phase-entrainment dynamics of visually coupled rhythmic movements

Biol. Cybern. 70, 369-376 (1994) Bbk;gN Cybe ne �9 Springer-Verlag 1994

Phase-entrainment dynamics of visually coupled rhythmic movements R. C. Sehmidt 1,2, M. T. Turvey 2,3

i Department of Psychology, 2007 Percival Stern Hall, Tulane University, New Orleans, LA 70118, USA 2 Center for the Ecological Study of Perception and Action, University of Connecticut, Storrs, CT 06268, USA 3 Haskins Laboratories, New Haven, CT 06511, USA

Received: 21 July 1992/Accepted in revised form: 21 July 1993

Abstract. Do interlimb rhythmic coordinations between individuals exhibit the same relations among the same observable quantities as interlimb rhythmic coordination within an individual? The 1:1 frequency locking between the limbs of two people was investigated using a paradigm in which each person oscillated a hand-held pendulum, achieving and maintaining the mutual entrainment through vision. The intended coordination was antiphase, ~b = rr, and the difference between the uncoupled eigenfrequencies, Am, was manipulated through differences in the lengths of the two pendulums. The mean phase relation and its variance for visually coupled coordinations differing in Am were predicted by an order parameter equation developed by Haken et al. (1985) and Schrner et al. (1986) for the relative phase of correlated movements of limb segments. Specifically, the experiment revealed that: (1) the deviation of ~b from n increased with increasing deviation of Ao9 from 0; and (2) fluctuations in q~ increased with increasing deviation of Am from 0. With deviations of Ao~ from 0, new peaks were added at higher harmonics in q~'s power spectrum. These results were in agreement with previous research on the stable states of interlimb coordination within a person, mediated by mechanoreceptive rather than photoreceptive mechan- isms. Additionally, they were in agreement with previous research on phase transitions in interlimb coordination which have been shown to conform to the same order parameter dynamics whether the coupling be mechanoreceptively or photoreceptively based. It was suggested that phase entrainment in biological movement systems may abide by dynamical principles that are indifferent to the details of the coupling.

1 Introduction

According to a dictionary definition, to coordinate means "to bring parts into proper relation." In biological

Correspondence to: R. C. Schmidt

coordinations such as walking and running, the "parts" are the limbs and limb segments of the individual animal. Bringing them into "proper relation" exploits information that is carried in the patternings of tissue deforma- tion and detected by the haptic perceptual system with its repertoire of receptors embedded in the muscles, tendons, and joint capsules. In other biological coordinations, of the kind typified by the dance routines of humans and the mating rituals of insects, fish, and birds (e.g., Tinbergen 1951), the "parts" are the bodies and limbs of two or more individuals whose movements must relate spatially and temporally. Bringing these parts into "proper relation" exploits information that is carried in the patternings of light and sound distributions and detected, respectively, by the visual and auditory perceptual systems.

In the present article we pursue the hypothesis that phase entrainment in biological movement systems abides by very general dynamical principles that are indifferent to the physical details of the coupling. Specifi- cally, we determine whether interlimb rhythmic coordination between individuals exhibits the same relations among the same observable quantities as interlimb rhythmic coordination within an individual, despite the obvious differences between the two cases (e.g., the nervous systems of two individuals vs. the nervous system of one individual; coupling through the visual perceptual system in between-individuals coordination vs. coupling through the haptic perceptual system in within-individual coordination).

1.1 Dynamic modelino of interlimb rhythmic coordination

The starting point for the present research is the inves- tigation and modeling of spontaneous transitions in interlimb rhythmic patterns. The hallmark criteria of a non-equilibrium phase transition (divergence, hysteresis, critical slowing down, critical fluctuations) have been observed in a paradigm in which a person is required to oscillate the two index fingers (or two hands) at the coupled frequency coc, where coc, is varied by a metronome that the person tracks (e.g., Kelso 1984; Kelso et al. 1986; Scholz et al. 1987). Results show that there are only

