Biological Cybemetics

Biol. Cybern. 63, 257-270 (1990)

9 Sprinser-Verlag1990

A Dynamic Theory of Coordination of Discrete Movement G. Sch6ner* Center for Complex Systems, Florida Atlantic University, Boca Raton, FL 33431, USA Received September 9, 1989/Accepted February 26, 1990

Abstract. The concepts of pattern dynamics and their

adaptation through behavioral information, developed in the context of rhythmic movement coordination, are generalized to describe discrete movements of single components and the coordination of multiple components in discrete movement. In a first step we consider only one spatial component and study the temporal order inherent in discrete movement in terms of stable, reproducible space-time relationships. The coordination of discrete movement is captured in terms of relative timing. Using an exactly solvable nonlinear oscillator as a mathematical model, we show how the timing properties of discrete movement can be described by these pattern dynamics and discuss the relation of the pattern variables to observable end-effector movement. By coupling several such component dynamics in a fashion analogous to models of rhythmic movement coordination we capture the coordination of discrete movements of two components. We find the tendency to synchronize the component movements as the discrete analogon of in,phase locking and study its breakdown when the components become too different in their dynamic properties. The concept of temporal stability leads to the prediction that remote compensatory responses occur such as the restore synchronization when one component is perturbed. This prediction can be used to test the theory. We find that the discrete analogon to antiphase locking in rhythmic movement is a tendency to move sequentially, a finding that can also be subjected to empirical test.

1 Introduction

Recently, dynamic theories have been proposed to understand the patterns of coordination of rhythmic

Institut f~ir Neuroinformatik, Ruhr-Universit/it Bochum, ND04, Postfach 102148, D-4630 Bochum, Federal Republic of Germany * Present address:

movements (Kelso and Sch6ner 1987; Sch6ner and Kelso 1988a). The basic concepts, which are motivated by physical theories of pattern formation (Haken 1983), are the description of such coordination patterns by collective variables and the identification of pattern dynamics as equations of motion of these collective variables. In such a theory, the stability is an important property of a pattern and the various measures of stability have led to prediction, in particular near phase transitions involving loss of stability of a coordination pattern. In a second step, the concept of behavioral information was introduced (Sch6ner and Kelso 1988b, c) to express the adaptation of patterns of coordination to various behavioral requirements (environmental, memorized, intentional) in terms of contributions to the pattern dynamics. The question we want to address in this article is how these concepts can be generalized to understand the coordination of discrete movement. At a first glance, discrete movement may seem quite different from rhythmic movement, and, in particular, the initiation of a discrete movement seems nontrivial within a dynamic theory. Empirical work on discrete movement has particularly stressed two aspects. On the one hand, the temporal order inherent in discrete movement has been described in terms of reproducible movement times as a function of various task parameters (e.g., Woodworth 1899; Fitts 1954; Beggs and Howarth 1972; Langolf et al. 1976), and in terms of reproducible peak velocity-amplitude relationships (e.g., Wadman et al. 1980; Milner 1986) or, more generally, in terms of the shape of position and velocity time functions (e.g., Woodworth 1903; Jeannerod 1984; Morasso 1981; Hogan and Flash 1987). On the other hand, much work has been done on relationships between the various spatial components of a discrete movement (the path) (e.g., Soechting and Lacquaniti 1981; Morasso 1981; Abend et al. 1982). Theoretical work has invoked concepts of information theory (Fitts 1954), extremal principles (e.g., Flash and Hogan 1985; Hasan 1986; Uno et al. 1989), and learning and neural network theory (e.g., Bullock and Grossberg 1988; Kawato et al. 1987).

258 Our aim in this article is to provide a unified description of posture, discrete and rhythmic movement in terms of pattern dynamics. The stress will lie on temporal order in such movements, described in terms of stable and reproducible timing. In single component movements we address movement time-amplitude relationships (the spatial aspect of Fitts' law). By restricting the theory to one spatial dimension per component we avoid the problem of spatial trajectory formation in a first step. Moreover, we shall not be very careful about describing the exact shape of the position-time or velocity-time functions in quantitative detail, and shall be satisfied if global parameters such as movement time and amplitude are correct. Coordination of discrete movement is then captured in terms of relative timing of the movements of two components. This is motivated by experiments (Kelso et al. 1979a) that found a tendency to synchronize discrete movements of two limbs as a characteristic signature of coordination (see also, Kelso et al. 1979b; Kelso et al. 1983; Marteniuk et al. 1984; Corcos 1984). The key step in achieving a dynamic description is to identify intrinsic dynamics in terms of initial and target postural states. These intrinsic dynamics determine the timing properties of discrete movement. The intention to move is captured as behavioral information and stabilizes a position-velocity curve, but does not completely specify timing. Coordination of discrete movement can then be understood as the result of coupling the dynamics of different components in a fashion entirely analogous to models of rhythmic movement coordination (Haken et al. 1985). The central concept of pattern stability gives rise to predictions on remote compensatory reactions to perturbations in one component. As a new aspect we find sequential movement as the discrete analogon of anti-phase locking of rhythmic movement. The article is organized as follows: In Sect. 2 we briefly review the theoretical framework and note the model dynamics in the rhythmic case. Here and in Sect. 4 we discuss the nature of the collective variables on which the theory is based. The model dynamics for discrete movement of a single end-effector are introduced building from local models of postural states in Sect. 3. We analyze the resulting movement patterns in Sect. 4, providing numerical values for model parameters. Coordination of discrete movement of two components is modelled in Sect. 5. The resulting coordination patterns are discussed and predictions are made based on the stability of these coordination patterns. In the Conclusion (Sect. 6) we point to possible and necessary generalizations of the present theory. 2 The Theoretical Framework

Viewing patterns of coordination in terms of their dynamics (i.e., their equation of motion) the key step is to characterize coordination patterns by collective variables so that the pattern dynamics can be determined as

equations of motion of these collective variables. The collective variables that describe the patterns of coordination must be identified empirically, including a choice of level of observation. These collective variables capture the order or relations among components at the chosen level of description and thus reflect the underlying neural organization. The choice of collective variables is closely linked to the identification of the dynamics of these variables. Such dynamics must be modeled theoretically based on several experimental constraints. In particular, observable (i.e., reproducible, stationary over a certain time scale) patterns are mapped onto attractors of the collective variable dynamics. The parameters of the dynamics depend on biological boundary conditions (e.g., environmental, task, or energetic constraints) and can be determined on the basis of the dependence of the observed patterns (and their stability) on these boundary conditions. In as far as these parameters are unspecific, that is, do not prescribe a particular coordination pattern, we may call the pattern dynamics intrinsic dynamics. Other aspects of the theory are fluctuations, which are conceptually important as a stability measure and because they determine important time scales; time scales relations, which determine how the dynamics can be observed; multistability and switching among multiple coordination patterns, which show the essential nonlinearity of the pattern dynamics; and loss of stability, which leads to switching of pattern (see Kelso et al. 1987; Schrner and Kelso 1988a, for details). It is important to note that the pattern dynamics determined in this fashion is function-specific. The same structure may be governed by different dynamics in other functions and other structures may be governed by very similar dynamics in the same function. In the present study we shall deal with posture, discrete and rhythmic movement of a single end-effector in one spatial dimension. Our chosen level of observation will be that of the end-effector kinematics. In particular, we shall attempt to describe the movement patterns by end-effector position, x, and velocity, v. The idea is that dynamics in a two-dimensional phase space will be sufficient to describe the observed types of trajectories. Note, that these variables are collective with respect to the underlying coordinated activity of multiple muscles. These variables must actually be interpreted more abstractly as expressing the coordination activity of the central nervous system in a given function. The relation between the kinematic variables and these abstract variables is close and unique in a stable coordination pattern, but less clear when kinematic variables are perturbed externally (see discussion at the end of Sect. 4 and in Sect. 6). Self-paced rhythmic movement can be described by dynamics on this level of description (Kay et al. 1987). In this case, the observed pattern is mapped onto a limit cycle attractor of nonlinear second order dynamics of x. The dynamics can be modeled as (Haken et al. 1985; Kay et al. 1987): .~ + t~.~ + O)2X -'[- fl.~3 + y.~X 2 = 0

