Method for determining a coupling function in coupled oscillators with application to Belousov-Zhabotinsky oscillators

Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators

The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for anti-phase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Dynamics of Reaction-Diffusion Oscillators in Star and other Networks with Cyclic Symmetries Exhibiting Multiple Clusters

We experimentally and theoretically investigate the dynamics of inhibitory coupled self-driven oscillators on a star network in which a single central hub node is connected to k peripheral arm nodes. The system consists of water-in-oil Belousov-Zhabotinsky ∼100  μm emulsion drops contained in storage wells etched in silicon wafers. We observed three dynamical attractors by varying the number of arms in the star graph and the coupling strength: (i) unlocked, uncorrelated phase shifts between all oscillators; (ii) locked, arm hubs synchronized in phase with a k-dependent phase shift between the arm and central hub; and (iii) center silent, a central hub stopped oscillating and the arm hubs oscillated without synchrony. We compare experiment to theory. For case (ii), we identified a logarithmic dependence of the phase shift on star degree, and were able to discriminate between contributions to the phase shift arising from star topology and oscillator chemistry.

Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

The Oregonator demonstrates that quorum sensing in populations of Belousov–Zhabotinsky oscillators arises from modification of the stoichiometry by catalyst confinement.

Synchronization of electrochemical oscillators with differential coupling

Experiments are presented to describe the effect of capacitive coupling of two electrochemical oscillators during Ni dissolution in sulfuric acid solution. Equivalent circuit analysis shows that the coupling between the oscillators occurs through the difference between the differentials of the electrode potentials. The differential nature of the coupling introduces strong negative nonisochronicity (i.e., phase shear, strong dependence of the period on the amplitude) in the coupling mechanism with smooth oscillators (under conditions just above a Hopf bifurcation point). Because of the negative nonisochronicity, asymmetrically coupled oscillators exhibit anomalous phase synchronization in the form of frequency difference enhancement. At strong coupling bistability is observed between in-phase and antiphase synchronized states. With relaxation oscillators, in contrast to the resistive coupling where antiphase synchronization can occur, the typical system response with weak coupling is out-of-phase synchronization. When the capacitance is applied on the individual resistors attached to the electrodes the oscillators exhibit weak positive nonisochronicity; this is in contrast with the strong negative nonisochronicity obtained with cross coupling. The proposed coupling configurations reveal the importance of the nonisochronicity level of oscillations for the experimentally observed synchronization patterns and also provide efficient ways of tuning the nonisochronicity level of the oscillations. This latter feature can be exploited to design synchronization features with a combination of resistive (difference) and capacitive (differential) coupling.

Clustering in delay-coupled smooth and relaxational chemical oscillators

We investigate cluster synchronization in networks of nonlinear systems with time-delayed coupling. Using a generic model for a system close to the Hopf bifurcation, we predict the order of appearance of different cluster states and their corresponding common frequencies depending upon coupling delay. We may tune the delay time in order to ensure the existence and stability of a specific cluster state. We qualitatively and quantitatively confirm these results in experiments with chemical oscillators. The experiments also exhibit strongly nonlinear relaxation oscillations as we increase the voltage, i.e., go further away from the Hopf bifurcation. In this regime, we find secondary cluster states with delay-dependent phase lags. These cluster states appear in addition to primary states with delay-independent phase lags observed near the Hopf bifurcation. Extending the theory on Hopf normal-form oscillators, we are able to account for realistic interaction functions, yielding good agreement with experimental findings.

Engineering the synchronization of neuron action potentials using global time-delayed feedback stimulation

We experimentally demonstrate the use of continuous, time-delayed, feedback stimulation for controlling the synchronization of neuron action potentials. Phase-based models were experimentally constructed from a single synaptically isolated cultured hippocampal neuron. These models were used to determine the stimulation parameters necessary to produce the desired synchronization behavior in the action potentials of a pair of neurons coupled through a global time-delayed interaction. Measurements made using a dynamic clamp system confirm the generation of the synchronized states predicted by the experimentally constructed phase model. This model was then utilized to extrapolate the feedback stimulation parameters necessary to disrupt the action potential synchronization of a large population of globally interacting neurons.

Synchronization engineering: tuning the phase relationship between dissimilar oscillators using nonlinear feedback

A mild, nonlinear, time-delayed feedback signal was applied to two heterogeneous oscillators in order to synchronize their frequencies with an arbitrary and controllable phase difference. The feedback was designed using phase models constructed from experimental measurements of the intrinsic dynamical properties of the oscillators. The feedback signal produced an interaction function that corresponds to the desired collective behaviour. The synchronized phase difference between the elements can be tuned to any value on the interval 0 and 2pi by shifting the phase of the interaction function using the feedback delay. Numerical simulations were conducted and experiments carried out with electrochemical oscillators.

Control of individual phase relationship between coupled oscillators using multilinear feedback

Due to various technological and medical demands, several methods for controlling the dynamical behavior of coupled oscillators have been developed. In the present study, we develop a method to control the individual phase relationship between coupled oscillators, in which multilinear feedback is used to modify the interaction between the oscillators. By carrying out a simulation, we show that the phase relationship can be well controlled by using the proposed method and the control is particularly robust when the target coupling function is selected properly.

Modeling synchronized calling behavior of Japanese tree frogs

We experimentally observed synchronized calling behavior of male Japanese tree frogs Hyla japonica; namely, while isolated single frogs called nearly periodically, a pair of interacting frogs called synchronously almost in antiphase or inphase. In this study, we propose two types of phase-oscillator models on different degrees of approximations, which can quantitatively explain the phase and frequency properties in the experiment. Moreover, it should be noted that, although the second model is obtained by fitting to the experimental data of the two synchronized states, the model can also explain the transitory dynamics in the interactive calling behavior, namely, the shift from a transient inphase state to a stable antiphase state. We also discuss the biological relevance of the estimated parameter values to calling behavior of Japanese tree frogs and the possible biological meanings of the synchronized calling behavior.

Stationary pattern formation in a discrete excitable system with strong inhibitory coupling

We study a discrete model described by coupled excitable elements following the monostable FitzHugh-Nagumo equations. Our model has a weakly coupled activator and a strongly coupled inhibitor. For two-coupled excitable elements, we show that the trivial state always exists stably, while nontrivial stable states appear depending on the coupling strengths. In a one-dimensional array, only the elements near the initial condition step remain at nontrivial states. We discuss stationary pattern formation in a one-dimensional array and a two-dimensional lattice using the analytical results of a two-coupled system.

Method to control the coupling function using multilinear feedback

Methods to control the dynamics of coupled oscillators have been developed owing to various medical and technological demands. In this study, we develop a method to control coupled oscillators in which the coupling function expressed in a phase model is regulated by the multilinear feedback. The present method has wide applicability because we do not need to measure an individual output from each oscillator, but only measure the sum of the outputs from all the oscillators. Moreover, it allows us to easily control the coupling function up to higher harmonics. The validity of the present method is confirmed through a simulation.

Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity

The dynamics of a chemical pulse at the interface of excitable media having different diffusion properties is presented experimentally using the Belousov-Zhabotinsky reaction system. When a chemical pulse propagates from a larger diffusion region to a smaller diffusion region, it stops and forms a steady excitation band at the interface. Furthermore, a pulse train stimulated by this band appears subsequently. A simple one-dimensional model with discontinuous diffusion is proposed, and a numerical simulation is performed that shows good agreement with the experiment. Unidirectional pulse propagation across the interface is strongly suggested on the basis of the proposed model.