Biological cybernetics
Biol. Cybern. 67, 223-231 (1992)
9 Springer-Verlag 1992
Average phase difference theory and 1:1 phase entrainment in interlimb coordination Dagmar Sternad ~, M. T. Turvey ~, and R. C. Schmidt ~,2 ] Center for the Ecological Study of Perception and Action, University of Connecticut, Storrs, CT 06268, USA 2 Tulane University, New Orleans, LA 70115, USA Received January 14, 1992/Accepted in revised form February 25, 1992
Abstract. The dynamics of coupled biological oscillators can be modeled by averaging the effects of coupling over each oscillatory cycle so that the coupling depends on the phase difference q~ between the two oscillators and not on their specific states. Average phase difference theory claims that mode locking phenomena can be predicted by the average effects of the coupling influences. As a starting point for both empirical and theoretical investigations, Rand et al. (1988) have proposed dfb/dt = Ao~ - K sin ~b, with phase-locked solutions ~b =arcsin(Aco/K), where Ao9 is the difference between the uncoupled frequencies and K is the coupling strength. Phase-locking was evaluated in three experiments using an interlimb coordination paradigm in which a person oscillates hand-held pendulums. Ao9 was controlled through length differences in the left and right pendulums. The coupled frequency O~cwas varied by a metronome, and scaled to the eigenfrequency o9~ of the coupled system; K was assumed to vary inversely with coc. The results indicate that: (1)Aco and K contribute multiplicatively to ~b; (2) ~b = 0 or ~b = n regardless of K when Aco = 0; (3) ~b ~ 0 or ~b ~ r~ regardless of Ao9 when K is large (relative to Ao~); (4) results (1) to (3) hold identically for both in phase and antiphase coordination. The results also indicate that the relevant frequency is o~c/o~v rather than co~. Discussion highlighted the significance of confirming q~ = arcsin(Aog/K) for more general treatments of phase-locking, such as circle map dynamics, and for the 1:1 phase-entrainment which characterizes biological movement systems.
1 Introduction Von Hoist (1937/1973) addressed the issue of movement coordination with medulla transected Labrus, a fish that swims with its main body axis immobile. Following
Correspondence to: M. T. Turvey
surgery, the fins oscillate autonomously for hours. With this preparation von Hoist observed the waxing and waning of phase-locking. Two different sized fins, with different uncoupled periods, would oscillate with a fixed phase relation and a common coupled frequency- a state von Hoist referred to as absolute coordination. Or they would oscillate without phase and frequency locking - a state yon Hoist referred to as relative coordination. Both states would be seen intermittently, and whenever one dominated, signs of the other would be visible. Von Hoist concluded that even when absolute coordination was achieved, interfin competition (each attempting to proceed at its own pace) remained, and even when relative coordination was occurring, interfin cooperation (each attempting to proceed at the pace of the other) was still in evidence. He referred to the competitive aspect as the "maintenance tendency" and the cooperative aspect as the "magnet effect." Von Hoist's research suggests four major requirements for an experimental paradigm directed at the dynamics of the interlimb rhythmic coordinations typifying locomotion. First, the studied movement pattern should be analogous to the locomotory pattern without engendering movement of the body relative to the environment - a fictive or mimed locomotion. Second, the eigenvalues of the individual rhythmic movement units should be manipulable and easily quantified. Third, the interlimb system should be easily prepared in one of the two basic patterns of in phase and antiphase. And fourth, the focus of measurement and dynamical modeling should be on the interactions of phase. An experimental paradigm satisfying the first three requirements has been developed by Kugler and Turvey (1987) and used to examine a variety of coordination phenomena (Bingham et al. 1991; Rosenblum and Turvey et al. 1989; Schmidt (submitted); Schmidt et al. (in press); Turvey et al. 1988 (in press). Dynamical models satisfying the fourth requirement have been developed most notably by Kopell (1988a), Rand et al. (1988); Murray (1990); Haken et al. (1985). Characteristic of these various models is the effort to reduce complicated (high dimensional) dynamics to simpler (lower dimensional)
224
dynamics by averaging the effects of coupling over each oscillatory cycle. This renders interoscillator coupling dependent on ~b = (01 - 02) (where Oi is the phase of an individual oscillator and ~b, therefore, is the phase difference), and not on 01 and 02. The proposal is that even though the coupling influences may occur at varied times in an oscillator's cycle, and even though their effects depend on more than phase differences, mode locking phenomena can be predicted by analyzing the averaged effects of the coupling influences (Kopell 1988a, b). In sum, phase locking behavior of the full dynamics may, under some circumstances, be determined completely by the dynamics of the averaged system (Kopell 1988b). Kopell (1988a) has suggested that the results of pursuing this modeling strategy be referred to as Average Phase Difference (APD) Theory. In what follows we identify the major details of the experimental procedure satisfying the aforementioned requirements, provide an overview of the basic assumptions and predictions of APD dynamics, and summarize previous results that comport with these predictions. The experiments reported in the present article use the experimental procedure of Kugler and Turvey (1987) to further evaluate APD theory predictions about 1:1 frequency locking of biological oscillators.
1.1 Interlimb pendular rhythmic movements Figure 1 depicts the basic methodology. A person is shown seated and holding a pendulum in each hand. The pendulums can vary physically in shaft length and/or the mass of the attached bob. Because of these physical magnitudes, a person's comfortable swinging of an individual pendulum about an axis in the wrist (with other joints essentially immobile), parallel to the sagittal plane, will tend to a particular frequency and a particular amplitude (Kugler and Turvey 1987). Viewing the neuromuscular process as driving the motion, it is reasonable to assume that the eigenfrequency of a "wrist-pendulum system" is the eigenfrequency of the equivalent simple gravitational pendulum, co = (g/L) ~/2, where L is the simple pendulum length and g is the constant acceleration due to gravity. The quantity L is calculated from the magnitudes of shaft length, added mass, and hand mass, through the standard methods for representing any arbitrary rigid body oscillating about a fixed point as a simple pendulum (den Hartog 1948; Kugler and Turvey 1987). For interlimb coordination, if the pendulums held in each hand differ in physical dimensions (length, mass), then their eigenfrequencies will not correspond. The component rhythmic units will be in frequency competition. With humans as the subjects, the implementation of an in phase or antiphase relation between the rhythmic units is readily achieved by instruction. The effect of a given frequency competition can be evaluated, therefore, for an interlimb coordination "prepared" in phase or antiphase.