(1)

259 where ~ > 0, to > 0. fl > 0 and F > 0 are model parameters. Note, that the concrete functional form of (t) is based on simplicity arguments only and is not of primary importance. Experimental observations on reproducible amplitude-frequency relationships and on relaxation after mechanical perturbations of the movement are accounted for by the radial stability of the limit cycle oscillator. Phase resetting analysis of such perturbation experiments (Kay et al. 1990) reveals that the phase does not, in general, relax to its unperturbed value, which is evidence for the assumed autonomy of the underlying dynamics (1). The details of the phase relaxation show, however, that the degree to which mechanical perturbations affect the underlying oscillatory dynamics varies according to phase and strength of the perturbation. This hints that a more complete dynamic description, that explicitly addresses the coupling of mechanical degrees of freedom to the nervous system level of coordination, may be necessary to understand phase stability. The coordination of rhythmic movements of two homologous limbs has been studied within the present theoretical framework in a series of experiments and model calculations (Kelso 1984; Haken et al. 1985). Experimentally, only two forms of movement, phaselocked in-phase or anti-phase are stable. The anti-phase pattern loses stability at higher movement frequencies and a spontaneous switch to in-phase occurs. Theoretically, these patterns can be accounted for by coupling the oscillator dynamics of the two component system, for example (Haken et al. 1985): "JC1"Jr 0~'~1 "JI-toEx I -Jr fl~3 At- ~)~lX2 =

g(Xl, )CI;

X23r

~2 + ~Yc2+ to2x2 + [~5c3 + 7Yc2x~ = g(x2, 5c2; x~, ~1)

(2)

with g(Xl, "~1; X2,-~2) = k(-~l--3c2) q-l(-~l--)c2)(Xl--x2) 2In this case it is possible to derive a closed dynamics of the phase difference, ~b, of these two oscillators in the limit of weak nonlinearity: q~ = - a sin(~b) - 2 b sin(2~b)

(3)

where a = - ( k + 2lr2o) > 0, and b =/r2/2 > 0. A series of experimental and theoretical studies based on these relative phase dynamics showed close match of model and experiments, including quantitative predictions (Schrner et al. 1986; Kelso et al. 1986; Scholz et al. 1987). This model system can serve as a reference point because both its experimental and theoretical analyses are fairly complete. More generally, of course, the tendency to synchronize different rhythmically moving components has been established in numerous experiments dating back to yon Hoist (1939/1973), who discovered this tendency in an impressive series of studies in a variety of species and introduced the concept of the M-effect to account for the phase attraction between different movement rhythms. Coordinated movement is not only stable under given biological boundary conditions, but can also adapt to specific requirements. In the present theoretical framework the concept of behavioral information expresses such requirements as required patterns, which

are described by the same types of collective variables as the coordination patterns themselves (Schrner and Kelso 1988b, c). Behavioral information is then made part of the pattern dynamics as an additive contribution to the vector field of strength cinfothat attracts the coordination pattern to the required pattern. The specific behavioral requirements may arise from the environment via action-perception patterns (Schrner and Kelso 1988d), or from memory after learning and activating a particular movement pattern (Sch/Sner and Kelso 1988d; Schrner 1989). Importantly, an intention to change behavior can also be treated as behavioral information in this sense (Schrner and Kelso 1988c). Experiments in the model system of bimanual rhythmic movement coordination, for instance, showed that intentional switching from one pattern of coordination (either in-phase or anti-phase) to the other can be understood in this way (Kelso et al. 1988). 3 Development of the Model Dynamics for a Single Effector Discrete movement always involves intention in one form or another: the movement must be initiated at a discrete time. Our strategy is to treat the intention to move as behavioral information in the sense defined above. This means, that the intention to move is expressed as a part of the end-effector dynamics that stabilizes the intended coordination pattern, the position-velocity curve (as a piece of a limit cycle). However, the temporal order in discrete movement is accounted for by the intrinsic dynamics, that is, by the pattern dynamics in the absence of an intention to move. Both duration and strength of the behavioral information can be unspecific to the performed movement. The intrinsic dynamics are determined by the task definition, in particular, by the initial posture and the definition of the target of the discrete movement. Moreover, we require that rhythmic movement be a possible stable solution of the intrinsic dynamics. In this way we can both account for the stability of the curve in the (x, v)-plane during discrete movement as well as achieve a unified description of the various patterns of movement (posture, discrete and rhythmic movement) of a one-dimensional end-effector. A posture state of the end-effector can be experimentally defined in a variety of paradigms. For instance, an active, voluntary movement to a visually defined position (see, e.g., Feldman 1966; Georgopoulos et al. 1981) may give rise to a postural state in the end position. A posture position can also be learned and can be maintained even with reduced sensory feedback (e.g., Bizzi et al. 1976; Kelso 1977). Theoretically, we define a posture state as a fixed point attractor (x = xpost, v = 0) of the (x, v)-dynamics. In the vicinity of such a state we can approximate the dynamics by a local, linear model, i.e.: .X=V 2 /) : --0~postV -- ('Dpost(X -- Xpost) -1- N/~t

(4)

260

0~post and o9post are parameters specific to the posture state x = Xpost. Here, and below, we treat random fluctuations by adding a stochastic force that is modeled as gaussian white noise, ~t, of unit variance (cf. Sch6ner et al. 1986). The posture state is stable as measured, for instance, by the relaxation time to the posture state after a small perturbation: Tre~~ 1/~post, more precisely: where

= ~'2/Gtpost 2 2 Trel [2/{~po~t[1--X/1--4o9po~t/~post]}

for 2ogpost > 0~post for 2ogpost< ~post

(5) The fluctuations around the posture state are another measure of this stability (see, e.g., Wang and Uhlenbeck 1945, Sect. 10 there):

SDx = /2~postO2ost

(6)

By measuring the mean posture state, its variability and the relaxation time, all parameters of the local posture model can be estimated. (To obtain an estimate of U)post2 the power spectrum of x can be analyzed.) The second structure of the intrinsic dynamics describes a situation in which two posture states are defined and stable under the same conditions and at the same time. Experimentally such a situation can be defined as follows: one posture state, the initial position, is given as discussed above. A second state is defined as the target of a movement. This can occur via visual targetting (e.g., Fitts 1954; Beggs and Howarth 1972), by learning a target (e.g., Polit and Bizzi 1979; Georgopoulous et al. 1981), or by preselecting a target (e.g., Stelmach et al. 1975; Kelso 1977). Theoretically, the vector field is assumed to have two point attractors. While the two local models continue to be approximated by the linear Eq. (4), the complete dynamics is of necessity nonlinear. An approximation can be based on a power series expansion of the force f ( x , v) in:

section a limit cycle description of rhythmic movement is possible and consistent with various experimental facts (Kay et al. 1987). In terms of developing the model we need to add terms in the power series expansion that contain v. (The reason for this is that the linear term must change sign to destabilize fixed point solution, but in the absence of nonlinear damping terms the solutions become unbounded.) A simple, low-order example is:

f ( x , v) = otv + o92x + bl x3 Jr b2xZv + " "

