Remote synchronization in star networks

Directed ratchet transport in starlike networks of driven damped pendula by localized symmetry-breaking-inducing excitations

We study the effectiveness of locally controlling the breakage of a significant space-time symmetry by zero-average periodic excitations at inducing and suppressing directed ratchet transport (i.e., that induced without an applied net bias) and chaos in starlike networks of driven damped pendula. While the emergence of chaos mainly depends upon the impulse transmitted by the periodic excitations, directed ratchet transport does upon a subtle balance between energy transmitted through the pendula via excitations' impulse and degree of symmetry breaking, thus defining a physical criticality scenario. Optimal enhancement of directed ratchet transport is found to occur when the waveform of the periodic excitation matches as closely as possible to a universal waveform when all nodes are homogeneously driven. In the case of networks with heterogeneous distributions of the symmetry breaking, the net ratcheting effect of increasing the effective breakage by periodic excitations acting on a number of particular nodes strongly depends upon their number and degree of connectivity as well as the coupling strength, while showing self-organized criticality with respect to the maximum strength of directed ratchet transport.

Investigation of transient extreme events in a mutually coupled star network of theoretical Brusselator system

In this article, we present evidence of a distinct class of extreme events that occur during the transient chaotic state within network modeling using the Brusselator with a mutually coupled star network. We analyze the phenomenon of transient extreme events in the network by focusing on the lifetimes of chaotic states. These events are identified through the finite-time Lyapunov exponent and quantified using threshold and statistical methods, including the probability distribution function (PDF), generalized extreme value (GEV) distribution, and return period plots. We also evaluate the transitions of these extreme events by examining the average synchronization error and the system's energy function. Our findings, validated across networks of various sizes, demonstrate consistent patterns and behaviors, contributing to a deeper understanding of transient extreme events in complex networks.

Temperature effects on neuronal synchronization in seizures

We present a computational model of networked neurons developed to study the effect of temperature on neuronal synchronization in the brain in association with seizures. The network consists of a set of chaotic bursting neurons surrounding a core tonic neuron in a square lattice with periodic boundary conditions. Each neuron is reciprocally coupled to its four nearest neighbors via temperature dependent gap junctions. Incorporating temperature in the gap junctions makes the coupling stronger when temperature rises, resulting in higher likelihood for synchrony in the network. Raising the temperature eventually makes the network elicit waves of synchronization in circular ripples that propagate from the center outwardly. We suggest this process as a possible underlying mechanism for seizures induced by elevated brain temperatures.

Recurrent chaotic clustering and slow chaos in adaptive networks

Adaptive dynamical networks are network systems in which the structure co-evolves and interacts with the dynamical state of the nodes. We study an adaptive dynamical network in which the structure changes on a slower time scale relative to the fast dynamics of the nodes. We identify a phenomenon we refer to as recurrent adaptive chaotic clustering (RACC), in which chaos is observed on a slow time scale, while the fast time scale exhibits regular dynamics. Such slow chaos is further characterized by long (relative to the fast time scale) regimes of frequency clusters or frequency-synchronized dynamics, interrupted by fast jumps between these regimes. We also determine parameter values where the time intervals between jumps are chaotic and show that such a state is robust to changes in parameters and initial conditions.

Complex localization mechanisms in networks of coupled oscillators: Two case studies

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

Endowing networks with desired symmetries and modular behavior

Symmetries in a network regulate its organization into functional clustered states. Given a generic ensemble of nodes and a desirable cluster (or group of clusters), we exploit the direct connection between the elements of the eigenvector centrality and the graph symmetries to generate a network equipped with the desired cluster(s), with such a synthetical structure being furthermore perfectly reflected in the modular organization of the network's functioning. Our results solve a relevant problem of designing a desired set of clusters and are of generic application in all cases where a desired parallel functioning needs to be blueprinted.

