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Phase-Locked Modes, Phase Transitions and Component Oscillators in Biological Motion
Physica Scripta
Abstract
We review the results of joint experimental and theoretical work on coordinated biological motion demonstrating the close alliance between our observations and other nonequilibrium phase transitions in nature (e.g., the presence of critical fluctuations, critical slowing down). Order parameters are empirically determined and their (low-dimensional) dynamics used in order to explain specific pattern formation in movement, including stability and loss of stability leading to behavioral change, phase-locked modes and entrainment. The system's components and their dynamics are identified and it is shown how these may be coupled to produce observed cooperative states. This "phenomenological synergetics" approach is minimalist and operational in strategy, and may be used to understand other systems (e.g., speech), other levels (e.g., neural) and the linkage among levels. It also promotes the search for additional forms of order in multi-component, multi-stable systems.
... Qualitative change does not mean that quantification is impossible. To the contrary, qualitative change is at the heart of pattern formation and, provided care is taken to evaluate system timescales (e. g., how quickly the control parameter is changed relative to the typical time of the system to react to perturbations; see [121]) quantitative predictions ensue that can be tested experimentally (see Sect. "The Theoretical Modeling Strategy of Coordination Dynamics: Symmetry and Bifurcations"). ...
... (2) and (3) ( [175,210,211] see Chap. 11 in [74] and [101,121] for Coordination Dynamics C 1549 a thorough discussion) allows key predictions of coordination dynamics to be tested and quantitatively evaluated [113,121,172]. Critical slowing is easy to understand from Fig. 3 (top). ...
... (2) and (3) ( [175,210,211] see Chap. 11 in [74] and [101,121] for Coordination Dynamics C 1549 a thorough discussion) allows key predictions of coordination dynamics to be tested and quantitatively evaluated [113,121,172]. Critical slowing is easy to understand from Fig. 3 (top). ...
Encyclopedia Article on Coordination Dynamics, the science of coordination.
... As the critical frequency was approached, the relaxation time was found to increase monotonically -and to drop, of course, when the subject switched into the in-phase mode [91]. Using another measure of relaxation time, the inverse of the line width of the power spectrum of relative phase [92], strong enhancement of relaxation time, was clearly observed. ...
... In this case, it is easy to calculate the time between the relative phase immediately before the transition and the value assumed immediately after the transition. The match between theoretically predicted and empirically observed switching time distributions was impressive, to say the least [92,94]. This aspect is particularly interesting, because it shows that the switching process itself is closely captured by a specific model of stochastic nonlinear HKB dynamics. ...
... Such analyses are useful, even important, for example, for heartbeat monitoring and other dynamic diseases or for distinguishing human performance in different task conditions over the long term (e.g., [100][101][102][103]) -or even individual differences between people in performing such tasks. One should be conscious, however, of the ancient saying attributed to Heraclitus; ''no man ever steps in the same river twice, for it's not the same river and he's not the same 8) At an early presentation of this work [92], the late Rolf Landauer (1927-1999) suggested that we call it a limiting case of a first-order transition. This is a technical point. ...
... In the coordination of two rhythmically moving limbs, two patterns of relative phasing of the two limbs are characteristically more stable than others. Although people are able to learn other patterns with conceited practice (Schoner, Zanone, & Kelso, 1992;, generally speaking, people move either in an in-phase pattern, in which the limbs move in a symmetrical fashion, or in an antiphase pattern, in which the limbs move in an alternating fashion (Kelso, 1984;Kelso, 1995;Kelso et al., 1981;Kelso, Schoner, Scholz, & Haken, 1987;Schmidt, Shaw, & Turvey, 1993;Tuller & Kelso, 1989;Turvey, Rosenblum, Schmidt, & Kugler, 1986;Wimmers, Beek, & van Wieringen, 1992;Yamanishi, Kawato, & Suzuki, 1980). Furthermore, the in-phase pattern has been demonstrated to be more stable than the antiphase pattern. ...
... The differential stability of the in-phase and antiphase patterns has been shown in experiments in which participants were instructed to increase or decrease the (common) frequency of movement during a trial. When participants are instructed to start moving in an antiphase pattern, to move gradually faster and faster, and not to resist the urge to switch patterns, a transition from the antiphase to an in-phase pattern occurs (e.g., Kelso, 1984;Kelso et al., 1981;Kelso et al., 1987;Schmidt, Carello, & Turvey, 1990;Scholz & Kelso, 1989). In contrast, starting in an in-phase pattern does not lead to a transition. ...
