G. Schöner

Nonequilibrium phase transitions in coordinated biological motion: critical fluctuations

Abstract We report the results of experiments on biological motion demonstrating the presence of critical order parameter fluctuations as the system evolves from one coordinated state to another at a critical control parameter value. This is a key feature of nonequilibrium phase transitions.

A synergetic theory of quadrupedal gaits and gait transitions

We present a theoretical analysis of the patterns of interlimb co-ordination in the gaits of quadrupedal locomotion. Introducing as collective variables a set of relative phases that describe the co-ordination patterns, we classify gaits by their symmetry properties, which can be expressed as invariances under groups of transformations. We define dynamics of the collective variables, on which we impose symmetry restrictions. The stable observable gait patterns correspond to atractors of these dynamics. A non-trivial consequence of this theoretical viewpoint is that gait transitions can take the form of non-equilibrium phase transitions that are accompanied by loss of stability. We show how various types of such phase transitions involving hysteresis, slowing down and fluctuation enhancement can occur. Also the difference between smooth and abrupt transitions is given theoretical foundation. While existing experimental evidence is consistent with the theory developed here, we propose new experimental measures that can serve to test the present theoretical framework. Finally, the influence of underlying symmetries of the dynamics on the nature of the gait patterns and their stability is analyzed. For example, breaking of a front-hind symmetry can lead to a change from absolute to relative co-ordination in the sense of von Holst (1939, Ergebnisse der Physiologie 42, 228). Also, differential stability of straight and reverse gaits results from thus lowering the symmetry.

Self-organization of coordinative movement patterns ☆

Abstract We present a strictly operational approach — in which theoretical tools and experimental data are developed together — to the problem of understanding the coordination of movement patterns. The empirical aspects are guided by synergetics, a theory of spontaneous pattern formation in open systems. Following an outline of our theoretical strategy, recent experimental results are reviewed that demonstrate the validity of the approach. From these studies, it is possible to establish the linkage between order-order transitions in movement behavior and other nonequilibrium phase transitions in nature. Behavioral patterns (corresponding to low-dimensional attractors for collective variables) and their dynamics are shown to arise in a purely self-organized fashion from cooperative coupling among individual components. This step has been implemented analytically and computationally. The insights gained from the present research allow a generalization in the form of seven theoretical propositions that aim at characterizing pattern formation, stability and change in complex, biological systems. In turn, a number of new research directions emerge, including studies of the collective dynamics of the environment-movement system, learning, and multiple-limb coordination.

A synergetic theory of environmentally-specified and learned patterns of movement coordination

This paper outlines and applies a synergetic strategy to the coordination of human rhythmical movement. It extends earlier empirical and theoretical work to include the influence of specific environmental information and of memory on the dynamics of the collective variables (order parameters) that characterize the coordination patterns. Key ideas concern cooperative and competitive influences on the collective dynamics. Recent experiments on environmentally specific and learned rhythmic movement patterns are modeled explicitly on the level of the collective variable, relative phase. New predictions are presented and research directions proposed that follow directly from the present theoretical approach.

Nonequilibrium phase transitions in coordinated biological motion: Critical slowing down and switching time

Abstract In new experiments on coordinated biological motion we measure relaxation times and switching times as the system evolves from one coordinated state to another at a critical control parameter value. Deviations from the coordinated state are induced by mechanical perturbations and relative phase is used as an order parameter to monitor the dynamics of the collective state. Clear evidence for critical slowing down, a key feature of nonequilibrium phase transitions, is found. The mean and distribution of switching times closely match predictions from a stochastic dynamic theory. Together with earlier results on critical fluctuations these findings strongly favor an interpretation of coordinative change in biological systems as a nonequilibrium phase transition.

A dynamic theory of coordination of discrete movement

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

Phase-Locked Modes, Phase Transitions and Component Oscillators in Biological Motion

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 systems 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.

Dynamics governs switching among patterns of coordination in biological movement

Abstract In new experiments on coordinated human movement we demonstrate that the process of intentionally switching from one pattern of coordination to another is governed by the dynamics of the patterns themselves. In particular, the stability of the patterns as established in earlier experiments on instabilities of these coordination patterns, determines the nature of the transient switching process. Measures such as the length of the transient (or switching time) and its distribution closely match theoretical predictions.

A dynamic pattern theory of behavioral change

Intentional change of behavior is an essential phenomenon that theoretical biology cannot fail to address. Often, theoretical attempts to understand the problem and experimental study of behavioral change are quite unrelated to each other. Recent progress in formulating a strictly operational dynamic theory of behavioral patterns, however, offers a link between theory and experiment. Here the understanding of intentional change of behavioral pattern in this theoretical language is shown. The general formulation provides predictions on the relation between the dynamics of behavioral patterns and the nature of the process of behavioral change. Theoretically founded measures, including switching time and first exit time, are introduced that allow a characterization of this process. A concrete system involving temporally ordered behavior is modelled explictly on two experimentally accessible levels of observation. Switching time and first passage time measures are calculated from the theory and the results compared to recent experimental observations. We discuss the potential of the switching time measures for the more general study of behavioral patterns and their dynamics.

Learning and recall in a dynamic theory of coordination patterns

A dynamic theory of learning and recall of coordination patterns is developed in the context of relative timing skills. Characterizing the coordination patterns in such skills by the collective variable, relative phase, we choose a model system in which the intrinsic pattern dynamics as well as the influence of environmental and memorized information are well understood from previous experimental and theoretical work. To describe learning we endow memorized information with dynamics which is determined by a phenomenological strategy. Similarly, additional degrees of freedom must be introduced to understand recall. As such recall variables we choose the relative strengths with which each memorized pattern acts on the pattern dynamics and model their dynamics phenomenologically. The resulting dynamical system that resembles models used in pattern recognition theory is shown to adequately describe the learning and recall processes. Moreover, due to the operational character of the theory, several predictions emerge that are open to experimental test. In particular, we show under which conditions phase transitions occur in the dynamics of the coordination patterns during learning and during recall. Considering different time scales and their relations we demonstrate how these phase transitions can be identified and observed. Other predictions include the influence of the intrinsic pattern dynamics on the recall process and the existence of history and hysteresis effects in recall. We discuss different forms of “forgetting” and differentiation of memorized information. The results show how a new theoretical view of learning and recall as change of behavioral dynamics can lead to a different understanding of these processes by providing testable predictions.