1.2 APD theory Of the several varieties of APD models, that developed by Rand et al. (1988) has proven especially well suited
Fig. 1. Front (left figure) and side (right figure) views of the experimental arrangement in which the subject oscillates two hand-held pendulums. The schematic depicts the four microphones of an ultrasonic 3-space digitizer arranged at the corners of a horizontal square grid beneath the subject. An ultrasonic emitter is attached to the lower end of each pendulum. The time varying positions of the emitters are recorded by computer
to addressing the observed coordinative pattern for pendular rhythmic movements (Schmidt et al. in press). The model is meant to provide insight into interactions among biological oscillators. It makes no assumptions, however, about the "state space" containing the dynamics of the individual rhythmic unit. The modeling is very general, based only on the phenomenologically observed oscillations; it is not directed at the oscillator's neurobiological constituents. Nonetheless, it is supposed that the modeling will facilitate the study and understanding of the collective behavior of systems of biological oscillators whose neuronal composition is unknown (see Kopell 1988a). The assumptions governing the development of a coupling dynamics focus upon (a) the characterization of the individual oscillator, and (b) the characterization of two or more coupled oscillators (for present purposes, only two coupled oscillators will be considered). To begin with, it is assumed that an individual oscillator is on a steady state limit cycle at all times (a closed orbit in the phase space of velocity and displacement, to which the system returns following a perturbation). It is further assumed that the oscillator's state can be specified by a single variable O(t) representing the phase of the limit cycle, where 0 passes from 0 rad to 2n rad in one cycle, and where 0 can be made proportional to the fraction of the cycle period that has elapsed. Consequently, 0 =09
(1)
where co is the eigenfrequency of the oscillator. By integration, and recognizing that successive oscillations are not distinguished, (1) gives
O(t) = cot + 0(0)(mod 2r0
(2)
Turning now to the coupling of oscillators characterized by (2), it is assumed that the coupled system lives on a two-dimensional torus T 2, meaning that it is representable by the geometric object swept out by transporting the origin of the circle (0 to 2re) defining the phases 01(0 of oscillator 1 around the circle (0 to 2re) defining the phases 02(0 of oscillator 2. The limit cycles of the individual oscillators are assumed to continue after coupling, although they may change shape. The parameterizing of 0 to run from 0 rad to 2r~ rad in a cycle permits continued reference to the same T 2 to
225
describe the oscillators' motions, and an assumption of structural stability implies that for sufficiently weak coupling, stable oscillations will continue (Kopell 1988a; Murray 1990). Given the preceding, a first pass on the behavior of an individual oscillator under coupling yields
Oi = ~o, + ~I,j(o,, oj)
(3)
Further developments require determination of the coupling function H~j. Because 0i is the single state variable of the individual oscillator, the coupling function Hq can depend only on 0t and 02 and cannot depend, by assumption, on factors such as the size of the limit cycle or the wave form of the oscillation. Further, in order for the motion expressed by (3) to be defined uniquely on T 2, Hij must be 2zc periodic. This simply expresses the fact that the effect of the one oscillator on the same phase point of the other oscillator must always be the same. One move to resolve the form of H~.jis to require that the coupling goes to zero when 01 = 02. This simplifying assumption defines what is known as diffusive coupling (Kopell 1988a; Murray 1990; Rand et al. 1988). It allows that H12(01,02) = H,2(02 - 0 , ) = H , ~ ( ~ )
(4)
Armed with (4) and the understanding that H~j must be 2re periodic, the requisite coupling function can be approximated by taking the first terms of the Fourier series. Letting the coefficient on the cosine term be zero results in H I 2 ( 0 1 , 0 2 ) = k~2 sin(02 -- 01)
(5)
Thus, for the two oscillators the motion equations take the form 01 = 091 + k12 sin(02 - 0L) 02 ----O)2 -1- k21 sin(01 - 02)
dp = arcsin( Ae~ / K )
(7)
where Ao = (~oI - co2) and K = (k12 + k21). If A~o = 0, and if K is positive in sign, then (7) exhibits a negative sinusoidal form with a negative second derivative (indexing a stable point) at the intercept values of 0 and 2n and a positive second derivative (indexing an unstable point) at n. These conditions characterize inphase behavior (Rand et al. 1988; Schmidt et al. in press). Conversely, if the sign of K is negative, then the sinusoidal form is positive with a negative second derivative at the intercept value of ~z and a positive second derivative at 0 and 2n. These conditions characterize antiphase behavior (Rand et al. 1988; Schmidt et al. in press). If Ao9 ~ 0, then the stable points will deviate from 0 and it to a degree depending on the magnitude of IAe;[. A general expression for the varied equilibria of (7) can be determined under the assumption that, at equi-
(8)
Regardless of whether K is positive or negative, the values of the arcsine function will be the same due to its symmetry. This means that (8) gives the same absolute values for in phase and antiphase.
1.3 Investigating APD theory Schmidt et al. (in press) examined the validity of (7) for modeling the interlimb coordination of rhythmic movements using the wrist-pendulum paradigm. The measurable quantities relevant to evaluating the model are ~b, Aog, and o~c. The average value of ~b over a dynamic run is ~b,ve and the intended value is ~ ( w i t h ~ symbolizing "intended"). Ao9 is a control parameter that governs the degree of competition between the oscillators (to comport with von Holst's intuitions about the nature of interlimb coordination). During a trial in the wrist-pendulum paradigm, the frequency competition between the rhythmic subsystems will be constant. The common or coupled frequency coc of the rhythmic units is another control parameter. It likewise can be held constant during a trial (by means of a pacing metronome). Schmidt et al. (in press) found that (7) modeled equally the two modes ~b~,= 0 and the, = n and that ]q~aw- ~ [ scaled to Ao9 as interacted with Ao~ in determining kbaw- ~b~,l and q~ variability. The changes induced by increasing 09c expected from (8). It was also demonstrated that ~oc were those expected from decreasing coupling strength K in (7) and (8). The implication is that K can be systematically manipulated by manipulating coc. Equation (8) can be approximated, therefore, by ~b = arcsin(Aco)(og~)
(6)
Subtracting the motion equation for oscillator 2 from that of oscillator 1 provides us with a low dimensional dynamic in q~ q~ = A~o - K sin ~b
librium, the time derivative of ~ equals 0. Consequently, the equilibrium values of ~b are given by
(9)
This latter relation is particularly useful because a decrease in coupling strength has been a major assumption in the modeling of transitions, observed at higher frequencies, from antiphase to in phase limb coordination (Haken et al. 1985; Kelso et al. 1986; Schmidt et al. 1990).
