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Anticipatory dynamical systems, intrinsic pattern dynamics and skill learning
Human Movement Science 10 (1991) 93-111 North-Holland
eaction to
llock and
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rossberg,
1
Florida Atlantic University, Boom Raton, USA
The study of voluntary movement and posture has a long history, full of philosophicai and conceptual content. It was not so long ago, in fact before the Watershed caused by the Kepler-Galilean synthesis, that willed movement - or, as the ancients saw it, how the mind rules the body - was a model for celestial motion. Bullock and Grossberg’s paper is a significant contribution because it provides specific neural network modules for the hypothetical operations involved. As is characteristic of much of the author’s work (in Grossberg’s case spanning many years) great care is taken to synthesize selected neural and behavioral evidence into a single theoretical picture. It is enormously pleasing for this commentator to see, in his adopted field, how sophisticated mathematics and theory can be meshed with explicit computational models and experimental data in such a masterly fashion. My comments invite B&G to extend their scenario for posture and voluntary movement, to probe the generality of their formulation and to suggest, in certain cases, a need to supplement or even substitute existing architectures. I confine my remarks largely to the behavioral or
* Much of the research discussed herein is supported by NIMH (Neurosciences Research Branch) Grant MH42900, BRS Grant RR07258, and ONR Contract NOOO14-99-J1191. Author’s address: J.A.S. Kelso, Program in Complex Systems and Brain Sciences, Center for Complex Systems, Florida Atlantic University, Boca Raton, FL 33431, USA.
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functional level with the goa! of achieving a self-consistent description at that scale. My views are very much in the spirit of a ‘ psychologicai thermodynamics’ that Grossberg has championed in many of his talks and papers. One means here, of course, a search for the laws and principles of coordinated behavior. Elsewhere we have suggested tjhat behavioral laws, on a chosen level of description, can be founded upon, and methodologically grounded in the concepts and tools of synergetic phase transitions (see e.g. Haken and Stadler 1990; Kelso 1990).
&G’s orientation to the role of nonspcrcific signals because they play a remarkable role in vertebrate behavior, e.g. the multiple behaviors of the farmyard chicken (von Hoist and van St. Paul 1973), the multiple modes of locomotion in decerebrate and intact cats (see Skinner and Garcia-till (1990) for review). The brain&em nuclei, both cholinergic and catecholaminergic, prove to be indispensable in the nonspecific control of these behaviors. Actually, as shown in Synergetits (f&ken 1983) nonspecific signals play a major role in pattern formation and cooperative phenomena in many natural systems. For example, in open nonlinear dynamical. systems, spatial, temporal and functional patterns arise spontaneously in a self-organized fashion when nonspecific coatrol parameters are changed (for many examples, and a full theoretical treatment see Haken 1983; i4icohs and. Prigogine 1989). Control parameters are unspecific to the resulting patterns; they “control’ only in the sense of keeping the system in a stable operating range or by leading the system into instabilities where new patterns are formed. Around these instabilities many theoretically predicted fcatures have been observed in the perceptual-motor system (see e.g. Jeka and Kelso (1989) Schiiner and Kelso (1988a, b) for reviews). B&G’s predictions about nonspecific control signals lie at relatively high levels of the nervous system, but it is important to recognize that such inputs play a ro!e in self-organization and pattern formation at many scales of observation. In my view, the authors’ emphasis on nonspecific input in motor control is far from misplaced, often ignored in the literature, yet quite fundamental. 0ne or two questions arise concerning their GO signal, however (see below).
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The benefit of nonspecific control signals inducing qualitative changes in kinematic pattern is that they allow one to identify the essential dimension(s) of the pattern that remains ‘invariant’ in the linear range of the control parameter. In a complex system how does orie know which variables are relevant? Experimentally observed kinematic invariants offer a hint about the underlying control algorithms or design constraints that the CNS may exploit (e.g. Kelso et al. 1979; Viviani and Terzuolo 1980; Schmidt 1987). For example, the relative timing or phasing among active components can remain ‘invariant’ over changes in movement speed. However, emphasis on findin.g invariant properties raises a number of serious statistical issues (Tuller and Kelso 1990). How much deviation from invariance does one accept? I would argue, based on empirical and theoretical grounds, that a more appropriate concept in this case is stability rather than invariance. r For example, one well-known order parameter, relative phase, displays enhanced fluctuations as the pattern loses its stability. Note that if careful attention is paid to time scales, the first moment, the mean, does not change as the control parameter is varied except at transition points. The order parameter appears ‘invariant’ across changes in speed, but order parameter fluctuations are a conceptually crucial part of the dynamics, probing the stability of collective states and allowing the system to discover new patterns. We refer to the order pdra meter dynamics in the absence of specific parametric change as intrinsic pattern dynamics, reflecting the internal coordination constraints on the system. The great advantage of knowing these intrinsic dynamics is that one can specify what changes due, e.g. to environmental, learned or intentional influences. That is, infor’ This is not to deny the plJwerfu1 concept of invariance for coordinated behavior especially when linked to symmetry properties. For example, symmetry has been used to classify coordination patterns (hand movements, locomotory gaits) and to restrict the functional form of the pattern dynamics (Schijner J$tal. 1990; also Haken et al. 1985). Symmetries are expressed as invariance of pattern under a set (group) of transformations. The idealized walk, trot, gallop and jump are members of the same symmetry group, i.e. the only patterns that are invariant under left-right, front-hind and time inversion operations. Quantitative analysis of kinematic patterns, on the other hand, seems better tied to the concept of stabili!y, not (as in the motor control literature) the notion of invariance, because statistical fluctuations are a necessary part of the measurement process and prove to be conceptually relevant (see text).
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n-ration can be formally expressed as a contribution to, a force acting upon, or a perturbation of, the intrinsic dynamics. Although a behavioral description figures prominently in the B&G model, the authors do not really say what the intrinsic behavioral constraints are in the task of reaching to targets. How can these be found for multi-joint reaching tasks? In synergetic/dynamic pattern theory, the necessary methodological step is to experimentally induce qualitative behavioral changes. One clue comes from a simple study by Rosenbaum and colleagues (1990). The basic phenomenon is as follows: subjects are faced with a bar or dowel, the left half of which is black, the right half white. The bar hes in a horizontal position between two supports. The task is to reach and pick up the bar placing the black (or white) end up hen black is up (white down) the (or down) on a nearby surface. subjects invariably grasp with an overhand grip. When black is down (white up) an underlaand grip is chosen. Note, the task and its accomplishment are binary by definition. The measure that Rosenbaum et al. (1990) use is simply how many subjects do one thing or the other. My students and I (Kelso 1989; Kelso et al. 1990b) have adopted the synergetic/ dynamic pattern approach to understanding this pattern selection process. In our experiments, a stepping motor moves the bar around a clocklike surface in either a clockwise (cw) or counterclockwise (ccw) direction. Wrist/forearm rotation and limb projection are measured using infrared light emitting diodes placed on relevant points of the limb. A single trial consists of a set bar angle followed by a tone which signals the subject to reach for the bar. The task is to return the bar to the twelve o’clock position (90 degrees). Our study reveals very interesting constraints and dynamical effects on reaching and grasping for an object that we invite B&G to consider: (1) For an entire range of bar rotations, from approximately 1 o’clock to 5 o’clock (in the clockwise direction) the same basic pattern is adopted. The phase relation, or joint angle velocity ratio, between limb projection and wrist/forearm rotation is constant in time and consists of an underhand, supine grasping motion. (2) Near 6 o’clock, however (it varies a little among subjects), the pattern switches spontaneously; an overhand counterc&-kwise pronation motion emerges, approximately 180 degree anti-phase with the previously supine clockwise motion. The different phase relations for the patterns are quite understandable: the projection or transport component stays the same during reaching, only the direction of forearm-wrist rotation changes. (3) When the bar is
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rotated in the opposite (counterclockwise) direction the same basic patterns are seen hut the switching now occurs at a different spot. Tl;pt is, the pattern adopted depends on the direction of the control parameter change (bar rotation), a clear hysteresis eflect (4) When the bar is perturbed during the reach itself (analogous to the ‘target switching’ paradigm discussed by B&G) several things happen: (a) if the direction of the perturbation does not cross the critical region, the pattern’s topology is unchanged; (b) if the perturbation crosses the critical region, the first reaction is to keep the same pattern but to recruit new degrees of freedom (e.g. trunk tilt, shoulder lean). Not until the perturbation is quite large (well into the stable region of the other pattern) does one observe perturbation induced switching. Why are these results relevant? First, they reveal the existence of an equivalence class of patterns for a wide range of object orientations. You don’t have to ‘reprogram’ the pattern for every object orientation: the basic reaching pattern stays the same and can be metrically adjusted for different object orientations. The reaching act in this task involves a constrained temporal relationship between hand projection to the target and forearm/wrist rotation. Second, the flexibility of the system is revealed at critical values of the control parameter, when switching occurs from the prone to supine pattern. Third, nonlinear dynamical effects ‘are observed such as hysteresis, context-dependent perturbation effects and recruitment of new degrees of freedom at critical points. The latter result appears extremely interesting. Unlike typical nonequilibrium phase transitions in which the system’s dimensionality is reduced, in the present situation, additional degrees of freedom come in at critical points. It is not clear, however, that the dimensionality of the system in terms of attractors of the pattern dynamics necessarily changes. I mention these results here as more of a challenge to B&G’s model of voluntary motor control. I see the flexible task-determined recruitment of degrees of freedom as an essential, adaptive feature of real biological systems, the basis of which must be understood and incorporated into neural network models for animal and robot control. I invite ow would VITE and FLETE accommodate B&G to c0mmen.t. target-induced qualitative changes? Even without an externally defined target the underlying neuromuscular organization for multijoint limb trajectories seems more abstract ere I refer to experiments than VITE and FLETE presently allow.