(9)

One can show that the phase diagram of this system contains regions, where a single fixed point is stable, regions with two stable fixed points, as well as regions with a stable limit cycle (Holmes and Rand 1980). This simplest model is not necessarily the best. One question is whether it is possible to parameterize the system to account for observed relations between movement parameters. The most general approach would be to fit a general functional form of the vector-field to movement data constrained by stability requirements (for general approaches of that type see, e.g., Cremers and Hfibler 1987; Crutchfield and McNamara 1987). Our approach will be instead to provide concrete and simple functional forms within which the determination of parameters can be performed (in terms of orders of magnitude) by analytic approximation. We think that such an approach provides more insight, in particular with respect to the main question addressed here, that is, temporal order in single and coordinated discrete movement. For our purpose then, an exactly solvable nonlinear oscillator model with a phase diagram that also contains all dynamic structures discussed above, is most suitable (Gonzalez and Piro 1987). The model reads as follows:

= - ( a 2 + co2)x + 2av - 4bx2v

(10)

+ 2abx 3 - b2x 5 + x//Q~t ~=v

(~ = - f ( x , v) + v/Q~t

(7)

Because all terms containing v vanish at the fixed point attractor we cannot constrain their coefficients by information on the stationary states and therefore we discard these terms except for the linear damping, ~tv, as in the local posture models. Assuming symmetry of the initial position and the target we are left only with:

f ( x , v) = ~v + o92x + blx 3 + " "

(8)

In third order we already find two posture states for ogo2/b~ < 0 at Xpost= +_~/--o9g/bl. These are stable for ~oo2< 0, while a single posture state at x = 0 is stable for O9o 2 > 0 ( ~ > 0 in both cases). This can be seen by straightforward linear stability analysis. Again, parameter determination can occur via the two local models obained from (7) by linearization. The third structure that we would like to include in the intrinsic dynamics is rhythmic movement captured as limit cycle oscillations. As discussed in the previous

where a, b, and o92 are model parameters. This model differs from (9) only through an additional term of the order x 5 and a specialized parameterization that copes with only three model parameters (apart from the noise strength). A phase diagram of this system in the (a, o92/ Iog[)-plane is shown in Fig. 1 (cf. Fig. 3 of Gonzales and Piro 1987). We must distinguish five rrgimes: (a) and (b): a single, globally stable point attractor at x - - 0 exists (in (a) as a focus, in (b) as a sink); (c) and (d): two point attractors at: xpost = + x / ( a + Iogl)/b

(11)

exist (in addition to two saddle points and one source in (c) and one saddle point in (d)); (e) a limit cycle exists. These different rrgimes can therefore be used to model the three intrinsic behaviors discussed above: a single posture state ((a),(b)), two simultaneously defined posture states ((c), (d)), rhythmic movement (e). The usefulness of this exactly solvable model becomes apparent if we note the following parametric

261

tion, that is, as a part of the vectorfield of the pattern dynamics that stabilizes the phase plane curve:

v

.~=V

I) = - f ( x ,

v) + C~intem(/)fintent (X, t3) + , q / a ~ t

(15)

Here, the behavioral information is explicitly timedependent to capture on-set and off-set of an intention to move. In a first approximation we neglect the details of the underlying processes and describe this force as:

Cintent(1)fintent(X, V) = Z[to ' to+At](t)Cintentfiintent(X, 12)

(16)

where X is the indicator function of the time interval [to, to + At]: f~ ,x I'

Fig. 1. Phase diagram of (10) in the (a, oj2/lcoD-plane (cf. Fig. 3 of Gonzales and Piro 1987) Note that in our conventions o~2 can take on negative values corresponding to a purely imaginary ~. In five separate regions characteristic phase plane trajectories illustrate the nature of the phase portrait. The regions are bounded by the positive ~o2/Icol-axis, the a-axis, and the negative half-rays a = _+,o2/Io, I. See text for details. In a discrete movement the system is taken from r~gimes (c) or (d) into r~gime (e) (right arrow) by behavioral information for the duration of an intention to move, At. After that time, the system returns to its original parameter setting (left arrow) with two postural states

Zt,o. ,o + All(t) =

for t e [ t o , t o + A t ] elsewhere

The functional form of fntent will be such that the system is taken from the r~gimes (c) or (d) into the r~gime (e) where limit cycle movement is stable (see illustration in Fig. 1). Within a range of permissible values the resulting movement depends only little (see next section) on the exact duration, At, and strength, Cintent> 0, (i.e., to which exact point within rrgime (e) the system is pushed by the behavioral information). The simplest choice is therefore: fintent

=

-- X

(18)

so that the complete dynamics now read ~=v = -ftotal(x, v)

relationships (Gonzalez and Piro 1987) in the oscillatory rrgime, where analytical results are generally most wanting: The cycle time, T, of the globally stable limit cycle is T = 2n/~

(12)

The relaxation time to the limit cycle (inverse reciprocal of the non-trivial Floquet multiplier) is: l Zrel = 2--a

(13)

The amplitude of the limit cycle is A = 2x/~/b

(14).

Obviously, the relevant measures depend in a very simple way on the different model parameters, which can therefore easily be identified (see next section for details). Our view of discrete movement can now be formulated more precisely as follows: By task definition two posture states, the initial position and the target, are intrinsically stable. The intention to move stabilizes a limit cycle. This phase plane curve is stable only over a period of time that allows for approximately a halfcycle of movement. After that time, the intrinsic dynamics again dominate so that the system relaxes autonomously to the second (target) fixed point. The intention to move is expressed as behavioral informa-

(17)

(19)

= --{ a2 + ~ + Ztt0,,o + atl (t)cintent }x + 2av - 4 b x 2 v + 2abx 3 - bZx 5 + w/Q~t

For ~o2< 0, a > 0, and Cintent>--~02 the system is in r~gimes (c) or (d) outside the time window of the intentional information and in the rrgime (e) while the intentional information is active.

4 Model Solutions and Predictions for a Single Effector

As discussed above the different coordination patterns are contained as solutions in the same nonlinear dynamical structure Eq. (19). To describe a particular pattern the parameter values have to be chosen adequately. In addition, a coordinate transformation may be necessary to map the real behavior onto (19). To capture posture at a position Xvo~twe choose in (19) coordinates that map the posture state onto the origin: x' = x - Xpost

(20)

The parameters of (19) must be chosen in the regions (a) or (b). We recover a local model from (19) by linearizing around the origin of the (x', v)-plane and thus obtain (4) for the original coordinates with O~post ~ - - 2 a

>0

(-'/)post 2 = a 2 + o92 > 0

(21)

262

In region (a) we thus encounter underdamped relaxation (09post> %ost/2), while in the region (b) we find overdamped relaxation ( 0 9 p o s t < ~ p o s t / 2 ) . The noise strength can be determined as before by (6). Note, that through this local description the model parameters of the full dynamics can be determined except for b, which measures only nonlinear contributions. The case of two simultaneously defined posture states is captured by the regions (c) and (d) of the phase diagram Fig. I. If the two posture positions are x~ and x2, we use coordinates in (19), that are centered symmetrically about these two positions so that the assumption of functional equivalence of initial posture and target make sense: x' = x

x

2.5

0.0

-2.5 i

00

X 1 "~-X 2

(22) 2 The distance, A, between the two positions is then given by (cf. (ll)):