Detecting partial synchrony in a complex oscillatory network using pseudovortices

Partial synchronization is an important dynamical process of coupled oscillators on various natural and artificial networks, which can remain undetected due to the system complexity. With an analogy between pairwise asynchrony of oscillators and topological defects, i.e., vortices, in the two-dimensional XY model, we propose a robust and data-driven method to identify the partial synchronization on complex networks. The proposed method is based on an integer matrix whose element is pseudovorticity that discretely quantifies asynchronous phase dynamics in every two oscillators, which results in graphical and entropic representations of partial synchrony. As a first trial, we apply our method to 200 FitzHugh-Nagumo neurons on a complex small-world network. Partially synchronized chimera states are revealed by discriminating synchronized states even with phase lags. Such phase lags also appear in partial synchronization in chimera states. Our topological, graphical, and entropic method is implemented solely with measurable phase dynamics data, which will lead to a straightforward application to general oscillatory networks including neural networks in the brain.

Sensitive dynamics of brain cognitive networks and its resource constraints

It is well known that brain functions are closely related to the synchronization of brain networks, but the underlying mechanisms are still not completely understood. To study this problem, we here focus on the synchronization of cognitive networks, in contrast to that of a global brain network, as individual brain functions are in fact performed by different cognitive networks but not the global network. In detail, we consider four different levels of brain networks and two approaches, i.e., either with or without resource constraints. For the case of without resource constraints, we find that global brain networks have fundamentally different behaviors from that of the cognitive networks; i.e., the former has a continuous synchronization transition, while the latter shows a novel transition of oscillatory synchronization. This feature of oscillation comes from the sparse links among the communities of cognitive networks, resulting in coupling sensitive dynamics of brain cognitive networks. While for the case of resource constraints, we find that at the global level, the synchronization transition becomes explosive, in contrast to the continuous synchronization for the case of without resource constraints. At the level of cognitive networks, the transition also becomes explosive and the coupling sensitivity is significantly reduced, thus guaranteeing the robustness and fast switch of brain functions. Moreover, a brief theoretical analysis is provided.

Topologically induced suppression of explosive synchronization

Nowadays, explosive synchronization is a well-documented phenomenon consisting in a first-order transition that may coexist with classical synchronization. Typically, explosive synchronization occurs when the network structure is represented by the classical graph Laplacian, and the node frequency and its degree are correlated. Here, we answer the question on whether this phenomenon can be observed in networks when the oscillators are coupled via degree-biased Laplacian operators. We not only observe that this is the case but also that this new representation naturally controls the transition from explosive to standard synchronization in a network. We prove analytically that explosive synchronization emerges when using this theoretical setting in star-like networks. As soon as this star-like network is topologically converted into a network containing cycles, the explosive synchronization gives rise to classical synchronization. Finally, we hypothesize that this mechanism may play a role in switching from normal to explosive states in the brain, where explosive synchronization has been proposed to be related to some pathologies like epilepsy and fibromyalgia.

Collective behavior of identical Stuart-Landau oscillators in a star network with coupling asymmetry effects

In this study, we investigated the impact of the asymmetry of a coupling scheme on oscillator dynamics in a star network. We obtained stability conditions for the collective behavior of the systems, ranging from an equilibrium point over complete synchronization (CS) and quenched hub incoherence to remote synchronization states using both numerical and analytical methods. The coupling asymmetry factor α significantly influences and determines the stable parameter region of each state. For α ≠ 1, the equilibrium point can emerge when the Hopf bifurcation parameter a is positive, which is impossible for diffusive coupling. However, CS can occur even if a is negative under α < 1. Unlike diffusive coupling, we observe more behavior when α ≠ 1, including additional in-phase remote synchronization. These results are supported by theoretical analysis and validated through numerical simulations and independent of network size. The findings may offer practical methods for controlling, restoring, or obstructing specific collective behavior.

Phase frustration induced remote synchronization

Remote synchronization (RS) may take an important role in brain functioning and its study has attracted much attention in recent years. So far, most studies of RS are focused on the Stuart-Landau oscillators with mean-field coupling. However, realistic cases may have more complicated couplings and behaviors, such as the brain networks. To make the study of RS a substantial progress toward realistic situations, we here present a model of RS with phase frustration and show that RS can be induced for those systems where no RS exists when there is no phase frustration. By numerical simulations on both the Stuart-Landau and Kuramoto oscillators, we find that the optimal range of RS depends on the match of phase frustrations between the hub and leaf nodes and a fixed relationship of this match is figured out. While for the non-optimal range of RS, we find that RS exists only in a linear band between the phase frustrations of the hub and leaf nodes. A brief theoretical analysis is provided to explain these results.

