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Stability and flexibility in the temporal organisation of movements
Human Movement North-Holland
347
Science 8 (1989) 347-356
STABILITY AND FLEXIBILITY IN THE TEMPORAL ORGANISATION OF MOVEMENTS Reaction to G.J. van Ingen Schenau (1989) P.J. BEEK
*
Free Unicer.ri~y,Amsterdam. The Netherlands
W.J. BEEK Technrcal Unlr~ersiry.Del&, The Netherlands
Beek, P.J. and W.J. Beek, 1989. Stability and flexibility in the temporal organisation of movements (Reaction to G.J. van Ingen Schenau, 1989). Human Movement Science 8, 347-356.
‘From rotation to translation’ describes in a clear way why joint assemblies consisting of three components incorporating bi-articular muscles have a functional advantage in translatory movements resulting from rotations in joints. On the basis of an elegant biomechanical analysis, it is shown how energy, if geared by the appropriate muscles, can be released in places other than where it is generated. Furthermore, it is shown how the temporally ordered sequence of activation patterns of mono- and bi-articular muscles, as observed in jumping, skating and cycling, conforms to the identified functional characteristics of these two types of muscle. As far as we are able to see, no intrinsic criticism can be raised against these findings and their interpretation. The biomechanical approach adopted is followed rigorously and with creativity. Of course, this state of affairs restricts one’s possibilities to enhance discussion. * Requests for reprints should be sent to P.J. Beck, Dept. of Psychology. Faculty of Human van de Boechorststraat 9, 1981 BT Amsterdam. The Movement Sciences, Free University, Netherlands.
0167-9457/89/$3.50
a 1989, Elsevier Science
Publishers
B.V. (North-Holland)
34x
P.J. Beek, W.J. Beek / Srability
ot~d/le.xrh~lity
We will, therefore, attempt to offer a number of considerations and observations with respect to the coordination and control of movement that seem to fall outside the purview of biomechanics but are of major relevance to a more encompassing understanding of movement behaviour. We believe that the biomechanical research programme developed by Van Ingen Schenau has some inherent limitations for an adequate understanding of the whole realm of coordinated movement behaviour. For the present purposes it may be worthwhile to discuss these limitations and hint at an alternative strategy of dealing with them. In doing so, our concern will be with the topic of tempo& order, because this topic is a crucial one in the study of movement coordination, as is indeed reflected in Van Ingen Schenau’s paper.
Modulation. According to Van Ingen Schenau, the temporal order in the sequence of muscle activations depends primarily on local point values of relative phase, given that the identified constraints ‘are mainly a function of the angles in the joints in relation to the direction of the external force and not, or to a much lesser extent, to the velocity of the movement’ (p. 329). If however, in eqs. (1) and (2) both the distance between hip and ankle ( HA) and the relative phase (6) change in a cyclic fashion, local values of the relative phase will change during a cycle. Thus, the fact that not only relative phase, but also the change of relative phase determines temporal and spatial order is implicit in eqs. (1) and (2). For instance, assume that the motion of HA is periodic. What then is the behaviour of 0 with respect to time? A simple analysis, in which nothing is assumed beyond what is assumed in the equations, shows that in such a case, /3(l) will exhibit a second harmonic, in addition to the base frequency determined by HA, as well as modulation, unless some very unrealistic conditions are met. Still, for the type of task studied by Van Ingen Schenau such modulation may be neglected, because the goal is to jump as high, to throw as far, or to skate as fast as possible: the requirements are primarily energetic, while changes in the environment are relatively unimportant. Thus, under such conditions the system will exhibit behaviour that can adequately be described in terms of local point values only. Nevertheless, many activities require a temporal order that is not only stub/e but also flexible, in order to be able to change according to environmental demands.