370

two steady states: in phase and antiphase. With increasing me, antiphase switches rapidly to in-phase. In-phase, however, does not switch to antiphase, and the antiphase-to-in-phase transition is not reversed by a reduction in me. Further, relative phase 4) exhibits increases in relaxation time Zre~ and increases in its fluctuations as the transition point is approached. Similar investigations have been conducted on the spontaneous jumps in coordination when two limbs are connected optically between two people rather than anatomically within a person. In these experiments (Schmidt et al. 1990), two seated people each oscillated a leg with the goal of co- ordinating the two legs antiphase or in phase as m~ was increased. To satisfy the goal, the two people watched each other closely. As with the 1-person, 2-oscillators case, the 2-persons, 2-oscillators case exhibited a sudden behavioral transition from antiphase to in phase, but not vice versa; it showed divergence, hysteresis, and critical fluctuations (critical slowing down was not investigated). If the two people began their movements out of phase and increased limb frequency simultaneously at the same rate without watching each other, then no transition occurred. The phase transition depended on looking.

The fundamental aspects of the above behavioral transitions have been successfully modeled by the following order parameter equation (Haken et al. 1985; Sch6ner et al. 1986; Kelso et al. 1990):

4; = Am - a sin(q~) -- 2b sin (24)) + x /~{ (1)

In (1), Am (the zero order term) is the difference in eigenfrequencies of the two oscillators, a and b are coefficients such that b/a increases as mc decreases and coupling strength increases, and ~ is a gaussian white noise process (arising from the multiplicity of underlying subsystems) functioning as a stochastic force of strength Q. With respect to the dissolution of the antiphase interlimb pattern in the experiments summarized above, the dynamics expressed by (1) exhibit, for Am = 0, a global minimum at 4) = 0 and local minima at 4) = + r~ (for Am = 0) which are modulated by mc such that at a critical value the local minima are annihilated.

The dynamics expressed by (1) comport with von Hoist's (1937/1973) far-reaching insights on the forming and sustaining of coordinations among rhythmically moving appendages. For von Hoist, an interappendage coordination reflects the maintenance tendencies by which individual components continue to do what they prefer to do, identified by Am in (1), and the magnet effect by which individual components are drawn together as a single unit, identified by ( - asin4) - 2bsin 2 4)) in (1). In short, (1) provides a compact formulation of von Hoist's understanding that the phase-interaction patterns (4;) of rhythmically moving limbs are selected by the interplay of competitive (Am) and cooperative ( - as in4) - 2bsin 2 4)) processes.

For the present research the importance of (1) lies in its predictions of the stationary states of interlimb coordination when the symmetry of the coordination dynamics (in which 4) and - 4 ) are equivalent) is broken or lowered. The latter occurs when the individual biological

oscillators are no longer equivalent, as when the two rhythmically moving limb segments differ in eigenfrequencies, that is, Am 50. The stationary, states can be determined by solving (1) numerically for 4) = 0. Graphi- cally, solutions to (1) with 4)= 0 can be obtained by plotting the right-hand side of (1) (excluding the stochastic force) against 4) for various parameter values (Kelso et al. 1990, 1991). The stationary values are given where q~ crosses the. 4) axis. These stationary states are stable if the slope of 4) at the zero-crossing is negative, and unstable if it is positive. Numerical analysis reveals that (1) makes the following predictions about stationary states in interlimb 1:1 frequency locking:

i) When Am = 0, the attractors ofinterlimb coordination are at 4) = 0 and 4) = rt for all magnitudes of the coupling b/a satisfying Ibl > lal/4. For coupling satisfying Ibl < lal/4, the attractor at 4) = 7: becomes unstable, but the attractor at 4) = 0 remains, invariant over variations in the coupling strength.

ii) When IAml > 0 and Ibl > lal/4, the attractors of interlimb coordination are displaced from 4)= 0 and 4) = r~ with a near equal displacement from 0 and ;~ such that the two attractors remain a distance n apart.

iii) Am and b/a contribute multiplicatively to 4) such that for a given b/a satisfying Ibl > lal/4, the greater the magnitude of IAm[ > 0, the greater are the displacements of the attractors from 4)= 0 and 4)= re. Similarly, for a given IAmt > 0 and Ibl > tal/4, the smaller the magnitude of b/a, the greater are the displacements of the attractors from 4) = 0 and 4) = n.

