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Neuroscience Letters 438 (2008) 340–345
Contents lists available at ScienceDirect
Neuroscience Letters
journal homepage: www.elsevier.com/locate/neulet
Comparing the attractor strength of intra- and interpersonal interlimb
coordination using cross-recurrence analysis
Michael J. Richardson a,∗ , Stacy Lopresti-Goodman b , Marisa Mancini c , Bruce Kay b , R.C. Schmidt d
a
Colby College, United States
University of Connecticut, United States
c
Federal University of Minas Gerais, Brazil
d
College of the Holy Cross, United States
b
a r t i c l e
i n f o
Article history:
Received 4 April 2008
Accepted 17 April 2008
Keywords:
Interlimb coordination
Interpersonal coordination
Movement stability
Cross-recurrence analysis
a b s t r a c t
Previous research has demonstrated that intra- and interpersonal rhythmic interlimb coordination are
both constrained by the self-organizing entrainment process of coupled oscillators. Despite intra- and
interpersonal coordination exhibiting the same stable macroscopic movement patterns the variability of
the coordination is typically found to be much greater for inter- compared to intrapersonal coordination.
Researchers have assumed that this is due to the interpersonal visual-motor coupling producing a weaker
attractor dynamic than the intrapersonal neuromuscular coupling. To determine whether this assumption
is true, two experiments were conducted in which pairs of participants coordinated hand-held pendulums swung about the wrist, either intra- and interpersonally. Using the cross-recurrence statistics of
percent recurrence and maxline to independently index the level of noise and the attractor strength of the
coordination, respectively, the results confirmed that the attractor strength was significantly weaker for
inter- compared to intrapersonal coordination and that a similar magnitude of noise underlies both.
© 2008 Elsevier Ireland Ltd. All rights reserved.
Previous research has demonstrated that intra- and interpersonal rhythmic coordination are constrained by the self-organizing
entrainment processes of coupled oscillators [10,11,13] and exhibit
two stable macroscopic movement patterns (i.e., inphase [ = 0◦ ]
and antiphase [ = 180◦ ]). The variability of interpersonal coordination, however, is typically found to be much greater than
that observed for intrapersonal coordination. Researchers have
assumed that this is due to a difference in the strength of the
attractor dynamics that underlie these two forms of coordination. The interpersonal visual coupling of limbs is assumed to
result in a weaker attractor dynamics than the intrapersonal
neuromuscular coupling [2,12]. It is possible, however, that the
difference in variability might also be due to a difference in the
magnitude of noise inherent in the system. The variability of interlimb coordination, which is captured by the standard deviation
of relative phase (S.D.), is a function of both attractor strength
and noise [17]. No previous research has attempted to examine
whether the magnitude of noise inherent to intra- and interpersonal coordination systems is equivalent. Thus, the current study
used cross-recurrence quantification analysis (CRQA) to examine
∗ Corresponding author at: Colby College, Waterville, ME 04901, United States.
Tel.: +1 207 859 5582; fax: +1 207 859 5555.
E-mail address: [email protected] (M.J. Richardson).
0304-3940/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.neulet.2008.04.083
whether the difference in the variability of intra- and interpersonal
coordination is due to a difference in attractor strength or noise.
The dynamics of rhythmic interlimb coordination. For 1:1 frequency locking, the dynamic stabilities of rhythmic interlimb
coordination can be captured using the following equation of
motion for the difference in the phase angles of the left and right
limbs (relative phase, = L − R ):
˙ = ω − a sin − 2bsin 2 +
Q t
(1)
where is the rate of change of the relative phase [5], a and
b are coefficients whose magnitudes govern the strength of the
between-oscillator coupling, ω indexes the difference between
the oscillators’ inherent uncoupled frequencies [6], and t is a Gaussian white noise process that dictates a stochastic force of strength
Q [17].
The stable solutions for Eq. (1) at = 0◦ and 180◦ represent
relative phase attractors (*) that constrain the coordinated movements to inphase and antiphase, whereby the inphase attractor at
= 0◦ is stronger than the antiphase attractor at = 180◦ [5]. The
attractor strength of * for interlimb coordination can be calculated as:
d˙ =
d (2)
=∗
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M.J. Richardson et al. / Neuroscience Letters 438 (2008) 340–345
where is referred to as the near-equilibrium Lyapunov exponent and indexes the rate of return to the relative phase attractor
* after perturbation [3]. For Eq. (1), |0 | > |180 |.
