Symmetries and cluster synchronization in multilayer networks

Synchronization cluster bursting in adaptive oscillator networks

Adaptive dynamical networks are ubiquitous in real-world systems. This paper aims to explore the synchronization dynamics in networks of adaptive oscillators based on a paradigmatic system of adaptively coupled phase oscillators. Our numerical observations reveal the emergence of synchronization cluster bursting, characterized by periodic transitions between cluster synchronization and global synchronization. By investigating a reduced model, the mechanisms underlying synchronization cluster bursting are clarified. We show that a minimal model exhibiting this phenomenon can be reduced to a phase oscillator with complex-valued adaptation. Furthermore, the adaptivity of the system leads to the appearance of additional symmetries, and thus, to the coexistence of stable bursting solutions with very different Kuramoto order parameters.

Symmetry breaker governs synchrony patterns in neuronal inspired networks

Experiments in the human brain reveal switching between different activity patterns and functional network organization over time. Recently, multilayer modeling has been employed across multiple neurobiological levels (from spiking networks to brain regions) to unveil novel insights into the emergence and time evolution of synchrony patterns. We consider two layers with the top layer directly coupled to the bottom layer. When isolated, the bottom layer would remain in a specific stable pattern. However, in the presence of the top layer, the network exhibits spatiotemporal switching. The top layer in combination with the inter-layer coupling acts as a symmetry breaker, governing the bottom layer and restricting the number of allowed symmetry-induced patterns. This structure allows us to demonstrate the existence and stability of pattern states on the bottom layer, but most remarkably, it enables a simple mechanism for switching between patterns based on the unique symmetry-breaking role of the governing layer. We demonstrate that the symmetry breaker prevents complete synchronization in the bottom layer, a situation that would not be desirable in a normal functioning brain. We illustrate our findings using two layers of Hindmarsh-Rose (HR) oscillators, employing the Master Stability function approach in small networks to investigate the switching between patterns.

Node-layer duality in networked systems

Real-world networks typically exhibit several aspects, or layers, of interactions among their nodes. By permuting the role of the nodes and the layers, we establish a new criterion to construct the dual of a network. This approach allows to examine connectivity from either a node-centric or layer-centric viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrate that nodewise and layerwise connectivity measure different but related aspects of the same system. Leveraging node-layer duality provides complementary insights, enabling a deeper comprehension of diverse networks across social science, technology and biology. Taken together, these findings reveal previously unappreciated features of complex systems and provide a fresh tool for delving into their structure and dynamics.

The transition to synchronization of networked systems

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes

In this paper, we investigate synchronous solutions of coupled van der Pol oscillator systems with multiple coupling modes using the theory of rotating periodic solutions. Multiple coupling modes refer to two or three types of coupling modes in van der Pol oscillator networks, namely, position, velocity, and acceleration. Rotating periodic solutions can represent various types of synchronous solutions corresponding to different phase differences of coupled oscillators. When matrices representing the topology of different coupling modes have symmetry, the overall symmetry of the oscillator system depends on the intersection of the symmetries of the different topologies, determining the type of synchronous solutions for the coupled oscillator network. When matrices representing the topology of different coupling modes lack symmetry, if the adjacency matrices representing different coupling modes can be simplified into structurally identical quotient graphs (where weights can be proportional) through the same external equitable partition, the symmetry of the quotient graph determines the synchronization type of the original system. All these results are consistent with multi-layer networks where connections between different layers are one-to-one.

Mode-Mixed Effects Based Intralayer-Dependent Impulsive Synchronization for Multiple Mismatched Multilayer Neural Networks

This article focuses on the intralayer-dependent impulsive synchronization of multiple mismatched multilayer neural networks (NNs) with mode-mixed effects. Initially, a novel multilayer NN model that removes the one-to-one interlayer coupling constraint and introduces nonidentical model parameters is first established to meet diverse modeling requirements in complex applications. To help the multilayer target NNs with mismatched connection coefficients and time delays achieve synchronization, the hybrid controller is designed using intralayer-dependent impulsive control and switched feedback control approaches. Furthermore, the mode-mixed effects caused by the intralayer coupling delays and switched intralayer topologies are incorporated into the novel model and analysis method to ensure that the subsystems operating within the current switching interval can effectively use the topology information of the previous switching intervals. Then, a novel analysis framework including super-Laplacian matrix, augmented matrix, and mode-mixed methods is developed to derive the synchronization results. Finally, the main results are verified via the numerical simulation with secure communication.

Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapses

This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

Regulation of cluster synchronization in multilayer networks of delay coupled semiconductor lasers with the use of disjoint layer symmetry

In real-world complex systems, heterogeneous components often interact in complex connection patterns and could be schematized by a formalism of multilayer network. In this work, the synchronization characteristics of multilayer network composed of semiconductor lasers (SLs) are investigated systematically. It is demonstrated that the interplay between different layers plays an important role on the synchronization patterns. We elucidate that the performance of cluster synchronization could be facilitated effectively with the introduction of disjoint layer symmetry into network topology. Intertwined stability of clusters from different layers could be decoupled into independent, and the parameter spaces for stable synchronization are extended significantly. The robustness of our proposed regulation scheme on operation parameters is numerically evaluated. Furthermore, the generality of presented theoretical results is validated in networks with more complex topology and multiple layers.

Desynchrony induced by higher-order interactions in triplex metapopulations

In a predator-prey metapopulation, two traits are adversely related: synchronization and persistence. A decrease in synchrony apparently leads to an increase in persistence and, therefore, necessitates the study of desynchrony in a metapopulation. In this article, we study predator-prey patches that communicate with one another while being interconnected through distinct dispersal structures in the layers of a three-layer multiplex network. We investigate the synchronization phenomenon among the patches of the outer layers by introducing higher-order interactions (specifically three-body interactions) in the middle layer. We observe a decrease in the synchronous behavior or, alternatively, an increase in desynchrony due to the inclusion of group interactions among the patches of the middle layer. The advancement of desynchrony becomes more prominent with increasing strength and numbers of three-way interactions in the middle layer. We analytically validate our numerical results by performing a stability analysis of the referred synchronous solution using the master stability function approach. Additionally, we verify our findings by taking into account two distinct predator-prey models and dispersal topologies, which ultimately supports that the findings are generalizable across various models and dispersal structures.

The complementary contribution of each order topology into the synchronization of multi-order networks

Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.

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.

Generalized external synchronization of networks based on clustered pandemic systems-The approach of Covid-19 towards influenza

Real-world models, like those used in social studies, epidemiology, energy transport, engineering, and finance, are often called "multi-layer networks." In this work, we have described a controller that connects the paths of synchronized models that are grouped together in clusters. We did this using Lyapunov theory and a variety of coupled matrices to look into the link between the groups of chaotic systems based on influenza and covid-19. Our work also includes the use of external synchrony in biological systems. For example, we have explained in detail how the pandemic disease covid-19 will get weaker over time and become more like influenza. The analytical way to get these answers is to prove a theorem, and the numerical way is to use MATLAB to run numerical simulations.

Synchronization Induced by Layer Mismatch in Multiplex Networks

Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers.

Transitional cluster dynamics in a model for delay-coupled chemical oscillators

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.

Interlayer antisynchronization in degree-biased duplex networks

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Pinning control of networks: Dimensionality reduction through simultaneous block-diagonalization of matrices

In this paper, we study the network pinning control problem in the presence of two different types of coupling: (i) node-to-node coupling among the network nodes and (ii) input-to-node coupling from the source node to the "pinned nodes." Previous work has mainly focused on the case that (i) and (ii) are of the same type. We decouple the stability analysis of the target synchronous solution into subproblems of the lowest dimension by using the techniques of simultaneous block diagonalization of matrices. Interestingly, we obtain two different types of blocks, driven and undriven. The overall dimension of the driven blocks is equal to the dimension of an appropriately defined controllable subspace, while all the remaining undriven blocks are scalar. Our main result is a decomposition of the stability problem into four independent sets of equations, which we call quotient controllable, quotient uncontrollable, redundant controllable, and redundant uncontrollable. Our analysis shows that the number and location of the pinned nodes affect the number and the dimension of each set of equations. We also observe that in a large variety of complex networks, the stability of the target synchronous solution is de facto only determined by a single quotient controllable block.

