@article{VlasovPikovskij2013, author = {Vlasov, Vladimir and Pikovskij, Arkadij}, title = {Synchronization of a Josephson junction array in terms of global variables}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.022908}, pages = {5}, year = {2013}, abstract = {We consider an array of Josephson junctions with a common LCR load. Application of the Watanabe-Strogatz approach [Physica D 74, 197 (1994)] allows us to formulate the dynamics of the array via the global variables only. For identical junctions this is a finite set of equations, analysis of which reveals the regions of bistability of the synchronous and asynchronous states. For disordered arrays with distributed parameters of the junctions, the problem is formulated as an integro-differential equation for the global variables; here stability of the asynchronous states and the properties of the transition synchrony-asynchrony are established numerically.}, language = {en} } @article{SchwabedalPikovskij2013, author = {Schwabedal, Justus T. C. and Pikovskij, Arkadij}, title = {Phase description of stochastic oscillations}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {20}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.204102}, pages = {5}, year = {2013}, abstract = {We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows us to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine isophases of observed irregular oscillations, providing a basis for a refined phase description in data analysis.}, language = {en} } @article{Pikovskij2013, author = {Pikovskij, Arkadij}, title = {Robust synchronization of spin-torque oscillators with an LC R load}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.032812}, pages = {8}, year = {2013}, abstract = {We study dynamics of a serial array of spin-torque oscillators with a parallel inductor-capacitor-resistor (LC R) load. In a large range of parameters the fully synchronous regime, where all the oscillators have the same state and the output field is maximal, is shown to be stable. However, not always such a robust complete synchronization develops from a random initial state; in many cases nontrivial clustering is observed, with a partial synchronization resulting in a quasiperiodic or chaotic mean-field dynamics.}, language = {en} } @article{MulanskyPikovskij2013, author = {Mulansky, Mario and Pikovskij, Arkadij}, title = {Energy spreading in strongly nonlinear disordered lattices}, series = {New journal of physics : the open-access journal for physics}, volume = {15}, journal = {New journal of physics : the open-access journal for physics}, number = {5}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/15/5/053015}, pages = {23}, year = {2013}, abstract = {We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters.}, language = {en} } @article{MontaseriYazdanpanahPikovskijetal.2013, author = {Montaseri, Ghazal and Yazdanpanah, Mohammad Javad and Pikovskij, Arkadij and Rosenblum, Michael}, title = {Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {23}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {3}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.4817393}, pages = {12}, year = {2013}, abstract = {Synchronization and emergence of a collective mode is a general phenomenon, frequently observed in ensembles of coupled self-sustained oscillators of various natures. In several circumstances, in particular in cases of neurological pathologies, this state of the active medium is undesirable. Destruction of this state by a specially designed stimulation is a challenge of high clinical relevance. Typically, the precise effect of an external action on the ensemble is unknown, since the microscopic description of the oscillators and their interactions are not available. We show that, desynchronization in case of a large degree of uncertainty about important features of the system is nevertheless possible; it can be achieved by virtue of a feedback loop with an additional adaptation of parameters. The adaptation also ensures desynchronization of ensembles with non-stationary, time-varying parameters. We perform the stability analysis of the feedback-controlled system and demonstrate efficient destruction of synchrony for several models, including those of spiking and bursting neurons.}, language = {en} } @article{KuptsovKuznetsovPikovskij2013, author = {Kuptsov, Pavel V. and Kuznetsov, Sergey P. and Pikovskij, Arkadij}, title = {Hyperbolic chaos at blinking coupling of noisy oscillators}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.032912}, pages = {7}, year = {2013}, abstract = {We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where onecluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity manifests itself in phases of the complex mean field, which obey a strongly chaotic Bernoulli map. Hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant Lyapunov vectors and Lyapunov exponents. We show how the chaotic dynamics of the phases is slightly smeared by finite-size fluctuations. DOI: 10.1103/PhysRevE.87.032912}, language = {en} } @article{KrishnanBazhenovPikovskij2013, author = {Krishnan, Giri Panamoottil and Bazhenov, Maxim and Pikovskij, Arkadij}, title = {Multipulse phase resetting curves}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.042902}, pages = {9}, year = {2013}, abstract = {In this paper, we introduce and study systematically, in terms of phase response curves, the effect of dual-pulse excitation on the dynamics of an autonomous oscillator. Specifically, we test the deviations from linear summation of phase advances resulting from two small perturbations. We analytically derive a correction term, which generally appears for oscillators whose intrinsic dimensionality is >1. The nonlinear correction term is found to be proportional to the square of the perturbation. We demonstrate this effect in the Stuart-Landau model and in various higher dimensional neuronal models. This deviation from the superposition principle needs to be taken into account in studies of networks of pulse-coupled oscillators. Further, this deviation could be used in the verification of oscillator models via a dual-pulse excitation.