Perception of relative phase and phase variability may play a fundamental role in interlimb coordination. This study was designed to investigate the perception of relative phase and of phase variability and the stability of perception in each case. Observers judged the relative phasing of two circles rhythmically moving on a computer display. The circles moved from side to side, simulating movement in the frontoparallel plane, or increased and decreased in size, simulating movement in depth. Under each viewing condition, participants observed the same displays but were to judge either mean relative phase or phase variability. Phase variability interfered with the mean-relative-phase judgments, in particular when the mean relative phase was 0°. Judgments of phase variability varied as a function of mean relative phase. Furthermore, the stability of the judgments followed an asymmetric inverted U-shaped relation with mean relative phase, as predicted by the Haken–Kelso–Bunz model.
... "[T]he long sought-for link between neuronal activities (microscopic events) and behavior (macroscopic events) may actually reside in the coupling of dynamics on different levels." However, "the path from the microscopic dynamics to the collective order parameters is not readily accessible" (Kelso et al., 1987). Perhaps such inaccessibility points to something more profound. ...
Many theoretical studies defend the existence of ongoing phase transitions in the brain dynamics that could explain its enormous plasticity to cope with the environment. However, tackling the ever-changing landscapes of brain dynamics seems a hopeless task with complex models. This paper uses a simple Haken-Kelso-Bunz (HKB) model to illustrate how phase transitions that change the number of attractors in the landscape for the relative phase between two neural assemblies can occur, helping to explain a qualitative agreement with empirical decision-making measures. Additionally, the paper discusses the possibility of interpreting this agreement with the aid of Irruption Theory (IT). Being the effect of symmetry breakings and the emergence of non-linearities in the fundamental equations, the order parameter governing phase transitions may not have a complete microscopic determination. Hence, many requirements of IT, particularly the Participation Criterion, could be fulfilled by the HKB model and its extensions. Briefly stated, triggering phase transitions in the brain activity could thus be conceived of as a consequence of actual motivations or free will participating in decision-making processes.
... The quality of coordination is a feature invisible to naked eye. Kelso's group showed that coordination is reliably measured by the relative phase (RP; [95,96]), an index quantifying how much one partner lags behind the other in each movement cycle, and expressed in degrees (see Fig. 1, panel B). In dyadic coordination, when the co-actors are in synchrony RP is close to zero. ...
We designed a coordination–cooperation game dedicated to teaching the theory of mind (ToM) to children with autism spectrum disorder. Children interacted with either a robot or a human. They had to coordinate their gestures with the beats of a ditty sung by their partner (coordination), who then implicitly asked them for help (cooperation). Before and after this cooperation–coordination task, the children performed a helping task that assessed their ToM skills: the ability to infer social partners’ intentions. Despite the regularity and predictability of the robot, children made the most progress in the helping task after interacting with a human. Motor coupling was more stable in child–human than in child–robot dyads. The ability of the social partner to actively maintain a stable social coupling seems to be a primary factor inciting the child to learn and transfer the just-practiced social skills.
... In addition, a primary focus was to test predictions concerning the loss of stability of the D-mode pattern with the asymmetric system. A parallel increase in both the variability of relative phase and the time to return to a pattern after a small perturbation (i.e., relaxation time) was predicted as movement frequency increased from the stochastic version of Equation 1; this was experimentally confirmed in the symmetric, bimanual case (Kelso, Schoner, Scholz, & Haken, 1987). The behavior of relaxation time in the asymmetric case has not been studied. ...
J. A. S. Kelso and J. J. Jeka (1992) demonstrated that symmetry is a useful conceptual tool to distinguish the coordination between components with similar versus different anatomical properties. The present experiments studied human arm–leg patterns to test whether their coordinative asymmetry was changed by manipulating the inertial properties of a single limb. The results showed that (a) consistent with model predictions, adding weight to the arm or the leg minimized or enhanced coordinative asymmetry, respectively and (b) the response to a perturbation slowed as movement frequency increased but in a fashion that reflected the underlying coordinative asymmetry. The observed coordinative effects suggest the influence of neural phase relationships and emphasize that symmetry plays an important role in understanding coordination in systems in which control cannot be traced unequivocally to a single end-effector or a neurophysiological substrate.
... Following synergetics (Haken, 1983a(Haken, , 1983b, our approach to coordination Kelso, Schoner, Scholz, & Haken, 1987: Schoner & Kelso, 1988a is to identify order parameters or collective variables for coordinative patterns and to determine the dynamics of these patterns by studying their stability and change. Here we apply these theoretical concepts to a new experimental system that involves the coordination among multiple anatomically and biophysically different components. ...
The dynamics of pattern formation and change are studied in a complex multicomponent system, specifically the arms and legs of human Ss. Among the novel features observed are differential stability of coordinative modes produced by limbs moving in the same versus different directions (Experiment 1); transitions between coordinative modes preceded by a slow drift in relative phase (Experiments 1 and 2); bifurcations or phase transitions from 1 four-limb pattern to another (Experiment 2); and spontaneous emergence of non-1:1-frequency- and phase-locked patterns, in addition to periods of relative coordination (Experiment 3). All observed relative phasing patterns and their dynamics (stability, loss of stability, intermittency) are shown to arise from the same underlying nonlinear dynamical structure, an important feature of which is broken symmetry.