Structural position vectors and symmetries in complex networks

Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.

Stability of synchronization in simplicial complexes with multiple interaction layers

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

When multilayer links exchange their roles in synchronization

Real world networks contain multiple layers of links whose interactions can lead to extraordinary collective dynamics, including synchronization. The fundamental problem of assessing how network topology controls synchronization in multilayer networks remains open due to serious limitations of the existing stability methods. Towards removing this obstacle, we propose an approximation method which significantly enhances the predictive power of the master stability function for stable synchronization in multilayer networks. For a class of saddle-focus oscillators, including Rössler and piecewise linear systems, our method reduces the complex stability analysis to simply solving a set of linear algebraic equations. Using the method, we analytically predict surprising effects due to multilayer coupling. In particular, we prove that two coupling layers-one of which would alone hamper synchronization and the other would foster it-reverse their roles when used in a multilayer network. We also analytically demonstrate that increasing the size of a globally coupled layer, that in isolation would induce stable synchronization, makes the multilayer network unsynchronizable.

Supermodal Decomposition of the Linear Swing Equation for Multilayer Networks

We study the swing equation in the case of a multilayer network in which generators and motors are modeled differently; namely, the model for each generator is given by second order dynamics and the model for each motor is given by first order dynamics. We also remove the commonly used assumption of equal damping coefficients in the second order dynamics. Under these general conditions, we are able to obtain a decomposition of the linear swing equation into independent modes describing the propagation of small perturbations. In the process, we identify symmetries affecting the structure and dynamics of the multilayer network and derive an essential model based on a 'quotient network.' We then compare the dynamics of the full network and that of the quotient network and obtain a modal decomposition of the error dynamics. We also provide a method to quantify the steady-state error and the maximum overshoot error. Two case studies are presented to illustrate application of our method.

Lag Synchronization of Noisy and Nonnoisy Multiple Neurobiological Coupled FitzHugh–Nagumo Networks with and without Delayed Coupling

This paper presents a methodology for synchronizing noisy and nonnoisy multiple coupled neurobiological FitzHugh–Nagumo (FHN) drive and slave neural networks with and without delayed coupling, under external electrical stimulation (EES), external disturbance, and variable parameters for each state of both FHN networks. Each network of neurons was configured by considering all aspects of real neurons communications in the brain, i.e., synapse and gap junctions. Novel adaptive control laws were developed and proposed that guarantee the synchronization of FHN neural networks in different configurations. The Lyapunov stability theory was utilized to analytically derive the sufficient conditions that ensure the synchronization of the FHN networks. The effectiveness and robustness of the proposed control laws were shown through different numerical simulations.

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.

Looking beyond community structure leads to the discovery of dynamical communities in weighted networks

A fundamental question is whether groups of nodes of a complex network can possibly display long-term cluster-synchronized behavior. While this question has been addressed for the restricted classes of unweighted and labeled graphs, it remains an open problem for the more general class of weighted networks. The emergence of coordinated motion of nodes in natural and technological networks is directly related to the network structure through the concept of an equitable partition, which determines which nodes can show long-term synchronized behavior and which nodes cannot. We provide a method to detect the presence of nearly equitable partitions in weighted networks, based on minimal information about the network structure. With this approach we are able to discover the presence of dynamical communities in both synthetic and real technological, biological, and social networks, to a statistically significant level. We show that our approach based on dynamical communities is better at predicting the emergence of synchronized behavior than existing methods to detect community structure.

