Mode-locking in systems of globally coupled phase oscillators

We investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode-locking in laser and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonic in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode-locking, transitions to phase turbulence and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type-II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence

Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units

We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems

Leap-frog patterns in systems of two coupled FitzHugh--Nagumo units

We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems

Dynamics of a stochastic excitable system with slowly adapting feedback

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker-Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting

Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources

We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the population and the resources can give rise to collective activity bursting in the population. To explain the mechanisms behind this emergent phenomenon, we combine the Ott-Antonsen reduction for the collective dynamics of the population and singular perturbation theory to obtain a reduced system describing the interaction between the population mean field and the resources.Peer Reviewe

The link between coherence echoes and mode locking

We investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally- coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self- organized periodic pulsations of the mean field, called mode locking, have been described re- cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu- ral frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as the result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimula- tion induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the mode-locking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The non-monotonous behavior of echo amplitudes can be explained as a result of the linear dispersion within the self-organized mode-locked frequency comb. Fi- nally we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus

Dynamics of a stochastic excitable system with slowly adapting feedback

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker-Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting

Mode locking in systems of globally coupled phase oscillators

In systems of globally coupled phase oscillators with sufficiently structured natural frequencies, a new type of collective behavior is discovered that exists below the synchronization threshold. The solution type is distinguished by the appearance of sharp pulses in the mean field amplitude which imply a temporary high coherence among the phases. This is similar to a process known in lasers called mode locking that refers to the formation of optical pulsed by the interaction through a nonlinear optical medium. General features of mode-locked solutions of coupled phase oscillators are identified and a classification of the different solution types is provided. The ability of phase oscillator systems to perform mode locking is investigated with respect to the interaction function, the system size and the realization of the natural frequencies. It was found that higher harmonics in the Fourier series of the interaction function play an influential role in the self-organization of mode-locked solutions. For the simple sinusoidal coupling of the Kuramoto model, self-organized mode locking could not be observed, though, mode-locked solutions are found that coexist with phase turbulence. The chaotic transients that precede mode locking are examined with respect to the interaction function and the system size revealing a supertransient behavior of type-II, i.e. average transient length grows exponentially with the system size. The stability and bifurcation scenarios of mode-locked solutions are studied, displaying an involved picture of the local stability and revealing intermittency as the typical route from mode locking to phase turbulence. Close to the stability boundaries of mode-locked solutions, low-dimensional chaotic attractors can be found that maintain the pulsed behavior with a jittering of the inter-pulse intervals and pulse heights. In large oscillator ensembles with a modal structure in the natural frequencies, mode-locked solutions generally arise in a two-stage process of inner-modal synchronization and inter-modal locking. Aside from the modal dynamics, which is covered by the introduced modal order parameters, the mode-locked solutions in large ensembles are found to share the characteristics regarding transient behavior and mean-field dynamics. The notion of mode locking is applied to intuitively explain the occurrence of coherence echoes that stem from the application of two consecutive stimuli to a population of oscillators. It is shown that with repetitive periodic stimulation, fully mode-locked states can be established that depend substantially on the interaction function. The non-monotonic behavior of the magnitude of the echoes is revealed and explained by the evolution of the modal order parameters for a synthetic, fully mode-locked initial state

Mode Locking in Systems of Globally-Coupled Phase Oscillators

In systems of globally coupled phase oscillators with sufficiently structured natural frequencies, a new type of collective behavior is discovered that exists below the synchronization threshold. The solution type is distinguished by the appearance of sharp pulses in the mean field amplitude which imply a temporary high coherence among the phases. This is similar to a process known in lasers called mode locking that refers to the formation of optical pulsed by the interaction through a nonlinear optical medium. General features of mode-locked solutions of coupled phase oscillators are identified and a classification of the different solution types is provided. The ability of phase oscillator systems to perform mode locking is investigated with respect to the interaction function, the system size and the realization of the natural frequencies. It was found that higher harmonics in the Fourier series of the interaction function play an influential role in the self-organization of mode-locked solutions. For the simple sinusoidal coupling of the Kuramoto model, self-organized mode locking could not be observed, though, mode-locked solutions are found that coexist with phase turbulence. The chaotic transients that precede mode locking are examined with respect to the interaction function and the system size revealing a supertransient behavior of type-II, i.e. average transient length grows exponentially with the system size. The stability and bifurcation scenarios of mode-locked solutions are studied, displaying an involved picture of the local stability and revealing intermittency as the typical route from mode locking to phase turbulence. Close to the stability boundaries of mode-locked solutions, low-dimensional chaotic attractors can be found that maintain the pulsed behavior with a jittering of the inter-pulse intervals and pulse heights. In large oscillator ensembles with a modal structure in the natural frequencies, mode-locked solutions generally arise in a two-stage process of inner-modal synchronization and inter-modal locking. Aside from the modal dynamics, which is covered by the introduced modal order parameters, the mode-locked solutions in large ensembles are found to share the characteristics regarding transient behavior and mean-field dynamics. The notion of mode locking is applied to intuitively explain the occurrence of coherence echoes that stem from the application of two consecutive stimuli to a population of oscillators. It is shown that with repetitive periodic stimulation, fully mode-locked states can be established that depend substantially on the interaction function. The non-monotonic behavior of the magnitude of the echoes is revealed and explained by the evolution of the modal order parameters for a synthetic, fully mode-locked initial state