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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.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The widely cited Haken–Kelso–Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself. [ABSTRACT FROM AUTHOR]

We study multifrequency Hebbian plasticity by analyzing phenomenological models of weakly connected neural networks. We start with an analysis of a model for single-frequency networks previously shown to learn and memorize phase differences between component oscillators. We then study a model for gradient frequency neural networks (GrFNNs) which extends the single-frequency model by introducing frequency detuning and nonlinear coupling terms for multifrequency interactions. Our analysis focuses on models of two coupled oscillators and examines the dynamics of steady-state behaviors in multiple parameter regimes available to the models. We find that the model for two distinct frequencies shares essential dynamical properties with the single-frequency model and that Hebbian learning results in stronger connections for simple frequency ratios than for complex ratios. We then compare the analysis of the two-frequency model with numerical simulations of the GrFNN model and show that Hebbian plasticity in the latter is locally dominated by a nonlinear resonance captured by the two-frequency model. [ABSTRACT FROM AUTHOR]

The hybrid rhythmic ball-bouncing task considered in this study requires a participant to hit a ball in a virtual environment by moving a paddle in the real environment. It allows for investigation of the online visual control of action in humans. Changes in gravity acceleration in the virtual environment affect the ball dynamics and modify the ball-paddle system limit cycle. These changes are shown to be accurately reproduced through simulation by a model integrating continuous information-movement couplings between the ball trajectory and the paddle trajectory, giving rise to a resonance-tuning phenomenon. On the contrary, the tested models integrating only intermittent sensorimotor couplings were unable to replicate the observed human behavior. Results suggest that the visual control of action is achieved online, in a prospective way. Human rhythmic motor control would benefit from the timing and phase control emerging from the low-level continuous coupling between the central pattern generator and the visual perception of the ball trajectory. This control strategy, which precludes the need for internal clock and explicit environmental representation, is also able to explain the empirical result that the bounces tend to converge toward a passive stability regime during human ball bouncing. [ABSTRACT FROM AUTHOR]

The problem of phase regulation for a population of oscillating systems is considered. The proposed control strategy is based on a phase response curve (PRC) model of an oscillator (the first-order reduced model obtained for linearized system and inputs with infinitesimal amplitude). It is proven that the control provides phase resetting for the original nonlinear system. Next, the problem of phase resetting for a network of oscillators is considered when applying a common control input. Performance of the obtained solutions is demonstrated via computer simulation for three different models of circadian/neural oscillators. [ABSTRACT FROM AUTHOR]

Oscillatory activity is ubiquitous in nervous systems, with solid evidence that synchronisation mechanisms underpin cognitive processes. Nevertheless, its informational content and relationship with behaviour are still to be fully understood. In addition, cognitive systems cannot be properly appreciated without taking into account brain-body- environment interactions. In this paper, we developed a model based on the Kuramoto Model of coupled phase oscillators to explore the role of neural synchronisation in the performance of a simulated robotic agent in two different minimally cognitive tasks. We show that there is a statistically significant difference in performance and evolvability depending on the synchronisation regime of the network. In both tasks, a combination of information flow and dynamical analyses show that networks with a definite, but not too strong, propensity for synchronisation are more able to reconfigure, to organise themselves functionally and to adapt to different behavioural conditions. The results highlight the asymmetry of information flow and its behavioural correspondence. Importantly, it also shows that neural synchronisation dynamics, when suitably flexible and reconfigurable, can generate minimally cognitive embodied behaviour. [ABSTRACT FROM AUTHOR]

This study proposes a method for the automatic classification of nonlinear interactions between a strictly periodical event series modelling the activity of an exogenous oscillator working at a fixed and well-known rate and an event series modelling the activity of a self-sustained oscillator forced by the exogenous one. The method is based on a combination of several well-known tools (probability density function of the cyclic relative phase, probability density function of the count of forced events per forcing cycle, conditional entropy of the cyclic relative phase sequence and a surrogate data approach). Classification is reached via a sequence of easily applicable decision rules, thus rendering classification virtually user-independent and fully reproducible. The method classifies four types of dynamics: full uncoupling, quasiperiodicity, phase locking and aperiodicity. In the case of phase locking, the coupling ratio (i.e.n:m) and the strength of the coupling are calculated. The method, validated on simulations of simple and complex phase-locking dynamics corrupted by different levels of noise, is applied to data derived from one anesthetized and artificially ventilated rat to classify the nonlinear interactions between mechanical ventilation and: (1) the discharges of two (contemporaneously recorded) single postganglionic sympathetic neurons innervating the caudal ventral artery in the tail and (2) arterial blood pressure. Under central apnea, the activity of the underlying sympathetic oscillators is perturbed by means of five different lung inflation rates (0.58, 0.64, 0.76, 0.95, 1.99 Hz). While ventilation and arterial pressure are fully uncoupled, ventilation is capable of phase locking sympathetic discharges, thus producing 40% of phase-locked patterns (one case of 2:5, 1:1, 3:2 and 2:2) and 40% of aperiodic dynamics. In the case of phase-locked patterns, the coupling strength is low, thus demonstrating that this pattern is sliding. Non-stationary interactions are observed in 20% of cases. The two discharges behave differently, suggesting the presence of a population of sympathetic oscillators working at different frequencies. [ABSTRACT FROM AUTHOR]