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that demonstrate that the orderly coordination among the joints of a single limb depends on the spatial orientation of the forearm, not on preferred coupling among muscle groups (Kelso 1990; Kelso et al., to elationships between elbow- and wrist appear). Stable joint to be direction-specific (see also Baldissera flexion and extension pr et al. 1982). Thus, when t e forearm is supine (palm facing up), flexion flexion (extension) of the elbow exhibit a (extension) of the wrist (see also Kots and Syrovegin 1966; stable relative phase r When the forearm is prone (palm facing Soechting and Terzuol nspecific changes in two control paramedown) the opposite is true. ters, frequency of motioc d spatial orientation, are capable of effecting changes from 0 nematic (and electromyographic) pattern to the other, i.e. the dyn_tics are minimally codimension two. An interesting point for $3 is that lesions of parietal cortex, which t+tic information, produce dramatic changes effectively eliminate in coordinative performan between elbow and wrist joints. Performance of single joint movements, however, can be carried out easily and accurately (Kots et al. 1971; see also Kelso 1977; Polit and Bizzi 1979), suggesting that propriospecific information is crucial to sustaining coordinative function between joints. What impact might knowledge of these constraints have on B&G’s model? I acknowled leve their FLETE module aims at a detailed structure-function ever, in the spirit of generalization the authors might conjecture how these nonspecific signals at a behavioral level influence compliance signals, Renshaw cells, Ia interneurons, muscle spindles, etc. As one can set, the behavioral pattern dynamics for single, nmltijoint limb trajectories are already quite
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In a similar vein, B model, they claim, is vastly superior to variational schemes such as ogan and Flash’s (1987) ‘minimum jerk model. In their preferred ‘ ‘mum torque change’ model, Uno et al. (1989) have also shown that joint torques need to be taken into account in two degree of freedom pl motions, especially when subjects must bring the band close to the dy or extend the limb to the limits of work space. ere again I t these debates between different minimization schemes are somewhat moot, in part because the paradigms
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restrict the conclusions (see also Kelso et al. 1984). For example, if an object is placed near the limits of -work space, additional degrees of freedom (forward body lean) are spontaneously recruited provided the motion is not restricted. Similarly, if a subject must move through a via point close to his or her chest (cf. Uno et al. 1989), the spontaneous response of the biological system is to lean back, not dumbly follow some minimization scheme. At these perceived transition regions, additional degrees of freedom again appear and are assimilated into the existing structure. The synergetic/dynamic pattern theory offers a natural language for understanding these effects in terms of order-order transitions. How do B&G accommodate such results on the behavioral level? (I am not asking for a description of correlated activity in neural structures.) y point is that when we open up our paradigms, complexify them so to speak, then often we see elegantly simple solutions emerge. When we restrict our paradigms, for instance in order to test different variational schemes for two degree of freedom motions, the results may be too local and restrictive to mean very much. I cannot get too excited over the degree of asymmetry in end effector velocity profiles, or that one gets two bumps in the velocity profile (Uno et al. 1989) rather than one. For what it is worth, my own view is that the two bumps would disappear if the subject was not forced by the apparatus to limit his/her reaching to two mechanical degrees of freedom.
(I) Anticipatory Dynamical Systems (ADS) A nice feature of the VITE model is that kinematic invariants emerge through network interactions rather than an explicitly precomputed command. Automatic processes include computation of a Present Position Command (PPC) and a Difference Vector (DV) between PPC and the Target Position Command (TPC). The PPC is gradually updated by integrating the DV through time. The time varying ~30 signal is independently controlled, acting multiplicatively on the before it is integrated by the PPC. As I understand it (fig. 2) the TPC does not interact directly with the GO signal. A key aspect of the VITE
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circuit is that the PPC is computed by outflow, iced forward signals rather than inflow, feedback signals. 1 wonder whether a slightly different circuit might be envisaged. A crucial feature of biological systems is their ability to anticipate future states of affairs (see e.g. Rosen 1985). In stable environments, the goal of an act, the future as it were, acts back i.n a locally time reversible fashion on the system’s parameter settings for the accomplishment of the goal. It seems poss;ible that this may be a ‘one shot’ rather than a continuous updating process: given some intrinsic dynamics (the order parameter equation of motion), the goal (e.g. a required final position) acts on those dynamics setting the parameters for goal achievement. For a neural network, the requirement to continuously update weights ropagation, or to compute the DV continuously in time tituted by an Anticipatory Dynamical System which adjusts or perturbs the parameters of its intrinsic dynamics (representative of the system’s existing pattern generating capabilities) before any action whatsoever. The evidence for this kind of Grcuit, like that cited by B&C, is mostly correlative. 9n the behavioral side the enhanced accuracy of voluntary (s&ject-defined) over constrained (e defined) movements - the so-called preselection effect and well-studied Stelmach et al. 1975) - was one of the most reliab effects in the motor behavior area in the 70’s (see elso and Wallace (1978) for a review). The kinematics reflect anticipatory control from the very beginning of preselected, compared to other kinds of movement, and can be quantified in terms of an index of anticipation (Kelso et al. 1978). Evidence for prior parameter setting in the nervous system based on the anticipated consequences of action (a corollary discharge rather than an efference copy) exists at many levels, from the early studies of Teuber (1964) on frontal lobe deficits, to the modification of reflex excitability be ent, demonstrated in numerous Soviet studies (e.g. y suggestion implies a kind of nonlinear causality, in which the future - a particular goal - adjusts the parameters/initial conditions of the system specifically for the accomplishment of that goal. sic dynamics (Kelso Mentions modify and are modified by the int so 1988b), but can et al. 1988; Scholz and elso 1990; Schiiner and dynamical systems anticipate future states? The standard picture, notwithstanding the existence of deterministic chaos, is that given some initial conditions and a law, future states are determined. Nonlinear
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dynamical systems, however, display behavior that can be interpreted as anticipatory. Iterative maps, such as the phase attractive circle map which describes coordination in multi-frequency systems (DeGuzman and Kelso, in press; elso and DeGuzman 1988; see also Beek 1989) are a good illustration. Near tangent (inverse saddle node) bifurcations, for example, phase densities emerge that appear to ‘anticipate’ upcoming stable solutions. Thus, as the system approaches a critical point, slowing down occurs, t e phase gathers and it is easy to predict future states even though the tion is quasiregular. The so-called ‘forward orbit at the critical point’ has all the appearance of an anticipatory dynamical system (see figs. 1 and 2 plus captions). Strictly speaking re is no DV, adaptive linearization for outflow-inflow mismatches llock and Grossberg 1988) and so forth. Like B&G, however, kinematic properties emerge from the underlying dynamics - a consistent, but unevenly interpreted, theme in our work (e.g. Kelso and Holt 1980; Kugler et al. 1980). Without detracting in any way from their VITE module, might the authors consider an ADS (Anticipatory Dynamical System) module as an alternative or complement to VITE? Let me pursue this notion a bit futher and consider another kind of targeting behavior, synchroruzing a discrete movement with a periodic signal whose frequency may be varied. The relationship between environmental event and the hand is again captured by a collective variable, relative phase, whose dynamics can be explicitly formulated and derived (Kelso et al. 1990a). Notice that if the time between events is too long, the syste s behavior is reactive, and characterized by reaction time delays. owever, over a wide range of environmental periodicities, phase and frequency entrainment curs. That is, the system’s dynamics ars: predictive or anticipatory. ere the relationship between environment and action systems may be theoretically understood as a pattern formation process, in which a single collective variable spans both components. Different kinds of experimentally observable patterns (entrainment, loss of entrainment, relative coordination, etc.) correspond to different regimes of the same underlying dynamics (Kelso et al. 1990a) Although it is possible to describe this predominantly temporal targeting behavior in terms of feed forward, predictive and feedback, reactive loops, the dynamical model seems much more parsimonious. Notice the concepts are different also, yet the equation of motion for the pattern dynamics clearly constitutes an ADS.
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Why is the behavioral description of reaching for a static object so different (and apparently more complicated) than the case of targeting behavior analyzed above? The latter seems simple, once it is captured in terms of patterned modes of entrainment between environment and action components. Yet the behavior itself, involving anticipation, prediction and intention is quite deep. The former is typically bound up in all kinds of difficult concepts and operations, even at the phenomenological level discussed by B&G, never mind underlying neural processes. To find the laws of reaching it might be useful to consider a higher level of analysis that incorporates the intrinsic constraints of the actor and environmental information in terms of a unified spatiotemporal pattern (for a start on this problem see Wallace et al. 1990; and the work of e.g. Jeannerod, Wing, and Soechting cited therein). In the dynamic pattern view, information (a) has no meaning outside its influence on the collective variables that characterize the repertoire of behavioral patterns; and (b) thereby defies an attractor of the pattern dynamics. The point that I invite B&G to consider is that the pattern dynamics for the control of reaching (analogous to our ADS module
Fig. 1. (a) Iterations
of the phase attractive
circle map:
1-c A COS(~~~,)] sin(2?r&), +h.kl=+“+~-g[
mod i
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for A - 0, D = 0.131, and K = 0.80. The map is a simple theoretical model of coupled, nonlinear osci!lators where K is tbe strength of the nonlinearity, B is tFe frequency ratio among the oscillatory components and A expresses the bistability of relative phase (coexistence of inphase and anti-phase). Near the tangent bifurcation shown here, very close to a fixed point (period 1) solution, the phase gathers and slows. The reason is th.at the iterations are compressed into the narrow space between the function and the 45 degree line. The message of this simple model, which accommodates a number of experimental observations is that nonlinear dynamical systems can display ‘anticipatory’ features. Note that the discontinuity of the function is not rea:, but due to rectangular coordinates parameterized in the interval [0, 11. The function is an invertible mapping of the circle onto itself (a Poincare section of the flow on the torus). (b) The corresponding bifurcation diagram where K is the bifurcation parameter (0.4 > K -C1.0). A = 0 and 52 = 0.131. The relative phase, $I is on the Y axis. The fuzzy area corresponds to quasiperiodic motion because the frequency ratio between the components is irrational. Note, however, that the darkening region ‘anticipates’ the tipccming stable sohnion (the single line) which indicates the system is trapped or mode-locked 1: 1. Much fancier dynamics, of course, are possible as K is increased beyond the critical suriace where the map lases invertibility. The main point is that the. system spends more time in a particular phase as it approaches the critical point. It behaves as E-I Anticipatory Dynamical System (ADS). ADS stay in contact with the future by Living near critical points.