}, language = {en} } @article{KralemannPikovskijRosenblum2013, author = {Kralemann, Bj{\"o}rn and Pikovskij, Arkadij and Rosenblum, Michael}, title = {Detecting triplet locking by triplet synchronization indices}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.052904}, pages = {6}, year = {2013}, abstract = {We discuss the effect of triplet synchrony in oscillatory networks. In this state the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. We suggest an easy to compute measure, a triplet synchronization index, which can be used to detect such states from experimental data.}, language = {en} } @article{KralemannFruehwirthPikovskijetal.2013, author = {Kralemann, Bjoern and Fruehwirth, Matthias and Pikovskij, Arkadij and Rosenblum, Michael and Kenner, Thomas and Schaefer, Jochen and Moser, Maximilian}, title = {In vivo cardiac phase response curve elucidates human respiratory heart rate variability}, series = {Nature Communications}, volume = {4}, journal = {Nature Communications}, publisher = {Nature Publ. Group}, address = {London}, issn = {2041-1723}, doi = {10.1038/ncomms3418}, pages = {9}, year = {2013}, abstract = {Recovering interaction of endogenous rhythms from observations is challenging, especially if a mathematical model explaining the behaviour of the system is unknown. The decisive information for successful reconstruction of the dynamics is the sensitivity of an oscillator to external influences, which is quantified by its phase response curve. Here we present a technique that allows the extraction of the phase response curve from a non-invasive observation of a system consisting of two interacting oscillators-in this case heartbeat and respiration-in its natural environment and under free-running conditions. We use this method to obtain the phase-coupling functions describing cardiorespiratory interactions and the phase response curve of 17 healthy humans. We show for the first time the phase at which the cardiac beat is susceptible to respiratory drive and extract the respiratory-related component of heart rate variability. This non-invasive method for the determination of phase response curves of coupled oscillators may find application in many scientific disciplines.}, language = {en} } @article{KomarovPikovskij2013, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Multiplicity of singular synchronous States in the kuramoto model of coupled oscillators}, series = {Physical review letters}, volume = {111}, journal = {Physical review letters}, number = {20}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.111.204101}, pages = {5}, year = {2013}, abstract = {We study the Kuramoto model of globally coupled oscillators with a biharmonic coupling function. We develop an analytic self-consistency approach to find stationary synchronous states in the thermodynamic limit and demonstrate that there is a huge multiplicity of such states, which differ microscopically in the distributions of locked phases. These synchronous regimes already exist prior to the linear instability transition of the fully asynchronous state. In the presence of white Gaussian noise, the multiplicity is lifted, but the dependence of the order parameters on coupling constants remains nontrivial.}, language = {en} } @article{KomarovPikovskij2013, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Dynamics of multifrequency oscillator communities}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {13}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.134101}, pages = {5}, year = {2013}, abstract = {We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The simplest situation of three resonantly interacting groups is analyzed in detail. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations and present examples where the interaction between communities leads to their synchrony or to a partially asynchronous state or to a chaotic dynamics of order parameters. DOI: 10.1103/PhysRevLett.110.134101}, language = {en} } @article{EhrichPikovskijRosenblum2013, author = {Ehrich, Sebastian and Pikovskij, Arkadij and Rosenblum, Michael}, title = {From complete to modulated synchrony in networks of identical Hindmarsh-Rose neurons}, series = {European physical journal special topics}, volume = {222}, journal = {European physical journal special topics}, number = {10}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2013-02025-8}, pages = {2407 -- 2416}, year = {2013}, abstract = {In most cases tendency to synchrony in networks of oscillatory units increases with the coupling strength. Using the popular Hindmarsh-Rose neuronal model, we demonstrate that even for identical neurons and simple coupling the dynamics can be more complicated. Our numerical analysis for globally coupled systems and oscillator lattices reveals a new scenario of synchrony breaking with the increase of coupling, resulting in a quasiperiodic, modulated synchronous state.}, language = {en} }