Synchronization properties of network motifs: influence of coupling delay and symmetry

A mathematical model for pancreatic cancer during intraepithelial neoplasia

Cancer is the result of complex interactions of intrinsic and extrinsic cell processes, which promote sustained proliferation, resistance to apoptosis, reprogramming and reorganization. The evolution of any type of cancer emerges from the role of the microenvironmental conditions and their impact of some molecular complexes on certain signalling pathways. The understanding of the early onset of cancer requires a multiscale analysis of the cellular microenvironment. In this paper, we analyse a qualitative multiscale model of pancreatic adenocarcinoma by modelling the cellular microenvironment through elastic cell interactions and their intercellular communication mechanisms, such as growth factors and cytokines. We focus on the low-grade dysplasia (PanIN 1) and moderate dysplasia (PanIN 2) stages of pancreatic adenocarcinoma. To this end, we propose a gene-regulatory network associated with the processes of proliferation and apoptosis of pancreatic cells and its kinetics in terms of delayed differential equations to mimic cell development. Likewise, we couple the cell cycle with the spatial distribution of cells and the transport of growth factors to show that the adenocarcinoma evolution is triggered by inflammatory processes. We show that the oncogene RAS may be an important target for developing anti-inflammatory strategies that limit the emergence of more aggressive adenocarcinomas.

Synchronization of delayed coupled neurons with multiple synaptic connections

Synchronization is a key feature of the brain dynamics and is necessary for information transmission across brain regions and in higher brain functions like cognition, learning and memory. Experimental findings demonstrated that in cortical microcircuits there are multiple synapses between pairs of connected neurons. Synchronization of neurons in the presence of multiple synaptic connections may be relevant for optimal learning and memory, however, its effect on the dynamics of the neurons is not adequately studied. Here, we address the question that how changes in the strength of the synaptic connections and transmission delays between neurons impact synchronization in a two-neuron system with multiple synapses. To this end, we analytically and computationally investigated synchronization dynamics by considering both phase oscillator model and conductance-based Hodgkin-Huxley (HH) model. Our results show that symmetry/asymmetry of feedforward and feedback connections crucially determines stability of the phase locking of the system based on the strength of connections and delays. In both models, the two-neuron system with multiple synapses achieves in-phase synchrony in the presence of small and large delays, whereas an anti-phase synchronization state is favored for median delays. Our findings can expand the understanding of the functional role of multisynaptic contacts in neuronal synchronization and may shed light on the dynamical consequences of pathological multisynaptic connectivity in a number of brain disorders.

Adaptive myelination causes slow oscillations in recurrent neural loops

The brain is known to be plastic, i.e., capable of changing and reorganizing as it develops and accumulates experience. Recently, a novel form of brain plasticity was described which is activity-dependent myelination of nerve fibers. Since the speed of propagation of action potentials along axons depends significantly on their degree of myelination, this process leads to adaptive change of axonal delays depending on the neural activity. To understand the possible influence of the adaptive delays on the behavior of neural networks, we consider a simple setup, a neuronal oscillator with delayed feedback. We show that introducing the delay plasticity into this circuit can lead to the occurrence of slow oscillations which are impossible with a constant delay.

Fundamental patterns of signal propagation in complex networks

Various disasters stem from minor perturbations, such as the spread of infectious diseases and cascading failure in power grids. Analyzing perturbations is crucial for both theoretical and application fields. Previous researchers have proposed basic propagation patterns for perturbation and explored the impact of basic network motifs on the collective response to these perturbations. However, the current framework is limited in its ability to decouple interactions and, therefore, cannot analyze more complex structures. In this article, we establish an effective, robust, and powerful propagation framework under a general dynamic model. This framework reveals classical and dense network motifs that exert critical acceleration on signal propagation, often reducing orders of magnitude compared with conclusions generated by previous work. Moreover, our framework provides a new approach to understand the fundamental principles of complex systems and the negative feedback mechanism, which is of great significance for researching system controlling and network resilience.

