@article{NagornovOsipoyKomarovetal.2016, author = {Nagornov, Roman and Osipoy, Grigory and Komarov, Maxim and Pikovskij, Arkadij and Shilnikov, Andrey}, title = {Mixed-mode synchronization between two inhibitory neurons with post-inhibitory rebound}, series = {Communications in nonlinear science \& numerical simulation}, volume = {36}, journal = {Communications in nonlinear science \& numerical simulation}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1007-5704}, doi = {10.1016/j.cnsns.2015.11.024}, pages = {175 -- 191}, year = {2016}, abstract = {We study an array of activity rhythms generated by a half-center oscillator (HCO), represented by a pair of reciprocally coupled neurons with post-inhibitory rebounds (PIR). Such coupling induced bursting possesses two time scales, one for fast spiking and another for slow quiescent periods, is shown to exhibit an array of synchronization properties. We discuss several HCO configurations constituted by two endogenous bursters, by tonic-spiking and quiescent neurons, as well as mixed-mode configurations composed of neurons of different type. We demonstrate that burst synchronization can be accompanied by complex, often chaotic, interactions of fast spikes within synchronized bursts. (C) 2015 Elsevier B.V. All rights reserved.}, language = {en} } @article{ZhangPikovskijLiu2017, author = {Zhang, Xiyun and Pikovskij, Arkadij and Liu, Zonghua}, title = {Dynamics of oscillators globally coupled via two mean fields}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-02283-1}, pages = {16}, year = {2017}, abstract = {Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branches which bifurcate from the incoherent state in a usual way, and anomalous branches, appearance of which cannot be described as a bifurcation. Furthermore, hybrid branches combining properties of both are described. In the situations where no stable traveling wave exists, modulated quasiperiodic in time dynamics is observed. Our results indicate that a competition between two coupling channels can lead to a complex system behavior, providing a potential generalized framework for understanding of complex phenomena in natural oscillatory systems.}, language = {en} } @article{ZaksPikovskij2017, author = {Zaks, Michael and Pikovskij, Arkadij}, title = {Chimeras and complex cluster states in arrays of spin-torque oscillators}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-04918-9}, pages = {10}, year = {2017}, abstract = {We consider synchronization properties of arrays of spin-torque nano-oscillators coupled via an RC load. We show that while the fully synchronized state of identical oscillators may be locally stable in some parameter range, this synchrony is not globally attracting. Instead, regimes of different levels of compositional complexity are observed. These include chimera states (a part of the array forms a cluster while other units are desynchronized), clustered chimeras (several clusters plus desynchronized oscillators), cluster state (all oscillators form several clusters), and partial synchronization (no clusters but a nonvanishing mean field). Dynamically, these states are also complex, demonstrating irregular and close to quasiperiodic modulation. Remarkably, when heterogeneity of spin-torque oscillators is taken into account, dynamical complexity even increases: close to the onset of a macroscopic mean field, the dynamics of this field is rather irregular.}, language = {en} } @article{PolitiPikovskijUllner2017, author = {Politi, Antonio and Pikovskij, Arkadij and Ullner, Ekkehard}, title = {Chaotic macroscopic phases in one-dimensional oscillators}, series = {European physical journal special topics}, volume = {226}, journal = {European physical journal special topics}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2017-70056-4}, pages = {1791 -- 1810}, year = {2017}, abstract = {The connection between the macroscopic description of collective chaos and the underlying microscopic dynamics is thoroughly analysed in mean-field models of one-dimensional oscillators. We investigate to what extent infinitesimal perturbations of the microscopic configurations can provide information also on the stability of the corresponding macroscopic phase. In ensembles of identical one-dimensional dynamical units, it is possible to represent the microscopic configurations so as to make transparent their connection with the macroscopic world. As a result, we find evidence of an intermediate, mesoscopic, range of distances, over which the instability is neither controlled by the microscopic equations nor by the macroscopic ones. We examine a whole series of indicators, ranging from the usual microscopic Lyapunov exponents, to the collective ones, including finite-amplitude exponents. A system of pulse-coupled oscillators is also briefly reviewed as an example of non-identical phase oscillators where collective chaos spontaneously emerges.}, language = {en} } @article{GoldobinPimenovaRosenblumetal.2017, author = {Goldobin, Denis S. and Pimenova, Anastasiya V. and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Competing influence of common noise and desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble}, series = {European physical journal special topics}, volume = {226}, journal = {European physical journal special topics}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2017-70039-y}, pages = {1921 -- 1937}, year = {2017}, abstract = {We describe analytically synchronization and desynchronization effects in an ensemble of phase oscillators driven by common noise and by global coupling. Adopting the Ott-Antonsen ansatz, we reduce the dynamics to closed stochastic equations for the order parameters, and study these equations for the cases of populations of identical and nonidentical oscillators. For nonidentical oscillators we demonstrate a counterintuitive effect of divergence of individual frequencies for moderate repulsive coupling, while the order parameter remains large.}, language = {en} } @article{Pikovskij2016, author = {Pikovskij, Arkadij}, title = {Reconstruction of a neural network from a time series of firing rates}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {93}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.93.062313}, pages = {4}, year = {2016}, abstract = {Randomly coupled neural fields demonstrate irregular variation of firing rates, if the coupling is strong enough, as has been shown by Sompolinsky et al. [Phys. Rev. Lett. 61, 259 (1988)]. We present a method for reconstruction of the coupling matrix from a time series of irregular firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set and a small measurement noise, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function.}, language = {en} } @article{Pikovskij2017, author = {Pikovskij, Arkadij}, title = {Reconstruction of a scalar voltage-based neural field network from observed time series}, series = {epl : a letters journal exploring the frontiers of physics}, volume = {119}, journal = {epl : a letters journal exploring the frontiers of physics}, publisher = {EDP Sciences}, address = {Mulhouse}, issn = {0295-5075}, doi = {10.1209/0295-5075/119/30004}, pages = {5}, year = {2017}, language = {en} } @article{GongZhengToenjesetal.2019, author = {Gong, Chen Chris and Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij}, title = {Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {3}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5084144}, pages = {11}, year = {2019}, abstract = {We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. Published under license by AIP Publishing.}, language = {en} } @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{PeterGongPikovskij2019, author = {Peter, Franziska and Gong, Chen Chris and Pikovskij, Arkadij}, title = {Microscopic correlations in the finite-size Kuramoto model of coupled oscillators}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.032210}, pages = {6}, year = {2019}, abstract = {Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators-at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. This publication demonstrates this effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean-field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and subcritical coupling.}, language = {en} }