... Coordination dynamics refers to the patterns that the system is capable of producing spontaneously at a given point in time along with the attractor landscape that defines the relative stability of these patterns as intrinsic dynamics [13]. Intrinsic dynamics is important to know because it influences what can be changed or modified by new experiences and how such change occurs (e.g., whether change is smooth or abrupt) [43,44]. The eminent philosopher and evolutionary biologist, Maxine Sheets-Johnstone, has repeatedly pointed to, and provided evidence for, the primacy of movement as 'the mother of all cognition', presaging every conscious mind that ever said 'I'. 'Spontaneous movement' argues Sheets-Johnstone [45] 'is the constitutive source of agency, of subjecthood, of selfhood, the dynamic core of ourselves as agents, subjects, selves'. ...
The question of agency and directedness in living systems has puzzled philosophers and scientists for centuries. What principles and mechanisms underlie the emergence of agency? Analysis and dynamical modeling of experiments on human infants suggest that the birth of agency is due to a eureka-like, pattern-forming phase transition in which the infant suddenly realizes it can make things happen in the world. The main mechanism involves positive feedback: when the baby's initially spontaneous movements cause the world to change, their perceived consequences have a sudden and sustained amplifying effect on the baby's further actions. The baby discovers itself as a causal agent. Some implications of this theory are discussed. What Is this 'I'? We humans tend to believe that we are agents, masters and mistresses of our fate, that our deeds and desires are our destiny. Yet, despite a sizeable literature on 'the sense of agency' and its behavioral and neuroimaging correlates (see [1,2] for recent reviews), the scientific basis of causal agency and how we come to experience ourselves as agents is lacking. Agency means action towards an end. When it comes to the behavior of living things, our inability to understand end-directedness forces us to posit (often implicitly) an intelligent agent residing somewhere inside the system that is responsible for the end-directed behavior we observe. The self as a causal agent remains a ghost in the machine awaiting exorcism, perhaps by new insights from the brain and cognitive sciences. Charles Darwin, in On the Origin of Species, touched only briefly on the topic of agency, although he noted how 'admirably adapted' was the woodpecker to catch insects under the bark of trees and how mistletoe 'absolutely' required the agency of certain insects to bring pollen from one flower to another ([3] p.12). His later work on the habits of worms notwithstanding [4], Darwin admitted 'I must promise that I have nothing to do with the origin of the primary mental powers, any more than I have with life itself' ([3] p.189). In the introduction to his remarkable history of physiological psychology, Franklin Fearing [5] noted that 'Even before man speculated about the nature and source of his own experiences, he was probably curious about the agencies by which animal motion was effected' ([5] p.1). Life and motion, Fearing remarks, are almost synonymous terms. In his famous book What Is Life?, Erwin Schrödinger [6], one of the chief architects of quantum mechanics and the author of the famous equation that bears his name, proposed an 'order from order' principle as the physical basis of life. Schrödinger speculated that this new kind of order took the form of an aperiodic crystal, later exposed as the beautiful double helical structure of the DNA molecule [7]. Not much more was said about Schrödinger's order from order principle or his call for 'new laws to be expected in the organism' (but see [8,9]). Still less truck was given to the question raised by Schrödinger in the final chapter of his small book. Each of us, says Schrödinger, has the indisputable impression that the sum total of our own experience and Trends Over the past 30 years, higher-order principles of self-organizing dynamical systems have influenced our understanding of brain, cognition, and behavior. They might also offer insights into age-old puzzles about the origins of agency and directedness in living things. Experiments and observations of human infants combined with theoretical modeling suggest that the birth of agency corresponds to a eureka-like phase transition in a coupled dynamical system whose key variables span the interaction between the baby and its environment. Analysis shows that the main mechanism underlying the emergence of agency is autocatalytic and involves positive feedback. When the baby's initially spontaneous movements cause the world to change, their perceived consequences have a sudden and sustained amplifying effect on the baby's further actions. The prelinguistic baby realizes it can make things happen!
... They are often simply encapsulated in terms like "Dynamical Systems Theory" (DST). As an established acronym, DST is comfortably presented as if it is a theory, rather than a set of mathematical concepts and tools that needs to be turned into a specific research program (for an early example of such, see Kelso & Schöner, 1987). As a result of HKB, it became readily--perhaps too readily-apparent that under the convenient banner of DST, the ideas introduced in HKB could be applied to all kinds of different settings. ...