Modelling the perception of music in brain network dynamics

We analyze the influence of music in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report an increase of coherence between the global dynamics in our network and the input signal induced by a specific music song. We show that the level of coherence depends crucially on the frequency band. We compare our results with experimental data, which also describe global neural synchronization between different brain regions in the gamma-band range in a time-dependent manner correlated with musical large-scale form, showing increased synchronization just before transitions between different parts in a musical piece (musical high-level events). The results also suggest a separation in musical form-related brain synchronization between high brain frequencies, associated with neocortical activity, and low frequencies in the range of dance movements, associated with interactivity between cortical and subcortical regions.

Metamorphoses and explosively remote synchronization in dynamical networks

We uncover a phenomenon in coupled nonlinear networks with a symmetry: as a bifurcation parameter changes through a critical value, synchronization among a subset of nodes can deteriorate abruptly, and, simultaneously, perfect synchronization emerges suddenly among a different subset of nodes that are not directly connected. This is a synchronization metamorphosis leading to an explosive transition to remote synchronization. The finding demonstrates that an explosive onset of synchrony and remote synchronization, two phenomena that have been studied separately, can arise in the same system due to symmetry, providing another proof that the interplay between nonlinear dynamics and symmetry can lead to a surprising phenomenon in physical systems.

Two mechanisms of remote synchronization in a chain of Stuart-Landau oscillators

Remote synchronization implies that oscillators interacting not directly but via an additional unit (hub) adjust their frequencies and exhibit frequency locking while the hub remains asynchronous. In this paper, we analyze the mechanisms of remote synchrony in a small network of three coupled Stuart-Landau oscillators using recent results on higher-order phase reduction. We analytically demonstrate the role of two factors promoting remote synchrony. These factors are the nonisochronicity of oscillators and the coupling terms appearing in the second-order phase approximation. We show a good correspondence between our theory and numerical results for small and moderate coupling strengths.

Catalytic feed-forward explosive synchronization in multilayer networks

Inhibitory couplings are crucial for the normal functioning of many real-world complex systems. Inhibition in one layer has been shown to induce explosive synchronization in another excitatory (or positive) layer of duplex networks. By extending this framework to multiplex networks, this article shows that inhibition in a single layer can act as a catalyst, leading to explosive synchronization transitions in the rest of the layers feed-forwarded through intermediate layer(s). Considering a multiplex network of coupled Kuramoto oscillators, we demonstrate that the characteristics of the transition emergent in a layer can be entirely controlled by the intra-layer coupling of other layers and the multiplexing strengths. The results presented here are essential to fathom the synchronization behavior of coupled dynamical units in multi-layer systems possessing inhibitory coupling in one of its layers, representing the importance of multiplexing.

Relay Synchronization in a Weighted Triplex Network

Relay synchronization in multi-layer networks implies inter-layer synchronization between two indirectly connected layers through a relay layer. In this work, we study the relay synchronization in a three-layer multiplex network by introducing degree-based weighting mechanisms. The mechanism of within-layer connectivity may be hubs-repelling or hubs-attracting whenever low-degree or high-degree nodes receive strong influence. We adjust the remote layers to hubs-attracting coupling, whereas the relay layer may be unweighted, hubs-repelling, or hubs-attracting network. We establish that relay synchronization is improved when the relay layer is hubs-repelling compared to the other cases. We determine analytically necessary stability conditions of relay synchronization state using the master stability function approach. Finally, we explore the relation between synchronization and the topological property of the relay layer. We find that a higher clustering coefficient hinders synchronizability, and vice versa. We also look into the intra-layer synchronization in the proposed weighted triplex network and establish that intra-layer synchronization occurs in a wider range when relay layer is hubs-attracting.