In approximate form, these predictions can be seen intuitively in (1) when restricted to the in-phase mode. Because the system in this case is close to 4) = 0, (1) may be linearized (4) is sufficiently small so that .sinq5 = 4)) (Haken 1988; Sch6ner et al. 1986). Then, for 4) = 0 and ignoring the noise term, (1) becomes

4)s~aao,arr = Ao)/(a + 4b) (2)

Further, predictions similar in form to i-iii above follow from the nonlinear coupled oscillator dynamics proposed by Rand et al. (1988) to address rhythmic coordination in the lamprey and by Haken (1983, Sect. 8.6) to address the bifurcation to frequency locking, namely,

q~ = Am - Ksin4) (3)

4) ~,a,io,~rr = arcsin( A m / K ). (4)

where K is the coupling strength. The similarity of (4) to (2) is self-evident. In contrast to (1), (3) and (1)'s linearized form about 4) = 0 (if the latter was to be taken as the general case) assume an equivalency of in-phase and antiphase and the influence of only one of the two attractors in any given dynamical run from an initial in-phase or antiphase state. To accommodate fluctuations in coordination, a stochastic force of the kind identified in (1) can be added to (3) (Turvey and Schmidt 1993).

Equation (1) can also be expressed as

= - (0 V/~4)) + ,,/Q~ (5)

Eo

-3 -2 -1 0 1 2 3

AC..O (tad/s)

-G-

O

-3 -2 -1 0 1 2 3

Af.0 (rad/s)

Fig. 1. Upper: Deviat ion of average q~ f rom 4}, = g as a function of Ao~, as predicted by (1). Lower: Fluctuat ions in q~ as a function of Am for q~, = rc as predicted by (6)

where V is the potential function

V(~b) = - Amq~ - acos(~b) - bcos(2q~) (6)

Through (6), the degree of stability of ~stationary c a n be determined. Figure 1 (top) gives ~b~,ao,~,r as a function of Am for the antiphase coordination under fixed values of a and b. Figure 1 (bottom) gives the corresponding potential values expressed as expected fluctuations. In sum, (1) permits the determination of which value of q~ will be stable for a given Am, and (6) permits the determination of the degree of stability associated with that value.

1.2 Experimental evaluation of (1) in i-person (haptically coupled) interlimb coordination

Predictions i-iii receive support from experiments in which ~bq, (intended coordination of either in-phase where ~b = 0 or antiphase where 4~ = n), Am, and m~ are manipulated within a procedure that involves a seated person oscillating hand-held pendulums parallel to the sagittal plane about an axis in the wrist (with other joints essen- tially immobile) (Kugler and Turvey 1987). The pendulums can vary physically in shaft length and/or the mass of the attached bob. The eigenfrequency of an individual "wrist-pendulum system" can be estimated as the eigenfrequency of the equivalent simple gravitational pendulum, m = (o/L) ~/2, where L is the simple pendulum length and g is the constant acceleration due to gravity (Kugler and Turvey 1987; Turvey et al. 1988). The quantity L is calculable from the magnitudes of shaft length, added mass, and hand mass, through the standard methods for

371

representing any arbitrary rigid body oscillating about a fixed point as a simple pendulum. Alternatively, the eigenfrequency of a given wrist-pendulum system can be measured empirically by having the subject swing the pendulum alone, that is, uncoupled. Patently, if the pendulums oscillated in each hand differ in physical dimen- sions (length, mass), then their eigenfrequencies (their maintenance tendencies in von Hoist's terms) will not correspond. The component rhythmic units will be in frequency competition.