-50

1.0 i

/

A = IXl - x21 = 2x/a +b1091

(23)

By linearizing around each posture state we obtain again the local posture model Eq. (4) with Xpost= Xl or X p o s t = X 2 and: %ost = 2a + 41091> 0

(,Opos t2

=

4[091(a + I091) > 0

(24)

Because 09post< ~post/2 we find that the local posture states are always overdamped in this model. We can use these local models to obtain estimates of the realistic orders of magnitude of the model parameters. Calculating the relaxation time to any of the posture states from (24) with (5) we find "~rel = 1/(21091). We estimate this time in experiments to be of the order of rrel "~ 0.5 s (cf., e.g., Feldman 1966). As a result we have 10921,-, 1 Hz 2. It is difficult to estimate the oscillatory component of the relaxation to a posture state. We assume that it is of the same order of magnitude as the relaxation time, so that we are close to critical damping. In this case we find 09post2~ 100 Hz 2, which leads to a ~ 10 Hz. The spatial range of movements is in most studies limited (in order of magnitude) to between 1 and 50 cm, which leads to a range of b between 0.005 and 50 Hz/cm z. These choices specify physical units of the collective variables (x, v) and time t, namely, cm, cm/s, and s, respectively. Note, that in this case all model parameters of the full dynamics can be determined by experimental information on the two posture states and their stability only. In region (e) of the phase diagram we obtain rhythmic movement. To be consistent with the symmetry assumption in (19) we need to center the phase plane trajectory by the transformation Eq. (22), where x~ and x2 denote the two turning points of the movement. The parameter determination can then be based on the cycle time, T, the relaxation time to the limit cycle, rrel, and the amplitude, A, of the movement (cf. (12) to (14)). Again realistic orders of magnitudes can be estimated based on experimental results (of., in particular, Kay et al. 1987). Assuming cycle times of order 0.5 s, relaxation times of the same order, and amplitudes of

2.0 i

time [s]l

f

Fig. 2. The position, x, and velocity, v, variables evolving in time during a discrete movement from an initial postural state at x = 3.43 cm to the target state at x = - 3 . 4 3 cm. The shown trajectories were obtained by a numeric solution of (19) at the standard parameter set Eq. (25). Behavioral information representing an intention to move is present in the time interval At = 0.3 s from t = 1 s to 1.3 s with a strength Cint~nt= 20 H z 2. The time To when half the distance is covered is used as a measure of the timing of the movement

order 10cm we find: 09 ,-~ 10Hz, a ,,~ 1 Hz, and b ,-~ 0.1 Hz/cm 2. Discrete movement is the result of the interplay of intrinsic dynamics with two stable posture states and behavioral information that stabilizes a limit cycle. The two new parameters are the duration, At, and strength, /2intent, of the behavioral information. Before we study in detail the parametric relationships, we demonstrate how discrete movement works in the present theory in a simple example 1. In Fig. 2, the parameters of the intrinsic dynamics were chosen as a=10Hz

b=lHz/cm 2 092=_3Hz 2

(25)

in the r6gime (d) of the phase diagram. These values will serve as the standard parameter set in further examples below. An intention to move occurred in the time window from 1. to 1.3 s with a strength Cintent = 20 HZ2 that is sufficient to shift the system to region (e) while behavioral information is present. Obviously, the system switches to the new state within approximately 1. s. Note, that the exact trajectory shape is not necessarily realistic. In particular, the velocity profile is not symmetric and does not approximate well the bellshape often observed in experiment. This is due to the particular functional form we chose for our model dynamics and can be remedied when an additional level of description expressing the linkage of nervous system control to kinematics is introduced (cf. Discussion). i Here and below the differential equations are solved numerically using a Runge-Kutta-Gills procedure

263

To describe the temporal order in discrete movement in more detail we characterize the duration of the movement by various measures, e.g., time for the initiation of movement 2 to time of peak velocity, or the time for initiation to completion of the discrete movement as judged by some criterion deviation (movement time). For the simulations we found the time, To, from movement initiation to the zero-crossing of the position variable, i.e., the time in which the first half of the distance is covered (see Fig. 2), to be a useful, parameter-free measure, and will report most results in terms of this measure. Rough analytical estimates of the movement time can be provided based on the properties of the limit cycle that is stabilized by behavioral information. While intention is "on", the half cycle time of the limit cycle movement is (12) T/2 = n/(x/co 2 + Cintent ). After the intention is turned off, the relaxation to the second posture position occurs on the time scale Zrel= 1/(21o~1). The movement time is therefore determined largely by T/2 if A t ,,~ T/2, and largely by A t + Zr,l if A t ~, T/2. Examining the influence of the intrinsic dynamics on the timing of a discrete movement we find that duration is independent of the nonlinearity parameter, b, as expected from the estimates. Likewise, the duration depends very little on the dissipation parameter, a, again consistent with the estimates: we found a change of To by a factor of 1.3 as a was changed by three orders of magnitude. Because we found that the noise level, Q, has little influence on the nature of the discrete movement as long as the noise is not excessive as judged by the fluctuations in the posture states we neglected noise in the subsequent numerical simulations. The intrinsic dynamics determine the timing of a discrete movement only through the parameter co2, which is why the particular functional form (10) of the intrinsic dynamics is so convenient for analysis (cf. also (12)). In Fig. 3a we have plotted the duration measure To as a function of 092 with all other parameters fixed at the standard values. For given parameters, At and Cintent, of the behavioral information, discrete movement is obtained only in a window of values for ~o2. Within this window, the movement time may vary by a factor of about 2 or more, as shown in the figure. The parameters of the behavioral information, At and cintr can also be varied within certain windows inside of which the solutions of the dynamics, (19), describe a single discrete movement. Typically, beyond one boundary of the window (e.g., At too short or Cintent tOO small), the system relaxes back to the initial posture state without achieving the target posture, while beyond the other boundary, the system may move back to the initial state after having reached the target or may even oscillate several times back and forth. Although these

2 Note, that in the model onset of behavioral information, to, and onset of movement are, in general, coincident. Delays between a "go" signal and onset of movement (reaction time) are related to the underlying processes that give rise to the behavioral information in (19). These processes have not been modeled here

"rdsl' 1.0 0.8 0.6 J

0.4 0.2

-J[:]

a 0.0 1.0

2.0

3.0

4.0

5.0

6.0

1.2 1.0 0.8 0.6 0.4 0.2

~t[~]~

0.0 0.2

0.4

0.6

0.8

1.0

%[s]' 1.4 1.2 1.0

0.8 0.6

Z

0.4 0.2 0.0

c

. 20

. 40

C i n t [HzZ]~ . . . . 60 80 100 120

Fig. 3 a-e. The estimate of movement time, To, (time in which half the distance is covered) is shown as a function of three model parameters: a -o~2; b At; e cintent. These data are obtained from simulations of the dynamics, (19)

windows depend on the parameters of the intrinsic dynamics, we found that their size remains roughly constant. Typical values are At ~ [0.20 s, 0.95 s] and cintent~[14Hz 2, 130Hz2]. These windows result if in each case the other parameters assume the values of the standard parameter set. In Fig. 3b and 3c we show the durations of the discrete movements for these parameter ranges. There is virtually no dependence of the movement time measure, To, on the duration, At, of the behavioral information "command" beyond the left boundary of the window of admissible values. (At that