1.4 Significance of the eigenfrequency of the coupled system When two pendular rhythmic units are mode locked, the system thus formed possesses, like the subsystems from which it is formed, an eigenfrequency. Kugler and Turvey (1987) presented arguments and data (see also Turvey et al. 1988) for characterizing a coupling of left and right wrist-pendulum systems as a single virtual system with an equivalent simple pendulum length. If the two wrist-pendulum systems were coupled such that 01 was always, at every instant, identically equal to 02, or to (02 + tO, then the two oscillators could be considered as rigidly connected. The simple pendulum equivalent Lequivalent of a compound pendulum so composed (that is, of two pendulums connected by a rigid bar) is given by Lequivalent
=
(ml l~ + m2lZ~)/(ml l~ + mfl2)
(10)
226 where mi and l,- refer to the mass and the equivalent simple pendulum length, respectively, of an individual (compound) pendulum system. (In words, the righthand-side of the equation reads: sum of the individual moments of inertia divided by the sum of the corresponding individual static moments.) By the use of (10), two coupled pendulums of lengths l~ and 12 are experessed as a virtual (v) pendulum of length L~. Consequently, the eigenfrequency of the system of coupled pendulums considered as a virtual single pendulum is o~v = (g/L~)~/2. In the same way that the eigenfrequencies ~o1 = ( g / l ~ ) ~/~ and ~o~ = ( g / l ~ ) ~/2 of the uncoupled wrist-pendulum systems can be taken as characterizing their respective maintenance tendencies, so it is the case that ~v can be taken as characterizing the maintenance tendency of the coupled wrist-pendulum system. For present purposes, the importance of the preceding is that the operating frequency of the coupled system ~oc should be e x p r e s s e d - by Kugler and Turvey's (1987) h y p o t h e s i s - in units of o~. That is, the control parameter is not purely e~ but Oc/o~. By this hypothesis, (9) becomes
and ~oc = 7.22 (rad/s) did not. It was the case, however, that the two frequency conditions ~0c = 5.84 (rad/s), and ~c =7.22 (rad/s) tended to be higher for all values of Aa~ than the frequency condition ~oc = ~ov. In agreement with APD theory, Schmidt et al. found a significant interaction between Aa~ and O9c, and no interactions with ~b~,, in the determination of tpave. The deviation of thaw from ~b~, increased with A~o and was increased further by ~oC. The coupled frequency conditions examined by Schmidt et al. approximated ~oc/o~v~> 1. The present experiment manipulates ~oc/~ov precisely and examines coupled frequency conditions satisfying o9c/~ov ~< 1 conditions where the frequency of oscillation is slower than the eigenfrequency of the coupled pendulum system. It does so for ~b~,= n with five values of Aco. Where Schmidt et al. had found an amplification of I~ave - - ~ l for a given IA ,I ~ 0 in the domain of O9c/ 09~ > 1, it was expected that the present experiment should reveal a reduction of Iq~av~- q~,l in the domain of r < 1.
~b = arcsin(A~o)(og~/o9~)
2.1 M e t h o d
(11)
1.5 Predictions
Three experiments on interlimb 1:1 frequency locking are reported in which Aa~, o9~, and o9~/o9v are manipulated. They are directed at providing further substantiation of the Rand et al. equations. There are three major predictions: (a) that the frequency at which a 2-oscillator coupled system is running, and the difference between the eigenfrequencies of the two oscillators, relate multiplicatively in determining ~b.... (b) that this relationship will hold for coordinations where tp~, = 0 and ~b~, = re, and (c) that the physically relevant measure of the coupling frequency is o9~/o~. More specifically, it is predicted from (11) that, for both tk~, = 0 and tp~, = re, when Aco = 0, deviations of tp,ve from tp~, will be zero independently of ~o~/co~. Further, for both ~b~,= 0 and q~, = re, when [Ao91~ 0, deviations of ~ba~ from ~b~,will be inversely dependent on o9c/~o~. Elaborating on the latter prediction, when ~Oc/CO~ < 1 (the coupled system is oscillating slower than its eigenfrequency), deviations of qb~w from ~b~, for a given IAo I ~ 0 will be less than when 09,/o9~ = 1 (the coupled system its oscillating at is eigenfrequency), and when o9~/a~ > 1 (the coupled system is oscillating faster than its eigenfrequency), deviations of tP~v~ from ~b~, for a given IAco[~ 0 will be greater than when ~o~/~o~= 1.
2 Experiment 1 Schmidt et al. (in press) investigated the predictions of APD theory for three values of ~o, : that at which the 1:1 frequency locking was most comfortable, o9~ = o9~, together with a~ = 5.84 (rad/s) and o9~ = 7.22 (rad/s). In the Schmidt et al. experiment, to~ was not scaled to e~,. Whereas the frequency condition coc = co~ varied with Aco, the frequency conditions ~o~ = 5.84 (rad/s),
Subjects. One woman and four men participated. Two participants were faculty at the University of Connecticut, and three participants were graduate students. The age range was 28 years to 49 years. All participants were right handed. Apparatus. The pendulums were constructed using the specifications described in Kugler and Turvey (1987). Each consisted of an ash dowel inserted into a bicycle hand grip. Weights were attached to a 10 cm long bolt that was drilled through the dowel at right angles 2 cm from the bottom. Four such pendulums were used, one of 74 cm, one of 48 cm, and two each of 31 cm. For all four pendulums the added mass was 200 g. There were five pairings that defined the coupled left-hand pendulum/right-hand pendulum systems: 74 cm/31 cm (Coupled System 1), 48cm/31 cm (Coupled System 2), 31 cm/31 cm (Coupled System 3), 31 cm/48 cm (Coupled System 4), and 31 cm/74 cm (Coupled System 5). The equivalent simple pendulum lengths li of the individual pendulums (consisting of the attached mass, the dowel, and the hand of the subject) were calculated according to procedures identified in Kugler and Turvey (1987), as was the equivalent simple pendulum length Lv of each of the five coupled systems (see (10)). The gravitational eigenfrequencies of the uncoupled systems, (/)left, O)right, the difference between time, (O~left- Ognght), and the gravitational eigenfrequency of the coupled system, ~ov, are displayed in Table 1 for each Coupled System 1-5. The subject sat in a specially designed chair with arm rests to support the forearms. The arm rests were designed to restrict oscillations to the wrist; the two forearms were kept in contact with the arm supports throughout a trial. The chair also provided a raised support for the subject's legs so that they did not interfere with the ultrasonic acquisition of the data (see below).