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for synchronization) are function-specific and thus may cover hand and target. Perhaps this point relates to their Vector Associative Map bu.t I am not sure. (2) The nonlinear dynamics of posture, discrete AND rhythmical movement
Returning specifically to posture and voluntary inovement (both discrete and rhythmic), Schijner and I (Kelso and Schiiner 1988; Schiiner 1990) have offered an explicit alternative to B&G at the level of end effector kinematics, i.e. position, x and velocity dx/dt. The approach is continuous with our treatment of intentional changes in relative timing behavior (Kelso et al. 1988; Scholz and Kelso 1990; Schijner and Kelso 1988b), and earlier ideas about targeting behavior (Kelso et al. 1980: 50; Kelso and Holt 1980). An intention to move is expressed as a part of the end effector dynamics that stabilizes the coordination pattern, in this case the position-velocity curve (as a piece of a limit cycle). The structure of the intrinsic dynamics, viz. the pattern dynamics in the absence of an intention to move, admits (a) postural stages as stable fixed point attractors; (b) two coexisting postural stages corresponding to the initial and target postures joined by a limit cycle trajectory (discrete movement); and (c) rhythmic motion, captured by limit cycle oscillations. The phase diagram of an exactly solvable nonlinear oscillator model (Gonzalez and Piro 1987) similar to one established by theoretical (Haken et al. 1985) and empirical work (Kay et al. 1987) contains all of these dynamical structures (see Schiiner 1990). Like B&G, Fitts Law relationships between amplitude and movement time can be accommodated in terms of relations among dynamic parameters. In addition, coordination of dis,Lrete movements of different amplitudes (Kelso et al. 1979) is accomplished by coupling individual components that have different intrinsic movement times in a fashion similar to the case of rhythmic
Fig. 2. The same picture as fig. 1 for different parameter values. The map (panel a) shows quasiperiodic motion near a period 3 (B = 0.355) where the component frequencies are almost related in a 3 : 1 ratio. The corresponding bifurcation diagram is shown in panel b. Exactly the same features as fig. lb are displayed. This time however, the phase densities increase as :he period 3 (single lines) is approached. Signature features, such as critical slowing are predictive of upcoming critical points where pattern switching occurs.
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coordination ( aken et al. 1985). The stability of the coordination pattern accommodates remote compensatory behavior when one comrturbed (Schiiner 1990; see also Saltzman and Kelso 1987). G, we interpret the pattern dynamics as representing the underlying neural organization, as long as the end effector kinematics are not strongly perturbed. When they are (cf. Kay et al., in press), it may be necessary to distingu.ish end effector position and velocity from the level of n ous system control of these variables (see Bizzi and t this is not the main point of discussion here. The Abend 1982). present ‘architecture’ possesses both similarities and differences to B&G, outside the issue of the data base. In the dynamic pattern model, the form, trajectory and temporal structure are determined by the intrinsic dynamics, which, once known, are ‘built in’ as a law. Movement initiation (maybe our equivalent of a GO signal) corresponds to perturbing the intrinsic dynamics, simply destabilizing one pattern -Y+h then switches to another. No explicit timing information is necessary. At the level of behavior, these dynamics contain posture, discrete and rhythmical movements of one or two end effecters, depending on parameters. The economy of description afforded by nonlinear dynamical models, and the contact between experiment and theoretical prediction is appealing. Neural networks that implement dynamical systems bviously constitute and active area of research (see e.g. Jordan 1990). &c’s VITE circuit simulates numerous properties d discrete movement. But how does it generate rhythmic suspect that the form of their GO s nal, which presently persists beyond the discrete movement time (see 6, fig. 5) must be modified considerably.
&G starts with the movement, grows during the n beyond the movement. If B&G are correct, why should the ‘will to act’ last longer than the movement? I am somewhat doubtful that G’s GO signal corresponds to the ‘will to act’. The will to act builds up over a long time before the movement begins. Why does it take so long and why is there so much activity in the nervous system before overt, volitional movement? The famous
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Bereitschaftpotential of Deecke and Komhuber is a case in point. The rise of this potential looks a lot like B&G’s GO function except that it occurs as much as a second before movement onset, and then changes sign. The sources of this readiness potential which occurs, to my knowledge, only in volitional movement (unlike, say, the CNV) are not entirely known, but the evidence, using all kinds of brain imaging and electrical recordings points strongly to supplementary motor cortex (Lang et al. 1988). At Neils Bohr’s institute in Copenhagen, when it was time to take a break, the physicists (including Bohr himself) would sometimes go to Western and gangster movies. In gunfights, Bohr noticed that the guy who drew his gun first always lost. Bohr’s collaborators (Gamow, Landau, Casimir et al.) were skeptical, but on submission to experimental test with cap pistols, Bohr’s observation was correct. According to Dirac (Blaedel 1988), Bohr formulated a new theory based on this finding. In Bohr’s theory, when the two cowboys face epch other in a gunfight all they can do is talk, because they both know that the one ,who draws first will die. The ‘will to act’ takes much longer to evolve than the reaction to a stimulus and is somehow slower. B&G’s GO signal which starts with the onset of movement is an unlikely candidate for the will to act, though it might help win a duel. But if Bohr is right, why are willed actions slower, or are they?
Lots of claims about the nature of skill learning are made by B&G. A main one, that early in skill learning movements are made with low compliance and low speed, appears more intuitive than factual. Given the controversies that have surrounded the cerebellum, it is probably not so useful to debate whether “it is now well-established that the cerebellum is a critical module for learning predictive movementcalibrating signals’. Such statements miss the point, as far as the present comments are concerned. Every major learning theorist has acknowledged that learning does not begin on an entirely blank slate, even in the neonate. The initial state is not a disordered random network, but an already highly organized system (e.g. Fitts 1964). Learning must therefore involve some kind of order-order transition, whether abrupt or continuous.
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Some of the scientific problems are (a) to quantitatively evaluate the initial structure prior to learning a new skill; and (b) to probe how that structure evolves in time as new tasks are learned. Predictions regarding an individual subject’s learning behavior follow, in our view, from knowledge of his/her intrinsic dynamics, i.e., the existing attractor layout or phase diagram. In the case of relative timing skills these predictions have been tested. For example, it can be demonstrated (Kelso 1990; Zanone and Kelso, to appear) that if the required task deviates from the intrinsic pattern dynamics, phase transitions in learning (qualitative changes in the phase diagram) occur due to competition (Schiiner and Kelso 1988~). This layer of behavioral pattern dynamics is missing in B&G, as it is in most formal neural network learning and pattern recognition schemes (cf. Schiiner 1989). Identification of the intrinsic dynamics and a metric for evaluating them would be useful, in my view, for the skills of interest to B&G. Put another way, there is more to learning to bring a glass of milk to the mouth than low speed, low compliance parameter setting and cerebellar computation.
Bald&era, F., P. Cavallari and P. Civischi, 1982. Preferential coupling between voluntary movements of ipsalateral limbs. Neuroscience Letters 34, 95-100. Beek, P.J., ..989. Juggling dynamics. Amsterdam: Free University Press. Bizzi, !Z. and W.A. Abend, 1982. ‘Posture control and trajectory formation in single and multiple jomt arm movements’. In: J.E. Desmedt (ed.), Brain and spinal mechanisms of movement control in man. New York: Raven Press. Blaedel, N., 1988. Harmony and unity. The life of Neils Bohr. Berlin: Springer-Vedag. Bullock, D. and S. Grossberg, 1988. ‘The VITE model: A neural command circuit for generating arm and articulator trajectories’. In: J.A.S. Kelso, A.J. Mandell and M.F. Shlesinger (eds.), Dynamic patterns in complex systems. Singapore: World Scientific. Bullock, D. and S. Grossberg, 1991. Adaptive neural networks for control of movement trajectories invariant under speed and force resealing (Target article). Human Movement Science 10, 3-53 (this issue). DeGuzman, G.C. and J.A.S. Kelso, in press. Multifrequency behavioral patterns and the phase attractive circle map. Biological Cybernetics. Fitts, P.M., 1964. ‘Perceptual-motor skill learning’. In: A.W. Melton (ed.), Categories of human learning. New York: Academic Press. pp. 254-285. Flash, T. and N. Hogm, 1985. The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neuroscience 5, 1688-1703. Gonzalez, D.L. and 0. Piro, 1987. Global bifurcations and phase portrait of an analytically solvable nonlinear oscillator: Relaxation oscillations and saddle node collisions. Physical Review A 36,4402-4410.
