Partially locked states in coupled oscillators due to inhomogeneous coupling

Synchronization in a system of Kuramoto oscillators with distributed Gaussian noise

We consider a system of globally coupled phase-only oscillators with distributed intrinsic frequencies and evolving in the presence of distributed Gaussian white noise, namely, a Gaussian white noise whose strength for every oscillator is a specified function of its intrinsic frequency. In the absence of noise, the model reduces to the celebrated Kuramoto model of spontaneous synchronization. For two specific forms of the mentioned functional dependence and for a symmetric and unimodal distribution of the intrinsic frequencies, we unveil the rich long-time behavior that the system exhibits, which stands in stark contrast to the case in which the noise strength is the same for all the oscillators, namely, in the studied dynamics, the system may exist in either a synchronized, or an incoherent, or a time-periodic state; interestingly, all these states also appear as long-time solutions of the Kuramoto dynamics for the case of bimodal frequency distributions, but in the absence of any noise in the dynamics.

Deterministic correlations enhance synchronization in oscillator populations with heterogeneous coupling

Synchronization is a critical phenomenon that displays a pivotal role in a wealth of dynamical processes ranging from natural to artificial systems. Here, we untangle the synchronization optimization in a system of globally coupled phase oscillators incorporating heterogeneous interactions encoded by the deterministic-random coupling. We uncover that, within the given restriction, the added deterministic correlations can profoundly enhance the synchronizability in comparison with the uncorrelated scenario. The critical points manifesting the onset of synchronization and desynchronization transitions, as well as the level of phase coherence, are significantly shaped by the increment of deterministic correlations. In particular, we provide an analytical treatment to properly ground the mechanism underlying synchronization enhancement and substantiate that the analytical predictions are in fair agreement with the numerical simulations. This study is a step forward in highlighting the importance of heterogeneous coupling among dynamical agents, which provides insights for control strategies of synchronization in complex systems.

Exponentially Long Transient Time to Synchronization of Coupled Chaotic Circle Maps in Dense Random Networks

We study the transition to synchronization in large, dense networks of chaotic circle maps, where an exact solution of the mean-field dynamics in the infinite network and all-to-all coupling limit is known. In dense networks of finite size and link probability of smaller than one, the incoherent state is meta-stable for coupling strengths that are larger than the mean-field critical coupling. We observe chaotic transients with exponentially distributed escape times and study the scaling behavior of the mean time to synchronization.

Graphop mean-field limits and synchronization for the stochastic Kuramoto model

Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator's phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov-Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence-coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.

Extended mean-field approach for chimera states in random complex networks

Identical oscillators in the chimera state exhibit a mixture of coherent and incoherent patterns simultaneously. Nonlocal interactions and phase lag are critical factors in forming a chimera state within the Kuramoto model in Euclidean space. Here, we investigate the contributions of nonlocal interactions and phase lag to the formation of the chimera state in random networks. By developing an extended mean-field approximation and using a numerical approach, we find that the emergence of a chimera state in the Erdös-Rényi network is due mainly to degree heterogeneity with nonzero phase lag. For a regularly random network, although all nodes have the same degree, we find that disordered connections may yield the chimera state in the presence of long-range interactions. Furthermore, we show a nontrivial dynamic state in which all the oscillators drift more slowly than a defined frequency due to connectivity disorder at large phase lags beyond the mean-field solutions.

Chimeras with uniformly distributed heterogeneity: Two coupled populations

Chimeras occur in networks of two coupled populations of oscillators when the oscillators in one population synchronize while those in the other are asynchronous. We consider chimeras of this form in networks of planar oscillators for which one parameter associated with the dynamics of an oscillator is randomly chosen from a uniform distribution. A generalization of the previous approach [Laing, Phys. Rev. E 100, 042211 (2019)2470-004510.1103/PhysRevE.100.042211], which dealt with identical oscillators, is used to investigate the existence and stability of chimeras for these heterogeneous networks in the limit of an infinite number of oscillators. In all cases, making the oscillators more heterogeneous destroys the stable chimera in a saddle-node bifurcation. The results help us understand the robustness of chimeras in networks of general oscillators to heterogeneity.