The variability of interlimb coordination, which is captured by
the standard deviation of relative phase (S.D.), is a function of both
attractor strength and the magnitude (Q) of the stochastic noise
that continuously operates to perturb the movements of the system
[17]:
S.D. =
Q
2 (3)
Since S.D. is dependent upon both and Q, observed differences
in S.D. across different movement conditions could be due to differences in attractor strength or differences in the magnitude of
noise Q. Thus, in order to interpret for any given movement conditions differences in S.D., one needs to be able to measure and Q
independently.
Cross-recurrence analysis and interlimb coordination. One way to
measure and Q independently is by means of cross-recurrence
analysis [7,9–11,19]. Cross-recurrence analysis involves picturing
the dynamic similarity of two system trajectories by determining
recurrent states in reconstructed phase space [8,22]. Drawing on
Taken’s (1981) [16] embedding theorem1 the dynamic structure of
each trajectory is first recovered by unfolding its measured scalar
sequence in a, higher dimensional, reconstructed phase space that
is isomorphic to the system’s true phase space [1]. Recurrences
between the two reconstructed phase space trajectories are then
plotted on a two-dimensional (N × N) array, where dots are used
to mark the recurrences and each axes (N in length) represents the
location in time along one of the two trajectories (Fig. 1). A point on
one system trajectory is considered to be recurrent with a point on
the other system trajectory when it falls within a designated sphere
of radius r (see [8,18,21]).
The patterns in cross-recurrence plots can be mathematically
quantified using cross-recurrence quantification analysis (CRQA;
[22]) and the relative change in the magnitude of the CRQA statistics
of maxline (Lmax ) and percent recurrence (%REC) have been shown
to index and Q, respectively [7,10,11]. Specifically, Lmax , which
is the longest diagonal line in a cross-recurrence plot, provides a
reliable index of attractor strength independent of Q, whereas,
%REC, which measures the probability of finding a recurrent point
in a cross-recurrence plot, is highly sensitive to stochastic processes
[20] and thus indexes Q relatively independently of [10,11].
Experiment 1: The current study is aimed at investigating
whether the difference in S.D. between intra- and interpersonal
coordination are the result of a decrease in attractor strength or an increase in amount of underlying noise Q, by means of Lmax
and %REC, respectively. If the higher S.D. for interpersonal coordination is due to weaker coupling, and therefore a weaker , then
Lmax should be significantly less for inter- compared to intrapersonal coordination. However, if the increase in S.D. is a result of
an increase in Q, then %REC should be less for inter- compared to
intrapersonal coordination.
Participants. Twelve students (six pairs) from the University of
Connecticut participated in the experiment.
Materials. Participants sat in chairs with two forearm supports
parallel to the ground and were positioned 1 m apart. For the
intrapersonal and interpersonal conditions, wrist-pendulums were
1
Taken’s (1981) embedding theorem states that information about the dynamics
of multidimensional systems can be uncovered through the measurement of a single
scalar time series assuming that interactions exist among the state variables of a
system.
341
coordinated in the sagittal plane using ulnar–radial deviations of
the wrist. The wrist-pendulums had a natural period of 1.10 s, and
were constructed using 46 cm lengths of aluminum tubing and had
a 150 g weight attached at the base. The movements of the participants’ wrists were recorded at 100 Hz using electrogoniometers
attached to the metacarpophalangeal joint of the middle finger of
each hand and extended to approximately 12 cm beyond the wrist
joint. A computer recorded the time series from both the left and
right wrists of each participant.
Procedure and design. This experiment was a 2 (phase mode:
inphase, antiphase) × 2 (coordination type: intrapersonal, interpersonal) repeated measures design. It was completed in two stages,
with participants performing intrapersonal trials before interpersonal. In each stage, participants completed four 100 s trials, two
inphase and two antiphase, with the order of these trials counterbalanced across participant pairs. For the intrapersonal trials,
participants were instructed to coordinate pendulums swung about
the left and right wrists at a self-selected comfortable tempo, while
a curtain hung between the two participants so that they could not
see each other’s movements. For the interpersonal trials, the curtain
was removed and participants were instructed to swing a single
pendulum about the wrist closest to their partner and to coordinate these motions in either an inphase or antiphase manner at a
self-selected tempo.