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Haken, H., 1977/1983. Synergetics, an introduction: Non-equilibrium phase transitions and self-organization in. physics, &nristry and biology (3rd ed.) Berlin: Sptinger-Verlag. Waken, H. and M. Stadler (eds.), 1990. Synergetics of cognition. Berlin: Springer-Verlag. Haken, H., J.A.S. Kelso and H. Bunz, 1985. A theoretical model of phase transitions in human hand movements. Biological Cytametics 39, 139-156. Hogan, N. and T. Flash, 1987. Moving gracefully: Quantitative theories of motor coordination. Trends in Neuroscience 10, 170-174. Jeannerod, M., 1984. The timing of natural prehensile movements. Journal of Motor Behavior 16, 235-254. Jeka, J.J. and J.A.S. Kelso, 1989. ‘The dynamic pattern approach to coordinated behavior: A tutorial review’. In: S.A. Wallace (ed.), Perspectives on the coordination of movement. Amsterdam: North-Holland. pp. 3-45. Jordan, MI., 1990. ‘Motor learning and the degrees of freedom problem’. In: M. Jeannerod (ed.) Attention and performance XII. Hillsdale, NJ: Erlbaum. pp. 796-836. Kay, B.A., J.A.S. Kel;o and E.L. Saltzman, in press. Steady-state and perturbed rhythical movements: A dynamical analysis. Journal of Experimental Psychology: Human Perception and Performance. Kay, B.A., J.A.S. Kelso, E.L. Sa!tzman and G. Schiiner, 1987. The space-time behavior of single and bimanual movements: data and model. Journal of experimental Psychology: Human Perception and Performance 13, 178-192. Kelsu, J.A.S., 1975. ‘Central and peripheral information in motor control’. In: J. King and W.W. Spirduso (eds.), Motor control symposium. Austin, TX: University of Texas Press. Kelso, J.A.S., 1977. Motor control mechanisms underlying human movement reproduction. Journal of Experimental Psychology: Human Perception and Performance 4, 529-543. Kelso. J.A.S., 1989. Haptics and sensory-guided motor control. Paper presented at ONR Cognitive and Neural Science Program, Woodshole, MA., September. Kelso, J.A.S., 1990. ‘Phase transitions: Foundations of behavior’. In: H. Haken and M. Stadler (eds.), Synergetics of cognition. Berlin Heidleberg, New York: Springer-Verlag. pp. 180-196. Kelso, J.A.S., J.J. Buchanan and S.A. Wallace, to appear. Order parameters for the neural organization of single, multijoint limb trajectories. Experimental Brain Research. Kelso, J.A.S., J.D. Delcolle and G. Schoner, 1990a. ‘Action-perception as a pattern fomtation process’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 139-169. Kelso, J.A.S., T. Murata, J.J. Buchanan and J.D. Delcolle, 199Oll. Spatially induced phase transitions in discrete reaching. Psychonomic Society, New Orleans, Nov. Kelso, J.A.S. and G. DeGuzman, 1988. ‘Order in time: How cooperation between the hands informs the design of the brain’. In: H. Haken (ed.), Neural and synergetic computers. Berlin: Springer-Verlag. pp. 180-196. Kelso, J.A.S. and K.G. Holt, 1980. Exploring a vibratory systems analysis of movement production. J. of Neurophysiology 43, 1183-1196. Kelso, J.A.S., KG. Holt, P.N. Kugler and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: Empirical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland: pp. 49-Z. Kelso, J.A.S., J.H. Pruitt and D. Goodman, 1978. ‘The anticipatory control of movement’. In: G.C. Roberts and KM. Newell (eds.), Psychology of motor behavior and sport. Champaign, IL: Human Kinetics. pp. 192-203. Kelso, J.A.S., J.P. Scholz and G. Schiiner, 1988. Dynamics governs switching among patterns of coordination in biological movement. Physics Letters A 134, 8-12. Kelso, J.A.S. and G. Schoner, 1988. Dynamic patterns of sensorimotor behavior: Biological control structures and constraints. Neural Networks 1, (Suppl. l), 343.
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K&o. J.A.S., D.L. Southard and D. Goodman, 1979. On the nature of human interlimb coordination. Science 203, 1029-1031. K&o, J.A.s., B. Tuller, E. Vatikiotis-Bateson and C. Fowler, 1984. Functionally specific articulatory cooperation following jaw perturbations during speech: Evidence for coordinative strttctures. Journal of Experimental Psychology: Human Perception and Performance 10, 812-832. K&o, J.A.S. and S.A. Wallace, 1978. ‘Conscious mechanisms in movement’. In: G.E. Stelmach (ed.), Information processing in motor control and learning. New York: Academic Press. pp. 79-116. Kots, Y.A.. 1977. The organization of vc!untary movement. New York: Plenum. Kots, Y.M.. V.I. Krinskiy, V.L. Naydin and M.L. Skik, 1971. ‘The control of movements of the joints and kinesthetic afferentation’. In: I.M. Gelfand, V.S. Gurfinkel, S.V. Fomin and M.L. Tstelin (eds.), Models of functional organization of ceriain biological systems. Cambridge, MA: MIT Press. pp. 371-381. Kots, Y.M. and A.V. Syrovegin, 1966. Fixed set of invariants of interaction of the muscles of two joints used in the execution of simple voluntary movements. Biofizika 11, 1061-1066. Kugler, P.N.. J.A.S. Kelso and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: Theoretical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.). Tutorials in motor behavior. Amsterdam: North Holland. pp. 3-47. Lang, W., M. Lang, F. Uhl, C. Koska, A. Kornhuber and L. Deecke, 1988. Negative cortical DC shifts preceeding and accompanying simultaneous and sequential finger movements. Experimental Brain Research 71, 579-587. Nicolis, G. and I. Prigogine, 1989. Exploring complexity. New York: W.H. Freeman. Polit, A. and E. Bizzi, 1979. Characteristics of motor programs underlying arm movements in monkeys. Journal of Neurophysiology 204, 443-460. Rosen, R., 1985. Anticipatory Systems. New York: Pergamon. Rosenbaum, D., F. Marchak, H.J. Barnes, J. Vaughan, J.D. Slotta and M.J. Jorgensen, 1990. ‘Constraints for action selection: Overhand versus underhand grips’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 321-342. Saltzman, E. and J.A.S. Kelso, 1987. Skilled actions: A task dynamic approach. Psychological Review 94, 84-106. Schmidt, R.A., 1987. Motor learning and control: A behavioral emphasis. Champaign, IL: Human Kinetics. Scholz, J.P. and J.A.S. Kelso. 1990. Intentional switching between patterns of bimanual coordination is dependent on the intrinsic dynamics of the patterns. Journal of Motor Behavior 22, 98- 124. Schiiner, G., 1989. Learning and recall in a dynamic theory of coordination patterns. Biological Cybernetics 62, 39-54. Schoner, G., 1990. A dynamic theory of coordination of discrete movement. Biclogical Cybernetics 63, 257-270. SchBner, G., W.Y. Jiang and J.A.S. Kelsn, 1990. A synergetic theory of quadrupedal gaits and gait transitions. Journal of Theoretical Biology 142, 359-393. Schoner. G. and J.A.S. Kelso, 1988a. Dynamic pattern generation in behavioral and neural systems. Science 239, 1513-1520. Schoner, 6. and J.A.S. Kelso, 1988h. A dynamic theory of behavioral change. J. Theor. Biology 135, 501-524. Schoner, G. and J.A.S. Kelso, 1988~. A synergetic theory of environmentally-specified and learned Patterns of movement coordination. 1. Relative phase dynamics. Biological Cybernetics 58, 71-80. Skinner, R.D. and E. GarciaRiB, 1990. ‘Brainstem modulation of rhythmic functions and behaviors’. In: W.R. Klemm York: Wiley.
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Soechting, J.F. and C.A. Terzuolo. 1986. An algorithm for the generation of curvilinear wrist motion in an arbitrary plane in three-dimensional space. Neuroscience 19, 1393-1406. Stelmach, G-E., J.A.S. Kelso and S.A. Wallace, 1975. Preselection in short term motor memory. Journal of Experimental Psychology: Human Learning Memory 1, 745-755. Teuber, H.L., 1964. ‘The riddle of frontal lobe function in man’. In: J.M. Warren and K. Akert (eds.), The frontal granular cortex and behavior. New York: McGraw-Hill. pp. 410-444. Tuller, B. and J.A.S. Kelso, 1990. ‘Phase transitions in speech productton and their perceptual consequences’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 429-452. Uno, Y., M. Kawato and R. Suzuki, 1989. Formation and control of optimal trajectory in human multijoint arm movement. Biol. Cybem. 61, 89-101. Viviani, P. and C. Terzuolo, 1980. ‘Space-time invariance in learned motor skills’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland. pp. 525-533. von Holst, E. and U. von Saint Paul, 1973. ‘On the functional organization of drives’. In: R. Martin (ed. and trans.), The collected papers of Erich von Holst: Vol. 1. The behavioral physiology of animals and man. Coral Gables, FL: Univ. of Miami. pp. 220-258. Wallace, S.A., D.L. Weeks an J.A.S. Kelso, 1990. Temporal constraints in reaching and grasping behavior. Human Movement Scie:: ‘e 9,69-93. Zanone, P.G. and J.A.S. Kelso, to appear. Intrinsic dynamics and their evolution with learning.
eaction to
llock and
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rossberg,
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Florida Atlantic University, Boom Raton, USA
The study of voluntary movement and posture has a long history, full of philosophicai and conceptual content. It was not so long ago, in fact before the Watershed caused by the Kepler-Galilean synthesis, that willed movement - or, as the ancients saw it, how the mind rules the body - was a model for celestial motion. Bullock and Grossberg’s paper is a significant contribution because it provides specific neural network modules for the hypothetical operations involved. As is characteristic of much of the author’s work (in Grossberg’s case spanning many years) great care is taken to synthesize selected neural and behavioral evidence into a single theoretical picture. It is enormously pleasing for this commentator to see, in his adopted field, how sophisticated mathematics and theory can be meshed with explicit computational models and experimental data in such a masterly fashion. My comments invite B&G to extend their scenario for posture and voluntary movement, to probe the generality of their formulation and to suggest, in certain cases, a need to supplement or even substitute existing architectures. I confine my remarks largely to the behavioral or
* Much of the research discussed herein is supported by NIMH (Neurosciences Research Branch) Grant MH42900, BRS Grant RR07258, and ONR Contract NOOO14-99-J1191. Author’s address: J.A.S. Kelso, Program in Complex Systems and Brain Sciences, Center for Complex Systems, Florida Atlantic University, Boca Raton, FL 33431, USA.
