Parametric stabilization of biological coordination: a theoretical model

In human coordination studies information from the environment may not only pace rhythmic behavior, but also contribute to the observed dynamics, e.g. aphenomenon known as anchoring in the literature. For the paradigmatic caseof bimanual coordination we study these contributions mathematically and develop a model of the interaction between the limb's intrinsic dynamics and environmental signals from a metronome in terms of oscillator equations. We discuss additive versus multiplicative metronomeimpact and show the latter to be more appropriate.Our model describes single limb-metronome interaction, as well as multilimb-metronome interaction. We establish a parametricstabilization term which preserves the characteristicsof bimanual coordination and additionally explains the varyingstability of movement under different metronome conditions, the frequency dependence of the amplitudes of finger movements, anchoring phenomena andgeometries of phase space trajectories. Predictions of our model are tested against experimental observations.     

Differing Dynamics of Intrapersonal and Interpersonal Coordination: Two-finger and Four-Finger Tapping Experiments

Finger-tapping experiments were conducted to examine whether the dynamics of intrapersonal and interpersonal coordination systems can be described equally by the Haken—Kelso—Bunz model, which describes inter-limb coordination dynamics. This article reports the results of finger-tapping experiments conducted in both systems. Two within-subject factors were investigated: the phase mode and the number of fingers. In the intrapersonal experiment (Experiment 1), the participants were asked to tap, paced by a gradually hastening auditory metronome, looking at their fingers moving, using the index finger in the two finger condition, or the index and middle finger in the four-finger condition. In the interpersonal experiment (Experiment 2), pairs of participants performed the task while each participant used the outside hand, tapping with the index finger in the two finger condition, or the index and middle finger in the four-finger condition. Some results did not agree with the HKB model predictions. First, from Experiment 1, no significant difference was observed in the movement stability between the in-phase and anti-phase modes in the two finger condition. Second, from Experiment 2, no significant difference was found in the movement stability between the in-phase and anti-phase mode in the four-finger condition. From these findings, different coordination dynamics were inferred between intrapersonal and interpersonal coordination systems against prediction from the previous studies. Results were discussed according to differences between intrapersonal and interpersonal coordination systems in the availability of perceptual information and the complexity in the interaction between limbs derived from a nested structure.     

A theoretical model of phase transitions in human hand movements

Earlier experimental studies by one of us (Kelso, 1981a, 1984) have shown that abrupt phase transitions occur in human hand movements under the influence of scalar changes in cycling frequency. Beyond a critical frequency the originally prepared out-of-phase, antisymmetric mode is replaced by a symmetrical, in-phase mode involving simultaneous activation of homologous muscle groups. Qualitavely, these phase transitions are analogous to gait shifts in animal locomotion as well as phenomena common to other physical and biological systems in which new modes or spatiotemporal patterns arise when the system is parametrically scaled beyond its equilibrium state (Haken, 1983). In this paper a theoretical model, using concepts central to the interdisciplinary field of synergetics and nonlinear oscillator theory, is developed, which reproduces (among other features) the dramatic change in coordinative pattern observed between the hands.     

Entrainment of leech swimming activity by the ventral stretch receptor

Rhythmic animal movements originate in CNS oscillator circuits; however, sensory inputs play an important role in shaping motor output. Our recent studies demonstrated that leeches with severed nerve cords swim with excellent coordination between the two ends, indicating that sensory inputs are sufficient for maintaining intersegmental coordination. In this study, we examined the neuronal substrates that underlie intersegmental coordination via sensory mechanisms. Among the identified sensory neurons in the leech, we found the ventral stretch receptor (VSR) to be the best candidate for our study because of its sensitivity to tension in longitudinal muscle. Our experiments demonstrate that (1) the membrane potential of the VSR is depolarized during swimming and oscillates with an amplitude of 1.5–5.0 mV, (2) rhythmic currents injected into the VSR can entrain ongoing swimming over a large frequency range (0.9–1.8 Hz), and (3) large current pulses injected into the VSR shift the phase of the swimming rhythm. These results suggest that VSRs play an important role in generating and modulating the swim rhythm. We propose that coordinated swimming in leech preparations with severed nerve cords results from mutual entrainment between the two ends of the leech mediated by stretch receptors.     