Shooting solitaries due to small-world connectivity in leaky integrate-and-fire networks

We study the synchronization properties in a network of leaky integrate-and-fire oscillators with nonlocal connectivity under probabilistic small-world rewiring. We demonstrate that the random links lead to the emergence of chimera-like states where the coherent regions are interrupted by scattered, short-lived solitaries; these are termed "shooting solitaries." Moreover, we provide evidence that random links enhance the appearance of chimera-like states for values of the parameter space that otherwise support synchronization. This last effect is counter-intuitive because by adding random links to the synchronous state, the system locally organizes into coherent and incoherent domains.

Coherence resonance in influencer networks

Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite the abundance of networks with influencers, little is known about their response to stochastic forcing. Here, for oscillatory dynamics on influencer networks, we show that subjecting influencers to an optimal intensity of noise can result in enhanced network synchronization. This new network dynamical effect, which we call coherence resonance in influencer networks, emerges from a synergy between network structure and stochasticity and is highly nonlinear, vanishing when the noise is too weak or too strong. Our results reveal that the influencer backbone can sharply increase the dynamical response in complex systems of coupled oscillators.

Phase and amplitude dynamics of coupled oscillator systems on complex networks

We investigated locking behaviors of coupled limit-cycle oscillators with phase and amplitude dynamics. We focused on how the dynamics are affected by inhomogeneous coupling strength and by angular and radial shifts in coupling functions. We performed mean-field analyses of oscillator systems with inhomogeneous coupling strength, testing Gaussian, power-law, and brain-like degree distributions. Even for oscillators with identical intrinsic frequencies and intrinsic amplitudes, we found that the coupling strength distribution and the coupling function generated a wide repertoire of phase and amplitude dynamics. These included fully and partially locked states in which high-degree or low-degree nodes would phase-lead the network. The mean-field analytical findings were confirmed via numerical simulations. The results suggest that, in oscillator systems in which individual nodes can independently vary their amplitude over time, qualitatively different dynamics can be produced via shifts in the coupling strength distribution and the coupling form. Of particular relevance to information flows in oscillator networks, changes in the non-specific drive to individual nodes can make high-degree nodes phase-lag or phase-lead the rest of the network.

A Brief Review of Chimera State in Empirical Brain Networks

Understanding the human brain and its functions has always been an interesting and challenging problem. Recently, a significant progress on this problem has been achieved on the aspect of chimera state where a coexistence of synchronized and unsynchronized states can be sustained in identical oscillators. This counterintuitive phenomenon is closely related to the unihemispheric sleep in some marine mammals and birds and has recently gotten a hot attention in neural systems, except the previous studies in non-neural systems such as phase oscillators. This review will briefly summarize the main results of chimera state in neuronal systems and pay special attention to the network of cerebral cortex, aiming to accelerate the study of chimera state in brain networks. Some outlooks are also discussed.

Chimera-like states induced by additional dynamic nonlocal wirings

We investigate the existence of chimera-like states in a small-world network of chaotically oscillating identical Rössler systems with an addition of randomly switching nonlocal links. By varying the small-world coupling strength, we observe no chimera-like state either in the absence of nonlocal wirings or with static nonlocal wirings. When we give an additional nonlocal wiring to randomly selected nodes and if we allow the random selection of nodes to change with time, we observe the onset of chimera-like states. Upon increasing the number of randomly selected nodes gradually, we find that the incoherent window keeps on shrinking, whereas the chimera-like window widens up. Moreover, the system attains a completely synchronized state comparatively sooner for a lower coupling strength. Also, we show that one can induce chimera-like states by a suitable choice of switching times, coupling strengths, and a number of nonlocal links. We extend the above-mentioned randomized injection of nonlocal wirings for the cases of globally coupled Rössler oscillators and a small-world network of coupled FitzHugh-Nagumo oscillators and obtain similar results.