264

boundary, A t ~ 0.1 s, the system is close to the separatrix of the two fixed point attractors when behavioral information "is turned off" leading to a very slow transient.) Likewise, the dependence on the strength, Cintent, of behavioral information is strong only at the left boundary of its admissible window (same reason). Beyond that boundary, the movement time depends weakly on the exact value of Cintcnt, as can be expected from above estimates. Thus, movement timing is largely determined by the intrinsic dynamics as defined by the two posture states, initial and target position. However, behavioral information can influence the timing to a certain degree, so that experimental parameters related to intention (such as instruction on speed of movement) can be expressed in this model. A consistent finding in studies of discrete movement is existence of a lawful relation between amplitude of movement and movement time. This relation can take the form of a monotonic increase in fast discrete movement (e.g., Fitts 1954; Langolf et al. 1976) as well as in rhythmic movements with a controlled frequency (e.g., Kay 1986). In more complex movements without explicit timing contraints, constancy of movement time as amplitude varies can be found (Viviani and Terzuolo 1980). If we exclude the precision requirement from consideration we can take Fitts' law (Fitts 1954) as a convenient summary of the relation between among movement time and amplitude in fast discrete movements. In our notation this law reduces to: T o = A F 'b OF

In A

(26)

where A F and Be are task dependent empirical constants whose order of magnitude is usually Ae ~ 0.1 and Be ~ 0.3 (note that Ar varies more strongly with the precision requirements; cf. Fitts 1954). Here we aim to show that our model can describe such a relation between movement parameters in terms of a corresponding relation of model parameters: In the model movement time is determined by the parameter 092 (cf. Fig. 3a). The parameter a determines relaxation properties of the system (cf. (13) and (24)). At present no systematic experimental assessment of these relaxation times as a function of movement amplitude exists. For simplicity we assume that relaxation does not depend on amplitude. Thus, the parameter b = b(co2) remains to encode the relation of (26). We can approximate the functional form of this relationship by finding a functional fit of the dependence of To on co2 (Fig. 3a) and matching this to (26). Using an exponential Ansatz we find b = exp[1.8 -- 0.2[~o21- 0.13[o~14] (if co is measured in Hz) where we have used the above orders of magnitude for the constants of Fitts' law and the data of Fig. 3a. Using this parameter relationship we find the movement time-amplitude relationship plotted in Fig. 4. Of course, this is only an example, which shows, however, how we can describe movement parameter relationships by relationships among model parameters. Subsequently we shall use this parameterization to model the coordination of discrete movements of two components with different intrinsic movement times (Sect. 5).

Wo[~] 1.0 0.8 0.6 0.4 0.2 ,

,

,

10

20

30

A[e m ]~

0.0 40

Fig. 4. The relationship between To and amplitude when the model parameter co2 is varied as in Fig. 3a and b = b(~ 2) as indicated in the text. This functional dependence roughly approximates the space-time part of Fitts' law (26)

An important concept of a dynamic theory of coordination patterns is that of temporal stability. For movement of a single component, we may view the underlying coordination of muscles, joints and nervous processes as a stable pattern. What does that imply for the macroscopic movement? Unfortunately, this is a relatively complicated question in the case of discrete movement, because the pattern that is stable may change over the course of the movement. Initially and toward the end of the movement, the corresponding posture states are stable patterns as discussed earlier (see Sect. 4). In the present model, we have assumed that the phase plane trajectory is stable during part of the discrete movement. This means, that the system is attracted to a limit during the period when behavioral information is present. The temporal stability of a pattern is responsible for its persistence under varying and fluctuating conditions. In the case of rhythmic movement, the degree of persistence, and thus, the degree of stability can be measured because the pattern is maintained over periods of time much larger than typical relaxation times. The same is not true for discrete movement, where the movement itself exists only over a time scale comparable to what can be expected as typical relaxation times, (i.e., ~0.5 s), based on measures in the rhythmic movement case (e.g., Kay et al. 1990). Therefore pattern fluctuations cannot be used to measure stability 3. For the same reason, the relaxation time itself as a direct measure of temporal stability may not be easy to implement except for the initial and final postural states: In principle, during the discrete movement a perturbation may relax only partially, that is, the relation between position and velocity may remain changed to the end of the movement. In the case of perturbations, an additional question arises as to what mechanical perturbations applied to an end-effector mean for the coordination system as 3 A possibility is, however, to interpret inter-trial variability of movement parameters as a measure of the underlying stability

265 described by the pattern dynamics, (19). This question was briefly discussed for rhythmic movement in Sect. 2 in the context of phase resetting analysis (Kay et al. 1990). In the case of discrete movement a perturbation of the coordination system Eq. (19) leads to relaxation back towards the phase plane curve (x, v) given by the limit cycle attractor solution in the presence of behavioral information and towards the target fixed point in the absence of behavioral information. The positional trajectory, x(t), itself is not stable, however. This is evident from the simulations shown in Fig. 5, where a perturbed trajectory is compared to an unperturbed one. While the phase plane trajectories of the two solutions converge, the spatial trajectories do not converge (due to a shift in the phase of the underlying oscillator). In experiments with monkeys who performed discrete movements towards learned visual targets Bizzi xa

2.5 0.0 -2.5 v

L

i

1.0 i

2.0 i

0.0 -50 -100

time [s I

50.0 ,-g,

E

and colleagues (Bizzi and Abend 1982; Bizzi et al. 1982) found that perturbations assisting the movement (similar as in the simulation of Fig. 5) can lead to relaxation back toward the typical unperturbed trajectory. (The effect was particularly clear in deafferented preparations.) In the present framework this means that the underlying coordination system was not perturbed, so that the end-effector relaxes to the trajectory defined by the coordination system. To address such effects we must explicitly model the relationship between endeffector and coordination system (cf. Discussion).

5 Coordinated Discrete Movement: Model and Results

An important experimental result on interlimb coordination of discrete movement can be expressed as a tendency to synchronize the movements of different limbs. Kelso et al. (1979a) studied a bimanual task, in which each hand moved from a home key to a target. By varying the distance and the precision requirements for each target, the index of difficulty in the sense of Fitts' law (Fitts' 1954) could be varied. As a result, the movement times in single hand movement could be varied by almost a factor of two. If the movements had to be performed with the two hands simultaneously, one hand to a difficult (far, small) target, the other to an easy (near, large) target, the movement times of both hands were almost identical. Essentially, the fast hand was slowed down to move synchronously with the slow hand. Further studies in similar paradigms confirmed this tendency to synchronize discrete movements of different limbs and showed the breakdown of synchronization when the intrinsic movement times of the two limbs become too dissimilar (Kelso et al. 1983; Marteniuk et al. 1984; Corcos 1984). To model the coordination of two end-effectors, we introduce a single effector description for each limb. Each structure's collective variables, (xi, vi) with i = l, 2, are assumed to be governed by dynamics of the form of (19):

0.0

Yc = v i

D = --f/, total(Xi,/)i) / ..'"