Novel modes of synchronization in star networks of coupled chemical oscillators

Photochemically coupled micro-oscillators are studied experimentally and computationally in star networks to investigate the modes and mechanisms of synchronization. The micro-oscillators are catalyst-loaded beads that are placed in catalyst-free Belousov-Zhabotinsky (BZ) solutions. The properties of the photochemical coupling between the oscillators are determined by the composition of the BZ reaction mixtures, and both excitatory coupling and inhibitory coupling are studied. Synchronization of peripheral oscillators coupled through a hub oscillator is exhibited at coupling strengths leading to novel modes of synchronization of the hub with the peripheral oscillators. A theoretical analysis provides insights into the mechanism of the synchronization. The heterogeneous peripheral oscillators have different phase velocities that give rise to a phase divergence; however, the perturbation from the hub acts to realign the phases by delaying the faster oscillators more than the slower oscillators.

Synchronization scenarios in three-layer networks with a hub

We study various relay synchronization scenarios in a three-layer network, where the middle (relay) layer is a single node, i.e., a hub. The two remote layers consist of non-locally coupled rings of FitzHugh-Nagumo oscillators modeling neuronal dynamics. All nodes of the remote layers are connected to the hub. The role of the hub and its importance for the existence of chimera states are investigated in dependence on the inter-layer coupling strength and inter-layer time delay. Tongue-like regions in the parameter plane exhibiting double chimeras, i.e., chimera states in the remote layers whose coherent cores are synchronized with each other, and salt-and-pepper states are found. At very low intra-layer coupling strength, when chimera states do not exist in single layers, these may be induced by the hub. Also, the influence of the dilution of links between the remote layers and the hub upon the dynamics is investigated. The greatest effect of dilution is observed when links to the coherent domain of the chimeras are removed.

Collective oscillations of coupled cell cycles

Problems with networks of coupled oscillators arise in multiple contexts, commonly leading to the question about the dependence of network dynamics on network structure. Previous work has addressed this question in Drosophila oogenesis, in which stable cytoplasmic bridges connect the future oocyte to the supporting nurse cells that supply the oocyte with molecules and organelles needed for its development. To increase their biosynthetic capacity, nurse cells enter the endoreplication program, a special form of the cell cycle formed by the iterated repetition of growth and synthesis phases without mitosis. Recent studies have revealed that the oocyte orchestrates nurse cell endoreplication cycles, based on retrograde (oocyte to nurse cells) transport of a cell cycle inhibitor produced by the nurse cells and localized to the oocyte. Furthermore, the joint dynamics of endocycles has been proposed to depend on the intercellular connectivity within the oocyte-nurse cell cluster. We use a computational model to argue that this connectivity guides, but does not uniquely determine the collective dynamics and identify several oscillatory regimes, depending on the timescale of intercellular transport. Our results provide insights into collective dynamics of coupled cell cycles and motivate future quantitative studies of intercellular communication in the germline cell clusters.

Remote firing propagation in the neural network of C. elegans

Understanding the mechanisms of firing propagation in brain networks has been a long-standing problem in the fields of nonlinear dynamics and network science. In general, it is believed that a specific firing in a brain network may be gradually propagated from a source node to its neighbors and then to the neighbors' neighbors and so on. Here, we explore firing propagation in the neural network of Caenorhabditis elegans and surprisingly find an abnormal phenomenon, i.e., remote firing propagation between two distant and indirectly connected nodes with the intermediate nodes being inactivated. This finding is robust to source nodes but depends on the topology of network such as the unidirectional couplings and heterogeneity of network. Further, a brief theoretical analysis is provided to explain its mechanism and a principle for remote firing propagation is figured out. This finding provides insights for us to understand how those cognitive subnetworks emerge in a brain network.

Spatial multi-scaled chimera states of cerebral cortex network and its inherent structure-dynamics relationship in human brain

Human cerebral cortex displays various dynamics patterns under different states, however the mechanism how such diverse patterns can be supported by the underlying brain network is still not well understood. Human brain has a unique network structure with different regions of interesting to perform cognitive tasks. Using coupled neural mass oscillators on human cortical network and paying attention to both global and local regions, we observe a new feature of chimera states with multiple spatial scales and a positive correlation between the synchronization preference of local region and the degree of symmetry of the connectivity of the region in the network. Further, we use the concept of effective symmetry in the network to build structural and dynamical hierarchical trees and find close matching between them. These results help to explain the multiple brain rhythms observed in experiments and suggest a generic principle for complex brain network as a structure substrate to support diverse functional patterns.

