Rhythmic Interlimb Coordination
Lyapunov exponent
(slope at
=
0
°
and
180
°
)
The Variability of Rhythmic Interlimb Coordination
*
*
(
1
,
Q
1
)
SD
(
1
,
Q
2
)

1
 = 
1

1
1
Q
1
<<
Q
2
Q
1
Q
2
Q
1
Q
1
Q
1
=
Q
1
1
2

1
 >> 
2

*
*
SD
(
1
,
Q
1
)
(
2
,
Q
1
)
The
same
SD
can
result
for
attractors
of
the
same
strength,
but
that
contain
different
magnitudes
Q
of
noise
.
The
greater
Q
the
more
is
perturbed
away
from
*
and
the
greater
the
SD
.
Intrapersonal
Interpersonal
Predicts
two
stable
phase
modes
:
Interlimb
coordination
dynamics
dynamics
of
coupled
oscillators
(Schmidt
et
al
.
,
1990
)
.
(Haken, Kelso, & Bunz, 1985)
z
w
Q
b
a
+


D
=
2
sin
2
sin
Rate of Change of
Relative Phase Angle
Detuning
Gaussian
Noise
Coupling Strength
Coefficients
=
0
°
or Inphase
=
180
°
or
Antiphase
Two
coordination
modes
:
Neuromuscular Coupling of
Visual Coupling of
SD
=
Q
2
Inphase Antiphase
Intrapersonal
Interpersonal
(SD
)
(SD
)
=
d
d
=
0
SD
confounds
λ
and
Q
. Cross

recurrence analysis can measure
λ
and
Q
independently (Richardson et al., 2007).
Index
λ
and
Q
using Cross

Recurrence Quantification Analysis
Cross

recurrence quantification analysis (CRQA) statistics of
Percent Recurrence
(%REC)
and
Maxline (L
max
)
can index the effects of
Q
and
λ
respectively.
L
max
(Longest diagonal line)
•
Sensitive to deterministic processes.
•
Provides a reliable index of attractor
strength
(
)
independent of
(
Q
)
(
Marwan, 2003;
Richardson et al., 2007
).
%REC
(Density of recurrent points)
•
Sensitive to stochastic processes.
•
Provides a reliable index of
Q
independent of
(
)
(Kudo et al., 2006;
Pellecchia et al, 2005; Zbilut &
Webber, 1992).
Cross

Recurrence Analysis
Determines recurrent states in reconstructed phase space (Marwan, 2003; Zbilut &
Webber, 1992).
Above,
the
measured
scalar
sequences
x(t)
and
y(t)
are
embedded
(unfolded)
into
a
phase
space
of
3

dimensions
using
time

delayed
(
)
copies
of
x
(
t
)
[
x
(t+
),
x
(t+
2
))
and
y
(
t
)
(
y
(t+
),
y
(t+
2
)]
as
the
surrogate
2
nd
and
3
rd
dimensions
.
In
the
corresponding
cross

recurrence
plot,
a
point
of
the
trajectory
at
y
j
is
considered
to
be
recurrent
with
a
point
x
i
,
when
y
j
falls
within
a
sphere
of
radius
r
about
x
i
.
1,2,3……
N
points
Right wrist
1,2,3……
N
points
y
Left wrist
x & y
(t)
x & y
(t +2
)
x & y
(t +
)
Threshol
d radius
(
r
)
Reconstructed Phase Space
x
i
y
j
Cross

Recurrence Plot
x
Method
Current Research Question:
Are differences in variability (SD
)
between intra

and
interpersonal coordination, as well as between inphase and
antiphase, due to differences in attractor strength (
) or
differences in magnitude (
Q
) of noise?
8 Pairs of University of Connecticut Graduate Students
Intrapersonal Coordination
(Without Vision)
Interpersonal Coordination
(With Vision)
Always 1
st
Block
2 trials per condition
= 8 trials per pair.
Inphase
Antiphase
Experimental Procedures
Conclusions:
Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand movements.
Biological
Cybernetics, 51
, 347

356.
Kudo, K., Park, H., Kay, B. A., & Turvey, M. T. (2006).
Environmental coupling modulates the attractors of rhythmic coordination
.
In press
.
Marwan, N. (2003).
Encounters with neighbors: Current developments of concepts based on recurrence plots and their applications
.
Doctoral Thesis, University of Potsdam.
Pellecchia, G. L., Shockley, K., & Turvey, M. T. (2005). Concurrent cognitive task modulates coordination dynamics.
Cognitive Science,
29,
531

557.
Richardson, M. J., Schmidt, R. C. & Kay, B. (2007). Distinguishing the noise and attractor strength of coordinated limb movem
ent
s using
recurrence analysis .
Biological Cybernetics, 96,
59

78.
Schmidt, R. C., Carello, C. & Turvey, M. T. (1990). Phase transitions and critical fluctuations in the visual coordination of
rh
ythmic
movements between people. .
JEP: HPP, 16
, 227

247.
Zbilut, J. P., & Webber, C. L., Jr. (1992). Embeddings and delays as derived from quantification of recurrence plots.
Physics Letters A,
171
, 199

203.
Hypotheses:
SD
(inphase)
<
SD
(antiphase)
L
max
(inphase)
>
L
max
(antiphase)
%REC
(inphase)
≈
%REC
(antiphase)
SD
(intrapersonal)
<
SD
(interpersonal)
L
max
(intrapersonal)
>
L
max
(interpersonal)
%REC
(intrapersonal)
≈
%REC
(interpersonal)
Results:
Eigenperiod = 1.1s
1 Beep/Cycle
Metronome
Set pace 1
st
15 s then stopped.
45 s continued to swing at
same pace.
Wrist movements recorded at 50 Hz with electrogoniometers.
Right wrist left person, left wrist right person for
interpersonal trials.
Phase:
Coordination:
SD
5
Inphase
Antiphase
Intra

Inter

25
20
15
10
L
max
0
400
800
1200
Inphase
Antiphase
Intra

Inter

%REC
0
4
8
12
Intra

Inter

Inphase
Antiphase
SD
(inphase)
<
SD
(antiphase)
L
max
(inphase)
>
L
max
(antiphase)
%REC
(inphase)
>
%REC
(antiphase)
SD
(intrapersonal)
<
SD
(interpersonal)
L
max
(intrapersonal)
>
L
max
(interpersonal)
%REC
(intrapersonal)
≈
%REC
(interpersonal)
Phase:
Coordination:
Differences in variability (SD
)
between intra

and interpersonal
coordination, as well as between in

and antiphase, are due to
differences in attractor strength (
) as indexed by
L
max
,
not
magnitude (
Q
) of noise, as indexed by
%REC
?
References:
Stacy Lopresti

Goodman, Marisa C. Mancini
1,2
, Richard C. Schmidt
1,3
,
Bruce Kay
1
, & Michael J. Richardson
14
1
Center for the Ecological Study of Perception and Action, University of Connecticut;
2
Federal University of Minas Gerais, Brazil;
3
College of the Holy Cross;
4
Colby College
Acknowledgments
.
This research was supported by grants from the National Science Foundation (BCS 02

40277 and BCS

02

402266) and
a CAPES Award from the Brazilian Ministry of Education (BEX 0330/05

1).
Comparing the Attractor Strength of Intra

and
Interpersonal Interlimb Coordination
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