Criticality as a Determinant of Integrated Information Φ in Human Brain Networks

Integrated information theory (IIT) describes consciousness as information integrated across highly differentiated but irreducible constituent parts in a system. However, in a complex dynamic system such as the brain, the optimal conditions for large integrated information systems have not been elucidated. In this study, we hypothesized that network criticality, a balanced state between a large variation in functional network configuration and a large constraint on structural network configuration, may be the basis of the emergence of a large Φ¯, a surrogate of integrated information. We also hypothesized that as consciousness diminishes, the brain loses network criticality and Φ¯ decreases. We tested these hypotheses with a large-scale brain network model and high-density electroencephalography (EEG) acquired during various levels of human consciousness under general anesthesia. In the modeling study, maximal criticality coincided with maximal Φ¯. The EEG study demonstrated an explicit relationship between Φ¯, criticality, and level of consciousness. The conscious resting state showed the largest Φ¯ and criticality, whereas the balance between variation and constraint in the brain network broke down as the response rate dwindled. The results suggest network criticality as a necessary condition of a large Φ¯ in the human brain.

Chimera states in coupled logistic maps with additional weak nonlocal topology

We demonstrate the occurrence of coexisting domains of partially coherent and incoherent patterns or simply known as chimera states in a network of globally coupled logistic maps upon addition of weak nonlocal topology. We find that the chimera states survive even after we disconnect nonlocal connections of some of the nodes in the network. Also, we show that the chimera states exist when we introduce symmetric gaps in the nonlocal coupling between predetermined nodes. We ascertain our results, for the existence of chimera states, by carrying out the recurrence quantification analysis and by computing the strength of incoherence. We extend our analysis for the case of small-world networks of coupled logistic maps and found the emergence of chimeralike states under the influence of weak nonlocal topology.

Weak multiplexing in neural networks: Switching between chimera and solitary states

We investigate spatio-temporal patterns occurring in a two-layer multiplex network of oscillatory FitzHugh-Nagumo neurons, where each layer is represented by a nonlocally coupled ring. We show that weak multiplexing, i.e., when the coupling between the layers is smaller than that within the layers, can have a significant impact on the dynamics of the neural network. We develop control strategies based on weak multiplexing and demonstrate how the desired state in one layer can be achieved without manipulating its parameters, but only by adjusting the other layer. We find that for coupling range mismatch, weak multiplexing leads to the appearance of chimera states with different shapes of the mean velocity profile for parameter ranges where they do not exist in isolation. Moreover, we show that introducing a coupling strength mismatch between the layers can suppress chimera states with one incoherent domain (one-headed chimeras) and induce various other regimes such as in-phase synchronization or two-headed chimeras. Interestingly, small intra-layer coupling strength mismatch allows to achieve solitary states throughout the whole network.

Nonautonomous driving induces stability in network of identical oscillators

Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilizing complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows as the amplitude of the frequency modulation is increased, through the birth of an intermittent synchronization regime. For repulsive couplings, we propose a control strategy to stabilize the dynamics by altering very slightly the network topology. We also show how, without changing the topology, time-variability in the driving frequency can itself stabilize the dynamics. As a byproduct of the analysis, we observe chimeralike states. We conclude that time-variability-induced stability phenomena are also present in networks, reinforcing the idea that this is a quite realistic scenario for living systems to use in maintaining their functioning in the face of ongoing perturbations.

Various synchronous states due to coupling strength inhomogeneity and coupling functions in systems of coupled identical oscillators

We study the effects of coupling strength inhomogeneity and coupling functions on locking behaviors of coupled identical oscillators, some of which are relatively weakly coupled to others while some are relatively strongly coupled. Through the stability analysis and numerical simulations, we show that several categories of fully locked or partially locked states can emerge and obtain the conditions for these categories. In this system with coupling strength inhomogeneity, locked and drifting subpopulations are determined by the coupling strength distribution and the shape of the coupling functions. Even the strongly coupled oscillators can drift while weakly coupled oscillators can be locked. The simulation results with Gaussian and power-law distributions for coupling strengths are compared and discussed.