P.J. Beck, W.J. Beek / Stability andflexibility
349
To obtain a complete picture of temporal order the constraints on the change of relative phase need to be disclosed, in addition to the constraints on relative phase identified by Van Ingen Schenau. In general, it is in the interplay of these two types of constraint that the variance and invariance of relative timing emerges. Or, to express this in less abstract language, if Van Ingen Schenau concludes that a change in relative time is needed when a skater changes the mean flexion of the knee joint (8) or when a jumper or runner carries extra mass (changing the total mass M), phase lags between characteristic angles of forces and EMD’s have to obey certain dynamical laws that are to still to be found. Our claim is that both frequency-doubling and modulation have to obey certain dynamical principles if coherent action, and not noise, is to be produced. These principles relate to a set of constraints that, in our view, are extremely relevant to understanding temporal order in movement, but seem to fall outside Van Ingen Schenau’s research programme. Temporal order in juggling. To illustrate how both the stable and flexible aspects of temporal order in movement can be understood at a macroscopic scale of observation, we think it is useful to discuss our present understanding of the activity of juggling in some detail. Clearly, the task demands of juggling are not primarily energetic. Rather, the first requirement for successful juggling is that the duration of the cycle time of the hands stands in a specific relation to the cycle time of the balls. Why this should be so can easily be made apparent by distinguishing three time components, namely the time that the hand is loaded with a ball (time loaded: t,), the time that the hand is moving freely (time unloaded: t,) and the flight time of the balls (time flight: t,), and define the cycle time of a hand as t, + t, and the cycle time of a ball as (t, + t,). This partitioning depends on the number of balls juggled (N) and the number of hands (H) that juggle. Now if we assume that the pattern is more or less periodic (in the sense that each configuration of balls recurs at certain time intervals) and that one hand never holds more than one ball, the total of flight time and hand-loaded time per ball must equal the total of hand-unloaded time and hand-loaded time per hand, regardless of the exact values of the absolute cycle frequencies. In other words, balls and hands must match a general timing requirement in that the ratio of the hand loop time and the cycle time of a ball during a loop equals the ratio of the
350
number
P.J. Beck, W.J. Beek / Stnbiiit.v und flexibilr~~
of hands (H)
t, + t, tr+ t, =
H
/
X-1 fi
of balls (N):
0)
N’
which can be rewritten
tUN t H+;
and the number
as:
=l. 1
Expression (2) shows how the relationship between the three component times of juggling can be represented as a relationship between two timing ratios. It follows from this temporal task constraint that if one ratio is fixed, the other ratio should be fixed as well. This allows one to raise the question how the juggler must fill up the loop time of a hand and the loop time of a ball with component times to ensure smooth juggling. The question, metaphorically, is how the juggler must ‘ tile’ the loop time of a hand and the loop time of a ball with component times to juggle comfortably? With respect to a fixed amount of 2-dimensional space, it is wellknown that filling the space completely (so that there are no cracks and gaps) can only be achieved through repetition of homomorphic 2-dimensional forms of a limited number of types (e.g., squares, triangles, hexagons, and the like). With respect to filling a fixed amount of time, defined by two ratios between three time components, there may well be similar limitations. The results of a study on timing and phase locking in juggling in a figure-eight pattern (Beek 1989) suggest that such dynamical constraints do indeed exist. Experiments conducted on four skilled jugglers, juggling 3 balls, revealed that, across three different rates of juggling, the ratios of loaded-time to hand loop time (t,/t,) were always close to l/3 without being exactly l/3. All jugglers tiled according to the same blueprint. In physical terms, the situation is one of near phase locking between two coupled oscillators. In non-linear dynamics, the stable frequency ratios that come into existence between two oscillatory processes are called ‘phase locks’. They are always rational numbers (meaning that the two numbers composing the ratio are, by definition, integers). Evidence was provided that the juggling hands can be conceived as a system of coupled (non-linear) oscillators,
P.J. Beek,
W.J. Beek / Stahhty andflexihilit~~
351
because they share the same dynamical characteristics. The integer proportionality t,: t,: tf is at the core of the temporal order of juggling, defining the potential minimum of complete phase locking. The observed deviations from this potential minimum could also be understood in a physical sense, namely as phase modulation. Thus, a single account could be given of both the stable and flexible properties of the temporal order emerging in juggling with the help of the tools of non-linear dynamics. The fact that jugglers do not precisely operate in the potential minimum associated with complete phase locking, but rather function in the neighbourhood of this minimum, can be theoretically understood as follows. A completely phase-locked system is a system for which the initial conditions are exactly repeatable from one instance of functioning to the next. Exact reproduction of both parameter values, however, is not characteristic of biological movement systems. They are information-driven systems, that are prohibited from residing in a potential trough by virtue of their informational basis. By the arguments advanced here, a system that relates to a potential function can hover about the vicinity of the potential minimum to the degree that it can register information specifying the minimum and the magnitude of departures from the minimum. There is a distinct benefit to this way of functioning, namely, phase lockings can be adjusted relatively freely to accommodate perturbations in the juggling routine. The juggler is not restricted to tiling the temporal spans of ball and hand trajectories with durations in integer ratios. Other durations can be created, suited to the moment-to-moment variations in the juggling conditions. Adaptability to varying circumstances, therefore, is a simple but highly significant consequence of the fact that coordinated movements are dynamical systems governed by information. Perceptual information. Whereas Van Ingen Schenau’s paper puts us in the middle of questions about various types of constraint on human action, informational constraints, as alluded to in the above, cannot be incorporated in a biomechanical account without introducing another discourse. Clearly, the identification and formalisation of constraints on coordinated movement depend on the approach adopted and the specific analytical tools applied. As such, Van Ingen Schenau’s distinction between constraints within the actor, the action and the environment, which runs parallel with Newell’s (1986) distinction between