In the experiments that bear upon the above predictions, Am was controlled in the manner suggested, namely, through differences in the lengths of the left and right pendulums, and m c (inversely related to b/a) was either the preferred frequency for a given pair of pendulums or another frequency determined by a metronome (Schmidt et al. 1991, 1993; Sternad et al. 1992; Turvey et al. 1986). Consistent with (1), the experiments revealed that: (a) when Am = 0, ~staao,a,y equals 0 or r~ over the range of mc values tested; (b) when A m r and me is the preferred value or close to it, q~stati . . . . y deviates from 0 and n by near equal amounts such that the stable phase relations remain a distance n apart; (c) Am and % contribute multiplicatively to Ck~aa .... r such that the lower the value of me, the more closely ~b~t,ao,~,y approximates 0 or n for any given Am ~ 0, and conversely, the higher the value of me, the less closely 4~,t,a . . . . y approximates 0 or n for any given Am ~ 0. In confirmation of (6), the experiments found that fluctuations in ~b~taao,~y are greater for a more nearly antiphase than in-phase coordination and increase as Am deviates from 0 and, therefore, as q~t~ao.a,r deviates from 0 or rc (Schmidt et al. 1991, 1993; Turvey et al. 1986), and that these fluctuations are ampli- fied further by increases in mc (Schmidt et al. 1993),

An important qualification to the preceding is that a given m~ cannot index the same b/a for wrist-pendulum systems characterized by different magnitudes and com- positions of Am = (mR -- mL); m~ needs to be scaled to the preferred time scale of the coupled system. It has been argued that the time scale in question is that of the right (R) and left (L) wrist-pendulum systems coupled such that 0R is always, at every instant, identically equal to 0L, or to (0L + z0 (Kugler and Turvey 1987). The latter ideal would be achieved if the coupling between the two oscillators was functionally equivalent to that of a rigid con- nection (Kugler and Turvey 1987). The simple pendulum equivalent Lequ~o,t~.t of a compound pendulum so com- posed (that is, of two pendulums connected by a rigid bar) is given by

Leq,i~ate.t = (mll~ + m2l~)/(mlll + m212) (7)

where mi and li refer to the mass and the equivalent simple pendulum length, respectively, of an individual (compound) pendulum system. Through (7), two coupled pendulums of lengths IR and lL can be interpreted as a virtual (v) pendulum of length Lv with an eigenfrequency my = (g/Lv) 1/2. When a subject is instructed to oscillate a pair of wrist-pendulum systems at the most comfortable frequency, it can be expected that the frequency so elected will correspond closely to my.

372

Accordingly, for several coupled wrist-pendulum systems distinguished by Aa~, the coupled frequency produced in each case under the "comfort" instructions will correspond closely to the coy for each case. In short, coupled systems that differ in Aco will oscillate at dynamically equivalent coupled frequencies, coo/co v ~ 1 and, therefore, with dynamically equivalent coupling values of b/a. The importance of this latter argument for evaluating the predictions in Fig. 1 is apparent. With respect to producing equivalent changes in b/a for systems that differ in Aco, metronome frequencies coc used to pace the coupled oscil- lations can be chosen so that coo/coy is identically greater or less than unity across the different systems. Evidence that equal values of O~c/COv ~ 1 define dynamically equivalent couplings over different magnitudes of A~o has been pro- vided by Sternad et al. (1992): b/a oc (o~dcov)-1.

1.3 Goal of the present research: experimental evaluation of (1) in 2-persons (visually coupled) interlimb coordination

As is evident from the preceding, the research on 1-person, 2-oscillator systems confirms the interlimb rhythmic-coordination relations identified in Fig. 2, viz., the dependencies of ~bstaa . . . . r and its fluctuations on Aog. These relations follow from (1). Would the same relations be in evidence if the two oscillators comprised not the right and left limbs of a person linked haptically but the right limb of one person and the left limb of another person linked visually? The question posed is one of the generality of (1). The observables of ~b(t), A~o, and ~oc can be defined in the wrist-pendulum paradigm for a 2-persons, 2-oscillator system in which the two persons in question maintain 1:1 frequency-locking by watching each other's movements. Moreover, a 2n-periodic coupling function is to be expected, despite the visual rather than haptic basis to the coupling, given that the task demand remains that of 1:1 frequency locking. Conse- quently, the dynamics of (1) are as formative, in principle, of the phase interactions between two oscillators in the 2-persons case as they are of the phase interactions between two oscillators in the 1-person case. That (1) ap- plies to visual coupling has been confirmed, as noted, for the spontaneous transition from antiphase to inphase interlimb coordination (Schmidt et al. 1990). The experiment reported in the present article tests the specific hypothesis that the relations depicted in Fig. 1, derived from (1) for the stationary states of 1:1 frequency locking, also hold for the visual coordination of wrist-pendular rhythmic movements between two people.