227 1. The gravitational eigenfrequencies(rad/s) of the uncoupled systems, coleft,c%~t, the differencebetweenthem, (Am = COleft coright), and the gravitational eigenfrequenciesof the coupled systems, coo TaMe
-
-
Coupled System
Oheft
(/)right
A(zI
(J)v
1 2 3 4 5
4.002 5.049 6.509 6.509 6.509
6.509 6.509 6.509 5.049 4.002
--2.507 -- 1.450 0 1.450 2.507
4.351 5.468 6.509 5.468 4.351
Pendular trajectories were measured using an Ultrasonic 3-Space Digitizer (SAC Coporation, Westport, CT). An ultrasound emitter was affixed to the end of each pendulum. An ultrasound "spark" was issued from each emitter at 90 Hz. The digitizer operates by registering each emission using any three of four microphones arranged to form a square grid. The digitizer calculates the distance of each emitter from each microphone, thereby pinpointing the position of the emitters in 3-space at the time of the emission. This slant range information was stored for later use on a 80286 based microcomputer using MASS digitizer software (En#heering Solutions, Columbus, Ohio). This software and analogous routines written on a Macintosh II use the slant range time series to calculate the primary angle of excursion of the pendulums and their relative phase angle q~. An electronic metronome was used to pace the pendulum oscillations at a preset frequency. Procedure and design. On a given trial, the subject was instructed to coordinate the two hand-held pendulums in antiphase (~b~,= re) at 1:1 frequency locking. The values of co,. at which a coupled system was 1:1 frequency locked were scaled to coy according to co~/ co~,= 1, coo/coy= 0.77, co~/cov= 0.63. Coupled Systems 1-5 differed in coy (see Table 1). Satisfying the preceding values of coc meant, therefore, that the three cocs differed for each coupled system. The 15 cocs were determined beforehand, given the preceding calculations of the respective covs, and were controlled during a trial by means of a metronome. The design of the experiment, therefore, was a factorial consisting of three values of coc/co~ and five values of Aco. There were two 32 s trials per condition for a total of 30 trials per subject. The experiment was divided into two blocks of 15 trials, each block including all conditions. To reduce the number of individual pendulum changes, the three coupled frequencies for a given coupled system were given in succession. In Block 1 the order was Coupled System 1 to Coupled System 5; in Block 2 the order was Coupled System 5 to Coupled System 1. Each subject was given intructions and allowed to practice before the beginning of a session. He or she was told to place the forearms squarely on the arm rests, to gaze straight ahead without looking at the pendulums, and to swing the pendulums smoothly back and forth. The subject was instructed to hold the
pendulums firmly in the hands so that rotation of the pendulum was generated about the wrist joint rather than about the finger joints. The motions of the pendulums were restricted to planes parallel to the subject's sagittal plane. The subject was allowed to practice the task in the various conditions prior to the experiment proper. Prefatory to a trial, the metronome was started, ticking at the rate designated for the particular conditions tested on that trial. The subject was allowed as much time as needed to achieve the essential antiphase pattern at the prescribed metronome frequency. Data recording began after the subject indicated that he or she was ready, which was usually within about 10 s. An experimental session lasted between 60 and 80 min. Data reduction. The digitized displacement time series of the pendulums were smoothed using a triangular moving average procedure with a window size of seven samples. Each trial was subjected to software analyses to determine the frequency of oscillation of each pendulum, the time series of the relative phase angle between the two pendulum systems, the power spectra of this relative phase time series, and the total power associated with each of these spectra. A peak picking algorithm was employed to determine the time of maximum forward extension of the pendular trajectories. From the peak extension times, the frequency of oscillation for the nth cycle was calculated as fe = 1/[(time of peak extensionn + l) -- (time of peak extensionn)]. The mean frequency of oscillation for a trial was calculated from these cycle frequencies. The phase angle of each pendulum (0i) was calculated for each sample (90/s) of the displacement time series to produce a time series of 0;. The phase angle was of wrist pendulum i at sample j(Oo) was calculated as 0;i = arctan(2 o ~Axe)
(12)
where the numerator on the right hand side is the velocity 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 displacement of the time series at sample j minus the average displacement for the trial. The relative phase angle (~b;) between the two pendulums was calculated for each sample as 0rightj- 0leftj- The ~b; that the subject intended was re. The mean of the ~b time series allows an evaluation of 9 how the subjected satisfied this task demand.
2.2 Results Three examples of the time series of ~b are shown in Fig. 2. Inspection reveals that ~b was not fixed but variable around q~avc in the manner of phase entrainment (Schmidt et al. in press) rather than phase locking; this is the usual observation for the interlimb coordination of pendular rhythmic movements (e.g., Schmidt et al. 1991). Comparison of actual coc to the metronome-specified coc revealed, across subjects and trials, a minimum average difference of the order of 1 ms and a maximum of the order of 6 ms (comparable values held for Experiments 2-3).
228
Experiment 1
3.6" 4"5 l 4.0
(Ov 3.4
3
3.2 3.0
2.5
2.8
2.0
. . . . . . . . . . . . . 5 10 15 20 25 30
0
35
2.6
40
-
-3
-2
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0
.4' .31 4.04"5l
.
.
1
-
2
,
3
Experiment 2
.21
"77 (Ov
>CO r
9 0.77(ov II .63O)v . , -3 -2 -1
-.2
;2t 0
%3
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5
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10
.
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36t %4
.
30
4.5
35
40
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0
1
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3.4
4.0
3.2J
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2.8 * 1.25~
~',,,.'~
I
2.5 9
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0
5
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_
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15
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.
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20 25 Time (s)
.
.
30
_
,
35
,
40
Fig. 2. Typical phase difference time series as a function of coupled frequency
The expectation f r o m (8), in the experimentally testable f o r m o f (11), is that average phase-locking behavior ~b~w is multiplicatively dependent on A~o and ~o~/ogv. The m a g n i t u d e o f ~b~vr as a function o f Ao~ and cor is presented in Fig. 3, averaged over the five subjects. (The value o f ~b~w was determined as the mean o f the two values o f qh~w obtained f r o m the two trials per condition.) Inspection reveals that the deviation o f ~bav~ f r o m n depended systematically on Aco when O~c/ ~ = 1; this increase o f ]~b~vr - n [ as a function o f IA~I concurs with previous observations for conditions in which ~ / ~ o ~ = 1 ( R o s e n b l u m and Turvey 1988; Schmidt et al. in press; Turvey et al. 1986). I m p o r tantly, inspection o f Fig 3 reveals that there was no
-2
-1
0 1 A(o(rad/s)
_
.
2
_
.