352
P.J. Beck, W.J. Beek / Stability and flexihilit)
organismic, task and environmental constraints, betrays an interest in how humans and animals perform meaningful actions in their natural environment. It follows, however, from the approach adopted that the constraints identified by Van Ingen Schenau are mainly organismic (anatomical, geometric) and only environmental to the extent to which these constraints can be captured in biomechanical terms (i.e. to the extent that there are forceful interactions with the environment). Thus, following such an approach it is possible to explain, for instance, how a table tennis player can attain maximal angular bat velocity during an attacking forehand drive, but not how critical parameters can be modulated in order to attune the movements with respect to the approaching ball (Bootsma and Van Wieringen in press). In many instances, the first requirements for successful actions are not exclusively in a sufficiently large delivery of either force or energy, but also, and foremost, in an optimal guidance of force and/or energy on the basis of perceptual information. The limits tc biomechanical explanations of coordinated behaviour becomes especially apparent in the temporal order that may be manifested in activities in which the moving limbs are only informationally (and not mechanically) coupled, such as in music-making, or in walking side-by-side and dancing. A remarkable example of temporal ordering in this type of activity is provided by Schmidt et al. (in press), who studied spontaneous phase transitions in between-person coordination of rhythmical leg movements. By watching each other’s lower oscillating leg, two seated people were instructed to keep a common tempo and to coordinate the two legs either anti-phase or in-phase as the frequency of the movement was increased. As in the Kelso experiments (see Kelso and Schoner 1988, for a review), the coordinated limbs changed suddenly from anti-phase to in-phase. The reverse transition, however, did not occur, nor was there a transition when subjects increased frequency at the same pace without viewing each other. As with the within-person case, the phenomenon satisfied criteria of a second-order phase transition (such as hysteresis and critical fluctuations). Clearly, it is impossible to explain the temporal phenomena encountered at this level of observation with the help of biomechanics, because they do not depend on the biomechanical properties of the oscillating legs. The theoretical significance of the experimental results of Schmidt et al. is that the general dynamical principles underlying phase transitions in coordinated biological motion appear to hold independently of the number of par-
P.J. Beck, W.J. Beek / Stability and flexibility
353
ticipating components (two nervous system vs. one nervous system) and the nature of the informational coupling (visual vs. haptic). Therefore, in addition to energetic and material constraints, those concerning the employment of information should also be considered. In this respect, we should acknowledge that the paper addresses the issue of coordination at a specific level of observation from which generalisations to other levels of observation cannot easily be made. To reiterate, the fact that biological movement systems are governed by information is reflected in the fact that they generate temporally ordered behaviour in both a stable and a flexible manner. Therefore, we seek to understand temporal order in terms of organisational principles, rather than mechanisms. To this aim, the tools of non-linear dynamics are very appropriate because they allow an understanding of both stability and flexibility in one common framework: the ‘potential landscape’ of an attractor state defines, at the same time, the stable and less stable regions of operation. Attractor
states.
Whereas the concepts of biomechanics refer to concrete observables, the concepts of dynamical systems theory are abstract in that they need not correspond to any observable biomechanical quantities. As has been emphasised by many authors (Kelso et al. 1980; Kugler et al. 1980; Kelso and Tuller 1984; Kay et al. 1987) the theoretical importance of attractor states in the study of movement coordination should not be sought in terms of the system’s material substrate and component structures but in their functional similarity with the macroscopic behaviour of a constrained collective of muscles, joints and nerves. Thus, these models, unlike the more mechanical interpretations of point attractors referred to by Van Ingen Schenau, are immune to particular observations regarding activity patterns in muscles and neurons. To an anatomist, a neurophysiologist or a biomechanicist this lack of specification at the neuro-muscular level may be an unevenly high price to pay, but what is gained is generality. The strength of nonlinear dynamics is that it allows one to see how a wide variety of systems exhibit the same type of behaviour and obey, in fact, the same type of dynamical laws, no matter how different their internal material structures may be. In other words, the thrust of dynamical systems theory is in its claim that the principles governing the co-operation of many subsystems, as manifested in their collective behaviour, are identical regardless of the structure of the subsystems involved.
354
P.J. Beck, W.J. Beek / Siuhht): andflexihilrt_~
Unification. To provide a final illustration of what is involved in such abstract understanding of cooperative phenomena, consider the following. In general, in periodic motion, the sum of potential energy (w’x’) and kinetic energy (dx/dt)* obeys rules of the following kind: td2x2+
($j2=f{cos(i
arc fg1;s‘)).