2 Methods

2.1 Subjects

Two research staff members from Haskins Laboratories (a man and a woman of 25 and 23 years of age, respectively) and a graduate student from the University of Connecticut (a man aged 24 years) served as subjects. They were paid $5 an hour for their participation. Each

subject coordinated movements with each other subject. All possible pairings of the subjects thus constituted three subject pairs.

2.2 Materials and procedure

The subjects of each pair sat on stools 1 m in height approximately 1 m from each other facing the same di- rection but turned in towards each other about 30 ~ (Fig. 2). Each subject was instructed to grasp the handle on the pendulum's shaft with his or her outside hand and to swing the pendulum in the anterior/posterior plane. The movement was to be executed using only the wrist joint. The forearm was to be kept continually parallel to the ground. The displacement of each subject's wrist- pendulum system was measured by a Teka PN-4 Polgon goniometer. Polarized light sensors were attached to the metal plate on the pendulum's shaft ( ~ 13 cm from the handle end) and on the subject's forearm ( ~ 7 cm from the wrist joint). The data were recorded onto FM tape and later digitized at 200 samples/s using a Datel ST- P D P 12 bit analogue-to-digital converter.

The pendulums were constructed using the specifica- tions described in Kugler and Turvey (1987). They con- sisted of an ash dowel with a bicycle hand grip attached to the top. Weights were attached to a 10 cm long bolt that was drilled through the dowel at a fight angle 2 cm from the bottom. Two sets of four different-sized pendulums were used; one set for each member of a subject pair. The magnitudes of the dowel lengths and attached mass- es for the four different-sized pendulums are identical to those used in the main experiment of Kugler and Turvey (1987). These mass and length magnitudes were, respectively, for pendulum A, 0.1 kg and 0.3 m, for pendulum B, 0.1 kg and 0.6 m, for pendulum C, 1.0 kg and 0.3 m and for pendulum D, 1.0kg and 0.6 m.

The subject pairs were instructed to swing the pendulums in two ways: uncoupled or coordinated. In both cases, the subjects swung the pendulums simultaneously. However, in the uncoupled case, each subject was instructed to look straight ahead and to swing the pendulum at his or her own preferred tempo. In the coordinated case, each subject was instructed to look at the

Fig. 2. The experimental arrangement for studying between-persons interlimb coordination. The functioning of the goniometers requires that the light sensors located on the subjects' wrists and pendulums be bathed in polarized light. The angular displacement between the sensors on the wrist and the pendulum was measured and converted into phase angle

373

other person's pendulum, to coordinate his or her oscillation with the other person's such that a mutually comfortable tempo was attained and to do so with an antiphase pattern (i.e., when one pendulum was forward, the other should be back). Because of how the subjects in a pair were seated, and because one subject oscillated with the left hand and the other with the right (see Fig. 2), it was the case that neither could see the motions of the pendulum that he or she oscillated, and each could see only the motions of the pendulum oscillated by the other person. There were four uncoupled conditions, one for each pendulum length. Further, there were 16 coordinated conditions, one for each possible pairing of the four pendulum lengths. Two 60-s trials were collected for each of the 20 conditions. The trials of the different conditions were randomized in two blocks of 20 trials each. A trial began with the subjects' searching for either a preferred mutual frequency (~o~) or a preferred non- coupled frequency (co~ where i = 1, 2 for the two subjects, respectively) depending upon the condition. In both the uncoupled and the coordinated trials, data collection began when the subjects both signaled that they had found the preferred period of oscillation. A subject pair's data were collected in a single 2.5 h session.