Data reduction and measures. The times series were down sampled from 100 to 50 Hz. The first 10 s of the participants’ motion
time series was removed to eliminate any transients, resulting in
90 s of analyzable data. The mean period, relative phase shift (PS;
deviation from intended relative phase angle), and S.D., were used
as the dependent measures of coordination stability. For the calculations of %REC and Lmax , CRQA was performed on the time series
from each trial using an embedding dimension of 5, a time lag of 12
samples (equal to a quarter cycle of the overall mean period, 1.01 s)
and a threshold radius equaling 10% of the maximum Euclidean distance separating points in reconstructed phase space (see [10,11] for
details). For the intrapersonal trials, the measures of period were
calculated for the left and right wrist of each participant and then
averaged across wrists to obtain one measure of period for each
participant. For the interpersonal trials, one value for each measure
of period was calculated by averaging across the single wrist time
series for each participant in a pair. PS, S.D., %REC, and Lmax were
calculated for each participant separately for the intrapersonal trials and for each pair for the interpersonal trials.
Coordination type was treated as a between-subjects variable
(12 intrapersonal subjects, 6 interpersonal pairs). Thus, for each
dependent measure a 2 × 2 mixed design ANOVA was conducted.
Period. Consistent with previous research [14,15], participants
produced a frequency of movement slightly faster than the eigenperiod of the wrist-pendulum system (M = 1.01 s). This increase
in frequency was more pronounced for interpersonal coordination, with the mean period (0.92 s) being significantly faster,
F(1, 16) = 5.25, p < .05, than that of intrapersonal coordination
(1.09 s). Anecdotal evidence (from post experimental interviews)
suggested that this difference was a result of participants compensating for the weaker visual coupling of interpersonal coordination
by increasing the neuromuscular control used to constrain the
motions of their respective wrist-pendulums. In other words, participant pairs appeared to increased the frequency-dependent
stiffness of their wrist-pendulum motions and as a result, the average cycle-to-cycle frequency of movement. There was no main
effect for phase mode, F(1, 16) = 3.98, p > .05, nor a phase mode by
coordination type interaction, F(1, 16) = 2.00, p > .05.
Relative phase. Analysis of PS revealed that participants performed the required relative phase for both intra- and interpersonal
coordination (average absolute PS = 1.18◦ ). Thus, phase mode, F(1,
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M.J. Richardson et al. / Neuroscience Letters 438 (2008) 340–345
Fig. 1. (a and b) Time series from a pair of coupled oscillators. (c and d) The measured scalar sequences x(t) and y(t) of each oscillator unfolded into a phase space of
three-dimensions using time-delayed () copies of x(t) (x(t + ), x(t + 2)) and y(t) (y(t + ), y(t + 2)) as the surrogate second and third dimensions. (e) The reconstructed phase
spaces of x(t) and y(t) overlapped with one another. (f) the corresponding cross-recurrence plot. A point of the trajectory yj is considered to be recurrent with a point xi ,
when yj falls within a sphere of radius r about xi . The black circle in (e) represents the sphere of radius r, the black point represents point yj and the gray point represents
point xi . The white dot represents another point of x that falls outside the sphere of radius r and is not recurrent with yj (adapted from [8,10,11]). The circle in (f) identifies
the recurrent point in the cross-recurrence plot for points xi and yj (e). Lmax for the cross-recurrence plot shown in (f) is highlighted by the ellipsoid.
16) < 1, coordination type, F(1, 16) = 1.29, p > .05, and the phase
mode by coordination type interaction, F(1, 16) < 1, were all nonsignificant.
As expected, the analysis of S.D. yielded main effects for phase
mode, F(1, 16) = 30.99, p < .01, and coordination type, F(1, 16) = 28.97,
p < .01. There was no phase mode by coordination type interaction,
F(1, 16) = 1.75, p > .05. The S.D. for both intra- and interpersonal
coordination was greater for antiphase than for inphase coordination and was also much greater for interpersonal than for
intrapersonal coordination (Fig. 2a). This increase in variability may,
in part, be due to the difference in movement period between
intrapersonal and interpersonal coordination, as S.D. increases
with increases in movement frequency [5].