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functional level with the goa! of achieving a self-consistent description at that scale. My views are very much in the spirit of a ‘ psychologicai thermodynamics’ that Grossberg has championed in many of his talks and papers. One means here, of course, a search for the laws and principles of coordinated behavior. Elsewhere we have suggested tjhat behavioral laws, on a chosen level of description, can be founded upon, and methodologically grounded in the concepts and tools of synergetic phase transitions (see e.g. Haken and Stadler 1990; Kelso 1990).
&G’s orientation to the role of nonspcrcific signals because they play a remarkable role in vertebrate behavior, e.g. the multiple behaviors of the farmyard chicken (von Hoist and van St. Paul 1973), the multiple modes of locomotion in decerebrate and intact cats (see Skinner and Garcia-till (1990) for review). The brain&em nuclei, both cholinergic and catecholaminergic, prove to be indispensable in the nonspecific control of these behaviors. Actually, as shown in Synergetits (f&ken 1983) nonspecific signals play a major role in pattern formation and cooperative phenomena in many natural systems. For example, in open nonlinear dynamical. systems, spatial, temporal and functional patterns arise spontaneously in a self-organized fashion when nonspecific coatrol parameters are changed (for many examples, and a full theoretical treatment see Haken 1983; i4icohs and. Prigogine 1989). Control parameters are unspecific to the resulting patterns; they “control’ only in the sense of keeping the system in a stable operating range or by leading the system into instabilities where new patterns are formed. Around these instabilities many theoretically predicted fcatures have been observed in the perceptual-motor system (see e.g. Jeka and Kelso (1989) Schiiner and Kelso (1988a, b) for reviews). B&G’s predictions about nonspecific control signals lie at relatively high levels of the nervous system, but it is important to recognize that such inputs play a ro!e in self-organization and pattern formation at many scales of observation. In my view, the authors’ emphasis on nonspecific input in motor control is far from misplaced, often ignored in the literature, yet quite fundamental. 0ne or two questions arise concerning their GO signal, however (see below).
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inv
The benefit of nonspecific control signals inducing qualitative changes in kinematic pattern is that they allow one to identify the essential dimension(s) of the pattern that remains ‘invariant’ in the linear range of the control parameter. In a complex system how does orie know which variables are relevant? Experimentally observed kinematic invariants offer a hint about the underlying control algorithms or design constraints that the CNS may exploit (e.g. Kelso et al. 1979; Viviani and Terzuolo 1980; Schmidt 1987). For example, the relative timing or phasing among active components can remain ‘invariant’ over changes in movement speed. However, emphasis on findin.g invariant properties raises a number of serious statistical issues (Tuller and Kelso 1990). How much deviation from invariance does one accept? I would argue, based on empirical and theoretical grounds, that a more appropriate concept in this case is stability rather than invariance. r For example, one well-known order parameter, relative phase, displays enhanced fluctuations as the pattern loses its stability. Note that if careful attention is paid to time scales, the first moment, the mean, does not change as the control parameter is varied except at transition points. The order parameter appears ‘invariant’ across changes in speed, but order parameter fluctuations are a conceptually crucial part of the dynamics, probing the stability of collective states and allowing the system to discover new patterns. We refer to the order pdra meter dynamics in the absence of specific parametric change as intrinsic pattern dynamics, reflecting the internal coordination constraints on the system. The great advantage of knowing these intrinsic dynamics is that one can specify what changes due, e.g. to environmental, learned or intentional influences. That is, infor’ This is not to deny the plJwerfu1 concept of invariance for coordinated behavior especially when linked to symmetry properties. For example, symmetry has been used to classify coordination patterns (hand movements, locomotory gaits) and to restrict the functional form of the pattern dynamics (Schijner J$tal. 1990; also Haken et al. 1985). Symmetries are expressed as invariance of pattern under a set (group) of transformations. The idealized walk, trot, gallop and jump are members of the same symmetry group, i.e. the only patterns that are invariant under left-right, front-hind and time inversion operations. Quantitative analysis of kinematic patterns, on the other hand, seems better tied to the concept of stabili!y, not (as in the motor control literature) the notion of invariance, because statistical fluctuations are a necessary part of the measurement process and prove to be conceptually relevant (see text).
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n-ration can be formally expressed as a contribution to, a force acting upon, or a perturbation of, the intrinsic dynamics. Although a behavioral description figures prominently in the B&G model, the authors do not really say what the intrinsic behavioral constraints are in the task of reaching to targets. How can these be found for multi-joint reaching tasks? In synergetic/dynamic pattern theory, the necessary methodological step is to experimentally induce qualitative behavioral changes. One clue comes from a simple study by Rosenbaum and colleagues (1990). The basic phenomenon is as follows: subjects are faced with a bar or dowel, the left half of which is black, the right half white. The bar hes in a horizontal position between two supports. The task is to reach and pick up the bar placing the black (or white) end up hen black is up (white down) the (or down) on a nearby surface. subjects invariably grasp with an overhand grip. When black is down (white up) an underlaand grip is chosen. Note, the task and its accomplishment are binary by definition. The measure that Rosenbaum et al. (1990) use is simply how many subjects do one thing or the other. My students and I (Kelso 1989; Kelso et al. 1990b) have adopted the synergetic/ dynamic pattern approach to understanding this pattern selection process. In our experiments, a stepping motor moves the bar around a clocklike surface in either a clockwise (cw) or counterclockwise (ccw) direction. Wrist/forearm rotation and limb projection are measured using infrared light emitting diodes placed on relevant points of the limb. A single trial consists of a set bar angle followed by a tone which signals the subject to reach for the bar. The task is to return the bar to the twelve o’clock position (90 degrees). Our study reveals very interesting constraints and dynamical effects on reaching and grasping for an object that we invite B&G to consider: (1) For an entire range of bar rotations, from approximately 1 o’clock to 5 o’clock (in the clockwise direction) the same basic pattern is adopted. The phase relation, or joint angle velocity ratio, between limb projection and wrist/forearm rotation is constant in time and consists of an underhand, supine grasping motion. (2) Near 6 o’clock, however (it varies a little among subjects), the pattern switches spontaneously; an overhand counterc&-kwise pronation motion emerges, approximately 180 degree anti-phase with the previously supine clockwise motion. The different phase relations for the patterns are quite understandable: the projection or transport component stays the same during reaching, only the direction of forearm-wrist rotation changes. (3) When the bar is
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rotated in the opposite (counterclockwise) direction the same basic patterns are seen hut the switching now occurs at a different spot. Tl;pt is, the pattern adopted depends on the direction of the control parameter change (bar rotation), a clear hysteresis eflect (4) When the bar is perturbed during the reach itself (analogous to the ‘target switching’ paradigm discussed by B&G) several things happen: (a) if the direction of the perturbation does not cross the critical region, the pattern’s topology is unchanged; (b) if the perturbation crosses the critical region, the first reaction is to keep the same pattern but to recruit new degrees of freedom (e.g. trunk tilt, shoulder lean). Not until the perturbation is quite large (well into the stable region of the other pattern) does one observe perturbation induced switching. Why are these results relevant? First, they reveal the existence of an equivalence class of patterns for a wide range of object orientations. You don’t have to ‘reprogram’ the pattern for every object orientation: the basic reaching pattern stays the same and can be metrically adjusted for different object orientations. The reaching act in this task involves a constrained temporal relationship between hand projection to the target and forearm/wrist rotation. Second, the flexibility of the system is revealed at critical values of the control parameter, when switching occurs from the prone to supine pattern. Third, nonlinear dynamical effects ‘are observed such as hysteresis, context-dependent perturbation effects and recruitment of new degrees of freedom at critical points. The latter result appears extremely interesting. Unlike typical nonequilibrium phase transitions in which the system’s dimensionality is reduced, in the present situation, additional degrees of freedom come in at critical points. It is not clear, however, that the dimensionality of the system in terms of attractors of the pattern dynamics necessarily changes. I mention these results here as more of a challenge to B&G’s model of voluntary motor control. I see the flexible task-determined recruitment of degrees of freedom as an essential, adaptive feature of real biological systems, the basis of which must be understood and incorporated into neural network models for animal and robot control. I invite ow would VITE and FLETE accommodate B&G to c0mmen.t. target-induced qualitative changes? Even without an externally defined target the underlying neuromuscular organization for multijoint limb trajectories seems more abstract ere I refer to experiments than VITE and FLETE presently allow.