(3)
where f is constrained to specific values of i: i = n,, n *, n3 . . . (where i is an integer). Coordination in such cases means that ratios like nk/nn, have to be fixed, which leads to a specific partitioning of characteristic times for the act under study. As we have discussed above, the presence of such ‘tessellation’ of time has been demonstrated in juggling N balls in a figure-eight pattern lying on its side: n&5,, represents here the fixed ratio, for a given N, of the times, in one hand loop, during which the hand is loaded with a ball and that the hand is unloaded (Beek 1989). Now, if f,, as defined by eq. (3), specifies a part of the system (e.g., a muscle) exhibitin g i-periodicity, and f/ specifies another part of the same system (e.g., another muscle), exhibiting j-periodicity, then any co-operation between the two parts obeying to a specification of the total behaviour as defined by F = f,( f,( f,- ‘)). leads to a main order of i-periodicity with possible sub- and superharmonics of order j + i and j - i-periodicity. Although this is a formal statement, it shows that order in parts creates overall order, whateoer the origin of the co-operation between the parts. It may be (bio-)mechanicaI, neurophysiological or informational, but whatever it is, is immaterial for this, dynamical, type of analysis. Of course, this does not imply that there are no urgent problems to be solved as to the relationships between observables at different levels of description. For instance, from a biomechanical point of view, the operator F is to be identified in biomechanical terms (e.g., by a certain force interaction between the constituent parts) or by principles of stereometric exclusion (i.e., where one muscle or bone is present another cannot be). If nervous coupling is the subject matter, then a neural network has to be identified in which the operator F is embedded. (By the way, there is growing evidence that neural patterns may be understood as non-linearly coupled oscillators, thus permitting a linkage among levels of
P.J. Beck, W.J. Beek / Stability and flexibility
355
observation in terms of non-linear dynamics.) Where perception is being studied, an informational quantity must be identified that gives rise to the F operator. This enterprise is especially intriguing in the case of between-person coordination, as in walking together with one common tempo different from the individually preferred frequencies, because in this case the F operator is defined over two nervous systems rather than one. We are convinced that the unifying concepts of nonlinear dynamics will become increasingly important in the sciences at large and biology in particular. Of course, our comprehension will not easily transcend the levels of observation associated with disciplines as biomechanics, neurophysiology, psychology, etc. For a long time to come, we will be far away from describing, let alone explaining, complete complex actions. But acknowledging the existence of encompassing concepts for studying action, general and potent enough to unify the sciences, is not only an antidote to excessive claims by the one or the other discipline, it is also a need for a branch of science whose eventual aim is to understand whole, coordinated actions.
References Beek, P.J., 1989. Timing and phase locking in cascade juggling. Ecological Psychology 1, 55-96. Bootsma, R.J. and P.C.W. van Wieringen, in press. Timing an attacking forehand drive in table tennis. Journal of Experimental Psychology: Human Perception and Performance. Kay, B.A., J.A.S. Kelso, E.L. Saltzman and G. Schoner, 1987. Space-time behavior of single and bimanual rhythmical movements: Data and limit cycle model. Journal of Experimental Psychology: Human Perception and Performance 13, 178-190. Kelso, J.A.S. and G. Schoner, 1988. Self-organisation of coordinative movement patterns. Human Movement Science 7. 21-46. Kelso, J.A.S. and B. Tuller, 1984. ‘A dynamical basis for action systems’. In: M.S. Gazzaniga (ed.). Handbook of cognitive neuroscience. New York: Plenum pp. 321-356. Kelso, J.A.S., K.G. Holt, P.N. Kugler and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipatives structures: II. Empirical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorial in motor behavior. Amsterdam: North-Holland. pp. 49970. Kugler, P.N. and M.T. Turvey, 1987. Information, natural law, and the self-assembly of rhythmic movement. Hillsdale, NJ: Erlbaum. Kugler, P.N., J.A.S. Kelso and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: I. Theoretical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland. pp. 3-47. Newell, K., 1986. ‘Constraints on the development of coordination’. In: M.G. Wade and H.T.A. Whiting (eds.), Motor development in children: Aspects of coordination and control. The Hague: Nijhoff. pp. 341-360.
356
P.J. Beck, W.J. Beek / Stabdity
and flexibility
Schmidt, R.C., C. Care110 and M.T. Turvey, in press. Phase transitions and critical fluctuations in the visual coordination of rhythmic movements between people. Journal of Experimental Psychology: Human Perception and Performance. Van Ingen Schenau, G.J., 1989. From rotation to translation: Constraints on multi-joint movements and the unique action of bi-articular muscles. Human Movement Science 8, 301-337 (this issue).
