Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

Phase-amplitude reduction and optimal phase locking of collectively oscillating networks

We present a phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems. The framework, which builds on the phase reduction method, takes into account not only the collective dynamics on the limit cycle but also deviations from it by introducing amplitude variables and using them with the phase variable. The framework allows us to study how networks react to applied inputs or coupling, including their synchronization and phase locking, while capturing the deviations of the network states from the unperturbed dynamics. Numerical simulations are used to demonstrate the effectiveness of the framework for networks composed of FitzHugh-Nagumo elements. The resulting phase-amplitude equations can be used in deriving optimal periodic waveforms or introducing feedback control for achieving fast phase locking while stabilizing the collective oscillations.

Generic Mechanism for Waveform Regulation and Synchronization of Oscillators: An Application for Robot Behavior Diversity Generation

While nonlinear oscillators have been widely used for central pattern generators to produce basic rhythmic signals for robot locomotion control, methods to shape and regulate the signal waveform without changing the characteristics of the oscillators have not been fully investigated, especially during the network synchronization process. To illustrate the principle and process of waveform regulation of nonlinear oscillators in detail and ensure that the influence can be controlled, we present a method for waveform regulation and synchronization and analyze the relationship of different factors (e.g., initial conditions, network parameters, phase, and waveform regulation factors) in synchronization deviation. Then, the method is indicated to be effective in other commonly used nonlinear oscillators and neural oscillators. As an example application, a three-layer behavioral control architecture for a legged robot is constructed based on the proposed method. Modules for the body behavior, leg coordination, and single-leg adjustment are established to realize diverse robot behaviors. The effectiveness of the method is validated by a series of experiments. The results prove that the method performs well in terms of signal control accuracy, behavior pattern diversity, and smooth motion transition.

Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the van der Pol and Willamowski-Rössler oscillators with real or complex Floquet exponents is analyzed as examples. It is demonstrated that the proposed methods can achieve considerably faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.

Sparse optimization of mutual synchronization in collectively oscillating networks

We consider a pair of collectively oscillating networks of dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed by Nakao et al. [Chaos 28, 045103 (2018)]. The dynamical equations describing a pair of weakly coupled networks are reduced to a pair of coupled phase equations, and the linear stability of the synchronized state between the networks is represented as a function of the internetwork coupling matrix. We seek the optimal coupling by minimizing the Frobenius and L₁ norms of the internetwork coupling matrix for the prescribed linear stability of the synchronized state. Depending on the norm, either a dense or sparse internetwork coupling yielding efficient mutual synchronization of the networks is obtained. In particular, a sparse yet resilient internetwork coupling is obtained by L₁-norm optimization with additional constraints on the individual connection weights.

Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators

Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase-reduction theory recently developed for quantum nonlinear oscillators [Y. Kato, N. Yamamoto, and H. Nakao, Phys. Rev. Res. 1, 033012 (2019)10.1103/PhysRevResearch.1.033012], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically analyzed. It is shown that the optimized waveform for the entrainment stability yields faster entrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from the sinusoidal case. These results are explained from the properties of the phase sensitivity function.

Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators

Optimization of mutual synchronization between a pair of limit-cycle oscillators with weak symmetric coupling is considered in the framework of the phase-reduction theory. By generalizing our previous study [S. Shirasaka, N. Watanabe, Y. Kawamura, and H. Nakao, Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling, Phys. Rev. E 96, 012223 (2017)2470-004510.1103/PhysRevE.96.012223] on the optimization of cross-diffusion coupling matrices between the oscillators, we consider optimization of mutual coupling signals to maximize the linear stability of the synchronized state, which are functionals of the past time sequences of the oscillator states. For the case of linear coupling, optimization of the delay time and linear filtering of coupling signals are considered. For the case of nonlinear coupling, general drive-response coupling is considered and the optimal response and driving functions are derived. The theoretical results are illustrated by numerical simulations.

Emergence of oscillatory coexistence with exponentially decayed waiting times in a coupled cyclic competition system

Interpatch migration between two environments is generally considered as a spatial concept and can affect species biodiversity in each patch by inducing flux of population such as inflow and outflow quantities of species. In this paper, we explore the effect of interpatch migration, which can be generally considered as a spatial concept and may affect species biodiversity between two different patches in the perspective of the macroscopic level by exploiting the coupling of two systems, where each patch is occupied by cyclically competing three species who can stably coexist by exhibiting periodic orbits. For two simple scenarios of interpatch migration either single or all species migration, we found that two systems with independently stable coexisting species in each patch are eventually synchronized, and oscillatory behaviors of species densities in two patches become identical, i.e., the synchronized coexistence emerges. In addition, we find that, whether single or all species interpatch migration occurs, the waiting time for the synchronization is exponentially decreasing as the coupling strength is intensified. Our findings suggest that the synchronized behavior of species as a result of migration between different patches can be easily predicted by the coupling of systems and additional information such as waiting times and sensitivity of initial densities.

Control of neural systems at multiple scales using model-free, deep reinforcement learning

Recent improvements in hardware and data collection have lowered the barrier to practical neural control. Most of the current contributions to the field have focus on model-based control, however, models of neural systems are quite complex and difficult to design. To circumvent these issues, we adapt a model-free method from the reinforcement learning literature, Deep Deterministic Policy Gradients (DDPG). Model-free reinforcement learning presents an attractive framework because of the flexibility it offers, allowing the user to avoid modeling system dynamics. We make use of this feature by applying DDPG to models of low-level and high-level neural dynamics. We show that while model-free, DDPG is able to solve more difficult problems than can be solved by current methods. These problems include the induction of global synchrony by entrainment of weakly coupled oscillators and the control of trajectories through a latent phase space of an underactuated network of neurons. While this work has been performed on simulated systems, it suggests that advances in modern reinforcement learning may enable the solution of fundamental problems in neural control and movement towards more complex objectives in real systems.

Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize the phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual synchronization between a pair of collectively oscillating networks of excitable and oscillatory FitzHugh-Nagumo elements with random coupling is studied.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system

We study the synchronization behaviors of two indirectly coupled mechanical oscillators of different frequencies in a doublecavity optomechanical system. It is found that quantum synchronization is roughly vanishing though classical synchronization seems rather good when each cavity mode is driven by an external field in the absence of temporal modulations. By periodically modulating cavity detunings or driving amplitudes, however, it is possible to observe greatly enhanced quantum synchronization accompanied with nearly perfect classical synchronization. The level of quantum synchronization observed here is, in particular, much higher than that for two directly coupled mechanical oscillators. Note also that the modulation on cavity detunings is more appealing than that on driving amplitudes when the robustness of quantum synchronization is examined against the bath’s mean temperature or the oscillators’ frequency difference.