j minus the average position for the trial. The relative phase angle (q~) between the two coordinated wrist-pendulum systems was calculated for each sample as the 02~- 01~. For the alternating coordination pattern, the

that the subject intended to produce was zc rad. The ~b time series allows an evaluation of how the subject satisfied this task demand. The evaluation was accomp- lished in a number of ways. First, the mean q~ was calculated for each trial and condition. Second, in order to determine the magnitude and patterning of the variability associated with this time series, a power spectral analysis was performed on the ~b time series. A 60-s time series was broken up into nine subsidiary time series each of 1024 samples and each overlapping by 512 samples. Each subsidiary time series was windowed using a Welch filter to reduce spectral leakage, and all of the nine spectra from a given trial were averaged to reduce the error of the spectral estimate (Press et al. 1988). Finally, the total power of ~b was calculated for each trial by summing the power at each frequency of the averaged spectra except the DC component at the zero frequency. This measure was used as a summary of the variability of ~b.

2.3 Data reduction

The digitized data were smoothed using a Bartlett (tri- angular) moving average procedure with a window size of 35 ms. The data records of the uncoupled trials were subjected to software analyses to determine the o9~ for each of the two subjects. The data records of the coordinated trials were subjected to software analyses to determine co~, the time series of the relative phase angle q~ between the two wrist-pendulum systems, the power spectra of this relative phase time series, and the total power associated with each of the spectra.

A peak picking algorithm was employed to determine the time of maximum forward extension (ulna extension) and maximum backward extension (ulna flexion) of the wrist-pendulum trajectories. From the peak forward extension times, the mean frequency of oscillation for the nth cycle was calculated as

f , = 1/(time of forward extension,

- time of forward extension, § 1). (8)

For both the uncoupled and coordinated conditions, the mean frequency of oscillation of a trial (either coi or o9~) was calculated from these cycle frequencies.

For the coordinated conditions, the phase angle of each wrist-pendulum system (0i) was calculated for each sample (200/s) of the displacement time series to produce a time series of 0i. The phase angles of wrist pendulum i at sample j(Oij) were calculated as

O~j = arctan(Scij/Axij) (9)

where the numerator on the right hand side is the velo- city of the time series of wrist-pendulum i at sample j divided by the mean angular frequency for the trial, and Ax~j is the position of the time series at sample

3 R e s u l t s

For a subject pairing, the task with each of the 16 combinations of wrist-pendulum systems was to oscillate the two systems at a comfortable, common frequency ego in antiphase even though the uncoupled comfortable frequencies a~i of the individual pendulums tended to differ from coc. All three subject pairs found the task easy to perform. The mean frequency locking ratios (frequency 1/frequency 2) of the three subject pairs (0.999, 1.001, and 1.001) indicated that they all performed a 1:1 frequency locking of their movements. For each subject pair, the frequency competition of each combination of wrist- pendulum systems was Am =col - (-02, the difference between the observed uncoupled frequencies of the two systems. Table 1 lists the average col and Aco values for the 16 combinations of wrist-pendulum systems.

3.1 Evaluating the basic predictions of (1) and (6)

Figure 3 displays ~b(t) from a representative subject pair under two values of Ao~. The impression from Fig. 3 is of two oscillators that stay roughly one-half cycle apart on average, though fluctuating about this phase relation as they push and pull each other through the entrained cycle. When dco ,~ 0 (Fig. 3, upper) the deviation of the average ~b (i.e., ~bstaao,a,r) from ~b~ = rr and the variability of ~b were smaller than when Aco ~ - 3 (Fig. 3, lower). Figure 4 (upper) reveals how the mean deviation from ~b~, = r~ changed with the mean Aco of the 16 pendulum combinations across the three subject pairs. The linear form of this plot (r2(1,14) = 0.85; y = 0.17x + 0.05; P < 0.001) concurs with that of the equilibrium values of ~b predicted by (1) under constant b/a (Fig. 1, upper). Also, as predicted from (1), the y-intercept of Fig. 4 (upper) does not differ significantly from 0 (P > 0.05).