347
Science 8 (1989) 347-356
STABILITY AND FLEXIBILITY IN THE TEMPORAL ORGANISATION OF MOVEMENTS Reaction to G.J. van Ingen Schenau (1989) P.J. BEEK
*
Free Unicer.ri~y,Amsterdam. The Netherlands
W.J. BEEK Technrcal Unlr~ersiry.Del&, The Netherlands
Beek, P.J. and W.J. Beek, 1989. Stability and flexibility in the temporal organisation of movements (Reaction to G.J. van Ingen Schenau, 1989). Human Movement Science 8, 347-356.
‘From rotation to translation’ describes in a clear way why joint assemblies consisting of three components incorporating bi-articular muscles have a functional advantage in translatory movements resulting from rotations in joints. On the basis of an elegant biomechanical analysis, it is shown how energy, if geared by the appropriate muscles, can be released in places other than where it is generated. Furthermore, it is shown how the temporally ordered sequence of activation patterns of mono- and bi-articular muscles, as observed in jumping, skating and cycling, conforms to the identified functional characteristics of these two types of muscle. As far as we are able to see, no intrinsic criticism can be raised against these findings and their interpretation. The biomechanical approach adopted is followed rigorously and with creativity. Of course, this state of affairs restricts one’s possibilities to enhance discussion. * Requests for reprints should be sent to P.J. Beck, Dept. of Psychology. Faculty of Human van de Boechorststraat 9, 1981 BT Amsterdam. The Movement Sciences, Free University, Netherlands.
0167-9457/89/$3.50
a 1989, Elsevier Science
Publishers
B.V. (North-Holland)
34x
P.J. Beek, W.J. Beek / Srability
ot~d/le.xrh~lity
We will, therefore, attempt to offer a number of considerations and observations with respect to the coordination and control of movement that seem to fall outside the purview of biomechanics but are of major relevance to a more encompassing understanding of movement behaviour. We believe that the biomechanical research programme developed by Van Ingen Schenau has some inherent limitations for an adequate understanding of the whole realm of coordinated movement behaviour. For the present purposes it may be worthwhile to discuss these limitations and hint at an alternative strategy of dealing with them. In doing so, our concern will be with the topic of tempo& order, because this topic is a crucial one in the study of movement coordination, as is indeed reflected in Van Ingen Schenau’s paper.
Modulation. According to Van Ingen Schenau, the temporal order in the sequence of muscle activations depends primarily on local point values of relative phase, given that the identified constraints ‘are mainly a function of the angles in the joints in relation to the direction of the external force and not, or to a much lesser extent, to the velocity of the movement’ (p. 329). If however, in eqs. (1) and (2) both the distance between hip and ankle ( HA) and the relative phase (6) change in a cyclic fashion, local values of the relative phase will change during a cycle. Thus, the fact that not only relative phase, but also the change of relative phase determines temporal and spatial order is implicit in eqs. (1) and (2). For instance, assume that the motion of HA is periodic. What then is the behaviour of 0 with respect to time? A simple analysis, in which nothing is assumed beyond what is assumed in the equations, shows that in such a case, /3(l) will exhibit a second harmonic, in addition to the base frequency determined by HA, as well as modulation, unless some very unrealistic conditions are met. Still, for the type of task studied by Van Ingen Schenau such modulation may be neglected, because the goal is to jump as high, to throw as far, or to skate as fast as possible: the requirements are primarily energetic, while changes in the environment are relatively unimportant. Thus, under such conditions the system will exhibit behaviour that can adequately be described in terms of local point values only. Nevertheless, many activities require a temporal order that is not only stub/e but also flexible, in order to be able to change according to environmental demands.