3
Fig. 3. Average phase difference ~bave as a function of the difference in uncoupled frequencies d~o and the relative coupled frequency o~/~% for Experiments I-3. In Experiments 1 and 3, the intended interlimb coordination is antiphase. In Experiment 2, the intended interlimb coordination is in phase. In Experiments 1 and 2, ~%/~o~~
Biol. Cybern. 67, 223-231 (1992)
9 Springer-Verlag 1992
Average phase difference theory and 1:1 phase entrainment in interlimb coordination Dagmar Sternad ~, M. T. Turvey ~, and R. C. Schmidt ~,2 ] Center for the Ecological Study of Perception and Action, University of Connecticut, Storrs, CT 06268, USA 2 Tulane University, New Orleans, LA 70115, USA Received January 14, 1992/Accepted in revised form February 25, 1992
Abstract. The dynamics of coupled biological oscillators can be modeled by averaging the effects of coupling over each oscillatory cycle so that the coupling depends on the phase difference q~ between the two oscillators and not on their specific states. Average phase difference theory claims that mode locking phenomena can be predicted by the average effects of the coupling influences. As a starting point for both empirical and theoretical investigations, Rand et al. (1988) have proposed dfb/dt = Ao~ - K sin ~b, with phase-locked solutions ~b =arcsin(Aco/K), where Ao9 is the difference between the uncoupled frequencies and K is the coupling strength. Phase-locking was evaluated in three experiments using an interlimb coordination paradigm in which a person oscillates hand-held pendulums. Ao9 was controlled through length differences in the left and right pendulums. The coupled frequency O~cwas varied by a metronome, and scaled to the eigenfrequency o9~ of the coupled system; K was assumed to vary inversely with coc. The results indicate that: (1)Aco and K contribute multiplicatively to ~b; (2) ~b = 0 or ~b = n regardless of K when Aco = 0; (3) ~b ~ 0 or ~b ~ r~ regardless of Ao9 when K is large (relative to Ao~); (4) results (1) to (3) hold identically for both in phase and antiphase coordination. The results also indicate that the relevant frequency is o~c/o~v rather than co~. Discussion highlighted the significance of confirming q~ = arcsin(Aog/K) for more general treatments of phase-locking, such as circle map dynamics, and for the 1:1 phase-entrainment which characterizes biological movement systems.
1 Introduction Von Hoist (1937/1973) addressed the issue of movement coordination with medulla transected Labrus, a fish that swims with its main body axis immobile. Following
Correspondence to: M. T. Turvey
surgery, the fins oscillate autonomously for hours. With this preparation von Hoist observed the waxing and waning of phase-locking. Two different sized fins, with different uncoupled periods, would oscillate with a fixed phase relation and a common coupled frequency- a state von Hoist referred to as absolute coordination. Or they would oscillate without phase and frequency locking - a state yon Hoist referred to as relative coordination. Both states would be seen intermittently, and whenever one dominated, signs of the other would be visible. Von Hoist concluded that even when absolute coordination was achieved, interfin competition (each attempting to proceed at its own pace) remained, and even when relative coordination was occurring, interfin cooperation (each attempting to proceed at the pace of the other) was still in evidence. He referred to the competitive aspect as the "maintenance tendency" and the cooperative aspect as the "magnet effect." Von Hoist's research suggests four major requirements for an experimental paradigm directed at the dynamics of the interlimb rhythmic coordinations typifying locomotion. First, the studied movement pattern should be analogous to the locomotory pattern without engendering movement of the body relative to the environment - a fictive or mimed locomotion. Second, the eigenvalues of the individual rhythmic movement units should be manipulable and easily quantified. Third, the interlimb system should be easily prepared in one of the two basic patterns of in phase and antiphase. And fourth, the focus of measurement and dynamical modeling should be on the interactions of phase. An experimental paradigm satisfying the first three requirements has been developed by Kugler and Turvey (1987) and used to examine a variety of coordination phenomena (Bingham et al. 1991; Rosenblum and Turvey et al. 1989; Schmidt (submitted); Schmidt et al. (in press); Turvey et al. 1988 (in press). Dynamical models satisfying the fourth requirement have been developed most notably by Kopell (1988a), Rand et al. (1988); Murray (1990); Haken et al. (1985). Characteristic of these various models is the effort to reduce complicated (high dimensional) dynamics to simpler (lower dimensional)
224
dynamics by averaging the effects of coupling over each oscillatory cycle. This renders interoscillator coupling dependent on ~b = (01 - 02) (where Oi is the phase of an individual oscillator and ~b, therefore, is the phase difference), and not on 01 and 02. The proposal is that even though the coupling influences may occur at varied times in an oscillator's cycle, and even though their effects depend on more than phase differences, mode locking phenomena can be predicted by analyzing the averaged effects of the coupling influences (Kopell 1988a, b). In sum, phase locking behavior of the full dynamics may, under some circumstances, be determined completely by the dynamics of the averaged system (Kopell 1988b). Kopell (1988a) has suggested that the results of pursuing this modeling strategy be referred to as Average Phase Difference (APD) Theory. In what follows we identify the major details of the experimental procedure satisfying the aforementioned requirements, provide an overview of the basic assumptions and predictions of APD dynamics, and summarize previous results that comport with these predictions. The experiments reported in the present article use the experimental procedure of Kugler and Turvey (1987) to further evaluate APD theory predictions about 1:1 frequency locking of biological oscillators.
1.1 Interlimb pendular rhythmic movements Figure 1 depicts the basic methodology. A person is shown seated and holding a pendulum in each hand. The pendulums can vary physically in shaft length and/or the mass of the attached bob. Because of these physical magnitudes, a person's comfortable swinging of an individual pendulum about an axis in the wrist (with other joints essentially immobile), parallel to the sagittal plane, will tend to a particular frequency and a particular amplitude (Kugler and Turvey 1987). Viewing the neuromuscular process as driving the motion, it is reasonable to assume that the eigenfrequency of a "wrist-pendulum system" is the eigenfrequency of the equivalent simple gravitational pendulum, co = (g/L) ~/2, where L is the simple pendulum length and g is the constant acceleration due to gravity. The quantity L is calculated from the magnitudes of shaft length, added mass, and hand mass, through the standard methods for representing any arbitrary rigid body oscillating about a fixed point as a simple pendulum (den Hartog 1948; Kugler and Turvey 1987). For interlimb coordination, if the pendulums held in each hand differ in physical dimensions (length, mass), then their eigenfrequencies will not correspond. The component rhythmic units will be in frequency competition. With humans as the subjects, the implementation of an in phase or antiphase relation between the rhythmic units is readily achieved by instruction. The effect of a given frequency competition can be evaluated, therefore, for an interlimb coordination "prepared" in phase or antiphase.