-50.0

9

-5.0

+ 2ajvi - 4bix2vi + 2aibix~ - b i x2i 5 + x / ~ i r

//"

~2.5

(27)

= - - { a 2 + co2i + •[q, ti+Ati](t)Cintent, i}Xi

".. _ .--

-100.0

-150.0

"'J

0.0

'2.5

5.0 x[cm]

Fig. 5a, b. Simulations of (19) at the standard parameter values (25). Here an unperturbed solution (solid lines) is shown together with a solution that was perturbed at t = 1.1 s to assist the movement (velocity advanced: v = - 2 . 3 - - + - 5 0 c r n / s ) . In panel a the spatial trajectories are shown together with their velocities. Note that there is no tendency to relax back to the unperturbed trajectory. By contrast, the phase plane trajectories shown in panel b converge due to relaxation of the perturbed solution

We introduce a different set of parameters for each end-effector to express different movement conditions of the two limbs. The coordination tendency is captured by coupling the two intrinsic dynamics. A similar coupling structure as used in the earlier work on rhythmic movements (Haken et al. 1985; cf. Sect. 2, (2)) can be used. Unlike in this earlier work, however, we treat here cases in which the two limbs are not equivalent but may, for example, differ in movement amplitude and limit cycle frequency. In such cases, the coupling functions as used in (2) are dominated by one component and do not lead to a synchronization tendency. Coupling functions

266

that account for the tendency of synchronization also in asymmetric cases can be obtained by normalizing each contribution. Such normalization can be based on the movement amplitude, Ai, and peak velocity, Vi.p, as estimated in the harmonic balance approximation: (i = 1, 2) Ae = ~/

b,

'

v,,, ~ Io~,lA,

(28)

x [cml

10

....... "'",

0.0

~

-10

', 1.0

or similar measures. The coupled system is then:

x [em]

.~] = v l

~)l=--fl,total(Xl'Vl)--k(V-~,pI)V~,) ;

10

i

....................... --4

Z Z/

3C2 = V2

(29)

where k and l are coupling constants. In the rrgime of stable limit cycle solutions, that is, when both oscillators are in part (e) of the phase diagram at moderate coupling strengths, the same results as obtained in the earlier work (Haken et al. t985) are reproduced. To address the coordination of discrete movement we study the case, where the two components have different movement amplitudes and movement times individually. In the model this corresponds to choosing the parameters of each component such that they correspond to different points within the parameterization of FiRs' law (Fig. 4). For example, by selecting a small ]coT[ value for x~ (with bl chosen correspondingly, cf. Sect. 4), we may endow this component with a small movement amplitude and a small movement time (see Fig. 6, top panel, solid line). The other component, x2, may make a large amplitude movement with larger movement time, if its og~-parameter is chosen corre, spondingly (Fig. 6, top panel, dashed line). When the two components are coupled, corresponding to the simultaneous performance of the discrete movements, we observe the synchronization of the movements (Fig. 6, bottom panel) although the movement amplitudes are unchanged and thus still different. This result is an account of Kelso et al. (1979a) observations. (We are taking into account only one spatial component of the movement, e.g., the projection of the movement onto the line connecting target and initial position. This is legitimate in the absence of any effect in the space curve of the movement.) We remark, that the tendency of synchronization persists when in the model the parameters of behavioral information, At; and Cintent' i are different for the two components (see Fig. 9). If we make the movement conditions more dissimilar, the tendency to synchronization is no longer strong enough to achieve complete simultaneity and a breakdown of synchronization is observed. In Fig. 7 the two

time [s]

2.0

time [s]

~,

-lo

iI 1.0

/)2: --f2, total(X2'V2)--k ( UV~,2p UU~I;)

i

"'"---...,

0.0

-t

2.0

I

Fig. 6. Top panel: Two components in discrete movement with different amplitudes and movement times. The solid line is a solution at the standard parameter set [(25), leading to movement amplitude A = 6.85cm and (half-distance) movement time To=0.53 sl. The dashed line is a solution with larger amplitude and movement time [m2= _ 5 H z 2 and b =0.0794Hz/cm 2, leading to A = 2 5 c m and To = 0.77 s]. Both movements are initiated at t = 1.0. The two components are uncoupled in the top panel corresponding to individual movements. In the bottom panel the same conditions prevail, but the components are coupled (k = - 4 0 H z 2 ) , corresponding to simultaneous performance of the movements. Note the almost perfect synchronization of the movements

x [ore 20 . . . . . . . . . . . . . . . . . . . . . . . . 10 0.0

'~

K_

-10 -20

1.0

2 0 ."....

time Is]

x Eoml 20

..........................

10

"",

0.0 -i0 -20

1.0

Fig. 7. Similar to Fig. 6, but the second component (dashed line) has parameter values ~o2 = - 5 . 5 Hz (b 2 = 0.036 Hz/cm2), so that its amplitude and movement time are even further removed from those o f the first component in individual movement (top panel). When coupled (bottom panel) with the same strength as in Fig. 6, the two components still tend to synchronize, but do not achieve the same degree of simultaneity

267

individual movements differ more strongly in movement amplitude and hence movement time (top panel) and evidently their synchronization is less complete (bottom panel) than in the case of Fig. 6. This accounts for the gradual breakdown of synchronization reported in Marteniuk et al. (1984) and Corcos (1984), as the movement conditions for the two components were • lO

0.0

-10 1.0

2.0

t i m e [s]

1.0

2.0

time [s]

1.0

2.0

time [s]

• 10

0.0

-10

Xltcm] 10

0.0

-10 1

•

3-

10 0.0

-10

1.0

2.0

t i m e [s]

Fig. 8a, b. The coordinated movements (with coupling) shown in Fig. 6 are reproduced as solid hnes: the small amplitude component in the top, and the large amplitude component in the bottom panel. The dashed lines show simulations at the same parameter values in which one component was perturbed at t = 1.2 s. In panel a of the figure the intrinsically faster component x~ (top) is perturbed to assist the movement (xl ~ 3 cm ~ 2 cm, v ~ - 2 cm/s ~ - 4 cm/s), leading to a speeding up of the unperturbed component to restore synchronization. In panel b the intrinsically slower component x2 (bottom) is perturbed to delay its movement (v I ~ - 8 em/s--, + 5 0 cm/s) leading to a slowing down of the unperturbed component again to restore synchronization

made more dissimilar. A similar breakdown of synchronization was observed by Kelso et al. (1983), when they placed an obstacle in the movement path of one component so that its space curve and hence movement time became much longer than that of the other hand. In the present theoretical account the tendency to synchronize discrete movement is due to the formation of a stable coordination pattern between the two components. This leads to the prediction that a remote compensatory reaction occurs in one component, if the other component is perturbed. (In light of previous discussion it is clear that the perturbation must be sufficiently strong to affect the nervous system level of coordination modelled here.) Moreover, the remote compensatory responses are such as to restore the coordination pattern, that is, to restore synchronization. We demonstrate this phenomenon for the model in Fig. 8. The two components perform movements with different movement amplitudes, but identical movement times due to coupling (solid lines of Fig. 8 are the same trajectories as shown before in the lower panel of Fig. 6). In Fig. 8a we perturb component x~ to assist its movement (top panel, dashed line). We observe an effect in the unperturbed component (bottom panel, dashed line): this component is also advanced in its movement, that means, the compensatory response tries to restore synchronization. Figure 8b shows the same effect when we perturb the slower component, x2, delaying its movement (lower panel, dashed line). In this case the effect in the other component is to move more slowly (top panel, dashed line), again in the direction of restoring synchronization. It would be very interesting to try to observe such a phenomenon in experiment, because this prediction is due to the conceptual structure of the present theory rather than to the detailed modelling assumptions. Experiments on perturbations of the movements of articulators in repetitive speech contain hints at such an effect. For example, Kelso et al. (1984) perturbed the jaw during various utterances and observed fast compensatory reactions in the upper lip if it was functionally necessary to achieve final lip closure. Recently, Gracco and Abbs (1988) showed, that in similar situations the compensatory movement of the remote articulator is such as to restore the normal relative timing of different articulators. Finally, we examine the consequences of the nonlinear coupling measured by the coupling coefficient/, that was introduced to account for anti-phase locking in rhythmic movement (see Haken et al. 1985). What could anti-phase locking mean in the case of discrete movement? The attraction to a relative timing corresponding to anti-phase locking leads to a tendency to perform two movements sequentially. Thus, if two discrete movements are initiated with sufficient delay (in the model: to, ~ different from to. 2), the movement time of the delayed movement increases to make the movement occur with less temporal overlap. In Fig. 9 we show two components moving with the same amplitude (standard parameter set, (25)), either individually (i.e., uncoupled: solid lines) or together (i.e., coupled: dashed

268

coordination pattern so that the predictions about remote compensatory responses arise analogously as for the synchronization tendency.