Self-organization of in vitro neuronal assemblies drives to complex network topology

Activity-dependent self-organization plays an important role in the formation of specific and stereotyped connectivity patterns in neural circuits. By combining neuronal cultures, and tools with approaches from network neuroscience and information theory, we can study how complex network topology emerges from local neuronal interactions. We constructed effective connectivity networks using a transfer entropy analysis of spike trains recorded from rat embryo dissociated hippocampal neuron cultures between 6 and 35 days in vitro to investigate how the topology evolves during maturation. The methodology for constructing the networks considered the synapse delay and addressed the influence of firing rate and population bursts as well as spurious effects on the inference of connections. We found that the number of links in the networks grew over the course of development, shifting from a segregated to a more integrated architecture. As part of this progression, three significant aspects of complex network topology emerged. In agreement with previous in silico and in vitro studies, a small-world architecture was detected, largely due to strong clustering among neurons. Additionally, the networks developed in a modular topology, with most modules comprising nearby neurons. Finally, highly active neurons acquired topological characteristics that made them important nodes to the network and integrators of modules. These findings leverage new insights into how neuronal effective network topology relates to neuronal assembly self-organization mechanisms.

Dynamics of coupled Kuramoto oscillators with distributed delays

This paper studies the effects of two different types of distributed-delay coupling in the system of two mutually coupled Kuramoto oscillators: one where the delay distribution is considered inside the coupling function and the other where the distribution enters outside the coupling function. In both cases, the existence and stability of phase-locked solutions is analyzed for uniform and gamma distribution kernels. The results show that while having the distribution inside the coupling function only changes parameter regions where phase-locked solutions exist, when the distribution is taken outside the coupling function, it affects both the existence, as well as stability properties of in- and anti-phase states. For both distribution types, various branches of phase-locked solutions are computed, and regions of their stability are identified for uniform, weak, and strong gamma distributions.

Information Transmission in Delay-Coupled Neuronal Circuits in the Presence of a Relay Population

Synchronization between neuronal populations is hypothesized to play a crucial role in the communication between brain networks. The binding of features, or the association of computations occurring in spatially segregated areas, is supposed to take place when a stable synchronization between cortical areas occurs. While a direct cortico-cortical connection typically fails to support this mechanism, the participation of a third area, a relay element, mediating in the communication was proposed to overcome this limitation. Among the different structures that could play the role of coordination during the binding process, the thalamus is the best placed region to carry out this task. In this paper we study how information flows in a canonical motif that mimics a cortico-thalamo-cortical circuit composed by three mutually coupled neuronal populations (also called the V-motif). Through extensive numerical simulations, we found that the amount of information transferred between the oscillating neuronal populations is determined by the delay in their connections and the mismatch in their oscillation frequencies (detuning). While the transmission from a cortical population is mostly restricted to positive detuning, transmission from the relay (thalamic) population to the cortical populations is robust for a broad range of detuning values, including negative values, while permitting feedback communication from the cortex at high frequencies, thus supporting robust bottom up and top down interaction. In this case, a strong feedback transmission between the cortex to thalamus supports the possibility of robust bottom-up and top-down interactions in this motif. Interestingly, adding a cortico-cortical bidirectional connection to the V-motif (C-motif) expands the dynamics of the system with distinct operation modes. While overall transmission efficiency is decreased, new communication channels establish cortico-thalamo-cortical association loops. Switching between operation modes depends on the synaptic strength of the cortico-cortical connections. Our results support a role of the transthalamic V-motif in the binding of spatially segregated cortical computations, and suggest an important regulatory role of the direct cortico-cortical connection.

Subgraphs of Interest Social Networks for Diffusion Dynamics Prediction

Finding the building blocks of real-world networks contributes to the understanding of their formation process and related dynamical processes, which is related to prediction and control tasks. We explore different types of social networks, demonstrating high structural variability, and aim to extract and see their minimal building blocks, which are able to reproduce supergraph structural and dynamical properties, so as to be appropriate for diffusion prediction for the whole graph on the base of its small subgraph. For this purpose, we determine topological and functional formal criteria and explore sampling techniques. Using the method that provides the best correspondence to both criteria, we explore the building blocks of interest networks. The best sampling method allows one to extract subgraphs of optimal 30 nodes, which reproduce path lengths, clustering, and degree particularities of an initial graph. The extracted subgraphs are different for the considered interest networks, and provide interesting material for the global dynamics exploration on the mesoscale base.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

Synchronization of complex human networks

The synchronization of human networks is essential for our civilization and understanding its dynamics is important to many aspects of our lives. Human ensembles were investigated, but in noisy environments and with limited control over the network parameters which govern the network dynamics. Specifically, research has focused predominantly on all-to-all coupling, whereas current social networks and human interactions are often based on complex coupling configurations. Here, we study the synchronization between violin players in complex networks with full and accurate control over the network connectivity, coupling strength, and delay. We show that the players can tune their playing period and delete connections by ignoring frustrating signals, to find a stable solution. These additional degrees of freedom enable new strategies and yield better solutions than are possible within current models such as the Kuramoto model. Our results may influence numerous fields, including traffic management, epidemic control, and stock market dynamics.

