The dimensionality of movement trajectories and the degrees of freedom problem: A tutorial

Kay, Bruce A.

doi:10.1016/0167-9457(88)90016-4

Cited by 114 publications

(98 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation (1) is not proposed as a model of a rhythmic limb, it simply identifies the class of dynamics involved and provides a framework for discussion. Research by Kay and colleagues (Kay 1988;Kay et al 1991) has confirmed that a rhythmically moving limb segment abides the limit cycle attractor dynamics of (1). Application of the spatial correlation procedure (Grassberger and Procaccio 1984) for computing the dimensionality of an attractor revealed a dimension closely approximating 1, the dimension of a limit cycle attractor.…”

Section: ;: + O~2x = F(x De) -2gy¢ = H(x De)mentioning

confidence: 75%

Constants underlying frequency changes in biological rhythmic movements

1993

View full text Add to dashboard Cite

When an animal increases or decreases the frequency of its limb motions, how should the transformation in timing be characterized? It has been hypothesized that the transformation is adiabatic, even though the biological conditions are nonconservative and non-rate-limited (Kugler and Turvey 1987). An adiabatic transformation requires that the rhythmic system's action (energy/frequency) and entropy production remain time-invariant throughout the transformation. The non-conservative adiabatic hypothesis was evaluated through an experiment on human rhythmic hand movements. On each trial, a subject began at a prescribed frequency and then, over a 30 s interval, increased (or decreased) the frequency continuously at will. For each subject, on each increasing and decreasing trial, cycle kinetic energy was a linear function of cycle frequency with a negative energy intercept. By the adiabatic hypothesis, the slope of the function defines the constant action and the intercept defines the constant dissipation - changes in cycle frequency incur no changes in energy dissipated per cycle. Slopes and intercepts were correlated suggesting a common basis for the two constants, and the variety of cycle amplitude-cycle duration relations were in agreement with the nonmonotonic, nonlinear space-time function predicted by the hypothesis. The possibilities of addressing aspects of the data through (a) muscle modeled as a continuum of Kelvin bodies with a continuous relaxation spectrum, and (b) various classes of autonomous differential equations, were discussed. Most importantly, the discussion focused on the puzzling independence of energy cost and speed exhibited by locomoting animals differing in morphology, physiology, size, and taxa. It was suggested that the independence may reflect a very general principle - adiabatic transformability of biological movement systems.

show abstract

Section: ;: + O~2x = F(x De) -2gy¢ = H(x De)mentioning

confidence: 75%

Constants underlying frequency changes in biological rhythmic movements

1993

View full text Add to dashboard Cite

show abstract

“…Other rhythmic limb movements have similarly been modeled as limit-cycle attractors (Kelso et al 1981;Kay et al 1987;Kay et al 1991). Once again, however, this is only an approximation, for the trajectory of a limb movement in the phase plane is not a single closed orbit, but rather a band of attraction due to slight modulations from cycle to cycle (Kay 1988). Such a limit-cycle oscillator, when forced, takes on the frequency of the driver under restricted conditions, but also exhibits N:M mode locking, quasiperiodicity, intermittency, and chaotic behavior.…”

Section: A Preliminary Modelmentioning

confidence: 99%

Coupling of posture and gait: mode locking and parametric excitation

Kay

Warren

2001

Biol Cybern

Self Cite

View full text Add to dashboard Cite

We investigate the temporal coordination of human gait and posture and infer the nature of their coupling. Participants viewed a sinusoidally oscillating visual display which induced medial-lateral postural sway during treadmill walking, while display frequency was varied (0.075-1.025 Hz). First, postural responses exhibited the usual low-pass characteristic but with an additional resonance peak near the preferred stride frequency, although shifted downward by 0.12 Hz; this provides evidence of a coupling from gait to posture. Second, the step cycle adapted to mode lock with the visual driver and postural sway, as well as displaying instances of intermittency (slipping in and out of phase) and quasiperiodicity (phase wandering); this provides evidence of a coupling from posture to gait. We observed a spectrum of integer mode locks, including a large 1:1 trapping region about the stride frequency and superharmonic entrainment (stride frequency > driver frequency) at lower driver frequencies. A coupled-oscillator model that incorporates a novel parametric coupling from posture to the gait "stiffness" term reproduces these features of the data, including the resonance peak shift. Biological coordination patterns may thus emerge naturally as properties of a system of appropriately coupled oscillators.

show abstract

“…In that regard, deterministic, autonomous, and timecontinuous systems are unambiguously described by their flow in phase (or state) space (or vector field), i.e., the space spanned by the system's state variables [19]. For movements along a single physical direction, as in a (sliding) Fitts' task, it is commonplace to use the movement's position and its time-derivative velocity as the state variables [11,20,21] (but see [22,23] for a critical discussion). The attractors that may live in such two-dimensional spaces are limited to (different kinds of ) fixed points (i.e., points where velocity and acceleration are zero) and limit cycles (nonlinear closed orbits), which are associated with discrete and rhythmic movements, respectively [24,25].…”

Section: Introductionmentioning

confidence: 99%

Does changing Fitts’ index of difficulty evoke transitions in movement dynamics?

Huys

Knol

Sleimen-Malkoun

et al. 2015

EPJ Nonlinear Biomed Phys

View full text Add to dashboard Cite

Background: The inverse relationship between movement speed and accuracy in goal-directed aiming is mostly investigated using the classic Fitts' paradigm. According to Fitts' law, movement time scales linearly with a single quantity, the index of difficulty (ID), which quantifies task difficulty through the quotient of target width and distance. Fitts' law remains silent, however, on how ID affects the dynamic and kinematic patterns (i.e., perceptual-motor system's organization) in goal-directed aiming, a question that is still partially answered only. Methods: Therefore, we here investigated the Fitts' task performed in a discrete as well as a cyclic task under seven IDs obtained either by scaling target width under constant amplitude or by scaling target distance under constant target width. Results: Under all experimental conditions Fitts' law approximately held. However, qualitative and quantitative dynamic as well as kinematic differences for a given ID were found in how the different task variants were performed. That is, while ID predicted movement time, its value in predicting movement organization appeared to be limited.

show abstract

The dimensionality of movement trajectories and the degrees of freedom problem: A tutorial

Cited by 114 publications

References 5 publications

Constants underlying frequency changes in biological rhythmic movements

Constants underlying frequency changes in biological rhythmic movements

Coupling of posture and gait: mode locking and parametric excitation

Does changing Fitts’ index of difficulty evoke transitions in movement dynamics?

Contact Info

Product

Resources

About