Xl[cml 5 0.0

6 Conclusion

-5

We have shown how a dynamic theory of discrete movement can be constructed on the basis of local dynamic descriptions of the initial and target postural states. The intention to move is captured as behavioral information, a part of the dynamics that stabilizes a limit cycle over a limited period of time. The timing of the movement is then largely determined by the intrinsic dynamics in the absence of an intention to move. We have shown that movement time-amplitude relationships can be captured in terms of relations among the parameters of the dynamics. We discussed the role of temporal stability in such a theory. The coordination of discrete movement was understood as the result of coupling different component dynamics in a fashion similar to the case of rhythmic movement coordination (Haken et al. 1985). Such coupling accounts for the experimentally established tendency to synchronize discrete movements even when the movements of the individual components have different intrinsic movement times. The stability of the coordination pattern in this theory gives rise to the prediction that remote compensatory responses occur when one component is perturbed. These responses will be such as to restore the stable relative timing of the two components. Furthermore, we proposed a discrete analogon to anti-phase locked rhythmic movement: the tendency to move sequentially is manifest if movement initiation of one component lags sufficiently that of the other component. We proposed an experiment to empirically identify such a tendency to movement sequentially. It may transpire from these results, that a unified dynamic theory of discrete and rhythmic movement can not only describe various relationships among movement parameters, but, through the important concept of temporal stability, enables prediction in suitable model systems. Comparison with experiment led us repeatedly to discuss the nature of the coordination pattern that we observe, and the nature of the collective variables on which the theory is founded. In particular, experiments on phase resetting in rhythmic movement (Kay et al. 1990) and on stability of the trajectory in discrete movement (Bizzi and Abend 1982; Bizzi et al. 1982) show that it is necessary to distinguish the level of end-effector position and velocity from the level of nervous system control of these variables. The collective variables that we use in the present dynamic theory must be correctly interpreted as representing the coordination activity of the nervous system. As long as the relation of the nervous system activity to the end-effector kinematics is unperturbed, these latter can be used to measure the former. When mechanical perturbations are applied, however, the two levels appear as different: sufficiently weak perturbations may not affect the nervous system level of control at all, leading to a relaxation of the

1.0

2.0

i

i

1.0

2.0

time [s]

~2[cm 5 0.0 -5 J

time [s]

L

Fig. 9. Two identical components perform movements with the same amplitude (standard parameter set, but At = 0.5 to allow for larger delays). The solid lines represent the same solution with movement initiation at to, L= 1.0 s (top) and to, 2 = 1.2 s (left trace in bottom panel). A second simulation is shown in the same panels with identical first component and movement initiation of the second component at t~, 2 = 1.4 s (right trace in bottom panel). If the two components are coupled (dashed lines) with the full nonlinear coupling (k = - 10 Hz 2, l = 10 Hz2), the tendencies to synchronization or to sequentialization can be observed depending on the delay between the movement initiations of the two components. For small delay ( = 0 . 2 s: left dashed traces) the synchronization tendency prevails leading to shorter movement time o f the delayed component. For large delay ( = 0 . 4 s: right dashed traces) the sequentialization tendency prevails leading to larger movement time of the delayed component

lines). If the onset of behavioral information of the second component (bottom panel) is delayed by a small amount compared to the first component (bottom panel, left solid line), the synchronization tendency shortens the movement time of the second component to restore synchronizity (bottom panel, left dashed line). If this delay becomes sufficiently large (bottom panel, right solid line), however, the tendency to sequential movement leads to a larger movement time of the second component, so that this movement overlaps little with the movement of the first component. The numerical simulation suggests an experimental paradigm to test whether such a coordination tendency of sequencial movement exists: In bimanual, fast movement, "go" signals are given separately for each limb. The delay between these go signals is manipulated systematically and movement time is measured. The prediction is that for small delay, the movements will tend to synchronize, while for larger delay, the movement times will adjust so as to produce more sequential movement. (Because movement times and reaction times can be measured separately, complications relating to the information processing, e.g., psychological refractory period, may be kept separate from the issue of coordination.) In this theory, such a tendency to move sequentially is also due to the formation of a

269 end-effectors to the centrally defined trajectory. W h e n the p e r t u r b a t i o n is sufficiently strong, however, the n e r v o u s system m a y be p e r t u r b e d as well a n d we m a y see its relaxation to its stable c o o r d i n a t i o n patterns. W i t h respect to such p e r t u r b a t i o n s , therefore, a m o r e complete theory which explicitly describes the two relev a n t levels is necessary. Such a theory can be based, for example, o n the f u n c t i o n a l m o d e l ( F e l d m a n 1966, 1974) o f muscle control, such that the trajectory as described b y p a t t e r n d y n a m i c s Eq. (19) is essentially a trajectory o f e q u i l i b r i u m points (cf. also Flash 1987). W e have s h o w n how d y n a m i c s a n d stability are i m p o r t a n t concepts to u n d e r s t a n d the c o o r d i n a t i o n activity o f the central n e r v o u s system in discrete movement. M a y b e , this could be viewed as a n element o f w h a t Erich v o n Hoist called "die M a t h e m a t i k der nerv6sen O r d n u n g s l e i s t u n g " (the m a t h e m a t i c s o f the ordering influence of the n e r v o u s system; see v o n Hoist 1948). Acknowledgements. Research supported by NIMH (Neurosciences

Research Branch) Grant MH 42900-01 and the U.S. Office of Naval Research (Grant No. N00014-88-J-1191). I would like to thank Pier Zanone for a critical reading of the manuscript as well as Anatol Feldman and Scott Kelso for discussion.

References Abend W, Bizzi E, Morasso P (1982) Human arm trajectory formation. Brain 105:331-348 Beggs WDA, Howarth CI (1972) The movement of the hand towards a target. Q J Exp Psychol 24:448-453 Bizzi E, Abend W (1982) Posture control and trajectory formation in single and multiple joint ann movements. In: Desmedt JE (ed) Brain and spinal mechanisms of movement control in man. Raven Press, New York Bizzi E, Polit A, Morasso P (1976) Mechanisms underlying achievement of final head position. J Neurophysiol 39:435-444 Bizzi E, Accornero N, Chapple N, Hogan N (1982) Arm trajectory formation in monkeys. Exp Brain Res 46:139-143 Bullock D, Grossberg D (1988) The VITE model: A neural command circuit for generating arm and articulator trajectories. In: Kelso JAS, Mandell AJ, Shlesinger MF (eds) Dynamic patterns in complex systems. World Scientific, Singapore, pp 305326 Corcos DM (1984) Two-handed movement control. Res Q Exercise Sport 55:117-122 Cremers J, Hiibler A (1987) Construction of differential equations from experimental data. Z Naturforsch 42a:797-802 Crutchfield JP, McNamara BS (1987) Equations of motion from a data series. Complex Syst 1:417-452 Feldman AG (1966) Functional tuning of the nervous system during control of movement or maintenance of a steady posture. III. Mechnographic analysis of the execution by man of the simplest motor tasks. Biofizika 11:667-675 Feldman AG (1974) Change in the length of the muscle as a consequence of a shift in equilibrium in the muscle-load system. Biophysics 11:565-578 Fitts PM (1954) The information capacity of the human motor system in controlling the amplitude of movement. J Exp Psychol 47:381-391 Flash T (1987) The control of hand equilibrium trajectories in multi-joint arm movements. Biol Cybern 57:257-274 Flash T, Hogan N (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J Neurosci 5:1688-1703 Georgopoulos AP, Kalaska JF, Massey JT (198l) Spatial trajectories and reaction times of aimed movements: effects of practice,