How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales

In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős-Rényi network. This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole.

Emergence of global synchronization in directed excitatory networks of type I neurons

The collective behaviour of neural networks depends on the cellular and synaptic properties of the neurons. The phase-response curve (PRC) is an experimentally obtainable measure of cellular properties that quantifies the shift in the next spike time of a neuron as a function of the phase at which stimulus is delivered to that neuron. The neuronal PRCs can be classified as having either purely positive values (type I) or distinct positive and negative regions (type II). Networks of type 1 PRCs tend not to synchronize via mutual excitatory synaptic connections. We study the synchronization properties of identical type I and type II neurons, assuming unidirectional synapses. Performing the linear stability analysis and the numerical simulation of the extended Kuramoto model, we show that feedforward loop motifs favour synchronization of type I excitatory and inhibitory neurons, while feedback loop motifs destroy their synchronization tendency. Moreover, large directed networks, either without feedback motifs or with many of them, have been constructed from the same undirected backbones, and a high synchronization level is observed for directed acyclic graphs with type I neurons. It has been shown that, the synchronizability of type I neurons depends on both the directionality of the network connectivity and the topology of its undirected backbone. The abundance of feedforward motifs enhances the synchronizability of the directed acyclic graphs.

Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh-Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Stochastic bursting in unidirectionally delay-coupled noisy excitable systems

We show that "stochastic bursting" is observed in a ring of unidirectional delay-coupled noisy excitable systems, thanks to the combinational action of time-delayed coupling and noise. Under the approximation of timescale separation, i.e., when the time delays in each connection are much larger than the characteristic duration of the spikes, the observed rather coherent spike pattern can be described by an idealized coupled point process with a leader-follower relationship. We derive analytically the statistics of the spikes in each unit, the pairwise correlations between any two units, and the spectrum of the total output from the network. Theory is in good agreement with the simulations with a network of theta-neurons.

Dendritic and Axonal Propagation Delays May Shape Neuronal Networks With Plastic Synapses

Biological neuronal networks are highly adaptive and plastic. For instance, spike-timing-dependent plasticity (STDP) is a core mechanism which adapts the synaptic strengths based on the relative timing of pre- and postsynaptic spikes. In various fields of physiology, time delays cause a plethora of biologically relevant dynamical phenomena. However, time delays increase the complexity of model systems together with the computational and theoretical analysis burden. Accordingly, in computational neuronal network studies propagation delays were often neglected. As a downside, a classic STDP rule in oscillatory neurons without propagation delays is unable to give rise to bidirectional synaptic couplings, i.e., loops or uncoupled states. This is at variance with basic experimental results. In this mini review, we focus on recent theoretical studies focusing on how things change in the presence of propagation delays. Realistic propagation delays may lead to the emergence of neuronal activity and synaptic connectivity patterns, which cannot be captured by classic STDP models. In fact, propagation delays determine the inventory of attractor states and shape their basins of attractions. The results reviewed here enable to overcome fundamental discrepancies between theory and experiments. Furthermore, these findings are relevant for the development of therapeutic brain stimulation techniques aiming at shifting the diseased brain to more favorable attractor states.

Propagation delays determine neuronal activity and synaptic connectivity patterns emerging in plastic neuronal networks

In plastic neuronal networks, the synaptic strengths are adapted to the neuronal activity. Specifically, spike-timing-dependent plasticity (STDP) is a fundamental mechanism that modifies the synaptic strengths based on the relative timing of pre- and postsynaptic spikes, taking into account the spikes' temporal order. In many studies, propagation delays were neglected to avoid additional dynamic complexity or computational costs. So far, networks equipped with a classic STDP rule typically rule out bidirectional couplings (i.e., either loops or uncoupled states) and are, hence, not able to reproduce fundamental experimental findings. In this review paper, we consider additional features, e.g., extensions of the classic STDP rule or additional aspects like noise, in order to overcome the contradictions between theory and experiment. In addition, we review in detail recent studies showing that a classic STDP rule combined with realistic propagation patterns is able to capture relevant experimental findings. In two coupled oscillatory neurons with propagation delays, bidirectional synapses can be preserved and potentiated. This result also holds for large networks of type-II phase oscillators. In addition, not only the mean of the initial distribution of synaptic weights, but also its standard deviation crucially determines the emergent structural connectivity, i.e., the mean final synaptic weight, the number of two-neuron loops, and the symmetry of the final connectivity pattern. The latter is affected by the firing rates, where more symmetric synaptic configurations emerge at higher firing rates. Finally, we discuss these findings in the context of the computational neuroscience-based development of desynchronizing brain stimulation techniques.