Cross-recurrence quantification analysis. As expected, the analysis
of %REC yielded no effect for phase mode, F(1, 16) = 2.85, p > .05,
whereas the values of Lmax were found to be significantly larger, F(1,
16) = 7.60, p < .05, for inphase than for antiphase. This was true for
both intra- and interpersonal coordination and is consistent with
the assumptions that the S.D. is greater for antiphase than for
inphase because |0◦ | |180◦ | and Q0◦ ≈ Q180◦ .
Consistent with |intra | |inter |, coordination type was found
to significantly influence Lmax , F(1, 16) = 10.31, p < .01, with a greater
magnitude of Lmax found for intrapersonal coordination compared
to interpersonal (Fig. 2a). Interestingly, coordination type was also
found to significantly influence %REC, F(1, 16) = 4.75, p < .05, suggesting that the difference in S.D. for intra- and interpersonal
coordination might also be due to a change in Q. That is, the level
of noise that emerges from the interpersonal system might be
magnified in comparison to the intrapersonal system. However,
while the relative change in the magnitude of %REC is typically
able to provide an effective measure of Q for interlimb coordination, %REC is not completely independent of Lmax , due to the
fact that an increase in the length of Lmax results in an increase
in %REC. Note, that this dependence is not reciprocal—an increase
or decrease in %REC does not necessarily result in a change in the
length of Lmax [10,11]. Thus, the small, yet significant difference in
%REC for the intra- and interpersonal coordination might not be
a result of change in Q, but simply the result of the small dependence of %REC on changes in Lmax . As an additional test of whether
the differences in S.D. between intra- and interpersonal coordina-
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M.J. Richardson et al. / Neuroscience Letters 438 (2008) 340–345
343
Fig. 2. S.D., %REC, and Lmax as a function of phase mode and coordination type for (a) Experiment 1 and (b) Experiment 2.
tion (and between inphase and antiphase) were due to a difference
in attractor strength rather than a difference in noise level Q,
a stepwise regression analysis was performed with S.D. as the
dependent variable and %REC and Lmax as the predictor variables.
The regression resulted in an R2 of .429 (p < .01), with Lmax being the
only significant predictor. Note that neither the %REC nor the Lmax ,
ANOVA produced a phase mode by coordination type interaction
(F(1, 16) = 1.19, p > .05 and Lmax , F(1, 16) = 2.65, p > .05).
In summary, the results of Experiment 1 indicate that inphase
coordination was found to have a stronger attractor than antiphase
coordination. This was true for both intra- and interpersonal
coordination and highlights that the patterns of interpersonal coordination are constrained by the same coupled oscillator dynamic as
intrapersonal coordination. With respect to the primary aim of the
current experiment—determining the degree to which the differ-
ence in S.D. for intra- and interpersonal coordination is due to a
difference in attractor strength or the magnitude of noise Q—the
results appear to be consistent with the assumption that the relative
phase attractors are weaker for inter- than for intrapersonal coordination [2,12]. However, finding that participants produced a faster
movement frequency when performing interpersonal coordination
compared to intrapersonal presents a possible confound. Given that
increases in movement frequency operate to weaken the relative
strengths of inphase and antiphase coordination [5], the difference
in the magnitude of Lmax (and S.D.) for intra- and interpersonal
coordination may be due in part to the difference in movement frequency. To eliminate the possible confounding effects of differences
in movement frequency, a second experiment was conducted. In
this second experiment, the frequency of oscillation for both intraand interpersonal conditions was controlled using a continuation
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M.J. Richardson et al. / Neuroscience Letters 438 (2008) 340–345
paradigm, in which an auditory metronome was employed to set
the pace of the participants’ movements at the beginning of each
trial.
Experiment 2: Participants. Sixteen students (eight pairs) from
the University of Connecticut participated in the experiment.
Materials and procedures. The same materials and similar procedures from Experiment 1 were used. In order to control for the
frequency of movement, an auditory metronome with a period of
1.10 s set the pace for the first 15 s of each trial for a total trial length
of 65 s. Participants were instructed to continue swinging at this
pace once the metronome stopped. The time series were down sampled from 100 to 50 Hz and the last 45 s of each trial was analyzed.