P.J. Beck, W.J. Beek / Stability andflexibility
349
To obtain a complete picture of temporal order the constraints on the change of relative phase need to be disclosed, in addition to the constraints on relative phase identified by Van Ingen Schenau. In general, it is in the interplay of these two types of constraint that the variance and invariance of relative timing emerges. Or, to express this in less abstract language, if Van Ingen Schenau concludes that a change in relative time is needed when a skater changes the mean flexion of the knee joint (8) or when a jumper or runner carries extra mass (changing the total mass M), phase lags between characteristic angles of forces and EMD’s have to obey certain dynamical laws that are to still to be found. Our claim is that both frequency-doubling and modulation have to obey certain dynamical principles if coherent action, and not noise, is to be produced. These principles relate to a set of constraints that, in our view, are extremely relevant to understanding temporal order in movement, but seem to fall outside Van Ingen Schenau’s research programme. Temporal order in juggling. To illustrate how both the stable and flexible aspects of temporal order in movement can be understood at a macroscopic scale of observation, we think it is useful to discuss our present understanding of the activity of juggling in some detail. Clearly, the task demands of juggling are not primarily energetic. Rather, the first requirement for successful juggling is that the duration of the cycle time of the hands stands in a specific relation to the cycle time of the balls. Why this should be so can easily be made apparent by distinguishing three time components, namely the time that the hand is loaded with a ball (time loaded: t,), the time that the hand is moving freely (time unloaded: t,) and the flight time of the balls (time flight: t,), and define the cycle time of a hand as t, + t, and the cycle time of a ball as (t, + t,). This partitioning depends on the number of balls juggled (N) and the number of hands (H) that juggle. Now if we assume that the pattern is more or less periodic (in the sense that each configuration of balls recurs at certain time intervals) and that one hand never holds more than one ball, the total of flight time and hand-loaded time per ball must equal the total of hand-unloaded time and hand-loaded time per hand, regardless of the exact values of the absolute cycle frequencies. In other words, balls and hands must match a general timing requirement in that the ratio of the hand loop time and the cycle time of a ball during a loop equals the ratio of the
350
number
P.J. Beck, W.J. Beek / Stnbiiit.v und flexibilr~~
of hands (H)
t, + t, tr+ t, =
H
/
X-1 fi
of balls (N):
0)
N’
which can be rewritten
tUN t H+;
and the number
as:
=l. 1
Expression (2) shows how the relationship between the three component times of juggling can be represented as a relationship between two timing ratios. It follows from this temporal task constraint that if one ratio is fixed, the other ratio should be fixed as well. This allows one to raise the question how the juggler must fill up the loop time of a hand and the loop time of a ball with component times to ensure smooth juggling. The question, metaphorically, is how the juggler must ‘ tile’ the loop time of a hand and the loop time of a ball with component times to juggle comfortably? With respect to a fixed amount of 2-dimensional space, it is wellknown that filling the space completely (so that there are no cracks and gaps) can only be achieved through repetition of homomorphic 2-dimensional forms of a limited number of types (e.g., squares, triangles, hexagons, and the like). With respect to filling a fixed amount of time, defined by two ratios between three time components, there may well be similar limitations. The results of a study on timing and phase locking in juggling in a figure-eight pattern (Beek 1989) suggest that such dynamical constraints do indeed exist. Experiments conducted on four skilled jugglers, juggling 3 balls, revealed that, across three different rates of juggling, the ratios of loaded-time to hand loop time (t,/t,) were always close to l/3 without being exactly l/3. All jugglers tiled according to the same blueprint. In physical terms, the situation is one of near phase locking between two coupled oscillators. In non-linear dynamics, the stable frequency ratios that come into existence between two oscillatory processes are called ‘phase locks’. They are always rational numbers (meaning that the two numbers composing the ratio are, by definition, integers). Evidence was provided that the juggling hands can be conceived as a system of coupled (non-linear) oscillators,
P.J. Beek,
W.J. Beek / Stahhty andflexihilit~~
351
because they share the same dynamical characteristics. The integer proportionality t,: t,: tf is at the core of the temporal order of juggling, defining the potential minimum of complete phase locking. The observed deviations from this potential minimum could also be understood in a physical sense, namely as phase modulation. Thus, a single account could be given of both the stable and flexible properties of the temporal order emerging in juggling with the help of the tools of non-linear dynamics. The fact that jugglers do not precisely operate in the potential minimum associated with complete phase locking, but rather function in the neighbourhood of this minimum, can be theoretically understood as follows. A completely phase-locked system is a system for which the initial conditions are exactly repeatable from one instance of functioning to the next. Exact reproduction of both parameter values, however, is not characteristic of biological movement systems. They are information-driven systems, that are prohibited from residing in a potential trough by virtue of their informational basis. By the arguments advanced here, a system that relates to a potential function can hover about the vicinity of the potential minimum to the degree that it can register information specifying the minimum and the magnitude of departures from the minimum. There is a distinct benefit to this way of functioning, namely, phase lockings can be adjusted relatively freely to accommodate perturbations in the juggling routine. The juggler is not restricted to tiling the temporal spans of ball and hand trajectories with durations in integer ratios. Other durations can be created, suited to the moment-to-moment variations in the juggling conditions. Adaptability to varying circumstances, therefore, is a simple but highly significant consequence of the fact that coordinated movements are dynamical systems governed by information. Perceptual information. Whereas Van Ingen Schenau’s paper puts us in the middle of questions about various types of constraint on human action, informational constraints, as alluded to in the above, cannot be incorporated in a biomechanical account without introducing another discourse. Clearly, the identification and formalisation of constraints on coordinated movement depend on the approach adopted and the specific analytical tools applied. As such, Van Ingen Schenau’s distinction between constraints within the actor, the action and the environment, which runs parallel with Newell’s (1986) distinction between