Bistability in two simple symmetrically coupled oscillators with symmetry-broken amplitude- and phase-locking

In the model system of two instantaneously and symmetrically coupled identical Stuart-Landau oscillators, we demonstrate that there exist stable solutions with symmetry-broken amplitude- and phase-locking. These states are characterized by a non-trivial fixed phase or amplitude relationship between both oscillators, while simultaneously maintaining perfectly harmonic oscillations of the same frequency. While some of the surrounding bifurcations have been previously described, we present the first detailed analytical and numerical description of these states and present analytically and numerically how they are embedded in the bifurcation structure of the system, arising both from the in-phase and the anti-phase solutions, as well as through a saddle-node bifurcation. The dependence of both the amplitude and the phase on parameters can be expressed explicitly with analytic formulas. As opposed to the previous reports, we find that these symmetry-broken states are stable, which can even be shown analytically. As an example of symmetry-breaking solutions in a simple and symmetric system, these states have potential applications as bistable states for switches in a wide array of coupled oscillatory systems.

Anticipated and zero-lag synchronization in motifs of delay-coupled systems

Anticipated and zero-lag synchronization have been observed in different scientific fields. In the brain, they might play a fundamental role in information processing, temporal coding and spatial attention. Recent numerical work on anticipated and zero-lag synchronization studied the role of delays. However, an analytical understanding of the conditions for these phenomena remains elusive. In this paper, we study both phenomena in systems with small delays. By performing a phase reduction and studying phase locked solutions, we uncover the functional relation between the delay, excitation and inhibition for the onset of anticipated synchronization in a sender-receiver-interneuron motif. In the case of zero-lag synchronization in a chain motif, we determine the stability conditions. These analytical solutions provide an excellent prediction of the phase-locked regimes of Hodgkin-Huxley models and Roessler oscillators.

Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

We study the dynamics of an array of nearest-neighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that, unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.

Analysis and application of neuronal network controllability and observability

Controllability and observability analyses are important prerequisite for designing suitable neural control strategy, which can help lower the efforts required to control and observe the system dynamics. First, 3-neuron motifs including the excitatory motif, the inhibitory motif, and the mixed motif are constructed to investigate the effects of single neuron and synaptic dynamics on network controllability (observability). Simulation results demonstrate that for networks with the same topological structure, the controllability (observability) of the node always changes if the properties of neurons and synaptic coupling strengths vary. Besides, the inhibitory networks are more controllable (observable) than the excitatory networks when the coupling strengths are the same. Then, the numerically determined controllability results of 3-neuron excitatory motifs are generalized to the desynchronization control of the modular motif network. The control energy and neuronal synchrony measure indexes are used to quantify the controllability of each node in the modular network. The best driver node obtained in this way is the same as the deduced one from motif analysis.

Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators

In a recent study about chaos synchronization in complex networks [Nat. Commun. 5, 4079 (2014)NCAOBW2041-172310.1038/ncomms5079], it is shown that a stable synchronous cluster may coexist with vast asynchronous nodes, resembling the phenomenon of a chimera state observed in a regular network of coupled periodic oscillators. Although of practical significance, this new type of state, namely, the isolated-desynchronization state, is hardly observed in practice due to its strict requirements on the network topology. Here, by the strategy of pinning coupling, we propose an effective method for inducing isolated-desynchronization states in symmetric networks of coupled chaotic oscillators. Theoretical analysis based on eigenvalue analysis shows that, by pinning a group of symmetric nodes in the network, there exists a critical pinning strength beyond which the group of pinned nodes can completely be synchronized while the unpinned nodes remain asynchronous. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks with the numerical results in good agreement with the theoretical predictions.

Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry

We study the effect of time delay on the dynamics of a system of repulsively coupled nonlinear oscillators that are configured as a geometrically frustrated network. In the absence of time delay, frustrated systems are known to possess a high degree of multistability among a large number of coexisting collective states except for the fully synchronized state that is normally obtained for attractively coupled systems. Time delay in the coupling is found to remove this constraint and to lead to such a synchronized ground state over a range of parameter values. A quantitative study of the variation of frustration in a system with the amount of time delay has been made and a universal scaling behavior is found. The variation in frustration as a function of the product of time delay and the collective frequency of the system is seen to lie on a characteristic curve that is common for all natural frequencies of the identical oscillators and coupling strengths. Thus time delay can be used as a tuning parameter to control the amount of frustration in a system and thereby influence its collective behavior. Our results can be of potential use in a host of practical applications in physical and biological systems in which frustrated configurations and time delay are known to coexist.

Dynamics of three unidirectionally coupled autonomous Duffing oscillators and application to inchworm piezoelectric motors: Effects of the coupling coefficient and delay

This work deals with the dynamics of three unidirectionally coupled Duffing oscillators and that of three coupled piezoelectric actuators, considering the special case of inchworm motors. Two configurations of the network are studied: ring configuration and chain configuration. The effects of the coupling coefficient and the time delay are analyzed through different bifurcation diagrams and phase difference variation. It is shown that varying the coupling coefficient and the time delay leads to the appearance of different dynamical behaviors: steady states, periodic and quasiperiodic oscillations, chaos, and phase synchronization.

Graph partitions and cluster synchronization in networks of oscillators

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.

Travelling waves in arrays of delay-coupled phase oscillators

We consider the effects of several forms of delays on the existence and stability of travelling waves in non-locally coupled networks of Kuramoto-type phase oscillators and theta neurons. By passing to the continuum limit and using the Ott/Antonsen ansatz, we derive evolution equations for a spatially dependent order parameter. For phase oscillator networks, the travelling waves take the form of uniformly twisted waves, and these can often be characterised analytically. For networks of theta neurons, the waves are studied numerically.

Introduction to focus issue: Patterns of network synchronization

The study of synchronization of coupled systems is currently undergoing a major surge fueled by recent discoveries of new forms of collective dynamics and the development of techniques to characterize a myriad of new patterns of network synchronization. This includes chimera states, phenomena determined by symmetry, remote synchronization, and asymmetry-induced synchronization. This Focus Issue presents a selection of contributions at the forefront of these developments, to which this introduction is intended to offer an up-to-date foundation.

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.

Controlling synchronous patterns in complex networks

Although the set of permutation symmetries of a complex network could be very large, few of them give rise to stable synchronous patterns. Here we present a general framework and develop techniques for controlling synchronization patterns in complex network of coupled chaotic oscillators. Specifically, according to the network permutation symmetry, we design a small-size and weighted network, namely the control network, and use it to control the large-size complex network by means of pinning coupling. We argue mathematically that for any of the network symmetries, there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized. The feasibility of the control method is verified by numerical simulations of both artificial and real-world networks and demonstrated experimentally in systems of coupled chaotic circuits. Our studies show the controllability of synchronous patterns in complex networks of coupled chaotic oscillators.

Complete characterization of the stability of cluster synchronization in complex dynamical networks

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based on the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. Understanding how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an optoelectronic experiment on a five-node network that confirms the synchronization patterns predicted by the theory.

Experimental synchrony of semiconductor lasers in coupled networks

The emission and synchronization of mutually-coupled semiconductor lasers with short cavities has been already recorded, with transversely unstable solutions existing within the chaotic synchronization manifold. Noise and laser-mismatch induced instabilities cause short de-synchronization events within the overall generalized synchronization, influencing the pragmatism of using these signals in secure data exchange applications. However, such operation can be functional for user authentication and sensing applications by assessing a time-averaged performance of synchrony. Until now, this has not been examined either in large-scale laser network configurations or in large transmission coupling paths, as real network implementations oblige. Here we present the first implementation of a fully-coupled fiber network with up to 16 semiconductor lasers, independently controlled and coupled through long interacting cavities. High level of consistent global or cluster synchrony via chaotic signals is demonstrated among all devices of the same origin and under appropriate operation. Devices that are not identical fail to synchronize at any condition, when coupled to the network. Under multiplexed operation, groups of lasers that emit at spectral distances as low as 50pm are shown to preserve intra-cluster synchronization when transmitted in the same fiber-optic channel, despite their large bandwidth of emitted signals.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Analysis of cluster explosive synchronization in complex networks