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T a b l e 1. The mean eigenfrequencies (rad/s) of the uncoupled wrist- pendulum systems, col and ~o2, and their difference A~o for each pendulum combinat ion

C o m b i n a t i o n 0 1 0 ) 2 A O )

1 7.20 7.62 -- 0.42 2 5.47 7.62 - 2.15 3 6.44 7.62 -- 1.18 4 4.72 7.62 - 2.89 5 7.20 5.33 1.87 6 5.47 5.33 0.14 7 6.44 5.33 1.11 8 4.72 5.33 -- 0.60 9 7.20 6.92 0.28

I0 5.47 6.92 -- 1.45 11 6.44 6.92 -- 0.48 12 4.72 6.92 -- 2.19 13 7.20 4.86 2.34 14 5.47 4.86 0.60 15 6.44 4.86 1.57 16 4.72 4.86 -- 0.14

4 . 5 4 -

- ~ - 3.14

Ar 0

1.75 , ,

0 5.0 i i

I 0 . 0

4.54- Ace ~- - 3

- e - 3.14 ~

1.75 . . . . . . . 0 5.0 10 .0

Time (s)

Fig. "3. The time series of the relative phase angle ~b ( = 01 -- 02) for conditions under which the differences between the uncoupled frequencies (do~) are approximately 0 and - 3

Figure 4 (lower) displays how the variability of (/)stationary, estimated as the total power of the spectral analysis, changes with Aco. The quadratic form of this dependency, (r2(2,14) = 0.47; y = 0.04x 2 + 0.04x + 2.3; P < 0.02) is similar to that predicted by (6) (Fig. 1, lower). As Aco deviates from 0, the total power of ~tatlo,~,y increases. In sum, the results of the present experiment, reported in Fig. 4, were in agreement with the expected outcomes, as identified in Fig. 1.

3.2 Spectral composition of q5 (t)

Inspection of the time series presented in Fig. 3 suggests that the ~b fluctuations were not random but contained

E

=o ~ g C~

.6

.4

.2

0

-.2

~

".6 -3

r 2 = .85

o - , , , , ~ .

o !~

i o

. . . . . i . . . . . .

-2 -1 0 1 2 3

Af.o (rad/s)

~ 2 .7

2.6 .g

. ~ 2 .5

0 2 . 4

o

~ 2.3

"~ 2 . 2

0 ~ 2.1

-3

r - ~ = .47 o

o

~o

-2 -1 0 1 2 3

At.0 (rad/s) Fig. 4. The mean deviation of ~b from ~b, and the fluctuations in ~b as a function of Aco. These experimental outcomes are to be compared with the model predictions in Fig. 1

coherent periodicities. The spectra of ~b(0 verify the pres- ence of these periodicities, as shown in Fig. 5. They contain peaks at integer multiples of the coupled frequency cot that indicate that periodicities at these frequencies are latent in the ~b time series. Similar spectra of ~b have been observed in haptically coupled interlimb coordination (Schmidt et al. 1991; 1993). In addition, the previous 1-person, 2-oscillators research has demon- strated that the magnitudes of these periodicities (i.e., the heights of the peaks in the ~b spectrum) are (i) inversely related to the frequency of the periodicity and (ii) increase with A~o. These facts together suggest that new spectral peaks come into existence as lAw[ departs from 0.

Accordingly, the heights of the spectral peaks in the present 2-persons, 2-oscillators data were submitted to a multiple regression analysis in double logarithmic co- ordinates with independent variables of peak frequency and the ratio of uncoupled frequencies I2. (The latter measure was used instead of Aco to allow a comparison with the results of previous studies). As in the previous 1-person, 2-oscillators study, significant regressions (r 2 = 0.46, 0.32, and 0.44, respectively, for subject pairs 1-3) indicate that the heights of the peaks decreased with increases in the peak frequency (coefficients of - 1.25, - 0.99, and - 1.09 on log peak frequency) and increased

as g2 deviated from 1 (coefficients of 2.17, 1.54, 2.01 on ]log f2]). The structure of the latent periodicities in the fluctuations of q~ appear to have the same form in both the 1-person, haptically coupled and the 2-persons,

.30"

r, .25-

.2(]

O ~ .15

~ .10

l~ .05-

0 0

S1

k • ~ ~ 2 3 4 5

.3O

.20

.lo

~ . 0 5

0 - ,

0

.30

.25

.20

"~ .15.

