@article{RosenblumFruehwirthMoseretal.2019,
  author    = {Rosenblum, Michael and Fr{\"u}hwirth, Martha and Moser, Maximilian and Pikovskij, Arkadij},
  title     = {Dynamical disentanglement in an analysis of oscillatory systems: an application to respiratory sinus arrhythmia},
  series = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  volume    = {377},
  journal   = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  number    = {2160},
  publisher = {Royal Society},
  address   = {London},
  issn      = {1364-503X},
  doi       = {10.1098/rsta.2019.0045},
  pages     = {14},
  year      = {2019},
  abstract  = {We develop a technique for the multivariate data analysis of perturbed self-sustained oscillators. The approach is based on the reconstruction of the phase dynamics model from observations and on a subsequent exploration of this model. For the system, driven by several inputs, we suggest a dynamical disentanglement procedure, allowing us to reconstruct the variability of the system's output that is due to a particular observed input, or, alternatively, to reconstruct the variability which is caused by all the inputs except for the observed one. We focus on the application of the method to the vagal component of the heart rate variability caused by a respiratory influence. We develop an algorithm that extracts purely respiratory-related variability, using a respiratory trace and times of R-peaks in the electrocardiogram. The algorithm can be applied to other systems where the observed bivariate data can be represented as a point process and a slow continuous signal, e.g. for the analysis of neuronal spiking. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.},
  language  = {en}
}
@article{RosenblumKurthsPikovskij2001,
  author    = {Rosenblum, Michael and Kurths, J{\"u}rgen and Pikovskij, Arkadij},
  title     = {Comment on "Phase synchronization in discrete chaotic systems"},
  year      = {2001},
  abstract  = {Chen et al. [Phys. Rev. E 61, 2559 (2000)] recently proposed an extension of the concept of phase for discrete chaotic systems. Using the newly introduced definition of phase they studied the dynamics of coupled map lattices and compared these dynamics with phase synchronization of coupled continuous-time chaotic systems. In this paper we illustrate by two simple counterexamples that the angle variable introduced by Chen et al. fails to satisfy the basic requirements to the proper phase. Furthermore, we argue that an extension of the notion of phase synchronization to generic discrete maps is doubtful.},
  language  = {en}
}
@article{RosenblumKurthsPikovskijetal.1998,
  author    = {Rosenblum, Michael and Kurths, J{\"u}rgen and Pikovskij, Arkadij and Schafer, C. and Tass, Peter and Abel, Hans-Henning},
  title     = {Synchronization in Noisy Systems and Cardiorespiratory Interaction},
  year      = {1998},
  language  = {en}
}
@article{RosenblumOsipovPikovskijetal.1997,
  author    = {Rosenblum, Michael and Osipov, Grigory V. and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Phase synchronization of chaotic oscillators by external driving},
  year      = {1997},
  language  = {en}
}
@article{RosenblumPikovskij2019,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Nonlinear phase coupling functions: a numerical study},
  series = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  volume    = {377},
  journal   = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  number    = {2160},
  publisher = {Royal Society},
  address   = {London},
  issn      = {1364-503X},
  doi       = {10.1098/rsta.2019.0093},
  pages     = {12},
  year      = {2019},
  abstract  = {Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator.},
  language  = {en}
}
@article{RosenblumPikovskij2019,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Numerical phase reduction beyond the first order approximation},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {29},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {1},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5079617},
  pages     = {6},
  year      = {2019},
  abstract  = {We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach. Published under license by AIP Publishing.},
  language  = {en}
}
@article{RosenblumPikovskij2018,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Efficient determination of synchronization domains from observations of asynchronous dynamics},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {28},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {10},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5037012},
  pages     = {8},
  year      = {2018},
  abstract  = {We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized. Published by AIP Publishing.},
  language  = {en}
}
@article{RosenblumPikovskij2004,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Delayed feedback control of collective synchrony : an approach to suppression of pathological brain rhythms},
  issn      = {1063-651X},
  year      = {2004},
  abstract  = {We suggest a method for suppression of synchrony in a globally coupled oscillator network, based on the time- delayed feedback via the mean field. Having in mind possible applications for suppression of pathological rhythms in neural ensembles, we present numerical results for different models of coupled bursting neurons. A theory is developed based on the consideration of the synchronization transition as a Hopf bifurcation},
  language  = {en}
}
@article{RosenblumPikovskij2004,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Controlling synchronization in an ensemble of globally coupled oscillators},
  issn      = {0031-9007},
  year      = {2004},
  abstract  = {We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self- sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique},
  language  = {en}
}
