Comparing the Attractor Strength of Intra- and Interpersonal Interlimb Coordination

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