Phase-amplitude dynamics in terms of extended response functions: invariant curves and arnold tongues

Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,. . . ), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong enough stimuli break up the oscillatory dynamics and lead to phase-locking, which is well captured by the 2D entrainment map.Peer ReviewedPreprin

Optimal Subharmonic Entrainment

For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when N control cycles occur for every M cycles of a forced oscillator, is referred to as N:M entrainment. In many applications, entrainment must be established in an optimal manner, for example by minimizing control energy or the transient time to phase locking. We present a theory for deriving inputs that establish subharmonic N:M entrainment of general nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. Formal averaging and the calculus of variations are then applied to such reduced models in order to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which is readily extended to arbitrary oscillating systems

Neuro-mechanical entrainment in a bipedal robotic walking platform

In this study, we investigated the use of van der Pol oscillators in a 4-dof embodied bipedal robotic platform for the purposes of planar walking. The oscillator controlled the hip and knee joints of the robot and was capable of generating waveforms with the correct frequency and phase so as to entrain with the mechanical system. Lowering its oscillation frequency resulted in an increase to the walking pace, indicating exploitation of the global natural dynamics. This is verified by its operation in absence of entrainment, where faster limb motion results in a slower overall walking pace

Developmental acquisition of entrainment skills in robot swinging using van der Pol oscillators

In this study we investigated the effects of different morphological configurations on a robot swinging task using van der Pol oscillators. The task was examined using two separate degrees of freedom (DoF), both in the presence and absence of neural entrainment. Neural entrainment stabilises the system, reduces time-to-steady state and relaxes the requirement for a strong coupling with the environment in order to achieve mechanical entrainment. It was found that staged release of the distal DoF does not have any benefits over using both DoF from the onset of the experimentation. On the contrary, it is less efficient, both with respect to the time needed to reach a stable oscillatory regime and the maximum amplitude it can achieve. The same neural architecture is successful in achieving neuromechanical entrainment for a robotic walking task

Neuro-mechanical entrainment in a bipedal robotic walking platform

In this study, we investigated the use of van der Pol oscillators in a 4-dof embodied bipedal robotic platform for the purposes of planar walking. The oscillator controlled the hip and knee joints of the robot and was capable of generating waveforms with the correct frequency and phase so as to entrain with the mechanical system. Lowering its oscillation frequency resulted in an increase to the walking pace, indicating exploitation of the global natural dynamics. This is verified by its operation in absence of entrainment, where faster limb motion results in a slower overall walking pace

Sensitivity analysis of circadian entrainment in the space of phase response curves

Sensitivity analysis is a classical and fundamental tool to evaluate the role of a given parameter in a given system characteristic. Because the phase response curve is a fundamental input--output characteristic of oscillators, we developed a sensitivity analysis for oscillator models in the space of phase response curves. The proposed tool can be applied to high-dimensional oscillator models without facing the curse of dimensionality obstacle associated with numerical exploration of the parameter space. Application of this tool to a state-of-the-art model of circadian rhythms suggests that it can be useful and instrumental to biological investigations.Comment: 22 pages, 8 figures. Correction of a mistake in Definition 2.1. arXiv admin note: text overlap with arXiv:1206.414

Dynamical Phase Transitions In Driven Integrate-And-Fire Neurons

We explore the dynamics of an integrate-and-fire neuron with an oscillatory stimulus. The frustration due to the competition between the neuron's natural firing period and that of the oscillatory rhythm, leads to a rich structure of asymptotic phase locking patterns and ordering dynamics. The phase transitions between these states can be classified as either tangent or discontinuous bifurcations, each with its own characteristic scaling laws. The discontinuous bifurcations exhibit a new kind of phase transition that may be viewed as intermediate between continuous and first order, while tangent bifurcations behave like continuous transitions with a diverging coherence scale.Comment: 4 pages, 5 figure