Chimera states in neuronal networks: A review

Neuronal networks, similar to many other complex systems, self-organize into fascinating emergent states that are not only visually compelling, but also vital for the proper functioning of the brain. Synchronous spatiotemporal patterns, for example, play an important role in neuronal communication and plasticity, and in various cognitive processes. Recent research has shown that the coexistence of coherent and incoherent states, known as chimera states or simply chimeras, is particularly important and characteristic for neuronal systems. Chimeras have also been linked to the Parkinson's disease, epileptic seizures, and even to schizophrenia. The emergence of this unique collective behavior is due to diverse factors that characterize neuronal dynamics and the functioning of the brain in general, including neural bumps and unihemispheric slow-wave sleep in some aquatic mammals. Since their discovery, chimera states have attracted ample attention of researchers that work at the interface of physics and life sciences. We here review contemporary research dedicated to chimeras in neuronal networks, focusing on the relevance of different synaptic connections, and on the effects of different network structures and coupling setups. We also cover the emergence of different types of chimera states, we highlight their relevance in other related physical and biological systems, and we outline promising research directions for the future, including possibilities for experimental verification.

Phase-lags in large scale brain synchronization: Methodological considerations and in-silico analysis

Architecture of phase relationships among neural oscillations is central for their functional significance but has remained theoretically poorly understood. We use phenomenological model of delay-coupled oscillators with increasing degree of topological complexity to identify underlying principles by which the spatio-temporal structure of the brain governs the phase lags between oscillatory activity at distant regions. Phase relations and their regions of stability are derived and numerically confirmed for two oscillators and for networks with randomly distributed or clustered bimodal delays, as a first approximation for the brain structural connectivity. Besides in-phase, clustered delays can induce anti-phase synchronization for certain frequencies, while the sign of the lags is determined by the natural frequencies and by the inhomogeneous network interactions. For in-phase synchronization faster oscillators always phase lead, while stronger connected nodes lag behind the weaker during frequency depression, which consistently arises for in-silico results. If nodes are in anti-phase regime, then a distance π is added to the in-phase trends. The statistics of the phases is calculated from the phase locking values (PLV), as in many empirical studies, and we scrutinize the method’s impact. The choice of surrogates do not affects the mean of the observed phase lags, but higher significance levels that are generated by some surrogates, cause decreased variance and might fail to detect the generally weaker coherence of the interhemispheric links. These links are also affected by the non-stationary and intermittent synchronization, which causes multimodal phase lags that can be misleading if averaged. Taken together, the results describe quantitatively the impact of the spatio-temporal connectivity of the brain to the synchronization patterns between brain regions, and to uncover mechanisms through which the spatio-temporal structure of the brain renders phases to be distributed around 0 and π.

Trial registration: South African Clinical Trials Register: http://www.sanctr.gov.za/SAClinicalbrnbspTrials/tabid/169/Default.aspx, then link to respiratory tract then link to tuberculosis, pulmonary; and TASK Applied Sciences Clinical Trials, AP-TB-201-16 (ALOPEXX): https://task.org.za/clinical-trials/.

Functional connectivity, and in particular, phase coupling between distant brain regions may be fundamental in regulating neuronal processing and communication. However, phase relationships between the nodes of the brain and how they are confined by its spatio-temporal structure, have been mostly overlooked. We use a model of oscillatory dynamics superimposed on the space-time structure defined by the connectome, and we analyze the possible regimes of synchronization. Limitations of data analysis are also considered and we show that the choice of the significance threshold for coherence does not essentially impact the statistics of the observed phase lags, although it is crucial for the right detection of statistically significant coherence. Analytical insights are obtained for networks with heterogeneous time-delays, based on the empirical data from the connectome, and these are confirmed by numerical simulations, which show in- or anti-phase synchronization depending on the frequency and the distribution of time-delays. Phase lags are shown to result from inhomogeneous network interactions, so that stronger connected nodes generally phase lag behind the weaker.

Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps

We investigate the dynamics of nonlocally coupled time-discrete maps with emphasis on the occurrence and robustness of chimera states. These peculiar, hybrid states are characterized by a coexistence of coherent and incoherent regions. We consider logistic maps coupled on a one-dimensional ring with finite coupling radius. Domains of chimera existence form different tongues in the parameter space of coupling range and coupling strength. For a sufficiently large coupling strength, each tongue refers to a wave number describing the structure of the spatial profile. We also analyze the period-adding scheme within these tongues and multiplicity of period solutions. Furthermore, we study the robustness of chimeras with respect to parameter inhomogeneities and find that these states persist for different widths of the parameter distribution. Finally, we explore the spatial structure of the chimera using a spatial correlation function.

Structure Shapes Dynamics and Directionality in Diverse Brain Networks: Mathematical Principles and Empirical Confirmation in Three Species

Identifying how spatially distributed information becomes integrated in the brain is essential to understanding higher cognitive functions. Previous computational and empirical studies suggest a significant influence of brain network structure on brain network function. However, there have been few analytical approaches to explain the role of network structure in shaping regional activities and directionality patterns. In this study, analytical methods are applied to a coupled oscillator model implemented in inhomogeneous networks. We first derive a mathematical principle that explains the emergence of directionality from the underlying brain network structure. We then apply the analytical methods to the anatomical brain networks of human, macaque, and mouse, successfully predicting simulation and empirical electroencephalographic data. The results demonstrate that the global directionality patterns in resting state brain networks can be predicted solely by their unique network structures. This study forms a foundation for a more comprehensive understanding of how neural information is directed and integrated in complex brain networks.

Chimera-like states in structured heterogeneous networks

Chimera-like states are manifested through the coexistence of synchronous and asynchronous dynamics and have been observed in various systems. To analyze the role of network topology in giving rise to chimera-like states, we study a heterogeneous network model comprising two groups of nodes, of high and low degrees of connectivity. The architecture facilitates the analysis of the system, which separates into a densely connected coherent group of nodes, perturbed by their sparsely connected drifting neighbors. It describes a synchronous behavior of the densely connected group and scaling properties of the induced perturbations.

Chimera states and the interplay between initial conditions and non-local coupling

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We study chimera states in a network of non-locally coupled Stuart-Landau oscillators. We investigate the impact of initial conditions in combination with non-local coupling. Based on an analytical argument, we show how the coupling phase and the coupling strength are linked to the occurrence of chimera states, flipped profiles of the mean phase velocity, and the transition from a phase- to an amplitude-mediated chimera state.

Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

The Ott-Antonsen (OA) ansatz [Ott and Antonsen, Chaos 18, 037113 (2008); Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this, we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore, we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step, we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.

Optimization of noise-induced synchronization of oscillator networks

We investigate common-noise-induced synchronization between two identical networks of coupled phase oscillators exhibiting fully locked collective oscillations. Using the collective phase description method for fully locked oscillators, we demonstrate that two noninteracting networks of coupled phase oscillators can exhibit in-phase synchronization between the networks when driven by weak common noise. We derive the Lyapunov exponent characterizing the relaxation time for synchronization and develop a method of obtaining the optimal input pattern of common noise to achieve fast synchronization. We illustrate the theory using three representative networks with heterogeneous, global, and local coupling. The theoretical results are validated by direct numerical simulations.

Synchronization failure caused by interplay between noise and network heterogeneity

We investigate synchronization in complex networks of noisy phase oscillators. We find that, while too weak a coupling is not sufficient for the whole system to synchronize, too strong a coupling induces a nontrivial type of phase slip among oscillators, resulting in synchronization failure. Thus, an intermediate coupling range for synchronization exists, which becomes narrower when the network is more heterogeneous. Analyses of two noisy oscillators reveal that nontrivial phase slip is a generic phenomenon when noise is present and coupling is strong. Therefore, the low synchronizability of heterogeneous networks can be understood as a result of the difference in effective coupling strength among oscillators with different degrees; oscillators with high degrees tend to undergo phase slip while those with low degrees have weak coupling strengths that are insufficient for synchronization.