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that demonstrate that the orderly coordination among the joints of a single limb depends on the spatial orientation of the forearm, not on preferred coupling among muscle groups (Kelso 1990; Kelso et al., to elationships between elbow- and wrist appear). Stable joint to be direction-specific (see also Baldissera flexion and extension pr et al. 1982). Thus, when t e forearm is supine (palm facing up), flexion flexion (extension) of the elbow exhibit a (extension) of the wrist (see also Kots and Syrovegin 1966; stable relative phase r When the forearm is prone (palm facing Soechting and Terzuol nspecific changes in two control paramedown) the opposite is true. ters, frequency of motioc d spatial orientation, are capable of effecting changes from 0 nematic (and electromyographic) pattern to the other, i.e. the dyn_tics are minimally codimension two. An interesting point for $3 is that lesions of parietal cortex, which t+tic information, produce dramatic changes effectively eliminate in coordinative performan between elbow and wrist joints. Performance of single joint movements, however, can be carried out easily and accurately (Kots et al. 1971; see also Kelso 1977; Polit and Bizzi 1979), suggesting that propriospecific information is crucial to sustaining coordinative function between joints. What impact might knowledge of these constraints have on B&G’s model? I acknowled leve their FLETE module aims at a detailed structure-function ever, in the spirit of generalization the authors might conjecture how these nonspecific signals at a behavioral level influence compliance signals, Renshaw cells, Ia interneurons, muscle spindles, etc. As one can set, the behavioral pattern dynamics for single, nmltijoint limb trajectories are already quite
er
s
In a similar vein, B model, they claim, is vastly superior to variational schemes such as ogan and Flash’s (1987) ‘minimum jerk model. In their preferred ‘ ‘mum torque change’ model, Uno et al. (1989) have also shown that joint torques need to be taken into account in two degree of freedom pl motions, especially when subjects must bring the band close to the dy or extend the limb to the limits of work space. ere again I t these debates between different minimization schemes are somewhat moot, in part because the paradigms
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restrict the conclusions (see also Kelso et al. 1984). For example, if an object is placed near the limits of -work space, additional degrees of freedom (forward body lean) are spontaneously recruited provided the motion is not restricted. Similarly, if a subject must move through a via point close to his or her chest (cf. Uno et al. 1989), the spontaneous response of the biological system is to lean back, not dumbly follow some minimization scheme. At these perceived transition regions, additional degrees of freedom again appear and are assimilated into the existing structure. The synergetic/dynamic pattern theory offers a natural language for understanding these effects in terms of order-order transitions. How do B&G accommodate such results on the behavioral level? (I am not asking for a description of correlated activity in neural structures.) y point is that when we open up our paradigms, complexify them so to speak, then often we see elegantly simple solutions emerge. When we restrict our paradigms, for instance in order to test different variational schemes for two degree of freedom motions, the results may be too local and restrictive to mean very much. I cannot get too excited over the degree of asymmetry in end effector velocity profiles, or that one gets two bumps in the velocity profile (Uno et al. 1989) rather than one. For what it is worth, my own view is that the two bumps would disappear if the subject was not forced by the apparatus to limit his/her reaching to two mechanical degrees of freedom.
(I) Anticipatory Dynamical Systems (ADS) A nice feature of the VITE model is that kinematic invariants emerge through network interactions rather than an explicitly precomputed command. Automatic processes include computation of a Present Position Command (PPC) and a Difference Vector (DV) between PPC and the Target Position Command (TPC). The PPC is gradually updated by integrating the DV through time. The time varying ~30 signal is independently controlled, acting multiplicatively on the before it is integrated by the PPC. As I understand it (fig. 2) the TPC does not interact directly with the GO signal. A key aspect of the VITE
1n0
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circuit is that the PPC is computed by outflow, iced forward signals rather than inflow, feedback signals. 1 wonder whether a slightly different circuit might be envisaged. A crucial feature of biological systems is their ability to anticipate future states of affairs (see e.g. Rosen 1985). In stable environments, the goal of an act, the future as it were, acts back i.n a locally time reversible fashion on the system’s parameter settings for the accomplishment of the goal. It seems poss;ible that this may be a ‘one shot’ rather than a continuous updating process: given some intrinsic dynamics (the order parameter equation of motion), the goal (e.g. a required final position) acts on those dynamics setting the parameters for goal achievement. For a neural network, the requirement to continuously update weights ropagation, or to compute the DV continuously in time tituted by an Anticipatory Dynamical System which adjusts or perturbs the parameters of its intrinsic dynamics (representative of the system’s existing pattern generating capabilities) before any action whatsoever. The evidence for this kind of Grcuit, like that cited by B&C, is mostly correlative. 9n the behavioral side the enhanced accuracy of voluntary (s&ject-defined) over constrained (e defined) movements - the so-called preselection effect and well-studied Stelmach et al. 1975) - was one of the most reliab effects in the motor behavior area in the 70’s (see elso and Wallace (1978) for a review). The kinematics reflect anticipatory control from the very beginning of preselected, compared to other kinds of movement, and can be quantified in terms of an index of anticipation (Kelso et al. 1978). Evidence for prior parameter setting in the nervous system based on the anticipated consequences of action (a corollary discharge rather than an efference copy) exists at many levels, from the early studies of Teuber (1964) on frontal lobe deficits, to the modification of reflex excitability be ent, demonstrated in numerous Soviet studies (e.g. y suggestion implies a kind of nonlinear causality, in which the future - a particular goal - adjusts the parameters/initial conditions of the system specifically for the accomplishment of that goal. sic dynamics (Kelso Mentions modify and are modified by the int so 1988b), but can et al. 1988; Scholz and elso 1990; Schiiner and dynamical systems anticipate future states? The standard picture, notwithstanding the existence of deterministic chaos, is that given some initial conditions and a law, future states are determined. Nonlinear
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dynamical systems, however, display behavior that can be interpreted as anticipatory. Iterative maps, such as the phase attractive circle map which describes coordination in multi-frequency systems (DeGuzman and Kelso, in press; elso and DeGuzman 1988; see also Beek 1989) are a good illustration. Near tangent (inverse saddle node) bifurcations, for example, phase densities emerge that appear to ‘anticipate’ upcoming stable solutions. Thus, as the system approaches a critical point, slowing down occurs, t e phase gathers and it is easy to predict future states even though the tion is quasiregular. The so-called ‘forward orbit at the critical point’ has all the appearance of an anticipatory dynamical system (see figs. 1 and 2 plus captions). Strictly speaking re is no DV, adaptive linearization for outflow-inflow mismatches llock and Grossberg 1988) and so forth. Like B&G, however, kinematic properties emerge from the underlying dynamics - a consistent, but unevenly interpreted, theme in our work (e.g. Kelso and Holt 1980; Kugler et al. 1980). Without detracting in any way from their VITE module, might the authors consider an ADS (Anticipatory Dynamical System) module as an alternative or complement to VITE? Let me pursue this notion a bit futher and consider another kind of targeting behavior, synchroruzing a discrete movement with a periodic signal whose frequency may be varied. The relationship between environmental event and the hand is again captured by a collective variable, relative phase, whose dynamics can be explicitly formulated and derived (Kelso et al. 1990a). Notice that if the time between events is too long, the syste s behavior is reactive, and characterized by reaction time delays. owever, over a wide range of environmental periodicities, phase and frequency entrainment curs. That is, the system’s dynamics ars: predictive or anticipatory. ere the relationship between environment and action systems may be theoretically understood as a pattern formation process, in which a single collective variable spans both components. Different kinds of experimentally observable patterns (entrainment, loss of entrainment, relative coordination, etc.) correspond to different regimes of the same underlying dynamics (Kelso et al. 1990a) Although it is possible to describe this predominantly temporal targeting behavior in terms of feed forward, predictive and feedback, reactive loops, the dynamical model seems much more parsimonious. Notice the concepts are different also, yet the equation of motion for the pattern dynamics clearly constitutes an ADS.
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Why is the behavioral description of reaching for a static object so different (and apparently more complicated) than the case of targeting behavior analyzed above? The latter seems simple, once it is captured in terms of patterned modes of entrainment between environment and action components. Yet the behavior itself, involving anticipation, prediction and intention is quite deep. The former is typically bound up in all kinds of difficult concepts and operations, even at the phenomenological level discussed by B&G, never mind underlying neural processes. To find the laws of reaching it might be useful to consider a higher level of analysis that incorporates the intrinsic constraints of the actor and environmental information in terms of a unified spatiotemporal pattern (for a start on this problem see Wallace et al. 1990; and the work of e.g. Jeannerod, Wing, and Soechting cited therein). In the dynamic pattern view, information (a) has no meaning outside its influence on the collective variables that characterize the repertoire of behavioral patterns; and (b) thereby defies an attractor of the pattern dynamics. The point that I invite B&G to consider is that the pattern dynamics for the control of reaching (analogous to our ADS module
Fig. 1. (a) Iterations
of the phase attractive
circle map:
1-c A COS(~~~,)] sin(2?r&), +h.kl=+“+~-g[
mod i
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for A - 0, D = 0.131, and K = 0.80. The map is a simple theoretical model of coupled, nonlinear osci!lators where K is tbe strength of the nonlinearity, B is tFe frequency ratio among the oscillatory components and A expresses the bistability of relative phase (coexistence of inphase and anti-phase). Near the tangent bifurcation shown here, very close to a fixed point (period 1) solution, the phase gathers and slows. The reason is th.at the iterations are compressed into the narrow space between the function and the 45 degree line. The message of this simple model, which accommodates a number of experimental observations is that nonlinear dynamical systems can display ‘anticipatory’ features. Note that the discontinuity of the function is not rea:, but due to rectangular coordinates parameterized in the interval [0, 11. The function is an invertible mapping of the circle onto itself (a Poincare section of the flow on the torus). (b) The corresponding bifurcation diagram where K is the bifurcation parameter (0.4 > K -C1.0). A = 0 and 52 = 0.131. The relative phase, $I is on the Y axis. The fuzzy area corresponds to quasiperiodic motion because the frequency ratio between the components is irrational. Note, however, that the darkening region ‘anticipates’ the tipccming stable sohnion (the single line) which indicates the system is trapped or mode-locked 1: 1. Much fancier dynamics, of course, are possible as K is increased beyond the critical suriace where the map lases invertibility. The main point is that the. system spends more time in a particular phase as it approaches the critical point. It behaves as E-I Anticipatory Dynamical System (ADS). ADS stay in contact with the future by Living near critical points.