Intralayer and interlayer synchronization in multiplex network with higher-order interactions

Recent developments in complex systems have witnessed that many real-world scenarios, successfully represented as networks, are not always restricted to binary interactions but often include higher-order interactions among the nodes. These beyond pairwise interactions are preferably modeled by hypergraphs, where hyperedges represent higher-order interactions between a set of nodes. In this work, we consider a multiplex network where the intralayer connections are represented by hypergraphs, called the multiplex hypergraph. The hypergraph is constructed by mapping the maximal cliques of a scale-free network to hyperedges of suitable sizes. We investigate the intralayer and interlayer synchronizations of such multiplex structures. Our study unveils that the intralayer synchronization appreciably enhances when a higher-order structure is taken into consideration in spite of only pairwise connections. We derive the necessary condition for stable synchronization states by the master stability function approach, which perfectly agrees with the numerical results. We also explore the robustness of interlayer synchronization and find that for the multiplex structures with many-body interaction, the interlayer synchronization is more persistent than the multiplex networks with solely pairwise interaction.

A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling

A memristor is a vital circuit element that can mimic biological synapses. This paper proposes the memristive version of a recently proposed map neuron model based on the phase space. The dynamic of the memristive map model is investigated by using bifurcation and Lyapunov exponents’ diagrams. The results prove that the memristive map can present different behaviors such as spiking, periodic bursting, and chaotic bursting. Then, a ring network is constructed by hybrid electrical and chemical synapses, and the memristive neuron models are used to describe the nodes. The collective behavior of the network is studied. It is observed that chemical coupling plays a crucial role in synchronization. Different kinds of synchronization, such as imperfect synchronization, complete synchronization, solitary state, two-cluster synchronization, chimera, and nonstationary chimera, are identified by varying the coupling strengths.

Group synchrony, parameter mismatches, and intragroup connections

Group synchronization arises when two or more synchronization patterns coexist in a network formed of oscillators of different types, with the systems in each group synchronizing on the same time evolution, but systems in different groups synchronizing on distinct time evolutions. Group synchronization has been observed and characterized when the systems in each group are identical and the couplings between the systems satisfy specific conditions. By relaxing these constraints and allowing them to be satisfied in an approximate rather than exact way, we observe that stable group synchronization may still occur in the presence of small deviations of the parameters of the individual systems and of the couplings from their nominal values. We analyze this case and provide necessary and sufficient conditions for stability through a master stability function approach, which also allows us to quantify the synchronization error. We also investigate the stability of group synchronization in the presence of intragroup connections and for this case extend some of the existing results in the literature. Our analysis points out a broader class of matrices describing intragroup connections for which the stability problem can be reduced in a low-dimensional form.

Cluster synchronization of networks via a canonical transformation for simultaneous block diagonalization of matrices

We study cluster synchronization of networks and propose a canonical transformation for simultaneous block diagonalization of matrices that we use to analyze the stability of the cluster synchronous solution. Our approach has several advantages as it allows us to: (1) decouple the stability problem into subproblems of minimal dimensionality while preserving physically meaningful information, (2) study stability of both orbital and equitable partitions of the network nodes, and (3) obtain a parameterization of the problem in a small number of parameters. For the last point, we show how the canonical transformation decouples the problem into blocks that preserve key physical properties of the original system. We also apply our proposed algorithm to analyze several real networks of interest, and we find that it runs faster than alternative algorithms from the literature.

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.

Predicting synchronized gene coexpression patterns from fibration symmetries in gene regulatory networks in bacteria

BACKGROUND

Gene regulatory networks coordinate the expression of genes across physiological states and ensure a synchronized expression of genes in cellular subsystems, critical for the coherent functioning of cells. Here we address the question whether it is possible to predict gene synchronization from network structure alone. We have recently shown that synchronized gene expression can be predicted from symmetries in the gene regulatory networks described by the concept of symmetry fibrations. We showed that symmetry fibrations partition the genes into groups called fibers based on the symmetries of their 'input trees', the set of paths in the network through which signals can reach a gene. In idealized dynamic gene expression models, all genes in a fiber are perfectly synchronized, while less idealized models-with gene input functions differencing between genes-predict symmetry breaking and desynchronization.

