual transition speeds into
account. This methodological strategy proved indispensable as these gait characteristics differed
greatly across participants, and enabled us to find coherent characteristics of the coordination patterns
of walking and running as well as walk-to-run and run-to-walk transitions across participants. In the
following, we discuss those steady and transient coordination features in detail.
Four principal components proved sufficient to capture the essential dynamic features, or pure
coordination structures, of walking and running in all participants. The first component reflected
the fundamental frequency of locomotion that was clearly present in all markers (albeit along different dimensions). Since these movements were present in both gaits one can conclude that all body
segments moved in a coordinated manner at this fundamental frequency, variations in the phasing between the individual time signals notwithstanding.
In spite of the correspondence in the number of components and the amount of variance covered
by the four components (65% and 70%, respectively), speed affected the timing and variance of the
coordination pattern of walking and running differently. To pinpoint those differences, the phase relations between the components were analyzed in detail. Components 1 and 2 were characterized by a
jump in mean relative phase from antiphase coordination while walking to a 90° phase difference
while running. The phase relation was relatively stable across walking and running speeds, but the
‘‘strength” of the phase coupling increased significantly from walking to running, where the phase variability was close to zero. Since the first and second components predominantly represented body
oscillations in x- and y-direction, the relative phase changes implied a ‘‘tight coupling” of these oscillations during running.
Similar changes in phase couplings were observed between components 1 and 3 and components 2
and 3. Unlike components 1 and 2, however, component 3 reflected a marked contribution of vertical
oscillations and thorax–pelvis motion. Consistent with well-documented changes in thorax–pelvis
coordination (Lamoth, Beek, & Meijer, 2002; Lamoth, Meijer, Daffertshofer, Wuisman, & Beek, 2006)
as a function of walking speed, component 3 changed systematically across walking speeds (but less
so across running speeds), resulting in systematic variations in the mean and variance of its relative
phasing with both components 1 and 2.
The finding that the coordination patterns of walking and running were quite similar in terms of
their main principal components is consistent with the suggestion that the same set of central pattern
generators (CPGs) may underwrite the production of both gaits. The spontaneous switches, however,
hint at nontrivial links (i.e., nonlinear interactions) between these CPGs (Haken, 1996), which may imply that the underlying neural circuits are (partly) the same (Golubitsky et al., 1999; Pinto & Golubitsky, 2006; Zehr et al., 2007). That is, changes in control parameters like locomotion speed do not
simply yield a gradual increase (or decrease) of a given CPG, but at a critical value prompts a transition
from one activity state to another (like the switch from, e.g., trot to hop in quadrupeds). Hence, at least
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to a degree, it appears possible to account for greater locomotion speed in both gait forms in terms of
an increasing (central) neural drive.
Without nonlinear interactions between CPGs, however, this account fails to capture the transient
changes in coordination around gait transitions, which were clearly different between walk-to-run
and run-to-walk transitions, and which reflected specific, transition-type dependent adaptations in
the identified coordination patterns (see also Choi and Bastian (2007)), which may be interpreted
as signatures of nonequilibrium phase transitions.
As outlined in the Introduction, previous gait transition studies have reported mixed results with
regard to the question whether they should be conceived of as transitions between attractors. Differences between studies might be due to differences in protocol pertaining to the time spent at a given
speed level and the increase and/or decrease of speed. However, Seay and colleagues (2006) showed
that time spent at a speed level did not (overall) affect the gait dynamics and the degree of hysteresis
appeared to be unaffected by the type of protocol (i.e., continuous versus incremental) used to determine transition speeds (Hreljac et al., 2007b). There are, however, other methodological considerations. In particular, it may be that the a priori selected variables, be they global stride
characteristics like stride length or frequency, or coordinative features like knee–ankle coordination,
are not sensitive enough to consistently detect subtle features like critical fluctuations and hysteresis,
and that a methodology such as that employed here is needed for this purpose.
Indeed, the results of the more generic multivariate analysis clearly showed several features of socalled phase transitions in the phasing between principal components (i.e., components 1 vs. 2, 1 vs. 3,
and 2 vs. 3). First, a switch in relative phase between components was found at both walk-to-run and
run-to-walk transitions. Second, both transitions were preceded by an increase in variance of the
phase coupling between the modes, which dropped to near zero after the transition. Third, hysteresis
was present in that there was a tendency to remain in the previous gait mode, which was evidenced
by smaller phase variance when decelerating compared to when accelerating as well as differences in
the corresponding mean relative phases.
Stride characteristics (stride length, stride frequency) also changed systematically with speed in a
gait-dependent fashion. That is, stride length increased and step frequency increased with increasing
speeds. A major difference between increasing walking speed and increasing running speeds was
that during walking adjustment of both stride length and stride frequency increased about equally,
whereas running faster was accomplished by adjusting stride length while stride frequency remained
relatively constant across speed levels. For the speed preceding (following) the running speed the CV
of both stride parameters increased markedly, and was significantly higher when accelerating than
when decelerating. However no significant hysteresis effects were observed for averaged speed
levels.
We determined the ‘‘transition stride” and analyzed the 20 strides before and after this stride. As
observed previously (Segers, Lenoir et al., 2007; Segers et al., 2006), the change in coordination from
walking to running and vice versa occurred within a single stride cycle, albeit that gait transitions
were preceded by changes in step characteristics several steps beforehand. Furthermore, the stride
prior to a transition resembled the previous gait pattern, whereas the stride following the transition
corresponded to the new pattern. The present study goes beyond previous studies (Hreljac et al.,
2007a; Segers, Lenoir et al., 2007; Segers et al., 2006) by showing that the observed transient changes
in stride characteristics are matched by transient changes in whole-body coordination patterns. In
fact, it may well be that the stride characteristics are ‘‘mere” epiphenomena of self-organized gait
patterns.
Appendix A
COM removal
From 16 anatomical landmarks 13 distinct body segments segj(t) were defined (pelvis and hip
markers were combined so that 13 instead of 14 segments were used matching the original cluster
markers; see Fig. 1) and added with the individual segments’ relative masses mj as weighting factors;
C.J.C. Lamoth et al. / Human Movement Science 28 (2009) 371–386
385
P
COM’s time series was hence given as COMðtÞ ¼ 13
j¼1 mj seg j ðtÞ. Both mj and segj where given via
regression equations developed by McConville and Chruchill (Dumas, Cheze, & Verriest, 2007; McConville & Churchill, 1980). Following calculation of the COM trajectories, the resulting time series were
first attached to the 48 body segment time series, and then subtracted from the entire data set using
PCA. That is, here the PCA served as a data driven filter, with the filter characteristics determined by
the COM trajectories. In detail, the 48 + 3 = 51 dimensional set yielded 51 principal components in
which three (or more) contained the 3D trajectories of the COM. To guarantee that only three modes
contained the COM and that these three modes yielded the leading principal components, we multiplied the COM time series by a large factor (1000) prior to PCA. Hence, the COM contribution to the
total variance was far larger than that of the original time series. All modes whose PCA eigenvalues
were larger than 1% were considered as part of the COM and removed from the set (see Daffertshofer
et al. (2004) for more details about removal of a mode via projection).
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