For the calculations of %REC and Lmax , CRQA was performed using
an embedding dimension of 5, a time lag of 14 samples (equal to a
quarter cycle of the metronome period, 1.10s) and a threshold radius
equaling 10% of the maximum Euclidean distance separating points
in reconstructed phase space.
Period. Although there was a significant effect of coordination
type for period, F(1, 22) = 17.33, p < .01, the use of the metronome
resulted in very similar movement periods for both intra(M = 1.11 s) and interpersonal (M = 1.08 s) coordination near the
natural eigenfrequency of the wrist-pendulum system (M = 1.10 s).
There was no main effect for phase mode, F(1, 22) = 3.17, p > .05, nor
a phase mode by coordination type interaction, F(1, 22) < 1.
Relative phase. Consistent with the findings of Experiment 1, an
analysis of PS revealed that participants performed the required
relative phase for both intra- and interpersonal coordination (average absolute PS = 1.68◦ ). Thus, phase mode, F(1, 22) < 1.22, p > .05,
coordination type, F(1, 22) = 2.96, p > .05, and the phase mode
by coordination type interaction, F(1, 22) = < 1, were all nonsignificant. The analysis of S.D. revealed a significant effect of
both phase mode, F(1, 22) = 60.40, p < .01, and coordination type,
F(1, 22) = 16.63, p < .05, with antiphase and interpersonal coordination being more variable (Fig. 2b). There was no phase mode by
coordination type interaction, F(1, 22) < 1.
Cross-recurrence quantification analysis. Although Q is also
assumed to be constant across phase modes [17], the analysis of
%REC revealed a significant effect of phase mode, F(1, 22) = 5.43,
p < .05, with a slightly greater magnitude of %REC for inphase
compared to antiphase coordination. Again, however, given the
large effect of phase mode on Lmax , F(1, 22) = 9.82, p < .01, which is
consistent with inphase being inherently stronger than antiphase
coordination [5], the difference in the magnitude of %REC for
inphase and antiphase coordination is not seen as evidence of a
difference in Q, but rather the small dependence of %REC on Lmax
[10,11].
Consistent with Q being constant across intra- and interpersonal
coordination, the analysis of %REC revealed no effect of coordination type, F(1, 22) = 1.53, p > .05, with similar magnitudes of %REC
being observed for intra- and interpersonal coordination. Replicating the results of Experiment 1, Lmax was found to be significantly
greater for the intrapersonal compared to interpersonal trials, F(1,
22) = 4.09, p = .056, supporting the assumption that intrapersonal
coordination has a stronger attractor than interpersonal coordination. There were no phase mode by coordination type interactions
on %REC, F(1, 22) < 1, or Lmax , F(1, 22) < 1. As further confirmation that
the differences in S.D. between intra- and interpersonal coordination (and between inphase and antiphase) were due to a difference
in attractor strength rather than a difference in noise level Q a
stepwise regression performed with S.D. as the dependent variable and %REC and Lmax as the predictor variables resulted in an R2
of .46 (p < .01), with Lmax being the only significant predictor.
The results of the current study are consistent with claims
that the differences in the S.D. for intra- and interpersonal coordination are not due to noise differences but attractor strength
differences. The CRQA results suggest that the magnitude of noise
Q—as indexed by %REC—is similar for intra- and interpersonal
coordination whereas attractor strength—as indexed by Lmax —is
different for intra- and interpersonal coordination. Recall, that the
difference between the S.D. for intra- and interpersonal coordination has been assumed to be due to the visual coupling that
constrains interpersonal coordination being weaker than the neuromuscular coupling that constrains intrapersonal coordination
[12]. Despite the fact that the current results provide support for
this claim, they do not provide any indication as to why an informational linkage established across the optic array is less robust
than the informational linkage established across the neuromuscular system. It has been suggested, that this might be a consequence
of the lack of perceptual saliency in the optic array of the movement
kinematics relevant to interlimb coordination compared to the perceptual saliency of such information in the haptic array utilized
by the proprioceptive system for intrapersonal coordination [12].
Given that the current results provide further evidence that Lmax
can be used to index , it seems clear that future research could
employ CRQA to investigate such possibilities more directly. Not
only would research of this type provide insights about how information couples perception and action, but it would also indicate
how such couplings influence the stability of actor-actor coordination.
Acknowledgements
This research was supported by NSF Grants BCS-0240277, BCS0240266, and SBR 04-23036.
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