@article{Pikovskij2021, author = {Pikovskij, Arkadij}, title = {Chimeras on a social-type network}, series = {Mathematical modelling of natural phenomena : MMNP}, volume = {16}, journal = {Mathematical modelling of natural phenomena : MMNP}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/2021012}, pages = {9}, year = {2021}, abstract = {We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.}, language = {en} } @article{ZhengToenjesPikovskij2021, author = {Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij}, title = {Transition to synchrony in a three-dimensional swarming model with helical trajectories}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {104}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.104.014216}, pages = {7}, year = {2021}, abstract = {We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found that alignment of the velocities occurs discontinuously for arbitrarily small attractive coupling. Adding noise to the system resolves this singular limit and leads to a continuous transition, either to a directed collective motion or to center-of-mass rotations.}, language = {en} } @article{SmirnovOsipovPikovskij2017, author = {Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Chimera patterns in the Kuramoto-Battogtokh model}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa55f1}, pages = {10}, year = {2017}, abstract = {Kuramoto and Battogtokh (2002 Nonlinear Phenom. Complex Syst. 5 380) discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After a reformulation in terms of a local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating, spatially periodic chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in the form of one-and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only the simplest chimeras with one synchronous region are stable.}, language = {en} } @article{PetereitPikovskij2017, author = {Petereit, Johannes and Pikovskij, Arkadij}, title = {Chaos synchronization by nonlinear coupling}, series = {Communications in nonlinear science \& numerical simulation}, volume = {44}, journal = {Communications in nonlinear science \& numerical simulation}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1007-5704}, doi = {10.1016/j.cnsns.2016.09.002}, pages = {344 -- 351}, year = {2017}, abstract = {We study synchronization properties of three nonlinearly coupled chaotic maps. Coupling is introduced in such a way, that it cannot be reduced to pairwise terms, but includes combined action of all interacting units. For two models of nonlinear coupling we characterize the transition to complete synchrony, as well as partially synchronized states. Relation to hypernetworks of chaotic units is also discussed.}, language = {en} } @article{PopovychLysyanskyRosenblumetal.2017, author = {Popovych, Oleksandr V. and Lysyansky, Borys and Rosenblum, Michael and Pikovskij, Arkadij and Tass, Peter A.}, title = {Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation}, series = {PLoS one}, volume = {12}, journal = {PLoS one}, publisher = {PLoS}, address = {San Fransisco}, issn = {1932-6203}, doi = {10.1371/journal.pone.0173363}, pages = {29}, year = {2017}, abstract = {High-frequency (HF) deep brain stimulation (DBS) is the gold standard for the treatment of medically refractory movement disorders like Parkinson's disease, essential tremor, and dystonia, with a significant potential for application to other neurological diseases. The standard setup of HF DBS utilizes an open-loop stimulation protocol, where a permanent HF electrical pulse train is administered to the brain target areas irrespectively of the ongoing neuronal dynamics. Recent experimental and clinical studies demonstrate that a closed-loop, adaptive DBS might be superior to the open-loop setup. We here combine the notion of the adaptive high-frequency stimulation approach, that aims at delivering stimulation adapted to the extent of appropriately detected biomarkers, with specifically desynchronizing stimulation protocols. To this end, we extend the delayed feedback stimulation methods, which are intrinsically closed-loop techniques and specifically designed to desynchronize abnormal neuronal synchronization, to pulsatile electrical brain stimulation. We show that permanent pulsatile high-frequency stimulation subjected to an amplitude modulation by linear or nonlinear delayed feedback methods can effectively and robustly desynchronize a STN-GPe network of model neurons and suggest this approach for desynchronizing closed-loop DBS.}, 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{Pikovskij2018, author = {Pikovskij, Arkadij}, title = {Reconstruction of a random phase dynamics network from observations}, series = {Physics letters : A}, volume = {382}, journal = {Physics letters : A}, number = {4}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2017.11.012}, pages = {147 -- 152}, year = {2018}, abstract = {We consider networks of coupled phase oscillators of different complexity: Kuramoto-Daido-type networks, generalized Winfree networks, and hypernetworks with triple interactions. For these setups an inverse problem of reconstruction of the network connections and of the coupling function from the observations of the phase dynamics is addressed. We show how a reconstruction based on the minimization of the squared error can be implemented in all these cases. Examples include random networks with full disorder both in the connections and in the coupling functions, as well as networks where the coupling functions are taken from experimental data of electrochemical oscillators. The method can be directly applied to asynchronous dynamics of units, while in the case of synchrony, additional phase resettings are necessary for reconstruction.}, language = {en} } @article{BolotovSmirnovOsipovetal.2017, author = {Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Breathing chimera in a system of phase oscillators}, series = {JETP Letters}, volume = {106}, journal = {JETP Letters}, publisher = {Pleiades Publ.}, address = {New York}, issn = {0021-3640}, doi = {10.1134/S0021364017180059}, pages = {393 -- 399}, year = {2017}, abstract = {Chimera states consisting of synchronous and asynchronous domains in a medium of nonlinearly coupled phase oscillators have been considered. Stationary inhomogeneous solutions of the Ott-Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. The direct numerical simulation has shown that these structures under certain conditions are transformed to oscillatory (breathing) chimera regimes because of the development of instability.}, language = {en} } @article{PeterPikovskij2018, author = {Peter, Franziska and Pikovskij, Arkadij}, title = {Transition to collective oscillations in finite Kuramoto ensembles}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {97}, 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.97.032310}, pages = {10}, year = {2018}, abstract = {We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.}, language = {en} } @article{SysoevPonomarenkoPikovskij2017, author = {Sysoev, Ilya V. and Ponomarenko, Vladimir I. and Pikovskij, Arkadij}, title = {Reconstruction of coupling architecture of neural field networks from vector time series}, series = {Communications in nonlinear science \& numerical simulation}, volume = {57}, journal = {Communications in nonlinear science \& numerical simulation}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1007-5704}, doi = {10.1016/j.cnsns.2017.10.006}, pages = {342 -- 351}, year = {2017}, abstract = {We propose a method of reconstruction of the network coupling matrix for a basic voltage-model of the neural field dynamics. Assuming that the multivariate time series of observations from all nodes are available, we describe a technique to find coupling constants which is unbiased in the limit of long observations. Furthermore, the method is generalized for reconstruction of networks with time-delayed coupling, including the reconstruction of unknown time delays. The approach is compared with other recently proposed techniques.}, language = {en} }