This article presents a brief retrospective on the Haken-Kelso-Bunz (HKB) model of certain dynamical properties of human movement. Though unanticipated, HKB introduced, and demonstrated the power of, a new vocabulary for understanding behavior, cognition and the brain, revealed through a visually compelling mathematical picture that accommodated highly reproducible experimental facts and predicted new ones. HKB stands as a harbinger of paradigm change in several scientific fields, the effects of which are still being felt. In particular, HKB constitutes the foundation of a mechanistic science of coordination called Coordination Dynamics that extends from matter to movement to mind, and beyond.
The production of a “simple” utterance, such as the syllable /ba/, involves the cooperation of a large number of neuromuscular elements operating on different time scales, e.g., at respiratory, laryngeal, and supralaryngeal levels. Yet somehow, from this huge dimensionality, /ba/ emerges as a coherent and well-formed pattern. Similarly, were one to count the neurons, muscles, and joints that cooperate to produce the “simple” act of walking, literally thousands of degrees of freedom would be involved. Yet again, somehow walking emerges as a fundamentally low-dimensional cyclical pattern—in the language of dynamical systems, a periodic attractor. In physics, an infinite dimensional system, described by a complicated set of partial, nonlinear differential equations can be reduced—when probed experimentally or analyzed theoretically—to a low-dimensional description [1,2]. In all these cases, it seems, information about the system is compressed—from a microscopic basis of huge dimensionality—to a macroscopic basis of low dimensionality.
A great deal of the success of science has rested on its specific methods. One of which has been to start with the study of simple phenomena such as that of falling bodies, or to decompose systems into parts with well-defined properties simpler than those of the total system. In our time there is a growing awareness that in many cases of great practical or scientific interest, such as economics or the hu man brain, we have to deal with truly complex systems which cannot be decomposed into their parts without losing crucial properties of the total system. In addi tion, complex systems have many facets and can be looked at from many points of view. Whenever a complicated problem arises, some scientists or other people are ready to invent lots of beautiful words, or to quote Goethe "denn immer wo Begriffe feh len, dort stellt ein Wort zur rechten Zeit sich ein" ("whenever concepts are lack king, a word appears at the right time"). Quite often such a procedure gives not only the layman but also scientists working in fields different from that of the in ventor of these new words the impression that this problem has been solved, and I am occasionally shocked to see how influential this kind of "linguistics" has become.
In many disciplines of science we deal with systems composed of many subsystems. A few examples, mainly taken from topics in this book, are listed in Fig. 1. Very often the properties of the large system can not be explained by a mear random superposition of actions of the subsystems. Quite on the contrary the subsystems behave in a well organized manner, so that the total system is in an ordered state or shows actions which one might even call purposeful. Furthermore one often observes more or less abrupt changes between disorder and order or transitions between different states of order. Thus the question arises, who are the mysterious demons who tell the subsystems in which way to behave so to create order, or, in a more scientific language, which are the principles by which order is created.
Among the most striking features to emerge from 15 years of investigation of Helisoma are the tremendous adaptability and resiliency of its nervous system and the precision with which remodeling can occur. Somewhat ironically, Helisoma was ini-tally selected for investigation because of the existence of definable behavior, especially fixed action patterns, and a nervous system that was approachable because of its relative simplicity and assumed stability. However, as the analytical precision of investigations on this system improved, the view of a nervous system of immutable circuits has been refined to encompass a much more dynamic set of capabilities. It is apparent now that these more plastic attributes may be a general property of nervous systems. One might also speculate that the responses of perturbation-induced neural plasticity are actually an amplification of ongoing events normally present in all neuronal systems.
Let us first discuss why systems may be very complex. First of all, quite a number of systems contain very many elements. Examples are provided by the following table:
Table 1
brain
1011
1012 neurons
world
1010 people
laser
1018 atoms
fluid
1023 molecules/cm3
The presented book is a corrected printing of the 1980-edition, see the review Zbl 0464.92001. It deals with dynamics of processes that repeat themselves regularly. Such rhythmic return through a cycle of change is an ubiquitous principle of organization in living systems. In particular, attention is drawn to phase singularities which play an important role in self-organization of biological patterns in space and time. Corresponding to the title “Geometry of biological time” rhythmic changes not in space so much as in time are studied. A phase singularity is a point (in the state space) at which phase is ambiguous and near which phase takes on all values. In chapter 2, examples of phase singularities, also of living clocks, are described and in chapter 10 the physical nature of phase singularities is discussed. The first part of the book (10 chapters) sketches on a lightly abstract level theoretical aspects of clocks (circular logic, phase singularities, rules of the ring, ring populations, collective rhythmicity, attracting cycles, circadian clock, attractor cycle oscillators, breakdown of rhythmic organization). Particular experimental systems described in the second half of the book provide background facts about the organisms or phenomena mentioned in the first half. The book is intended primarily for research students.
