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This paper presents an approach for developing an experimentally validated dynamic multisegment model to simulate human flight-phase dynamics and multijoint control. Modeling and experimental techniques were integrated to systematically examine the contribution of multiple error sources to the accuracy of the model and to determine the complexity of a model that adequately emulates the dynamic behavior at the total-body and multijoint levels during flight. The accuracy of the model and of the experimental data was assessed using an inverse dynamics simulation of flight-phase motion for two representative cases: (i) a physical model released from a bar and (ii) a gymnast performing a layout dismount from a bar. Multijoint models with varying numbers of segments were assessed in order to determine the complexity of the model that adequately simulates the flight-phase task. A five-segment model was found to adequately simulate the layout dismount performed by the gymnast. The error introduced during modeling and digitizing contributed to an apparent violation of the conservation law manifested as large external forces acting on the nonactuated joints. These results demonstrate the need to reduce sources of error prior to testing hypotheses regarding feedforward and feedback components of the multijoint control system. The proposed approach for quantifying sources of error provides a crucial step that is required in the development of experimentally based dynamic models designed to examine and test hypotheses regarding multijoint control logic.

Schöner [Schöner 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.

Various stability features of bimanual rhythmic coordination, including phase transitions, have been modeled successfully by means of a one-dimensional equation of motion for relative phase obeying a gradient dynamics, the Haken-Kelso-Bunz model. The present study aimed at assessing pattern stability for stationary performance and estimating the model parameters (a, b, and Q) for the stochastic extension of this model. Estimates of a and b allowed for reconstruction of the potential defining the gradient dynamics. Two coordination patterns between the forearms (in-phase, antiphase) were performed at seven different frequencies. Model parameters were estimated on the basis of an exponential decay parameter describing the relaxation behavior of continuous relative phase following a mechanical perturbation. Variability of relative phase and relaxation time provided measures of pattern stability. Although the predicted inverse relation between pattern stability and movement frequency was observed for the lower tempo conditions, it was absent for the higher tempos, reflecting the influence of task constraints. No statistically significant differences in stability were observed between the two coordination modes, indicating the influence of intention. The reconstructed potential reflected the observed stability features, underscoring the adequacy of the parameter estimations. The relaxation process could not be captured adequately by means of a simple exponential decay function but required an additional oscillatory term. In accordance with previous assumptions, noise strength Q did not vary as a function of movement frequency. However, systematic differences in Q were observed between the two coordination modes. The advantages and (potential) pitfalls of using stationary performance of single patterns to examine the stability features of a bistable potential were discussed.

When standing human subjects are exposed to a moving visual environment, the induced postural sway displays varying degrees of coherence with the visual information. In our experiment we varied the frequency of an oscillatory visual display and analysed the temporal relationship between visual motion and sway. We found that subjects maintain sizeable sway amplitudes even as temporal coherence with the display is lost. Postural sway tended to phase lead (for frequencies below 0.2 Hz) or phase lag (above 0.3 Hz). However, we also observed at a fixed frequency, highly variable phase relationships in which a preferred range of phase lags is prevalent, but phase jumps occur that return the system into the preferred range after phase has begun drifting out of the preferred regime. By comparing the results quantitatively with a dynamical model (the sine-circle map), we show that this effect can be understood as a form of relative coordination and arises through an instability of the dynamics of the action-perception cycle. Because such instabilities cannot arise in passively driven systems, we conclude that postural sway in this situation is actively generated as rhythmic movement which is coupled dynamically to the visual motion.

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 specified 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.

We elucidate how coupling delays and noise impact phase and mutual information relationships between two stochastic brain rhythms. This impact depends on the dynamical regime of each PING-based rhythm, as well as on network heterogeneity and coupling asymmetry. The number of peaks at positive and negative time lags in the delayed mutual information between the two bi-directionally communicating rhythms defines our measure of flexibility of information sharing and reflects the number of ways in which the two networks can alternately lead one another. We identify two distinct mechanisms for the appearance of qualitatively similar flexible information sharing. The flexibility in the quasi-cycle regime arises from the coupling delay-induced bimodality of the phase difference distribution, and the related bimodal mutual information. It persists in the presence of asymmetric coupling and heterogeneity but is limited to two routes of information sharing. The second mechanism in noisy limit cycle regime is not induced by the delay. However, delay-coupling and heterogeneity enable communication routes at multiple time lags. Noise disrupts the shared compromise frequency, allowing the expression of individual network frequencies which leads to a slow beating pattern. Simulations of an envelope-phase description for delay-coupled quasi-cycles yield qualitatively similar properties as for the full system. Near the bifurcation from in-phase to out-of-phase behaviour, a single preferred phase difference can coexist with two information sharing routes; further, the phase laggard can be the mutual information leader, or vice versa. Overall, the coupling delay endows a two-rhythm system with an array of lead-lag relationships and mutual information resonances that exist in spite of the noise and across the Hopf bifurcation. These beg to be mapped out experimentally with the help of our predictions., (© 2022. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks., (© 2021. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

In this paper, we propose a framework for generating coordinated periodic movements of robotic systems with multiple external inputs. We developed an adaptive pattern generator model that is composed of a two-factor observation model with a style parameter and phase dynamics with a phase variable. The style parameter controls the spatial patterns of the generated trajectories, and the phase variable manages its temporal profiles. By exploiting the style-phase separation in the pattern generation, we can independently design adaptation schemes for the spatial and temporal profiles of the pattern generator to multiple external inputs. To validate the effectiveness of our proposed method, we applied it to a user-exoskeleton model to achieve user-adaptive walking assistance for which the exoskeleton robot's movements need to be coordinated with the user walking patterns and environment. As a result, the exoskeleton robot successfully performed stable biped walking behaviors for walking assistance even when the style of the observed walking pattern and the period were suddenly changed.

Weight-bearing tasks performed by humans consist of a series of phases with multiple objectives. Analysis of the relationship between control and dynamics during successive phases of the tasks is essential for improving performance without sustaining injury. Experimental evidence regarding foot landings suggests that the distribution of momentum among segments at contact influences stability during interaction with the landing surface. In this study, we hypothesized that modification of control in one subsystem, in our case shoulder torque, during the flight phase of an aerial task would enable the performer to maintain behavior of other subsystems (e.g.lower extremity kinematics) and initiate contact with momentum conditions consistent with successful task performance. To test this hypothesis, an experimentally validated multilink dynamic model that incorporated modifications in shoulder torque was used to simulate the flight phase dynamics of overrotated landings. The simulation results indicate that modification in shoulder torque during the flight phase enables gymnasts to maintain lower extremity kinematics and initiate contact with trunk angular velocities consistent with those observed during successful landings. These results suggest that modifications in the control logic of one subsystem may be sufficient for achieving both global and local task objectives of landing.

Based on neurophysiological evidence, theoretical studies have shown that locomotion is generated by mutual entrainment between the oscillatory activities of central pattern generators (CPGs) and body motion. However, it has also been shown that the time delay in the sensorimotor loop can destabilize mutual entrainment and result in the failure to walk. In this study, a new mechanism called flexible-phase locking is proposed to overcome the time delay. It is realized by employing the Bonhoeffer-Van der Pol formalism - well known as a physiologically faithful neuronal model - for neurons in the CPG. The formalism states that neurons modulate their phase according to the delay so that mutual entrainment is stabilized. Flexible-phase locking derives from the phase dynamics related to an asymptotically stable limit cycle of the neuron. The effectiveness of the mechanism is verified by computer simulations of a bipedal locomotion model.