352
P.J. Beck, W.J. Beek / Stability and flexihilit)
organismic, task and environmental constraints, betrays an interest in how humans and animals perform meaningful actions in their natural environment. It follows, however, from the approach adopted that the constraints identified by Van Ingen Schenau are mainly organismic (anatomical, geometric) and only environmental to the extent to which these constraints can be captured in biomechanical terms (i.e. to the extent that there are forceful interactions with the environment). Thus, following such an approach it is possible to explain, for instance, how a table tennis player can attain maximal angular bat velocity during an attacking forehand drive, but not how critical parameters can be modulated in order to attune the movements with respect to the approaching ball (Bootsma and Van Wieringen in press). In many instances, the first requirements for successful actions are not exclusively in a sufficiently large delivery of either force or energy, but also, and foremost, in an optimal guidance of force and/or energy on the basis of perceptual information. The limits tc biomechanical explanations of coordinated behaviour becomes especially apparent in the temporal order that may be manifested in activities in which the moving limbs are only informationally (and not mechanically) coupled, such as in music-making, or in walking side-by-side and dancing. A remarkable example of temporal ordering in this type of activity is provided by Schmidt et al. (in press), who studied spontaneous phase transitions in between-person coordination of rhythmical leg movements. By watching each other’s lower oscillating leg, two seated people were instructed to keep a common tempo and to coordinate the two legs either anti-phase or in-phase as the frequency of the movement was increased. As in the Kelso experiments (see Kelso and Schoner 1988, for a review), the coordinated limbs changed suddenly from anti-phase to in-phase. The reverse transition, however, did not occur, nor was there a transition when subjects increased frequency at the same pace without viewing each other. As with the within-person case, the phenomenon satisfied criteria of a second-order phase transition (such as hysteresis and critical fluctuations). Clearly, it is impossible to explain the temporal phenomena encountered at this level of observation with the help of biomechanics, because they do not depend on the biomechanical properties of the oscillating legs. The theoretical significance of the experimental results of Schmidt et al. is that the general dynamical principles underlying phase transitions in coordinated biological motion appear to hold independently of the number of par-
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ticipating components (two nervous system vs. one nervous system) and the nature of the informational coupling (visual vs. haptic). Therefore, in addition to energetic and material constraints, those concerning the employment of information should also be considered. In this respect, we should acknowledge that the paper addresses the issue of coordination at a specific level of observation from which generalisations to other levels of observation cannot easily be made. To reiterate, the fact that biological movement systems are governed by information is reflected in the fact that they generate temporally ordered behaviour in both a stable and a flexible manner. Therefore, we seek to understand temporal order in terms of organisational principles, rather than mechanisms. To this aim, the tools of non-linear dynamics are very appropriate because they allow an understanding of both stability and flexibility in one common framework: the ‘potential landscape’ of an attractor state defines, at the same time, the stable and less stable regions of operation. Attractor
states.
Whereas the concepts of biomechanics refer to concrete observables, the concepts of dynamical systems theory are abstract in that they need not correspond to any observable biomechanical quantities. As has been emphasised by many authors (Kelso et al. 1980; Kugler et al. 1980; Kelso and Tuller 1984; Kay et al. 1987) the theoretical importance of attractor states in the study of movement coordination should not be sought in terms of the system’s material substrate and component structures but in their functional similarity with the macroscopic behaviour of a constrained collective of muscles, joints and nerves. Thus, these models, unlike the more mechanical interpretations of point attractors referred to by Van Ingen Schenau, are immune to particular observations regarding activity patterns in muscles and neurons. To an anatomist, a neurophysiologist or a biomechanicist this lack of specification at the neuro-muscular level may be an unevenly high price to pay, but what is gained is generality. The strength of nonlinear dynamics is that it allows one to see how a wide variety of systems exhibit the same type of behaviour and obey, in fact, the same type of dynamical laws, no matter how different their internal material structures may be. In other words, the thrust of dynamical systems theory is in its claim that the principles governing the co-operation of many subsystems, as manifested in their collective behaviour, are identical regardless of the structure of the subsystems involved.
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Unification. To provide a final illustration of what is involved in such abstract understanding of cooperative phenomena, consider the following. In general, in periodic motion, the sum of potential energy (w’x’) and kinetic energy (dx/dt)* obeys rules of the following kind: td2x2+
($j2=f{cos(i
arc fg1;s‘)).