Relative phase dynamics in perturbed interlimb coordination: the effects of frequency and amplitude

Sch?ner [Sch?ner G (1995) Ecol Psychol 7: 291-314] argued that the relative phase dynamics of rhythmic interlimb coordination may be attributed to the timing level in that the stability properties of the relative phase are largely independent of dynamical principles operating at the goal level, such as those related to the maintenance of a particular amplitude or target position. Yet, according to the coupling functions in the coupled oscillator model proposed by Haken et al. [Haken H, Kelso JAS, Bunz H (1985) Biol Cybern 51: 347-356], the effect of frequency on the stability properties of relative phase is either wholly or partially mediated by frequency-induced changes in amplitude, implying that the relative phase dynamics strongly depends on spatial factors. In order to distinguish between these contrasting interpretations of the organizational principles underwriting the phase dynamics of interlimb coordination, an experiment was conducted in which the effects of frequency and amplitude on the stability of relative phase were separated. Six subjects performed both in-phase and antiphase coordination patterns at seven different frequencies and three different amplitudes. Two measures of pattern stability were used, the standard deviation of relative phase and the exponent of the relaxation process following phasic perturbations of relative phase. According to both measures, pattern stability decreased with increasing frequency, whereas the amplitude manipulation only had a significant effect on the standard deviation of relative phase. This result was interpreted to imply that the organizational principles at the (relative) timing level are affected only moderately by task constraints pertaining to the goal level, and that models of interlimb coordination in which amplitude coupling plays a partial or subordinate role should be preferred above models relying solely on amplitude coupling.     

A Neuro-Mechanical Model for Interpersonal Coordination

The present study investigates the coordination between two people oscillating handheld pendulums, with a special emphasis on the influence of the mechanical properties of the effector systems involved. The first part of the study is an experiment in which eight pairs of participants are asked to coordinate the oscillation of their pendulum with the other participant’s in an in-phase or antiphase fashion. Two types of pendulums, A and B, having different resonance frequencies (Freq A=0.98 Hz and Freq B=0.64 Hz), were used in different experimental combinations. Results confirm that the preferred frequencies produced by participants while manipulating each pendulum individually were close to the resonance frequencies of the pendulums. In their attempt to synchronize with one another, participants met at common frequencies that were influenced by the mechanical properties of the two pendulums involved. In agreement with previous studies, both the variability of the behavior and the shift in the intended relative phase were found to depend on the task-effector asymmetry, i.e., the difference between the mechanical properties of the effector systems involved. In the second part of the study, we propose a model to account for these results. The model consists of two cross-coupled neuro-mechanical units, each composed of a neural oscillator driving a wrist-pendulum system. Taken individually, each unit reproduced the natural tendency of the participants to freely oscillate a pendulum close to its resonance frequency. When cross-coupled through the vision of the pendulum of the other unit, the two units entrain each other and meet at a common frequency influenced by the mechanical properties of the two pendulums involved. The ability of the proposed model to address the other effects observed as a function of the different conditions of the pendulum and intended mode of coordination is discussed.     