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.

Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps

We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

Effect of topology upon relay synchronization in triplex neuronal networks

Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer networks. We show that including randomness in the connectivity structure either of the remote network layers or of the relay layer increases the range of interlayer coupling strength where relay synchronization can be observed.

Functionability in complex networks: Leading nodes for the transition from structural to functional networks through remote asynchronization

Complex networks are essentially heterogeneous not only in the basic properties of the constituent nodes, such as their degree, but also in the effects that these have on the global dynamical properties of the network. Networks of coupled identical phase oscillators are good examples for analyzing these effects, since an overall synchronized state can be considered a reference state. A small variation of intrinsic node parameters may cause the system to move away from synchronization, and a new phase-locked stationary state can be achieved. We propose a measure of phase dispersion that quantifies the functional response of the system to a given local perturbation. As a particular implementation, we propose a variation of the standard Kuramoto model in which the nodes of a complex network interact with their neighboring nodes, by including a node-dependent frustration parameter. The final stationary phase-locked state now depends on the particular frustration parameter at each node and also on the network topology. We exploit this scenario by introducing individual frustration parameters and measuring what their effect on the whole network is, measured in terms of the phase dispersion, which depends only on the topology of the network and on the choice of the particular node that is perturbed. This enables us to define a characteristic of the node, its functionability, that can be computed analytically in terms of the network topology. Finally, we provide a thorough comparison with other centrality measures.

Dynamics of Reaction-Diffusion Oscillators in Star and other Networks with Cyclic Symmetries Exhibiting Multiple Clusters

We experimentally and theoretically investigate the dynamics of inhibitory coupled self-driven oscillators on a star network in which a single central hub node is connected to k peripheral arm nodes. The system consists of water-in-oil Belousov-Zhabotinsky ∼100  μm emulsion drops contained in storage wells etched in silicon wafers. We observed three dynamical attractors by varying the number of arms in the star graph and the coupling strength: (i) unlocked, uncorrelated phase shifts between all oscillators; (ii) locked, arm hubs synchronized in phase with a k-dependent phase shift between the arm and central hub; and (iii) center silent, a central hub stopped oscillating and the arm hubs oscillated without synchrony. We compare experiment to theory. For case (ii), we identified a logarithmic dependence of the phase shift on star degree, and were able to discriminate between contributions to the phase shift arising from star topology and oscillator chemistry.

Master-slave synchronization of hyperchaotic systems through a linear dynamic coupling

The development of synchronization strategies for dynamical systems is an important research activity that can be applied in several different fields from locomotion control of multilimbed structures to secure communication. In the presence of chaotic systems, synchronization is more difficult to accomplish and there are different techniques that can be adopted. In this paper we considered a master-slave topology where the coupling mechanism is realized through a second-order linear dynamical system. This control scheme, recently applied to chaotic systems, is here analyzed in the presence of hyperchaotic dynamics that represent a more challenging scenario. The possibility to reach a complete synchronization and the range of allowable coupling strength is investigated comparing the effects of the dynamical coupling with a standard configuration characterized by a static gain. This methodology is also applied to weighted networks to reach synchronization regimes otherwise not obtainable with a static coupling.

Fading of remote synchronization in tree networks of Stuart-Landau oscillators

Remote synchronization (RS) is characterized by the appearance of phase coherence between oscillators that do not directly interact through a structural link in a network but exclusively through other units that are not synchronized or more weakly synchronized with them. This form of phase synchronization was observed initially in starlike motifs and later in random networks. In this paper, we report on an experimental setup for the analysis of RS in networks of Stuart-Landau oscillators and in particular investigate the behavior of tree structures focusing on the path to synchronization, that is, on the analysis of how synchronization emerges as the coupling strength increases from zero. We find that RS occurs in a region wherein further increases of the coupling strength lead to a direct transition to global synchronization but may also be observed in a second region, corresponding to lower coupling values, wherein it first emerges and then disappears, hallmarking a scenario that we denote as fading of remote synchronization. We show that this result is related to the behavior of pairs of remotely synchronized nodes observed in networks with more general topologies. Experiments are corroborated by numerical simulations confirming the major findings and providing further characterization of the phenomenon. We demonstrate that the distribution of natural oscillation frequencies and the parameter uncertainty in the links both play a fundamental role in shaping the behaviors observed.