(3)
where f is constrained to specific values of i: i = n,, n *, n3 . . . (where i is an integer). Coordination in such cases means that ratios like nk/nn, have to be fixed, which leads to a specific partitioning of characteristic times for the act under study. As we have discussed above, the presence of such ‘tessellation’ of time has been demonstrated in juggling N balls in a figure-eight pattern lying on its side: n&5,, represents here the fixed ratio, for a given N, of the times, in one hand loop, during which the hand is loaded with a ball and that the hand is unloaded (Beek 1989). Now, if f,, as defined by eq. (3), specifies a part of the system (e.g., a muscle) exhibitin g i-periodicity, and f/ specifies another part of the same system (e.g., another muscle), exhibiting j-periodicity, then any co-operation between the two parts obeying to a specification of the total behaviour as defined by F = f,( f,( f,- ‘)). leads to a main order of i-periodicity with possible sub- and superharmonics of order j + i and j - i-periodicity. Although this is a formal statement, it shows that order in parts creates overall order, whateoer the origin of the co-operation between the parts. It may be (bio-)mechanicaI, neurophysiological or informational, but whatever it is, is immaterial for this, dynamical, type of analysis. Of course, this does not imply that there are no urgent problems to be solved as to the relationships between observables at different levels of description. For instance, from a biomechanical point of view, the operator F is to be identified in biomechanical terms (e.g., by a certain force interaction between the constituent parts) or by principles of stereometric exclusion (i.e., where one muscle or bone is present another cannot be). If nervous coupling is the subject matter, then a neural network has to be identified in which the operator F is embedded. (By the way, there is growing evidence that neural patterns may be understood as non-linearly coupled oscillators, thus permitting a linkage among levels of
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observation in terms of non-linear dynamics.) Where perception is being studied, an informational quantity must be identified that gives rise to the F operator. This enterprise is especially intriguing in the case of between-person coordination, as in walking together with one common tempo different from the individually preferred frequencies, because in this case the F operator is defined over two nervous systems rather than one. We are convinced that the unifying concepts of nonlinear dynamics will become increasingly important in the sciences at large and biology in particular. Of course, our comprehension will not easily transcend the levels of observation associated with disciplines as biomechanics, neurophysiology, psychology, etc. For a long time to come, we will be far away from describing, let alone explaining, complete complex actions. But acknowledging the existence of encompassing concepts for studying action, general and potent enough to unify the sciences, is not only an antidote to excessive claims by the one or the other discipline, it is also a need for a branch of science whose eventual aim is to understand whole, coordinated actions.
References Beek, P.J., 1989. Timing and phase locking in cascade juggling. Ecological Psychology 1, 55-96. Bootsma, R.J. and P.C.W. van Wieringen, in press. Timing an attacking forehand drive in table tennis. Journal of Experimental Psychology: Human Perception and Performance. Kay, B.A., J.A.S. Kelso, E.L. Saltzman and G. Schoner, 1987. Space-time behavior of single and bimanual rhythmical movements: Data and limit cycle model. Journal of Experimental Psychology: Human Perception and Performance 13, 178-190. Kelso, J.A.S. and G. Schoner, 1988. Self-organisation of coordinative movement patterns. Human Movement Science 7. 21-46. Kelso, J.A.S. and B. Tuller, 1984. ‘A dynamical basis for action systems’. In: M.S. Gazzaniga (ed.). Handbook of cognitive neuroscience. New York: Plenum pp. 321-356. Kelso, J.A.S., K.G. Holt, P.N. Kugler and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipatives structures: II. Empirical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorial in motor behavior. Amsterdam: North-Holland. pp. 49970. Kugler, P.N. and M.T. Turvey, 1987. Information, natural law, and the self-assembly of rhythmic movement. Hillsdale, NJ: Erlbaum. Kugler, P.N., J.A.S. Kelso and M.T. Turvey, 1980. ‘On the concept of coordinative structures as dissipative structures: I. Theoretical lines of convergence’. In: G.E. Stelmach and J. Requin (eds.), Tutorials in motor behavior. Amsterdam: North-Holland. pp. 3-47. Newell, K., 1986. ‘Constraints on the development of coordination’. In: M.G. Wade and H.T.A. Whiting (eds.), Motor development in children: Aspects of coordination and control. The Hague: Nijhoff. pp. 341-360.
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Schmidt, R.C., C. Care110 and M.T. Turvey, in press. Phase transitions and critical fluctuations in the visual coordination of rhythmic movements between people. Journal of Experimental Psychology: Human Perception and Performance. Van Ingen Schenau, G.J., 1989. From rotation to translation: Constraints on multi-joint movements and the unique action of bi-articular muscles. Human Movement Science 8, 301-337 (this issue).