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for synchronization) are function-specific and thus may cover hand and target. Perhaps this point relates to their Vector Associative Map bu.t I am not sure. (2) The nonlinear dynamics of posture, discrete AND rhythmical movement
Returning specifically to posture and voluntary inovement (both discrete and rhythmic), Schijner and I (Kelso and Schiiner 1988; Schiiner 1990) have offered an explicit alternative to B&G at the level of end effector kinematics, i.e. position, x and velocity dx/dt. The approach is continuous with our treatment of intentional changes in relative timing behavior (Kelso et al. 1988; Scholz and Kelso 1990; Schijner and Kelso 1988b), and earlier ideas about targeting behavior (Kelso et al. 1980: 50; Kelso and Holt 1980). An intention to move is expressed as a part of the end effector dynamics that stabilizes the coordination pattern, in this case the position-velocity curve (as a piece of a limit cycle). The structure of the intrinsic dynamics, viz. the pattern dynamics in the absence of an intention to move, admits (a) postural stages as stable fixed point attractors; (b) two coexisting postural stages corresponding to the initial and target postures joined by a limit cycle trajectory (discrete movement); and (c) rhythmic motion, captured by limit cycle oscillations. The phase diagram of an exactly solvable nonlinear oscillator model (Gonzalez and Piro 1987) similar to one established by theoretical (Haken et al. 1985) and empirical work (Kay et al. 1987) contains all of these dynamical structures (see Schiiner 1990). Like B&G, Fitts Law relationships between amplitude and movement time can be accommodated in terms of relations among dynamic parameters. In addition, coordination of dis,Lrete movements of different amplitudes (Kelso et al. 1979) is accomplished by coupling individual components that have different intrinsic movement times in a fashion similar to the case of rhythmic
Fig. 2. The same picture as fig. 1 for different parameter values. The map (panel a) shows quasiperiodic motion near a period 3 (B = 0.355) where the component frequencies are almost related in a 3 : 1 ratio. The corresponding bifurcation diagram is shown in panel b. Exactly the same features as fig. lb are displayed. This time however, the phase densities increase as :he period 3 (single lines) is approached. Signature features, such as critical slowing are predictive of upcoming critical points where pattern switching occurs.
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coordination ( aken et al. 1985). The stability of the coordination pattern accommodates remote compensatory behavior when one comrturbed (Schiiner 1990; see also Saltzman and Kelso 1987). G, we interpret the pattern dynamics as representing the underlying neural organization, as long as the end effector kinematics are not strongly perturbed. When they are (cf. Kay et al., in press), it may be necessary to distingu.ish end effector position and velocity from the level of n ous system control of these variables (see Bizzi and t this is not the main point of discussion here. The Abend 1982). present ‘architecture’ possesses both similarities and differences to B&G, outside the issue of the data base. In the dynamic pattern model, the form, trajectory and temporal structure are determined by the intrinsic dynamics, which, once known, are ‘built in’ as a law. Movement initiation (maybe our equivalent of a GO signal) corresponds to perturbing the intrinsic dynamics, simply destabilizing one pattern -Y+h then switches to another. No explicit timing information is necessary. At the level of behavior, these dynamics contain posture, discrete and rhythmical movements of one or two end effecters, depending on parameters. The economy of description afforded by nonlinear dynamical models, and the contact between experiment and theoretical prediction is appealing. Neural networks that implement dynamical systems bviously constitute and active area of research (see e.g. Jordan 1990). &c’s VITE circuit simulates numerous properties d discrete movement. But how does it generate rhythmic suspect that the form of their GO s nal, which presently persists beyond the discrete movement time (see 6, fig. 5) must be modified considerably.
&G starts with the movement, grows during the n beyond the movement. If B&G are correct, why should the ‘will to act’ last longer than the movement? I am somewhat doubtful that G’s GO signal corresponds to the ‘will to act’. The will to act builds up over a long time before the movement begins. Why does it take so long and why is there so much activity in the nervous system before overt, volitional movement? The famous
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Bereitschaftpotential of Deecke and Komhuber is a case in point. The rise of this potential looks a lot like B&G’s GO function except that it occurs as much as a second before movement onset, and then changes sign. The sources of this readiness potential which occurs, to my knowledge, only in volitional movement (unlike, say, the CNV) are not entirely known, but the evidence, using all kinds of brain imaging and electrical recordings points strongly to supplementary motor cortex (Lang et al. 1988). At Neils Bohr’s institute in Copenhagen, when it was time to take a break, the physicists (including Bohr himself) would sometimes go to Western and gangster movies. In gunfights, Bohr noticed that the guy who drew his gun first always lost. Bohr’s collaborators (Gamow, Landau, Casimir et al.) were skeptical, but on submission to experimental test with cap pistols, Bohr’s observation was correct. According to Dirac (Blaedel 1988), Bohr formulated a new theory based on this finding. In Bohr’s theory, when the two cowboys face epch other in a gunfight all they can do is talk, because they both know that the one ,who draws first will die. The ‘will to act’ takes much longer to evolve than the reaction to a stimulus and is somehow slower. B&G’s GO signal which starts with the onset of movement is an unlikely candidate for the will to act, though it might help win a duel. But if Bohr is right, why are willed actions slower, or are they?
Lots of claims about the nature of skill learning are made by B&G. A main one, that early in skill learning movements are made with low compliance and low speed, appears more intuitive than factual. Given the controversies that have surrounded the cerebellum, it is probably not so useful to debate whether “it is now well-established that the cerebellum is a critical module for learning predictive movementcalibrating signals’. Such statements miss the point, as far as the present comments are concerned. Every major learning theorist has acknowledged that learning does not begin on an entirely blank slate, even in the neonate. The initial state is not a disordered random network, but an already highly organized system (e.g. Fitts 1964). Learning must therefore involve some kind of order-order transition, whether abrupt or continuous.
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Some of the scientific problems are (a) to quantitatively evaluate the initial structure prior to learning a new skill; and (b) to probe how that structure evolves in time as new tasks are learned. Predictions regarding an individual subject’s learning behavior follow, in our view, from knowledge of his/her intrinsic dynamics, i.e., the existing attractor layout or phase diagram. In the case of relative timing skills these predictions have been tested. For example, it can be demonstrated (Kelso 1990; Zanone and Kelso, to appear) that if the required task deviates from the intrinsic pattern dynamics, phase transitions in learning (qualitative changes in the phase diagram) occur due to competition (Schiiner and Kelso 1988~). This layer of behavioral pattern dynamics is missing in B&G, as it is in most formal neural network learning and pattern recognition schemes (cf. Schiiner 1989). Identification of the intrinsic dynamics and a metric for evaluating them would be useful, in my view, for the skills of interest to B&G. Put another way, there is more to learning to bring a glass of milk to the mouth than low speed, low compliance parameter setting and cerebellar computation.
Bald&era, F., P. Cavallari and P. Civischi, 1982. Preferential coupling between voluntary movements of ipsalateral limbs. Neuroscience Letters 34, 95-100. Beek, P.J., ..989. Juggling dynamics. Amsterdam: Free University Press. Bizzi, !Z. and W.A. Abend, 1982. ‘Posture control and trajectory formation in single and multiple jomt arm movements’. In: J.E. Desmedt (ed.), Brain and spinal mechanisms of movement control in man. New York: Raven Press. Blaedel, N., 1988. Harmony and unity. The life of Neils Bohr. Berlin: Springer-Vedag. Bullock, D. and S. Grossberg, 1988. ‘The VITE model: A neural command circuit for generating arm and articulator trajectories’. In: J.A.S. Kelso, A.J. Mandell and M.F. Shlesinger (eds.), Dynamic patterns in complex systems. Singapore: World Scientific. Bullock, D. and S. Grossberg, 1991. Adaptive neural networks for control of movement trajectories invariant under speed and force resealing (Target article). Human Movement Science 10, 3-53 (this issue). DeGuzman, G.C. and J.A.S. Kelso, in press. Multifrequency behavioral patterns and the phase attractive circle map. Biological Cybernetics. Fitts, P.M., 1964. ‘Perceptual-motor skill learning’. In: A.W. Melton (ed.), Categories of human learning. New York: Academic Press. pp. 254-285. Flash, T. and N. Hogm, 1985. The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neuroscience 5, 1688-1703. Gonzalez, D.L. and 0. Piro, 1987. Global bifurcations and phase portrait of an analytically solvable nonlinear oscillator: Relaxation oscillations and saddle node collisions. Physical Review A 36,4402-4410.