1.2 APD theory Of the several varieties of APD models, that developed by Rand et al. (1988) has proven especially well suited
Fig. 1. Front (left figure) and side (right figure) views of the experimental arrangement in which the subject oscillates two hand-held pendulums. The schematic depicts the four microphones of an ultrasonic 3-space digitizer arranged at the corners of a horizontal square grid beneath the subject. An ultrasonic emitter is attached to the lower end of each pendulum. The time varying positions of the emitters are recorded by computer
to addressing the observed coordinative pattern for pendular rhythmic movements (Schmidt et al. in press). The model is meant to provide insight into interactions among biological oscillators. It makes no assumptions, however, about the "state space" containing the dynamics of the individual rhythmic unit. The modeling is very general, based only on the phenomenologically observed oscillations; it is not directed at the oscillator's neurobiological constituents. Nonetheless, it is supposed that the modeling will facilitate the study and understanding of the collective behavior of systems of biological oscillators whose neuronal composition is unknown (see Kopell 1988a). The assumptions governing the development of a coupling dynamics focus upon (a) the characterization of the individual oscillator, and (b) the characterization of two or more coupled oscillators (for present purposes, only two coupled oscillators will be considered). To begin with, it is assumed that an individual oscillator is on a steady state limit cycle at all times (a closed orbit in the phase space of velocity and displacement, to which the system returns following a perturbation). It is further assumed that the oscillator's state can be specified by a single variable O(t) representing the phase of the limit cycle, where 0 passes from 0 rad to 2n rad in one cycle, and where 0 can be made proportional to the fraction of the cycle period that has elapsed. Consequently, 0 =09
(1)
where co is the eigenfrequency of the oscillator. By integration, and recognizing that successive oscillations are not distinguished, (1) gives
O(t) = cot + 0(0)(mod 2r0
(2)
Turning now to the coupling of oscillators characterized by (2), it is assumed that the coupled system lives on a two-dimensional torus T 2, meaning that it is representable by the geometric object swept out by transporting the origin of the circle (0 to 2re) defining the phases 01(0 of oscillator 1 around the circle (0 to 2re) defining the phases 02(0 of oscillator 2. The limit cycles of the individual oscillators are assumed to continue after coupling, although they may change shape. The parameterizing of 0 to run from 0 rad to 2r~ rad in a cycle permits continued reference to the same T 2 to
225
describe the oscillators' motions, and an assumption of structural stability implies that for sufficiently weak coupling, stable oscillations will continue (Kopell 1988a; Murray 1990). Given the preceding, a first pass on the behavior of an individual oscillator under coupling yields
Oi = ~o, + ~I,j(o,, oj)
(3)
Further developments require determination of the coupling function H~j. Because 0i is the single state variable of the individual oscillator, the coupling function Hq can depend only on 0t and 02 and cannot depend, by assumption, on factors such as the size of the limit cycle or the wave form of the oscillation. Further, in order for the motion expressed by (3) to be defined uniquely on T 2, Hij must be 2zc periodic. This simply expresses the fact that the effect of the one oscillator on the same phase point of the other oscillator must always be the same. One move to resolve the form of H~.jis to require that the coupling goes to zero when 01 = 02. This simplifying assumption defines what is known as diffusive coupling (Kopell 1988a; Murray 1990; Rand et al. 1988). It allows that H12(01,02) = H,2(02 - 0 , ) = H , ~ ( ~ )
(4)
Armed with (4) and the understanding that H~j must be 2re periodic, the requisite coupling function can be approximated by taking the first terms of the Fourier series. Letting the coefficient on the cosine term be zero results in H I 2 ( 0 1 , 0 2 ) = k~2 sin(02 -- 01)
(5)
Thus, for the two oscillators the motion equations take the form 01 = 091 + k12 sin(02 - 0L) 02 ----O)2 -1- k21 sin(01 - 02)
dp = arcsin( Ae~ / K )
(7)
where Ao = (~oI - co2) and K = (k12 + k21). If A~o = 0, and if K is positive in sign, then (7) exhibits a negative sinusoidal form with a negative second derivative (indexing a stable point) at the intercept values of 0 and 2n and a positive second derivative (indexing an unstable point) at n. These conditions characterize inphase behavior (Rand et al. 1988; Schmidt et al. in press). Conversely, if the sign of K is negative, then the sinusoidal form is positive with a negative second derivative at the intercept value of ~z and a positive second derivative at 0 and 2n. These conditions characterize antiphase behavior (Rand et al. 1988; Schmidt et al. in press). If Ao9 ~ 0, then the stable points will deviate from 0 and it to a degree depending on the magnitude of IAe;[. A general expression for the varied equilibria of (7) can be determined under the assumption that, at equi-
(8)
Regardless of whether K is positive or negative, the values of the arcsine function will be the same due to its symmetry. This means that (8) gives the same absolute values for in phase and antiphase.
1.3 Investigating APD theory Schmidt et al. (in press) examined the validity of (7) for modeling the interlimb coordination of rhythmic movements using the wrist-pendulum paradigm. The measurable quantities relevant to evaluating the model are ~b, Aog, and o~c. The average value of ~b over a dynamic run is ~b,ve and the intended value is ~ ( w i t h ~ symbolizing "intended"). Ao9 is a control parameter that governs the degree of competition between the oscillators (to comport with von Holst's intuitions about the nature of interlimb coordination). During a trial in the wrist-pendulum paradigm, the frequency competition between the rhythmic subsystems will be constant. The common or coupled frequency coc of the rhythmic units is another control parameter. It likewise can be held constant during a trial (by means of a pacing metronome). Schmidt et al. (in press) found that (7) modeled equally the two modes ~b~,= 0 and the, = n and that ]q~aw- ~ [ scaled to Ao9 as interacted with Ao~ in determining kbaw- ~b~,l and q~ variability. The changes induced by increasing 09c expected from (8). It was also demonstrated that ~oc were those expected from decreasing coupling strength K in (7) and (8). The implication is that K can be systematically manipulated by manipulating coc. Equation (8) can be approximated, therefore, by ~b = arcsin(Aco)(og~)
(6)
Subtracting the motion equation for oscillator 2 from that of oscillator 1 provides us with a low dimensional dynamic in q~ q~ = A~o - K sin ~b
librium, the time derivative of ~ equals 0. Consequently, the equilibrium values of ~b are given by
(9)
This latter relation is particularly useful because a decrease in coupling strength has been a major assumption in the modeling of transitions, observed at higher frequencies, from antiphase to in phase limb coordination (Haken et al. 1985; Kelso et al. 1986; Schmidt et al. 1990).