Cubic Intuitionistic q-Ideals of BCI-Algebras

In this paper, the notion of cubic intuitionistic q-ideals in BCI-algebras is introduced. A relationship between a cubic intuitionistic subalgebra, a cubic intuitionistic ideal, and a cubic intuitionistic q-ideal is discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic q-ideal are provided. Characterizations of a cubic intuitionistic q-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic q-ideal is established. Furthermore, the product of cubic intuitionistic subalgebras, ideals, and q-ideals are investigated.

Synchronous tonic-to-bursting transitions in a neuronal hub motif

We study a heterogeneous neuronal network motif where a central node (hub neuron) is connected via electrical synapses to other nodes (peripheral neurons). Our numerical simulations show that the networked neurons synchronize in three different states: (i) robust tonic, (ii) robust bursting, and (iii) tonic initially evolving to bursting through a period-doubling cascade and chaos transition. This third case displays interesting features, including the carrying on of a characteristic firing rate found in the single neuron tonic-to-bursting transition.

Relay synchronization in multiplex networks

Relay (or remote) synchronization between two not directly connected oscillators in a network is an important feature allowing distant coordination. In this work, we report a systematic study of this phenomenon in multiplex networks, where inter-layer synchronization occurs between distant layers mediated by a relay layer that acts as a transmitter. We show that this transmission can be extended to higher order relay configurations, provided symmetry conditions are preserved. By first order perturbative analysis, we identify the dynamical and topological dependencies of relay synchronization in a multiplex. We find that the relay synchronization threshold is considerably reduced in a multiplex configuration, and that such synchronous state is mostly supported by the lower degree nodes of the outer layers, while hubs can be de-multiplexed without affecting overall coherence. Finally, we experimentally validated the analytical and numerical findings by means of a multiplex of three layers of electronic circuits.

Apparent remote synchronization of amplitudes: A demodulation and interference effect

A form of "remote synchronization" was recently described, wherein amplitude fluctuations across a ring of non-identical, non-linear electronic oscillators become entrained into spatially-structured patterns. According to linear models and mutual information, synchronization and causality dip at a certain distance, then recover before eventually fading. Here, the underlying mechanism is finally elucidated through novel experiments and simulations. The system non-linearity is found to have a dual role: it supports chaotic dynamics, and it enables the energy exchange between the lower and higher sidebands of a predominant frequency. This frequency acts as carrier signal in an arrangement resembling standard amplitude modulation, wherein the lower sideband and the demodulated baseband signals spectrally overlap. Due to a spatially-dependent phase relationship, at a certain distance near-complete destructive interference occurs between them, causing the observed dip. Methods suitable for detecting non-trivial entrainment, such as transfer entropy and the auxiliary system approach, nevertheless, reveal that synchronization and causality actually decrease with distance monotonically. Remoteness is, therefore, arguably only apparent, as also reflected in the propagation of external perturbations. These results demonstrate a complex mechanism of dynamical interdependence, and exemplify how it can lead to incorrectly inferring synchronization and causality.

Hub-driven remote synchronization in brain networks

The phenomenon of “remote synchronization” (RS), first observed in a star network of oscillators, involves synchronization of unconnected peripheral nodes through a hub that maintains independent dynamics. In the RS regime the central hub was thought to serve as a passive gate for information transfer between nodes. Here, we investigate the physical origin of this phenomenon. Surprisingly, we find that a hub node can drive remote synchronization of peripheral oscillators even in the presence of a repulsive mean field, thus actively governing network dynamics while remaining asynchronous. We study this novel phenomenon in complex networks endowed with multiple hub-nodes, a ubiquitous feature of many real-world systems, including brain connectivity networks. We show that a change in the natural frequency of a single hub can alone reshape synchronization patterns across the entire network, and switch from direct to remote synchronization, or to hub-driven desynchronization. Hub-driven RS may provide a mechanism to account for the role of structural hubs in the organization of brain functional connectivity networks.