@article{RosenblumPikovskij2015,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Two types of quasiperiodic partial synchrony in oscillator ensembles},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {92},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {1},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.92.012919},
  pages     = {8},
  year      = {2015},
  abstract  = {We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the first case the time-averaged frequencies of oscillators and of the mean field differ, while in the second case they are equal, but the motion of oscillators is additionally modulated. We describe transitions from the synchronous state to both types of quasiperiodic dynamics, and a transition between two different quasiperiodic states. We present an example of a bifurcation diagram, where we show the borderlines for all these transitions, as well as domain of bistability.},
  language  = {en}
}
@article{RosenblumPikovskijKurths2004,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Synchronization approach to analysis of biological systems},
  issn      = {0219-4775},
  year      = {2004},
  abstract  = {In this article we review the application of the synchronization theory to the analysis of multivariate biological signals. We address the problem of phase estimation from data and detection and quantification of weak interaction, as well as quantification of the direction of coupling. We discuss the potentials as well as limitations and misinterpretations of the approach},
  language  = {en}
}
@article{RosenblumPikovskijKurths1997,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {From Phase to Lag Synchronization in Coupled Chaotic Oscillators},
  year      = {1997},
  abstract  = {We study synchronization transitions in a system of two coupled self-sustained chaotic oscillators. We demonstrate that with the increase of coupling strength the system first undergoes the transition to phase synchronization. With a further increase of coupling, a new synchronous regime is observed, where the states of two oscillators are nearly identical, but one system lags in time to the other. We describe thisregime as a state with correlated amplitudes and a constant phase shift. These transitions are traced in the Lyapunov spectrum.},
  language  = {en}
}
@article{RosenblumPikovskijKurths1997,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Phase synchronization in noisy and chaotic oscillators},
  year      = {1997},
  language  = {en}
}
@article{RosenblumPikovskijKurths1997,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Phase synchronization in driven and coupled chaotic oscillators},
  year      = {1997},
  language  = {en}
}
@article{RosenblumPikovskijKurths1997,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Effect of phase synchronization in driven chaotic oscillators},
  year      = {1997},
  language  = {en}
}
@article{RosenblumPikovskijKurthsetal.2002,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen and Osipov, Grigory V. and Kiss, Istvan Z. and Hudson, J. L.},
  title     = {Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics},
  year      = {2002},
  language  = {en}
}
@article{RosenblumPikovskijKuehnetal.2021,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij and K{\"u}hn, Andrea A. and Busch, Johannes Leon},
  title     = {Real-time estimation of phase and amplitude with application to neural data},
  series = {Scientific reports},
  volume    = {11},
  journal   = {Scientific reports},
  publisher = {Springer Nature},
  address   = {London},
  issn      = {2045-2322},
  doi       = {10.1038/s41598-021-97560-5},
  pages     = {11},
  year      = {2021},
  abstract  = {Computation of the instantaneous phase and amplitude via the Hilbert Transform is a powerful tool of data analysis. This approach finds many applications in various science and engineering branches but is not proper for causal estimation because it requires knowledge of the signal's past and future. However, several problems require real-time estimation of phase and amplitude; an illustrative example is phase-locked or amplitude-dependent stimulation in neuroscience. In this paper, we discuss and compare three causal algorithms that do not rely on the Hilbert Transform but exploit well-known physical phenomena, the synchronization and the resonance. After testing the algorithms on a synthetic data set, we illustrate their performance computing phase and amplitude for the accelerometer tremor measurements and a Parkinsonian patient's beta-band brain activity.},
  language  = {en}
}
@article{RoyPikovskij2012,
  author    = {Roy, S. and Pikovskij, Arkadij},
  title     = {Spreading of energy in the Ding-Dong model},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {22},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {2},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.3695369},
  pages     = {7},
  year      = {2012},
  abstract  = {We study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.},
  language  = {en}
}
@article{RuffoPikovskij1999,
  author    = {Ruffo, Stefano and Pikovskij, Arkadij},
  title     = {Finite-size effects in a population of interacting oscillators},
  year      = {1999},
  abstract  = {We consider a large population of globally coupled noisy phase oscillators. In the thermodynamic limit N this system exhibits a nonequilibrium phase transition, at which amacroscopic mean field appears. It is shown that for large but finite system size N the system can be described by the noisy Stuart-Landau equation, yielding scaling behavior of statistical characteristics of the macroscopic mean field with N. The predictions of the theory are checked numerically.},
  language  = {en}
}
@article{RuzickScheffczykPikovskijetal.1997,
  author    = {Ruzick, Oliver and Scheffczyk, Christian and Pikovskij, Arkadij and Kurths, J{\"u}rgen},
  title     = {Dynamics of chaos-order interface in coupled map lattices},
  year      = {1997},
  language  = {en}
}