Correlations between intrinsic dynamics and local topology have become a new trend in the study of synchronization in complex networks. In this paper, we investigate the influence of topology on the dynamics of networks made up of second-order Kuramoto oscillators. In particular, based on mean-field calculations, we provide a detailed investigation of cluster explosive synchronization (CES) [Phys. Rev. Lett. 110, 218701 (2013)] in scale-free networks as a function of several topological properties. Moreover, we investigate the robustness of discontinuous transitions by including an additional quenched disorder, and we show that the phase coherence decreases with increasing strength of the quenched disorder. These results complement the previous findings regarding CES and also fundamentally deepen the understanding of the interplay between topology and dynamics under the constraint of correlating natural frequencies and local structure.

Cyclic motifs as the governing topological factor in time-delayed oscillator networks

We identify the relative amount of short cyclic motifs as an important topological factor in networks of time-delayed Kuramoto oscillators. The patterns emerging from the cyclic motifs are most clearly distinguishable in the average frequency and the momentary frequency dispersion as a function of the time delay. In particular, the common distinction between bidirectional and unidirectional couplings is shown to have a decisive effect on the network dynamics. We argue that the behavior peculiar to the sparsely connected unidirectional random network can be described essentially as the lack of distinguishable patterns originating from cyclic motifs of any specific length.

Stochastic switching in delay-coupled oscillators

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.

Sensitivity to external signals and synchronization properties of a non-isochronous auto-oscillator with delayed feedback

For auto-oscillators of different nature (e.g. active cells in a human heart under the action of a pacemaker, neurons in brain, spin-torque nano-oscillators, micro and nano-mechanical oscillators, or generating Josephson junctions) a critically important property is their ability to synchronize with each other. The synchronization properties of an auto oscillator are directly related to its sensitivity to external signals. Here we demonstrate that a non-isochronous (having generation frequency dependent on the amplitude) auto-oscillator with delayed feedback can have an extremely high sensitivity to external signals and unusually large width of the phase-locking band near the boundary of the stable auto-oscillation regime. This property could be used for the development of synchronized arrays of non-isochronous auto-oscillators in physics and engineering, and, for instance, might bring a better fundamental understanding of ways to control a heart arrythmia in medicine.

Limits to detection of generalized synchronization in delay-coupled chaotic oscillators

We study how reliably generalized synchronization can be detected and characterized from time-series analysis. To that end, we analyze synchronization in a generalized sense of delay-coupled chaotic oscillators in unidirectional ring configurations. The generalized synchronization condition can be verified via the auxiliary system approach; however, in practice, this might not always be possible. Therefore, in this study, widely used indicators to directly quantify generalized and phase synchronization from noise-free time series of two oscillators are employed complementarily to the auxiliary system approach. In our analysis, none of the indices provide the consistent results of the auxiliary system approach. Our findings indicate that it is a major challenge to directly detect synchronization in a generalized sense between two oscillators that are connected via a chain of other oscillators, even if the oscillators are identical. This has major consequences for the interpretation of the dynamics of coupled systems and applications thereof.

Emergence of metastable state dynamics in interconnected cortical networks with propagation delays

The importance of the large number of thin-diameter and unmyelinated axons that connect different cortical areas is unknown. The pronounced propagation delays in these axons may prevent synchronization of cortical networks and therefore hinder efficient information integration and processing. Yet, such global information integration across cortical areas is vital for higher cognitive function. We hypothesized that delays in communication between cortical areas can disrupt synchronization and therefore enhance the set of activity trajectories and computations interconnected networks can perform. To evaluate this hypothesis, we studied the effect of long-range cortical projections with propagation delays in interconnected large-scale cortical networks that exhibited spontaneous rhythmic activity. Long-range connections with delays caused the emergence of metastable, spatio-temporally distinct activity states between which the networks spontaneously transitioned. Interestingly, the observed activity patterns correspond to macroscopic network dynamics such as globally synchronized activity, propagating wave fronts, and spiral waves that have been previously observed in neurophysiological recordings from humans and animal models. Transient perturbations with simulated transcranial alternating current stimulation (tACS) confirmed the multistability of the interconnected networks by switching the networks between these metastable states. Our model thus proposes that slower long-range connections enrich the landscape of activity states and represent a parsimonious mechanism for the emergence of multistability in cortical networks. These results further provide a mechanistic link between the known deficits in connectivity and cortical state dynamics in neuropsychiatric illnesses such as schizophrenia and autism, as well as suggest non-invasive brain stimulation as an effective treatment for these illnesses.