J.A.S. Kelso / Anticipatory
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109
Haken, H., 1977/1983. Synergetics, an introduction: Non-equilibrium phase transitions and self-organization in. physics, &nristry and biology (3rd ed.) Berlin: Sptinger-Verlag. Waken, H. and M. Stadler (eds.), 1990. Synergetics of cognition. Berlin: Springer-Verlag. Haken, H., J.A.S. Kelso and H. Bunz, 1985. A theoretical model of phase transitions in human hand movements. Biological Cytametics 39, 139-156. Hogan, N. and T. Flash, 1987. Moving gracefully: Quantitative theories of motor coordination. Trends in Neuroscience 10, 170-174. Jeannerod, M., 1984. The timing of natural prehensile movements. Journal of Motor Behavior 16, 235-254. Jeka, J.J. and J.A.S. Kelso, 1989. ‘The dynamic pattern approach to coordinated behavior: A tutorial review’. In: S.A. Wallace (ed.), Perspectives on the coordination of movement. Amsterdam: North-Holland. pp. 3-45. Jordan, MI., 1990. ‘Motor learning and the degrees of freedom problem’. In: M. Jeannerod (ed.) Attention and performance XII. Hillsdale, NJ: Erlbaum. pp. 796-836. Kay, B.A., J.A.S. Kel;o and E.L. Saltzman, in press. Steady-state and perturbed rhythical movements: A dynamical analysis. Journal of Experimental Psychology: Human Perception and Performance. Kay, B.A., J.A.S. Kelso, E.L. Sa!tzman and G. Schiiner, 1987. The space-time behavior of single and bimanual movements: data and model. Journal of experimental Psychology: Human Perception and Performance 13, 178-192. Kelsu, J.A.S., 1975. ‘Central and peripheral information in motor control’. In: J. King and W.W. Spirduso (eds.), Motor control symposium. Austin, TX: University of Texas Press. Kelso, J.A.S., 1977. Motor control mechanisms underlying human movement reproduction. Journal of Experimental Psychology: Human Perception and Performance 4, 529-543. Kelso. J.A.S., 1989. Haptics and sensory-guided motor control. Paper presented at ONR Cognitive and Neural Science Program, Woodshole, MA., September. Kelso, J.A.S., 1990. ‘Phase transitions: Foundations of behavior’. In: H. Haken and M. Stadler (eds.), Synergetics of cognition. Berlin Heidleberg, New York: Springer-Verlag. pp. 180-196. Kelso, J.A.S., J.J. Buchanan and S.A. Wallace, to appear. Order parameters for the neural organization of single, multijoint limb trajectories. Experimental Brain Research. Kelso, J.A.S., J.D. Delcolle and G. Schoner, 1990a. ‘Action-perception as a pattern fomtation process’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 139-169. Kelso, J.A.S., T. Murata, J.J. Buchanan and J.D. Delcolle, 199Oll. Spatially induced phase transitions in discrete reaching. Psychonomic Society, New Orleans, Nov. Kelso, J.A.S. and G. DeGuzman, 1988. ‘Order in time: How cooperation between the hands informs the design of the brain’. In: H. Haken (ed.), Neural and synergetic computers. Berlin: Springer-Verlag. pp. 180-196. Kelso, J.A.S. and K.G. Holt, 1980. Exploring a vibratory systems analysis of movement production. J. of Neurophysiology 43, 1183-1196. Kelso, J.A.S., KG. Holt, P.N. Kugler and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: Empirical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland: pp. 49-Z. Kelso, J.A.S., J.H. Pruitt and D. Goodman, 1978. ‘The anticipatory control of movement’. In: G.C. Roberts and KM. Newell (eds.), Psychology of motor behavior and sport. Champaign, IL: Human Kinetics. pp. 192-203. Kelso, J.A.S., J.P. Scholz and G. Schiiner, 1988. Dynamics governs switching among patterns of coordination in biological movement. Physics Letters A 134, 8-12. Kelso, J.A.S. and G. Schoner, 1988. Dynamic patterns of sensorimotor behavior: Biological control structures and constraints. Neural Networks 1, (Suppl. l), 343.
110
J.A.S. Kelso / Anticipatory dynamical systems
K&o. J.A.S., D.L. Southard and D. Goodman, 1979. On the nature of human interlimb coordination. Science 203, 1029-1031. K&o, J.A.s., B. Tuller, E. Vatikiotis-Bateson and C. Fowler, 1984. Functionally specific articulatory cooperation following jaw perturbations during speech: Evidence for coordinative strttctures. Journal of Experimental Psychology: Human Perception and Performance 10, 812-832. K&o, J.A.S. and S.A. Wallace, 1978. ‘Conscious mechanisms in movement’. In: G.E. Stelmach (ed.), Information processing in motor control and learning. New York: Academic Press. pp. 79-116. Kots, Y.A.. 1977. The organization of vc!untary movement. New York: Plenum. Kots, Y.M.. V.I. Krinskiy, V.L. Naydin and M.L. Skik, 1971. ‘The control of movements of the joints and kinesthetic afferentation’. In: I.M. Gelfand, V.S. Gurfinkel, S.V. Fomin and M.L. Tstelin (eds.), Models of functional organization of ceriain biological systems. Cambridge, MA: MIT Press. pp. 371-381. Kots, Y.M. and A.V. Syrovegin, 1966. Fixed set of invariants of interaction of the muscles of two joints used in the execution of simple voluntary movements. Biofizika 11, 1061-1066. Kugler, P.N.. J.A.S. Kelso and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: Theoretical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.). Tutorials in motor behavior. Amsterdam: North Holland. pp. 3-47. Lang, W., M. Lang, F. Uhl, C. Koska, A. Kornhuber and L. Deecke, 1988. Negative cortical DC shifts preceeding and accompanying simultaneous and sequential finger movements. Experimental Brain Research 71, 579-587. Nicolis, G. and I. Prigogine, 1989. Exploring complexity. New York: W.H. Freeman. Polit, A. and E. Bizzi, 1979. Characteristics of motor programs underlying arm movements in monkeys. Journal of Neurophysiology 204, 443-460. Rosen, R., 1985. Anticipatory Systems. New York: Pergamon. Rosenbaum, D., F. Marchak, H.J. Barnes, J. Vaughan, J.D. Slotta and M.J. Jorgensen, 1990. ‘Constraints for action selection: Overhand versus underhand grips’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 321-342. Saltzman, E. and J.A.S. Kelso, 1987. Skilled actions: A task dynamic approach. Psychological Review 94, 84-106. Schmidt, R.A., 1987. Motor learning and control: A behavioral emphasis. Champaign, IL: Human Kinetics. Scholz, J.P. and J.A.S. Kelso. 1990. Intentional switching between patterns of bimanual coordination is dependent on the intrinsic dynamics of the patterns. Journal of Motor Behavior 22, 98- 124. Schiiner, G., 1989. Learning and recall in a dynamic theory of coordination patterns. Biological Cybernetics 62, 39-54. Schoner, G., 1990. A dynamic theory of coordination of discrete movement. Biclogical Cybernetics 63, 257-270. SchBner, G., W.Y. Jiang and J.A.S. Kelsn, 1990. A synergetic theory of quadrupedal gaits and gait transitions. Journal of Theoretical Biology 142, 359-393. Schoner. G. and J.A.S. Kelso, 1988a. Dynamic pattern generation in behavioral and neural systems. Science 239, 1513-1520. Schoner, 6. and J.A.S. Kelso, 1988h. A dynamic theory of behavioral change. J. Theor. Biology 135, 501-524. Schoner, G. and J.A.S. Kelso, 1988~. A synergetic theory of environmentally-specified and learned Patterns of movement coordination. 1. Relative phase dynamics. Biological Cybernetics 58, 71-80. Skinner, R.D. and E. GarciaRiB, 1990. ‘Brainstem modulation of rhythmic functions and behaviors’. In: W.R. Klemm York: Wiley.
and R. Vertes (eds.), Brainstem
mechanisms
of behavior.
New
J.A.S. Kelso / Anticipatory dynnmical systems
111
Soechting, J.F. and C.A. Terzuolo. 1986. An algorithm for the generation of curvilinear wrist motion in an arbitrary plane in three-dimensional space. Neuroscience 19, 1393-1406. Stelmach, G-E., J.A.S. Kelso and S.A. Wallace, 1975. Preselection in short term motor memory. Journal of Experimental Psychology: Human Learning Memory 1, 745-755. Teuber, H.L., 1964. ‘The riddle of frontal lobe function in man’. In: J.M. Warren and K. Akert (eds.), The frontal granular cortex and behavior. New York: McGraw-Hill. pp. 410-444. Tuller, B. and J.A.S. Kelso, 1990. ‘Phase transitions in speech productton and their perceptual consequences’. In: M. Jeannerod (ed.), Attention and performance XIII. Hillsdale, NJ: Erlbaum. pp. 429-452. Uno, Y., M. Kawato and R. Suzuki, 1989. Formation and control of optimal trajectory in human multijoint arm movement. Biol. Cybem. 61, 89-101. Viviani, P. and C. Terzuolo, 1980. ‘Space-time invariance in learned motor skills’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland. pp. 525-533. von Holst, E. and U. von Saint Paul, 1973. ‘On the functional organization of drives’. In: R. Martin (ed. and trans.), The collected papers of Erich von Holst: Vol. 1. The behavioral physiology of animals and man. Coral Gables, FL: Univ. of Miami. pp. 220-258. Wallace, S.A., D.L. Weeks an J.A.S. Kelso, 1990. Temporal constraints in reaching and grasping behavior. Human Movement Scie:: ‘e 9,69-93. Zanone, P.G. and J.A.S. Kelso, to appear. Intrinsic dynamics and their evolution with learning.