Chimera states in networks of Van der Pol oscillators with hierarchical connectivities

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics.

Complex Rotating Waves and Long Transients in a Ring Network of Electrochemical Oscillators with Sparse Random Cross-Connections

We perform experiments and phase model simulations with a ring network of oscillatory electrochemical reactions to explore the effect of random connections and nonisochronicity of the interactions on the pattern formation. A few additional links facilitate the emergence of the fully synchronized state. With larger nonisochronicity, complex rotating waves or persistent irregular phase dynamics can derail the convergence to global synchronization. The observed long transients of irregular phase dynamics exemplify the possibility of a sudden onset of hypersynchronous behavior without any external stimulus or network reorganization.

Phase oscillators in modular networks: The effect of nonlocal coupling

We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.

Nonlinearity of local dynamics promotes multi-chimeras

Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

We study numerically the development of chimera states in networks of nonlocally coupled oscillators whose limit cycles emerge from a Hopf bifurcation. This dynamical system is inspired from population dynamics and consists of three interacting species in cyclic reactions. The complexity of the dynamics arises from the presence of a limit cycle and four fixed points. When the bifurcation parameter increases away from the Hopf bifurcation the trajectory approaches the heteroclinic invariant manifolds of the fixed points producing spikes, followed by long resting periods. We observe chimera states in this spiking regime as a coexistence of coherence (synchronization) and incoherence (desynchronization) in a one-dimensional ring with nonlocal coupling and demonstrate that their multiplicity depends on both the system and the coupling parameters. We also show that hierarchical (fractal) coupling topologies induce traveling multichimera states. The speed of motion of the coherent and incoherent parts along the ring is computed through the Fourier spectra of the corresponding dynamics.

General relationship of global topology, local dynamics, and directionality in large-scale brain networks

The balance of global integration and functional specialization is a critical feature of efficient brain networks, but the relationship of global topology, local node dynamics and information flow across networks has yet to be identified. One critical step in elucidating this relationship is the identification of governing principles underlying the directionality of interactions between nodes. Here, we demonstrate such principles through analytical solutions based on the phase lead/lag relationships of general oscillator models in networks. We confirm analytical results with computational simulations using general model networks and anatomical brain networks, as well as high-density electroencephalography collected from humans in the conscious and anesthetized states. Analytical, computational, and empirical results demonstrate that network nodes with more connections (i.e., higher degrees) have larger amplitudes and are directional targets (phase lag) rather than sources (phase lead). The relationship of node degree and directionality therefore appears to be a fundamental property of networks, with direct applicability to brain function. These results provide a foundation for a principled understanding of information transfer across networks and also demonstrate that changes in directionality patterns across states of human consciousness are driven by alterations of brain network topology.

Current brain connectome projects are attempting to construct a map of the structural and functional network connections in the brain. One goal of these projects is to understand how network organization determines local functions and information transfer patterns, which is essential to achieve higher cognitive brain functions. Because of the limitation of constructing all brain maps for all cognitive states, finding a general relationship of global topology, local dynamics and the directionality of information transfer in a network is crucial. In this study, we show that inter-node directionality arises naturally from the topology of the network. Analytical, computational, and empirical results all demonstrate that network nodes with more connections (i.e., higher degree) lag in phase, while lower-degree nodes lead. Our mathematical analysis allowed us to predict the directionality patterns in general model networks as well as human brain networks across different states of consciousness. These findings may provide more straightforward approaches to dissecting how directionality between interacting nodes is shaped in complex brain networks, providing a foundation for understanding principles of information transfer. Furthermore, the underlying mathematical relationship between node connections and directionality patterns has the potential to advance network science across numerous disciplines.