1.4 Significance of the eigenfrequency of the coupled system When two pendular rhythmic units are mode locked, the system thus formed possesses, like the subsystems from which it is formed, an eigenfrequency. Kugler and Turvey (1987) presented arguments and data (see also Turvey et al. 1988) for characterizing a coupling of left and right wrist-pendulum systems as a single virtual system with an equivalent simple pendulum length. If the two wrist-pendulum systems were coupled such that 01 was always, at every instant, identically equal to 02, or to (02 + tO, then the two oscillators could be considered as rigidly connected. The simple pendulum equivalent Lequivalent of a compound pendulum so composed (that is, of two pendulums connected by a rigid bar) is given by Lequivalent
=
(ml l~ + m2lZ~)/(ml l~ + mfl2)
(10)
226 where mi and l,- refer to the mass and the equivalent simple pendulum length, respectively, of an individual (compound) pendulum system. (In words, the righthand-side of the equation reads: sum of the individual moments of inertia divided by the sum of the corresponding individual static moments.) By the use of (10), two coupled pendulums of lengths l~ and 12 are experessed as a virtual (v) pendulum of length L~. Consequently, the eigenfrequency of the system of coupled pendulums considered as a virtual single pendulum is o~v = (g/L~)~/2. In the same way that the eigenfrequencies ~o1 = ( g / l ~ ) ~/~ and ~o~ = ( g / l ~ ) ~/2 of the uncoupled wrist-pendulum systems can be taken as characterizing their respective maintenance tendencies, so it is the case that ~v can be taken as characterizing the maintenance tendency of the coupled wrist-pendulum system. For present purposes, the importance of the preceding is that the operating frequency of the coupled system ~oc should be e x p r e s s e d - by Kugler and Turvey's (1987) h y p o t h e s i s - in units of o~. That is, the control parameter is not purely e~ but Oc/o~. By this hypothesis, (9) becomes
and ~oc = 7.22 (rad/s) did not. It was the case, however, that the two frequency conditions ~0c = 5.84 (rad/s), and ~c =7.22 (rad/s) tended to be higher for all values of Aa~ than the frequency condition ~oc = ~ov. In agreement with APD theory, Schmidt et al. found a significant interaction between Aa~ and O9c, and no interactions with ~b~,, in the determination of tpave. The deviation of thaw from ~b~, increased with A~o and was increased further by ~oC. The coupled frequency conditions examined by Schmidt et al. approximated ~oc/o~v~> 1. The present experiment manipulates ~oc/~ov precisely and examines coupled frequency conditions satisfying o9c/~ov ~< 1 conditions where the frequency of oscillation is slower than the eigenfrequency of the coupled pendulum system. It does so for ~b~,= n with five values of Aco. Where Schmidt et al. had found an amplification of I~ave - - ~ l for a given IA ,I ~ 0 in the domain of O9c/ 09~ > 1, it was expected that the present experiment should reveal a reduction of Iq~av~- q~,l in the domain of r < 1.
~b = arcsin(A~o)(og~/o9~)
2.1 M e t h o d
(11)
1.5 Predictions
Three experiments on interlimb 1:1 frequency locking are reported in which Aa~, o9~, and o9~/o9v are manipulated. They are directed at providing further substantiation of the Rand et al. equations. There are three major predictions: (a) that the frequency at which a 2-oscillator coupled system is running, and the difference between the eigenfrequencies of the two oscillators, relate multiplicatively in determining ~b.... (b) that this relationship will hold for coordinations where tp~, = 0 and ~b~, = re, and (c) that the physically relevant measure of the coupling frequency is o9~/o~. More specifically, it is predicted from (11) that, for both tk~, = 0 and tp~, = re, when Aco = 0, deviations of tp,ve from tp~, will be zero independently of ~o~/co~. Further, for both ~b~,= 0 and q~, = re, when [Ao91~ 0, deviations of ~ba~ from ~b~,will be inversely dependent on o9c/~o~. Elaborating on the latter prediction, when ~Oc/CO~ < 1 (the coupled system is oscillating slower than its eigenfrequency), deviations of qb~w from ~b~, for a given IAo I ~ 0 will be less than when 09,/o9~ = 1 (the coupled system its oscillating at is eigenfrequency), and when o9~/a~ > 1 (the coupled system is oscillating faster than its eigenfrequency), deviations of tP~v~ from ~b~, for a given IAco[~ 0 will be greater than when ~o~/~o~= 1.
2 Experiment 1 Schmidt et al. (in press) investigated the predictions of APD theory for three values of ~o, : that at which the 1:1 frequency locking was most comfortable, o9~ = o9~, together with a~ = 5.84 (rad/s) and o9~ = 7.22 (rad/s). In the Schmidt et al. experiment, to~ was not scaled to e~,. Whereas the frequency condition coc = co~ varied with Aco, the frequency conditions ~o~ = 5.84 (rad/s),
Subjects. One woman and four men participated. Two participants were faculty at the University of Connecticut, and three participants were graduate students. The age range was 28 years to 49 years. All participants were right handed. Apparatus. The pendulums were constructed using the specifications described in Kugler and Turvey (1987). Each consisted of an ash dowel inserted into a bicycle hand grip. Weights were attached to a 10 cm long bolt that was drilled through the dowel at right angles 2 cm from the bottom. Four such pendulums were used, one of 74 cm, one of 48 cm, and two each of 31 cm. For all four pendulums the added mass was 200 g. There were five pairings that defined the coupled left-hand pendulum/right-hand pendulum systems: 74 cm/31 cm (Coupled System 1), 48cm/31 cm (Coupled System 2), 31 cm/31 cm (Coupled System 3), 31 cm/48 cm (Coupled System 4), and 31 cm/74 cm (Coupled System 5). The equivalent simple pendulum lengths li of the individual pendulums (consisting of the attached mass, the dowel, and the hand of the subject) were calculated according to procedures identified in Kugler and Turvey (1987), as was the equivalent simple pendulum length Lv of each of the five coupled systems (see (10)). The gravitational eigenfrequencies of the uncoupled systems, (/)left, O)right, the difference between time, (O~left- Ognght), and the gravitational eigenfrequency of the coupled system, ~ov, are displayed in Table 1 for each Coupled System 1-5. The subject sat in a specially designed chair with arm rests to support the forearms. The arm rests were designed to restrict oscillations to the wrist; the two forearms were kept in contact with the arm supports throughout a trial. The chair also provided a raised support for the subject's legs so that they did not interfere with the ultrasonic acquisition of the data (see below).