~ . 1 0

.05

0 �9

0

$2

2 3 4 5

$3

•J• ~ ~ . 2 3 4

Frequency/~ e

Fig. 5. The spectra of the relative phase angle ~b for all 16 Ao values (top, subject pair 1; middle, subject pair 2, lower, subject pair 3)

optically coupled interlimb coordinations. For both forms of coupling, the periodicities in ~b(t) reinforce the impression that the coordination of the rhythmic subsystems is one of phase entrainment rather than phase locking: The phase angles of the two oscillators are attracted to one another but are not perfectly locked at a constant lag (Keith and Rand 1984). As emphasized by Kelso and colleagues (DeGuzman and Kelso 1991; Kelso et al. 1991), rigid mode locking which reflects a system's asymptotic approach to a well-defined frequency and phase relation may well be the exception in biological systems. More likely is a general disposition to phase attraction which promises both stability and flexibility (see also Beck 1989).

4 Discussion

Mutual entrainment is said to occur when two or more oscillators interact with each other so as to bring about

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a locking together of frequencies. It need not imply a fixed phase relation throughout the entrained cycles. Mutual entrainment appears to be a self-organizing principle of broad biological significance (e.g., Garfinkel 1987; Winfree 1980). Notable and well-studied examples of the principle in animal behavior are populations of crickets and fireflies that mutually entrain to chirp in synchrony and flash in synchrony, respectively (Winfree 1980). In the present article, we have examined pairs of people who mutually entrain to oscillate limb segments at the same tempo. Despite the explicit intention on the part of the subjects to achieve entrainment, and despite the visual basis for their achievement, the patternings of their resultant intersegmental coordinations were consistent with principles known to apply in other settings. Specifically, they conformed to (1). This low-dimensional, nonlinear dynamic has been shown empirically to predict both the prominent and subtle behavioral characteristics of the spontaneous transition from antiphase to in-phase coordination when the limb segments belong to one person and are coupled haptically (e.g., Kelso et al. 1986; Scholz et al. 1987) or belong to two persons and are coupled visually (Schmidt et al. 1990). In respect to interlimb coordination within a person, the expectations from (1) about the stationary characteristics of 1:1 frequency locking find support in a number of experiments (Schmidt et al. 1991, 1993; Sternad et al. 1992; Turvey et al. 1986). The present research shows that those expectations also receive support when the 1:1 frequency locking of limb segments is achieved visually between two people.

As noted in the Introduction, the coupling dynamics of (3) make predictions about stationary interlimb coordinations similar to those of(l). Equation (3) more so than (1) guided the experimental analyses of Schmidt et al. (1993) and Sternad et al. (1992). As a more general formulation, (3) is limited with respect to (1) in that it would need boundary conditions affixed that distinguish the in-phase and antiphase coordinations (see Schmidt et al. 1993). Some analyses of stationary behavior favor (3) over (1) (Schmidt et al. 1993). It will be a feature of future research to adjudicate more thoroughly between the two models.

In sum, there is growing evidence that phase entrainment in biological movement systems might abide by dynamical principles that are indifferent to the physical details of the coupling. This conclusion is in concert with current efforts to develop concepts and methods by which the functional orders character- istic of biological systems might be addressed through very general principles (e.g., Haken 1977/1983, 1988; Kugler and Turvey 1987; Schrner and Kelso 1988; Yates 1987).

Acknowledgements. The research was supported in part by a University of Connecticut Doctoral Fellowship awarded to the first author and an NIH grant (BRS-RR-05596) awarded to Haskins Laboratories. Data analyses and preparation of the manuscript were supported by NSF grants 88-11510 and 91-09880. The authors would like to thank Jeffrey Kinsella-Shaw, Claudia Carello, and Paula Fitzpatrick for their help at various stages in the conducting of the present research and in the preparation of the manuscript.

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