A coupled oscillator model of disordered interlimb coordination in patients with Parkinson's disease

Coordination between the left and right limbs during cyclic movements, which can be characterized by the amplitude of each limb's oscillatory movement and relative phase, is impaired in patients with Parkinson's disease (PD). A pedaling exercise on an ergometer in a recent clinical study revealed several types of coordination disorder in PD patients. These include an irregular and burst-like amplitude modulation with intermittent changes in its relative phase, a typical sign of chaotic behavior in nonlinear dynamical systems. This clinical observation leads us to hypothesize that emergence of the rhythmic motor behaviors might be concerned with nonlinearity of an underlying dynamical system. In order to gain insight into this hypothesis, we consider a simple hard-wired central pattern generator model consisting of two identical oscillators connected by reciprocal inhibition. In the model, each oscillator acts as a neural half-center controlling movement of a single limb, either left or right, and receives a control input modeling a flow of descending signals from higher motor centers. When these two control inputs are tonic-constant and identical, the model has left-right symmetry and basically exhibits ordered coordination with an alternating periodic oscillation. We show that, depending on the intensities of these two control inputs and on the difference between them that introduces asymmetry into the model, the model can reproduce several behaviors observed in the clinical study. Bifurcation analysis of the model clarifies two possible mechanisms for the generation of disordered coordination in the model: one is the spontaneous symmetry-breaking bifurcation in the model with the left-right symmetry. The other is related to the degree of asymmetry reflecting the difference between the two control inputs. Finally, clinical implications by the model's dynamics are briefly discussed.     

Amplitude model for the effects of mutations and temperature on period and phase resetting of the Neurospora circadian oscillator.

This paper analyzes published and unpublished data on phase resetting of the circadian oscillator in the fungus Neurospora crassa and demonstrates a correlation between period and resetting behavior in several mutants with altered periods: As the period increases, the apparent sensitivity to resetting by light and by cycloheximide decreases. Sensitivity to resetting by temperature pulses may also decrease. We suggest that these mutations affect the amplitude of the oscillator and that a change in amplitude is responsible for the observed changes in both period and resetting by several stimuli. As a secondary hypothesis, we propose that temperature compensation of period in Neurospora can be explained by changes in amplitude: As temperature increases, the compensation mechanism may increase the amplitude of the oscillator to maintain a constant period. A number of testable predictions arising from these two hypotheses are discussed. To demonstrate these hypotheses, a mathematical model of a time-delay oscillator is presented in which both period and amplitude can be increased by a change in a single parameter. The model exhibits the predicted resetting behavior: With a standard perturbation, a smaller amplitude produces type 0 resetting and a larger amplitude produces type 1 resetting. Correlations between period, amplitude, and resetting can also be demonstrated in other types of oscillators. Examples of correlated changes in period and resetting behavior in Drosophila and hamsters raise the possibility that amplitude changes are a general phenomenon in circadian oscillators.     

Haptic information stabilizes and destabilizes coordination dynamics

Goal-directed, coordinated movements in humans emerge from a variety of constraints that range from 'high-level' cognitive strategies based on perception of the task to 'low-level' neuromuscular-skeletal factors such as differential contributions to coordination from flexor and extensor muscles. There has been a tendency in the literature to dichotomize these sources of constraint, favouring one or the other rather than recognizing and understanding their mutual interplay. In this experiment, subjects were required to coordinate rhythmic flexion and extension movements with an auditory metronome, the rate of which was systematically increased. When subjects started in extension on the beat of the metronome, there was a small tendency to switch to flexion at higher rates, but not vice versa. When subjects were asked to contact a physical stop, the location of which was either coincident with or counterphase to the auditory stimulus, two effects occurred. When haptic contact was coincident with sound, coordination was stabilized for both flexion and extension. When haptic contact was counterphase to the metronome, coordination was actually destabilized, with transitions occurring from both extension to flexion on the beat and from flexion to extension on the beat. These results reveal the complementary nature of strategic and neuromuscular factors in sensorimotor coordination. They also suggest the presence of a multimodal neural integration process - which is parametrizable by rate and context - in which intentional movement, touch and sound are bound into a single, coherent unit.     