uncertainty and change in target location. J Neurophysiol 46:725-743 Gonzalez DL, Piro O (1987) Global bifurcations and phase portrait of an analytically solvable nonlinear oscillator: relaxation oscillations and saddle-node collisions. Phys Rev A36:44024410 Gracco VL, Abbs JH (1988) Central patterning of speech movements. Exp Brain Res 71:515-526 Haken H (1983) Synergetics--an introduction. Springer, Berlin Heidelberg New York Haken H, Kelso JAS, Bunz H (1985) A theoretical model of phase transitions in human movements. Biol Cybern 51:347-356 Hasan Z (1986i Optimized movement trajectories and joint stiffness in unperturbed, inertially loaded movements. Biol Cybern 53:1688-1703 Hogan N, Flash T (1987) Moving gracefully: quantitative theories of motor coordination. Trend Neurosci 10:170-174 Holmes PJ, Rand DA (1980) Phase portraits and bifurcations of the nonlinear oscillator .~ + (o~ + yx2)~ + px + 6x 3 = 0. Int J Nonlinear Mech 15:449-458 Holst E von (1939/1973) Die relative Koordination als Ph~inomen and als Methode zentral-nerv6ser Funktionsanalyse. Ergebnisse Physiol 42:228-306. (English translation in: Martin R (ed) The behavioral physiology of animal and man. University of Miami Press, Coral Gables (1973) Holst E von (1948) Von der Mathematik der nervSsen Ordnungsleistung. Experientia IV/10:374-381 Jeannerod M (1984) The timing of natural prehension movements. J Motor Behav 16:235-254 Kawato M, Furukawa K, Suzuki R (1987) A hierarchical neural-network model for control and learning of movement. Biol Cybern 57:169-185 Kay BA, Kelso JAS, Saltzman E, Sch6ner G (1987) The space-time behavior of single and bimanual rhythmical movements: data and model. J Exp Psychol: Hum Percept Perf 13:178-192 Kay BA, Saltzman E, Kelso JAS (1990) Steady-state and perturbed rhythmical movement: A dynamical analysis. J Exp Psychol: Human Percept Perf (in press) Kelso JAS (1977) Motor control mechanisms underlying human movement reproduction. J Exp Psychol: Human Percept Perf 3:529-543 Kelso JAS (1984) Phase transitions and critical behavior in human bimanual coordination. Am J Physiol: Reg Integ Comp 15:R1000-RI004 Kelso JAS, Sch6ner G (1987) Toward a physical (synergetic) theory of biological coordination. Springer Proc Phys 19:224-237 Kelso JAS, Southard DL, Goodman D (1979a) On the nature of human interlimb coordination. Science 203:1029-1031 Kelso JAS, Southard DL, Goodman D (1979b) On the coordination of two-handed movements. J Exp Psychol: Hum Percept Perf 5:229-238 Kelso JAS, Putnam CA, Goodman D (1983) On the space-time structure of human interlimb coordination. Q J Exp Psychol 35A: 347-375 Kelso JAS, Tuller B, Vatikiotis-Bateson E, Fowler CA (1984) Functionally specific articulatory cooperation followingjaw perturbations during speech: Evidence for coordinative structures. J Exp Psychol: Hum Percept Perf 10:812-832 Kelso JAS, Scholz JP, Sch6ner G (1986) Nonequilibrium phase transitions in coordinated biological motion: critical fluctuations. Phys Lett A118:279-284 Kelso JAS, Sch6ner G, Scholz JP, Haken H (1987) Phase-locked modes, phase transitions and component oscillators in biological motion. Phys Scr 35:79-87 Kelso JAS, Scholz JP, Sch6ner G (1988) Dynamics governs switching among patterns of coordination in biological movement. Phys Lett A134:8-12 Langolf GD, Chaffin DB, Foulke JA (1976) An investigation of Fitts' law. J Motor Behav 6:113-128 Marteniuk RG, MacKenzie CL, Baba DM (1984) Bimanual movement control: information processing and interaction effects. Q J Exp Psychol 36A:335-365 Milner, TE (1986) Controlling velocity in rapid movements. J Motor Behavior 18:147-161

270 Morasso P (1981) Spatial control of arm movements. Exp Brain Res 42:223 227 Polit A, Bizzi E (1979) Characteristics of the motor programs underlying arm movements in monkeys. J Neurophysiol 42:183194 Schrner G (1989) Learning and recall in dynamic theory of coordination patterns. Biol Cybern 62:39-54 Schfner G, Kelso JAS (1988a) Dynamic pattern generation in behavioral and neural systems. Science 239:1513-1520 Schrner G, Kelso JAS (1988b) Dynamic patterns in biological coordination: Theoretical strategy and new results. In: Kelso JAS, Mandell AJ, Shlesinger MF (ed) Dynamic patterns in complex systems. World Scientific, Singapore, pp 77-102 Schrner G, Kelso JAS (1988c) A dynamic theory of behavioral change. J Theor Biol 135:501-524 Schrner G, Kelso JAS (1988d) A synergetic theory of environmentally-specified and learned patterns of movement coordination. I. Relative phase dynamics. Biol Cybern 58:71-80; II. Component oscillator dynamics. Biol Cybern 58:81-89 Schrner G, Haken H, Kelso JAS (1986) A stochastic theory of phase transitions in human hand movement. Biol Cybern 53:247-257 Scholz JP, Kelso JAS, Schfner G (1987) Nonequilibrium phase transitions in coordinated biological motion: Critical slowing down and switching time. Phys Lett A123:390 394

Soechting JF, Lacquaniti F (1981) Invariant characteristics of a pointing movement in man. J Neurosci 1:710-720 Stelmach GE, Kelso JAS, Wallace SA (1975) Preselection in shortterm motor memory. J Exp Psychol: Hum Learn Mem 1:745-755 Uno Y, Kawato M, Suzuki R (1989) Formation and control of optimal trajectory in human multijoint arm movement. Biol Cybern 61:89-101 Viviani P, Terzuolo C (1980) Space-time invariance in learned motor skills. In: Stelmach GE, Requin J (eds) Tutorials in motor behavior. North-Holland, Amsterdam Wadman WJ, Dernier van der Gon JJ, Dereksen RJA (1980) Muscle activation patterns for fast goal-directed arm movements. J Hum Mov Stud 9:157-169 Wang MC, Uhlenbeck GE (1945) On the theory of Brownian motion II. Rev Mod Phys 17:323-342 Woodworth RS (1899) The accuracy of voluntary movement. Psyehol Rev Monogr 3, No 13 Woodworth RS (1903) Le mouvement. Doin, Paris Dr. Gregor Schrner Institut f'tir Neuroinformatik Ruhr-Universit~it Bochum ND 04 Postfach 102148 D-4630 Bochum 1 Federal Republic of Germany