227 1. The gravitational eigenfrequencies(rad/s) of the uncoupled systems, coleft,c%~t, the differencebetweenthem, (Am = COleft coright), and the gravitational eigenfrequenciesof the coupled systems, coo TaMe
-
-
Coupled System
Oheft
(/)right
A(zI
(J)v
1 2 3 4 5
4.002 5.049 6.509 6.509 6.509
6.509 6.509 6.509 5.049 4.002
--2.507 -- 1.450 0 1.450 2.507
4.351 5.468 6.509 5.468 4.351
Pendular trajectories were measured using an Ultrasonic 3-Space Digitizer (SAC Coporation, Westport, CT). An ultrasound emitter was affixed to the end of each pendulum. An ultrasound "spark" was issued from each emitter at 90 Hz. The digitizer operates by registering each emission using any three of four microphones arranged to form a square grid. The digitizer calculates the distance of each emitter from each microphone, thereby pinpointing the position of the emitters in 3-space at the time of the emission. This slant range information was stored for later use on a 80286 based microcomputer using MASS digitizer software (En#heering Solutions, Columbus, Ohio). This software and analogous routines written on a Macintosh II use the slant range time series to calculate the primary angle of excursion of the pendulums and their relative phase angle q~. An electronic metronome was used to pace the pendulum oscillations at a preset frequency. Procedure and design. On a given trial, the subject was instructed to coordinate the two hand-held pendulums in antiphase (~b~,= re) at 1:1 frequency locking. The values of co,. at which a coupled system was 1:1 frequency locked were scaled to coy according to co~/ co~,= 1, coo/coy= 0.77, co~/cov= 0.63. Coupled Systems 1-5 differed in coy (see Table 1). Satisfying the preceding values of coc meant, therefore, that the three cocs differed for each coupled system. The 15 cocs were determined beforehand, given the preceding calculations of the respective covs, and were controlled during a trial by means of a metronome. The design of the experiment, therefore, was a factorial consisting of three values of coc/co~ and five values of Aco. There were two 32 s trials per condition for a total of 30 trials per subject. The experiment was divided into two blocks of 15 trials, each block including all conditions. To reduce the number of individual pendulum changes, the three coupled frequencies for a given coupled system were given in succession. In Block 1 the order was Coupled System 1 to Coupled System 5; in Block 2 the order was Coupled System 5 to Coupled System 1. Each subject was given intructions and allowed to practice before the beginning of a session. He or she was told to place the forearms squarely on the arm rests, to gaze straight ahead without looking at the pendulums, and to swing the pendulums smoothly back and forth. The subject was instructed to hold the
pendulums firmly in the hands so that rotation of the pendulum was generated about the wrist joint rather than about the finger joints. The motions of the pendulums were restricted to planes parallel to the subject's sagittal plane. The subject was allowed to practice the task in the various conditions prior to the experiment proper. Prefatory to a trial, the metronome was started, ticking at the rate designated for the particular conditions tested on that trial. The subject was allowed as much time as needed to achieve the essential antiphase pattern at the prescribed metronome frequency. Data recording began after the subject indicated that he or she was ready, which was usually within about 10 s. An experimental session lasted between 60 and 80 min. Data reduction. The digitized displacement time series of the pendulums were smoothed using a triangular moving average procedure with a window size of seven samples. Each trial was subjected to software analyses to determine the frequency of oscillation of each pendulum, the time series of the relative phase angle between the two pendulum systems, the power spectra of this relative phase time series, and the total power associated with each of these spectra. A peak picking algorithm was employed to determine the time of maximum forward extension of the pendular trajectories. From the peak extension times, the frequency of oscillation for the nth cycle was calculated as fe = 1/[(time of peak extensionn + l) -- (time of peak extensionn)]. The mean frequency of oscillation for a trial was calculated from these cycle frequencies. The phase angle of each pendulum (0i) was calculated for each sample (90/s) of the displacement time series to produce a time series of 0;. The phase angle was of wrist pendulum i at sample j(Oo) was calculated as 0;i = arctan(2 o ~Axe)
(12)
where the numerator on the right hand side is the velocity 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 displacement of the time series at sample j minus the average displacement for the trial. The relative phase angle (~b;) between the two pendulums was calculated for each sample as 0rightj- 0leftj- The ~b; that the subject intended was re. The mean of the ~b time series allows an evaluation of 9 how the subjected satisfied this task demand.
2.2 Results Three examples of the time series of ~b are shown in Fig. 2. Inspection reveals that ~b was not fixed but variable around q~avc in the manner of phase entrainment (Schmidt et al. in press) rather than phase locking; this is the usual observation for the interlimb coordination of pendular rhythmic movements (e.g., Schmidt et al. 1991). Comparison of actual coc to the metronome-specified coc revealed, across subjects and trials, a minimum average difference of the order of 1 ms and a maximum of the order of 6 ms (comparable values held for Experiments 2-3).
228
Experiment 1
3.6" 4"5 l 4.0
(Ov 3.4
3
3.2 3.0
2.5
2.8
2.0
. . . . . . . . . . . . . 5 10 15 20 25 30
0
35
2.6
40
-
-3
-2
-1
0
.4' .31 4.04"5l
.
.
1
-
2
,
3
Experiment 2
.21
"77 (Ov
>CO r
9 0.77(ov II .63O)v . , -3 -2 -1
-.2
;2t 0
%3
.
.
.
5
.
.
.
10
.
.
.
15
.
20
.
.
.
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25
36t %4
.
30
4.5
35
40
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0
1
2
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3.4
4.0
3.2J
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3.0
2.8 * 1.25~
~',,,.'~
I
2.5 9
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0
5
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10
_
.
15
_
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,
20 25 Time (s)
.
.
30
_
,
35
,
40
Fig. 2. Typical phase difference time series as a function of coupled frequency
The expectation f r o m (8), in the experimentally testable f o r m o f (11), is that average phase-locking behavior ~b~w is multiplicatively dependent on A~o and ~o~/ogv. The m a g n i t u d e o f ~b~vr as a function o f Ao~ and cor is presented in Fig. 3, averaged over the five subjects. (The value o f ~b~w was determined as the mean o f the two values o f qh~w obtained f r o m the two trials per condition.) Inspection reveals that the deviation o f ~bav~ f r o m n depended systematically on Aco when O~c/ ~ = 1; this increase o f ]~b~vr - n [ as a function o f IA~I concurs with previous observations for conditions in which ~ / ~ o ~ = 1 ( R o s e n b l u m and Turvey 1988; Schmidt et al. in press; Turvey et al. 1986). I m p o r tantly, inspection o f Fig 3 reveals that there was no
-2
-1
0 1 A(o(rad/s)
_
.
2
_
.
3
Fig. 3. Average phase difference ~bave as a function of the difference in uncoupled frequencies d~o and the relative coupled frequency o~/~% for Experiments I-3. In Experiments 1 and 3, the intended interlimb coordination is antiphase. In Experiment 2, the intended interlimb coordination is in phase. In Experiments 1 and 2, ~%/~o~~