New determinants of firing rates and patterns of vasopressinergic magnocellular neurons: predictions using a mathematical model of osmodetection

Arginine vasopressin (AVP), one of the most important hormones involved in hydromineral homeostasis, is secreted by hypothalamic magnocellular neurons (MCNs). Here, we implemented two critical parameters for MCN physiology into a Hodgkin-Huxley simulation of the MCN. By incorporating the mechanosensitive channel (MSC) responsible for osmodetection and the synaptic inputs whose frequencies are modulated by changes in ambient osmolality into our model, we were able to develop an improved model with increased physiological relevance and gain new insight into the determinants of the firing patterns of AVP magnocellular neurons. Our results with this MCN model predict that 1) a single MCN is able to display all the firing patterns experimentally observed: silent, irregular, phasic and continuous firing patterns; 2) under conditions of hyperosmolality, burst durations are regulated by the frequency-dependent fatigue of dynorphin secretion; and 3) the transitions between firing patterns are controlled by EPSP and IPSP frequencies (0, 2, 4, 8, 16, 32, 64 and 128 Hz). Moreover, this simulation predicts that EPSPs and IPSPs do not modify the spiking frequency (SF) of phasic firing patterns (0.0034 Hz/Hz [EPSP]; 0.0012 Hz/Hz [IPSP]). Rather, these afferents strongly regulate SF during irregular and continuous firing patterns (0.075 Hz/Hz [EPSP]; 0.027 Hz/Hz [IPSP]). The use of the realistic MCN model developed here allows for an improved understanding of the determinants driving the firing patterns and spiking frequencies of vasopressinergic magnocellular neurons.     

Coupling of posture and gait: mode locking and parametric excitation

 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. Received: 23 June 1999 / Accepted in revised form: 10 January 2001     

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation

In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships – with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits. Received: 20 November 1996 / Accepted: 29 July 1997     

Important relation between EEG and brain evoked potentials

In this study, we analyze the important relation between the spontaneous and evoked activities of the substructures of the cat brain, such as the reticular formation, hippocampus, inferior colliculus, medial geniculate nucleus and acoustical cortex, with an ensemble of systems theory methods consisting of the following steps: (1) single auditory and/or visual evoked potentials (EPs) and the spontaneous activities (EEG) just preceding the stimuli are recorded from the brain center under study; (2) selectively averaged evoked potentials (SAEPs) are obtained from the recorded EPs; (3) amplitude frequency characteristics are computed from the AAEPs by means of Fourier transform; (4) the single EEG-EP sweeps are theoretically pass-band filtered with adequate band limits determined according to the selectivities revealed by the amplitude characteristics; (5) the EEG and EP components obtained in this way are compared with regard to the amplification in the population response upon the application of the stimulus. The results of this analysis support quantitatively our prediction of various types of resonance phenomena in a number of nuclei in the cat brain and in a large scale of frequencies from 1 Hz to 1000 Hz and show that the amplification factor related to resonance phenomena has probabilistic nature. Therefore, the analogy which we have recently drawn between the behaviors of a neural population and a random-phase probabilistic harmonic oscillator is extended by assigning also the amplitude and the frequency of the oscillations as random variables. A working hypothesis for the dynamics of neuronal populations is elaborated accordingly.Presented in Part at the Third European Meeting on Cybernetics and Systems Research 1976 in Vienna, April 20–23, 1976Supported by Grant No. TAG-345 of the Scientific and Technical Research Council of Turkey     

Coordination dynamics: (in)stability and metastability in the behavioural and neural systems