Emergence of chaos in starlike networks of dissipative nonlinear oscillators by localized parametric excitations

We study the effectiveness of locally controlling the impulse transmitted by parametric periodic excitations at inducing and suppressing chaos in starlike networks of driven damped pendula, leading to asynchronous chaotic states and equilibria, respectively. We found that the inducing (suppressor) effect of increasing (decreasing) the impulse transmitted by the parametric excitations acting on particular nodes depends strongly on their number and degree of connectivity as well as the coupling strength. Additionally, we provide a theoretical analysis explaining the basic physical mechanisms of the emergence and suppression of chaos as well as the main features of the chaos-control scenario. Our findings constitute proof of the impulse-induced control of chaos in a simple model of complex networks, thus opening the way to its application to real-world networks.

Incoherence-Mediated Remote Synchronization

In previously identified forms of remote synchronization between two nodes, the intermediate portion of the network connecting the two nodes is not synchronized with them but generally exhibits some coherent dynamics. Here we report on a network phenomenon we call incoherence-mediated remote synchronization (IMRS), in which two noncontiguous parts of the network are identically synchronized while the dynamics of the intermediate part is statistically and information-theoretically incoherent. We identify mirror symmetry in the network structure as a mechanism allowing for such behavior, and show that IMRS is robust against dynamical noise as well as against parameter changes. IMRS may underlie neuronal information processing and potentially lead to network solutions for encryption key distribution and secure communication.

Diversity of coupled oscillators can enhance their synchronization

The heterogeneity of coupled oscillators is important for the degree of their synchronization. According to the classical Kuramoto model, larger heterogeneity reduces synchronization. Here, we show that in a model for coupled pancreatic β-cells, higher diversity of the cells induces higher synchrony. We find that any system of coupled oscillators that oscillates on two time scales and in which heterogeneity causes a transition from chaotic to damped oscillations on the fast time scale exhibits this property. Thus, synchronization of a subset of oscillating systems can be enhanced by increasing the heterogeneity of the system constituents.

Network-complement transitions, symmetries, and cluster synchronization

Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here, we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.

Impulse-induced localized control of chaos in starlike networks

Locally decreasing the impulse transmitted by periodic pulses is shown to be a reliable method of taming chaos in starlike networks of dissipative nonlinear oscillators, leading to both synchronous periodic states and equilibria (oscillation death). Specifically, the paradigmatic model of damped kicked rotators is studied in which it is assumed that when the rotators are driven synchronously, i.e., all driving pulses transmit the same impulse, the networks display chaotic dynamics. It is found that the taming effect of decreasing the impulse transmitted by the pulses acting on particular nodes strongly depends on their number and degree of connectivity. A theoretical analysis is given explaining the basic physical mechanism as well as the main features of the chaos-control scenario.

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

Amplitude dynamics favors synchronization in complex networks

In this paper we study phase synchronization in random complex networks of coupled periodic oscillators. In particular, we show that, when amplitude dynamics is not negligible, phase synchronization may be enhanced. To illustrate this, we compare the behavior of heterogeneous units with both amplitude and phase dynamics and pure (Kuramoto) phase oscillators. We find that in small network motifs the behavior crucially depends on the topology and on the node frequency distribution. Surprisingly, the microscopic structures for which the amplitude dynamics improves synchronization are those that are statistically more abundant in random complex networks. Thus, amplitude dynamics leads to a general lowering of the synchronization threshold in arbitrary random topologies. Finally, we show that this synchronization enhancement is generic of oscillators close to Hopf bifurcations. To this aim we consider coupled FitzHugh-Nagumo units modeling neuron dynamics.

Inhomogeneity induces relay synchronization in complex networks

Relay synchronization is a collective state, originally found in chains of interacting oscillators, in which uncoupled dynamical units synchronize through the action of mismatched inner nodes that relay the information but do not synchronize with them. It is demonstrated herein that relay synchronization is not limited to such simple motifs, rather it can emerge in larger and arbitrary network topologies. In particular, we show how this phenomenon can be observed in networks of chaotic systems in the presence of some mismatched units, the relay nodes, and how it is actually responsible for an enhancement of synchronization in the network.