RESULTS

To study the functional role of gene fibers and to test whether some of the fiber-induced coexpression remains in reality, we analyze gene fibrations for the gene regulatory networks of E. coli and B. subtilis and confront them with expression data. We find approximate gene coexpression patterns consistent with symmetry fibrations with idealized gene expression dynamics. This shows that network structure alone provides useful information about gene synchronization, and suggest that gene input functions within fibers may be further streamlined by evolutionary pressures to realize a coexpression of genes.

CONCLUSIONS

Thus, gene fibrations provide a sound conceptual tool to describe tunable coexpression induced by network topology and shaped by mechanistic details of gene expression.

One-way dependent clusters and stability of cluster synchronization in directed networks

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.

Mechanisms of cluster formation in networks with directed links differ from those in undirected networks. Lodi et al. propose a method to compute interdependencies among clusters of nodes in directed networks. They show that clusters can be one-way dependent, as found in social and neural networks.

Computational exploration of dynamic mechanisms of steady state visual evoked potentials at the whole brain level

Periodic visual stimulation can induce stable steady-state visual evoked potentials (SSVEPs) distributed in multiple brain regions and has potential applications in both neural engineering and cognitive neuroscience. However, the underlying dynamic mechanisms of SSVEPs at the whole-brain level are still not completely understood. Here, we addressed this issue by simulating the rich dynamics of SSVEPs with a large-scale brain model designed with constraints of neuroimaging data acquired from the human brain. By eliciting activity of the occipital areas using an external periodic stimulus, our model was capable of replicating both the spatial distributions and response features of SSVEPs that were observed in experiments. In particular, we confirmed that alpha-band (8-12 Hz) stimulation could evoke stronger SSVEP responses; this frequency sensitivity was due to nonlinear entrainment and resonance, and could be modulated by endogenous factors in the brain. Interestingly, the stimulus-evoked brain networks also exhibited significant superiority in topological properties near this frequency-sensitivity range, and stronger SSVEP responses were demonstrated to be supported by more efficient functional connectivity at the neural activity level. These findings not only provide insights into the mechanistic understanding of SSVEPs at the whole-brain level but also indicate a bright future for large-scale brain modeling in characterizing the complicated dynamics and functions of the brain.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Lag synchronization of coupled time-delayed FitzHugh-Nagumo neural networks via feedback control

Synchronization plays a significant role in information transfer and decision-making by neurons and brain neural networks. The development of control strategies for synchronizing a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is an essential and very challenging task. Researchers have extensively studied the synchronization mechanism of two coupled time-delayed neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigates the synchronization problem in a network of coupled FitzHugh-Nagumo (FHN) neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in the mathematical models. More specifically, this study investigates the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN neuronal systems with and without external noise. Different gap junctions and delay parameters are used to incorporate time-delay dynamics in both neuronal networks. We also investigate the influence of the time delays between connected neurons on synchronization conditions. Further, to ensure the synchronization of the time-delayed FHN neuronal networks, different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks. In addition, necessary and sufficient conditions to achieve synchronization are provided by employing the Lyapunov stability theory. The results of numerical simulations conducted for different-sized multiple networks of time-delayed FHN neurons verify the effectiveness of the proposed adaptive control schemes.

Analysis of input-induced oscillations using the isostable coordinate framework

Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.

Delays induced cluster synchronization in chaotic networks

We study networks of coupled oscillators and analyze the role of coupling delays in determining the emergence of cluster synchronization. Given a network topology and a particular arrangement of the coupling delays over the network connections, different patterns of cluster synchronization may emerge. We focus on a simple ring network of six bidirectionally coupled identical oscillators, for which with two different values of the delays, a total of eight cluster synchronization patterns may emerge, depending on the assignment of the delays to the ring connections. We analyze stability of each of the patterns and find that for large enough coupling strength and specific values of the delays, they can all be stabilized. We construct an experimental ring of six bidirectionally coupled Colpitts oscillators, with delayed connections obtained by coupling the oscillators via RF cables of appropriate length. We find that experimental observations of cluster synchronization are in essential agreement with theoretical predictions. We also verify our theory in a fully connected network of fifty nodes for which connections are randomly assigned to be either undelayed or delayed with a given probability.

Analyzing synchronized clusters in neuron networks

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.