For more than 20 years, coordination dynamics have provided research on human movement science with new views about the nonlinear relationships between behavioral and neural dynamics. A number of studies across various experimental settings including bimanual, postural or interpersonal coordination, and also coordination between movements of a limb and an external event in the environment revealed the self-organized nature of human coordination. Here we review an extensive body of literature - in the human movement science and the neuroscience fields - that has investigated the coordination dynamics of brain and behavior when individuals are involved in two rhythmic coordination patterns: synchronization (on-the-beat movements) and syncopation (in-between beats movements). When the frequency of movement approaches 2 Hz, the syncopation mode is destabilized and synchronization is spontaneously adopted. The abrupt change between the two patterns illustrates a phenomenon known as non-equilibrium phase transition. Phase transitions offer a novel entry point into the investigation of pattern formation (and dissolution) at both the behavioral and the cerebral levels as they illustrate the loss of stability of the system. Brain imaging methods (MEG, EEG and fMRI) were used to reveal the neural signatures of (in)stability underlying the differences between behavioral coordination patterns, and pointed at the role of self-organization and metastability principles in brain functioning. Relationships between behavioral and brain dynamics can therefore be investigated within a unified empirical and theoretical framework.     

Changes in posture alter the attentional demands of voluntary movement.

Two simple experiments reveal that the ease with which an action is performed by the neuromuscular-skeletal system determines the attentional resources devoted to the movement. Participants were required to perform a primary task, consisting of rhythmic flexion and extension movements of the index finger, while being paced by an auditory metronome, in one of two modes of coordination: flex on the beat or extend on the beat. Using a classical dual-task methodology, we demonstrated that the time taken to react to an unpredictable visual probe stimulus (the secondary task) by means of a pedal response was greater when the extension phase of the finger movement sequence was made on the beat of the metronome than when the flexion phase was coordinated with the beat. In a second experiment, the posture of the wrist was manipulated in order to alter the operating lengths of muscles that flex and extend the index finger. The attentional demands of maintaining the extend-on-the-beat pattern of coordination were altered in a systematic fashion by changes in wrist posture, even though the effector used to respond to the visual probe stimulus was unaffected.     

Hard-wired central pattern generators for quadrupedal locomotion

Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

 We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment. Received: 30 December 1994/Accepted in revised form: 27 June 1995     

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

Rhythmic movement patterns have served as a model case for developing a synergetic theory of biological coordination. In part I of this work we extended the approach to environmentally-specified and learned movement patterns on the level of the collective variable relative phase. Here we show that an identical strategy may be applied to the same problem at the level of the component oscillators. Coordinative patterns and their dynamics are derived from the coupled component dynamics and their interaction with the environment. Thus, behavioral patterns are shown to arise in a purely self-organized fashion. New directions for further research (e.g. dynamics of action-perception systems) follow from the oscillator theory. Finally the relationship between our approach and other kinds of analyses of temporal order (e.g. phase resetting) is addressed.     

Stability of Phase Relationships While Coordinating Arm Reaches with Whole Body Motion

The human movement repertoire is characterized by the smooth coordination of several body parts, including arm movements and whole body motion. The neural control of this coordination is quite complex because the various body parts have their own kinematic and dynamic properties. Behavioral inferences about the neural solution to the coordination problem could be obtained by examining the emerging phase relationship and its stability. Here, we studied the phase relationships that characterize the coordination of arm-reaching movements with passively-induced whole-body motion. Participants were laterally translated using a vestibular chair that oscillated at a fixed frequency of 0.83 Hz. They were instructed to reach between two targets that were aligned either parallel or orthogonal to the whole body motion. During the first cycles of body motion, a metronome entrained either an in-phase or an anti-phase relationship between hand and body motion, which was released at later cycles to test phase stability. Results suggest that inertial forces play an important role when coordinating reaches with cyclic whole-body motion. For parallel reaches, we found a stable in-phase and an unstable anti-phase relationship. When the latter was imposed, it readily transitioned or drifted back toward an in-phase relationship at cycles without metronomic entrainment. For orthogonal reaches, we did not find a clear difference in stability between in-phase and anti-phase relationships. Computer simulations further show that cost models that minimize energy expenditure (i.e. net torques) or endpoint variance of the reach cannot fully explain the observed coordination patterns. We discuss how predictive control and impedance control processes could be considered important mechanisms underlying the rhythmic coordination of arm reaches and body motion.