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Phase ordering in coupled noisy bistable systems on scale-free networks

We study a system consisting of diffusively coupled noisy bistable elements on a scale-free random network. This system exhibits an order-disorder phase transition as the noise intensity is varied. The phase ordering process takes place consecutively and in order of the degrees, reflecting strong degree heterogeneity of the scale-free network. A nonlinear Fokker-Planck equation describing the network dynamics is derived under mean-field approximation of the network, and is used to explain the phase ordering dynamics of the system.

Persistent fluctuations in synchronization rate in globally coupled oscillators with periodic external forcing

A system of phase oscillators with repulsive global coupling and periodic external forcing undergoing asynchronous rotation is considered. The synchronization rate of the system can exhibit persistent fluctuations depending on parameters and initial phase distributions, and the amplitude of the fluctuations scales with the system size for uniformly random initial phase distributions. Using the Watanabe-Strogatz transformation that reduces the original system to low-dimensional macroscopic equations, we show that the fluctuations are collective dynamics of the system corresponding to low-dimensional trajectories of the reduced equations. It is argued that the amplitude of the fluctuations is determined by the inhomogeneity of the initial phase distribution, resulting in system-size scaling for the random case.

Self-organized network of phase oscillators coupled by activity-dependent interactions

We investigate a network of coupled phase oscillators whose interactions evolve dynamically depending on the relative phases between the oscillators. We found that this coevolving dynamical system robustly yields three basic states of collective behavior with their self-organized interactions. The first is the two-cluster state, in which the oscillators are organized into two synchronized groups. The second is the coherent state, in which the oscillators are arranged sequentially in time. The third is the chaotic state, in which the relative phases between oscillators and their coupling weights are chaotically shuffled. Furthermore, we demonstrate that self-assembled multiclusters can be designed by controlling the weight dynamics. Note that the phase patterns of the oscillators and the weighted network of interactions between them are simultaneously organized through this coevolving dynamics. We expect that these results will provide new insight into self-assembly mechanisms by which the collective behavior of a rhythmic system emerges as a result of the dynamics of adaptive interactions.

Sparse gamma rhythms arising through clustering in adapting neuronal networks

Gamma rhythms (30-100 Hz) are an extensively studied synchronous brain state responsible for a number of sensory, memory, and motor processes. Experimental evidence suggests that fast-spiking interneurons are responsible for carrying the high frequency components of the rhythm, while regular-spiking pyramidal neurons fire sparsely. We propose that a combination of spike frequency adaptation and global inhibition may be responsible for this behavior. Excitatory neurons form several clusters that fire every few cycles of the fast oscillation. This is first shown in a detailed biophysical network model and then analyzed thoroughly in an idealized model. We exploit the fact that the timescale of adaptation is much slower than that of the other variables. Singular perturbation theory is used to derive an approximate periodic solution for a single spiking unit. This is then used to predict the relationship between the number of clusters arising spontaneously in the network as it relates to the adaptation time constant. We compare this to a complementary analysis that employs a weak coupling assumption to predict the first Fourier mode to destabilize from the incoherent state of an associated phase model as the external noise is reduced. Both approaches predict the same scaling of cluster number with respect to the adaptation time constant, which is corroborated in numerical simulations of the full system. Thus, we develop several testable predictions regarding the formation and characteristics of gamma rhythms with sparsely firing excitatory neurons.

Shear diversity prevents collective synchronization

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold.

Diffusion-induced instability and chaos in random oscillator networks

We demonstrate that diffusively coupled limit-cycle oscillators on random networks can exhibit various complex dynamical patterns. Reducing the system to a network analog of the complex Ginzburg-Landau equation, we argue that uniform oscillations can be linearly unstable with respect to spontaneous phase modulations due to diffusional coupling-the effect corresponding to the Benjamin-Feir instability in continuous media. Numerical investigations under this instability in random scale-free networks reveal a wealth of complex dynamical regimes, including partial amplitude death, clustering, and chaos. A dynamic mean-field theory explaining different kinds of nonlinear dynamics is constructed.