TY - GEN A1 - Cestnik, Rok A1 - Abel, Markus T1 - Erratum: Inferring the dynamics of oscillatory systems using recurrent neural networks (Chaos : an interdisciplinary journal of nonlinear science. - 29 (2019) 063128) T2 - Chaos : an interdisciplinary journal of nonlinear science Y1 - 2019 U6 - https://doi.org/10.1063/1.5122803 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 8 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Cestnik, Rok A1 - Abel, Markus T1 - Inferring the dynamics of oscillatory systems using recurrent neural networks JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit. Published under license by AIP Publishing. Y1 - 2019 U6 - https://doi.org/10.1063/1.5096918 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 6 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Cestnik, Rok A1 - Pikovskij, Arkadij T1 - Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories JF - Chaos : an interdisciplinary journal of nonlinear science N2 - Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles. Published under an exclusive license by AIP Publishing. Y1 - 2022 U6 - https://doi.org/10.1063/5.0106171 SN - 1054-1500 SN - 1089-7682 VL - 32 IS - 11 PB - AIP CY - Melville ER - TY - JOUR A1 - Cestnik, Rok A1 - Pikovsky, Arkady T1 - Hierarchy of exact low-dimensional reductions for populations of coupled oscillators JF - Physical review letters N2 - We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an Ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the Ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)]. Dynamics on the extended manifolds facilitate higher-dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevLett.128.054101 SN - 0031-9007 SN - 1079-7114 VL - 128 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Cestnik, Rok A1 - Rosenblum, Michael T1 - Inferring the phase response curve from observation of a continuously perturbed oscillator JF - Scientific reports N2 - Phase response curves are important for analysis and modeling of oscillatory dynamics in various applications, particularly in neuroscience. Standard experimental technique for determining them requires isolation of the system and application of a specifically designed input. However, isolation is not always feasible and we are compelled to observe the system in its natural environment under free-running conditions. To that end we propose an approach relying only on passive observations of the system and its input. We illustrate it with simulation results of an oscillator driven by a stochastic force. Y1 - 2018 U6 - https://doi.org/10.1038/s41598-018-32069-y SN - 2045-2322 VL - 8 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Cestnik, Rok A1 - Rosenblum, Michael T1 - Inferring the phase response curve from observation of a continuously perturbed oscillator JF - Scientific Reports N2 - Phase response curves are important for analysis and modeling of oscillatory dynamics in various applications, particularly in neuroscience. Standard experimental technique for determining them requires isolation of the system and application of a specifically designed input. However, isolation is not always feasible and we are compelled to observe the system in its natural environment under free-running conditions. To that end we propose an approach relying only on passive observations of the system and its input. We illustrate it with simulation results of an oscillator driven by a stochastic force. Y1 - 2018 U6 - https://doi.org/10.1038/s41598-018-32069-y SN - 2045-2322 VL - 8 SP - 1 EP - 10 PB - Nature Publishing Group CY - London ER - TY - JOUR A1 - Cestnik, Rok A1 - Rosenblum, Michael T1 - Reconstructing networks of pulse-coupled oscillators from spike trains JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We present an approach for reconstructing networks of pulse-coupled neuronlike oscillators from passive observation of pulse trains of all nodes. It is assumed that units are described by their phase response curves and that their phases are instantaneously reset by incoming pulses. Using an iterative procedure, we recover the properties of all nodes, namely their phase response curves and natural frequencies, as well as strengths of all directed connections. Y1 - 2017 U6 - https://doi.org/10.1103/PhysRevE.96.012209 SN - 2470-0045 SN - 2470-0053 VL - 96 SP - 3455 EP - 3461 PB - American Physical Society CY - College Park ER -