Emergence and coherence of oscillations in star networks of stochastic excitable elements

We study the emergence and coherence of stochastic oscillations in star networks of excitable elements in which peripheral nodes receive independent random inputs. A biophysical model of a distal branch of sensory neuron in which peripheral nodes of Ranvier are coupled to a central node by myelinated cable segments is used along with a generic model of networked stochastic active rotators. We show that coherent oscillations can emerge due to stochastic synchronization of peripheral nodes and that the degree of coherence can be maximized by tuning the coupling strength and the size of the network. Analytical results are obtained for the strong-coupling regime of the active rotator network. In particular, we show that in the strong-coupling regime, the network dynamics can be described by an effective single active rotator with rescaled parameters and noise.

Remote synchronization of amplitudes across an experimental ring of non-linear oscillators

In this paper, the emergence of remote synchronization in a ring of 32 unidirectionally coupled non-linear oscillators is reported. Each oscillator consists of 3 negative voltage gain stages connected in a loop to which two integrators are superimposed and receives input from its preceding neighbour via a "mixing" stage whose gains form the main system control parameters. Collective behaviour of the network is investigated numerically and experimentally, based on a custom-designed circuit board featuring 32 field-programmable analog arrays. A diverse set of synchronization patterns is observed depending on the control parameters. While phase synchronization ensues globally, albeit imperfectly, for certain control parameter values, amplitudes delineate subsets of non-adjacent but preferentially synchronized nodes; this cannot be trivially explained by synchronization paths along sequences of structurally connected nodes and is therefore interpreted as representing a form of remote synchronization. Complex topology of functional synchronization thus emerges from underlying elementary structural connectivity. In addition to the Kuramoto order parameter and cross-correlation coefficient, other synchronization measures are considered, and preliminary findings suggest that generalized synchronization may identify functional relationships across nodes otherwise not visible. Further work elucidating the mechanism underlying this observation of remote synchronization is necessary, to support which experimental data and board design materials have been made freely downloadable.

Experimental study of synchronization of coupled electrical self-oscillators and comparison to the Sakaguchi-Kuramoto model

We explore the collective phase dynamics of Wien-bridge oscillators coupled resistively. We carefully analyze the behavior of two coupled oscillators, obtaining a transformation from voltage to effective phase. From the phase dynamics we show that the coupling can be quantitatively described by Sakaguchi's modification to the Kuramoto model. We also examine an ensemble of oscillators whose frequencies are taken from a flat distribution within a fixed frequency interval. We characterize in detail the synchronized cluster, its initial formation, as well as its effect on unsynchronized oscillators, all as a function of a global coupling strength.

Variety of regimes of starlike networks of Hénon maps

In this paper we categorize dynamical regimes demonstrated by starlike networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a starlike structure is the main motif of them. We analyze starlike networks of Hénon maps. They are found to demonstrate a huge diversity of regimes. Varying the coupling strength we reveal chaos, quasiperiodicity, and periodicity. The nodes can be both fully and phase synchronized. The hub node can be either synchronized with the subordinate nodes or oscillate separately from fully synchronized subordinates. There is a range of wild multistability where the zoo of regimes is the most various. One can hardly predict here even a qualitative nature of the expected solution, since each perturbation of the coupling strength or initial conditions results in a new character of dynamics.

Explosive synchronization is discontinuous

Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated with the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out of) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite-size graphs are well described by our predictions.

Impact of a leader on cluster synchronization

We study the mechanisms of frequency-synchronized cluster formation in coupled nonidentical oscillators and investigate the impact of presence of a leader on the cluster synchronization. We find that the introduction of a leader, a node having large parameter mismatch, induces a profound change in the cluster pattern as well as in the mechanism of the cluster formation. The emergence of a leader generates a transition from the driven to the mixed cluster state. The frequency mismatch turns out to be responsible for this transition. Additionally, for a chaotic evolution, the driven mechanism stands as a primary mechanism for the cluster formation, whereas for a periodic evolution the self-organization mechanism becomes equally responsible.