Coupled lasers: phase versus chaos synchronization

The synchronization of chaotic lasers and the optical phase synchronization of light originating in multiple coupled lasers have both been extensively studied. However, the interplay between these two phenomena, especially at the network level, is unexplored. Here, we experimentally compare these phenomena by controlling the heterogeneity of the coupling delay times of two lasers. While chaotic lasers exhibit deterioration in synchronization as the time delay heterogeneity increases, phase synchronization is found to be independent of heterogeneity. The experimental results are found to be in agreement with numerical simulations for semiconductor lasers.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.

Synchronization of delayed coupled neurons in presence of inhomogeneity

In principle, two directly coupled limit cycle oscillators can overcome mismatch in intrinsic rates and match their frequencies, but zero phase lag synchronization is just achievable in the limit of zero mismatch, i.e., with identical oscillators. Delay in communication, on the other hand, can exert phase shift in the activity of the coupled oscillators. In this study, we address the question of how phase locked, and in particular zero phase lag synchronization, can be achieved for a heterogeneous system of two delayed coupled neurons. We have analytically studied the possibility of inphase synchronization and near inphase synchronization when the neurons are not identical or the connections are not exactly symmetric. We have shown that while any single source of inhomogeneity can violate isochronous synchrony, multiple sources of inhomogeneity can compensate for each other and maintain synchrony. Numeric studies on biologically plausible models also support the analytic results.

Experiments on autonomous Boolean networks

We realize autonomous Boolean networks by using logic gates in their autonomous mode of operation on a field-programmable gate array. This allows us to implement time-continuous systems with complex dynamical behaviors that can be conveniently interconnected into large-scale networks with flexible topologies that consist of time-delay links and a large number of nodes. We demonstrate how we realize networks with periodic, chaotic, and excitable dynamics and study their properties. Field-programmable gate arrays define a new experimental paradigm that holds great potential to test a large body of theoretical results on the dynamics of complex networks, which has been beyond reach of traditional experimental approaches.

Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices

We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block diagonalization of matrices. We obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower-dimensional problems and test the predictions of our low-dimensional analysis through numerical simulations. Under certain conditions, this technique may yield a substantial reduction of the dimensionality of the problem. For example, for a class of dynamical hypernetworks analyzed in the paper, we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2. We apply our reduction technique to a number of different examples, including the class of undirected unweighted hypermotifs with 3 nodes.

Cluster and group synchronization in delay-coupled networks

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.

Spectral and correlation properties of rings of delay-coupled elements: comparing linear and nonlinear systems

The dynamical properties of delay-coupled systems are currently of great interest. So far the analysis has concentrated primarily on identical synchronization properties. Here we study the dynamics of rings of delay-coupled nodes, a topology that cannot show identical synchronization, and compare its properties to those of linear stochastic maps. We find that, in the long delay limit, the correlation functions and spectra of delay-coupled rings of nonlinear systems obey the same scaling laws as linear systems, indicating that important properties of the emerging solution result from network topology.

An optimally evolved connective ratio of neural networks that maximizes the occurrence of synchronized bursting behavior

Background

Synchronized bursting activity (SBA) is a remarkable dynamical behavior in both ex vivo and in vivo neural networks. Investigations of the underlying structural characteristics associated with SBA are crucial to understanding the system-level regulatory mechanism of neural network behaviors.

Results

In this study, artificial pulsed neural networks were established using spike response models to capture fundamental dynamics of large scale ex vivo cortical networks. Network simulations with synaptic parameter perturbations showed the following two findings. (i) In a network with an excitatory ratio (ER) of 80-90%, its connective ratio (CR) was within a range of 10-30% when the occurrence of SBA reached the highest expectation. This result was consistent with the experimental observation in ex vivo neuronal networks, which were reported to possess a matured inhibitory synaptic ratio of 10-20% and a CR of 10-30%. (ii) No SBA occurred when a network does not contain any all-positive-interaction feedback loop (APFL) motif. In a neural network containing APFLs, the number of APFLs presented an optimal range corresponding to the maximal occurrence of SBA, which was very similar to the optimal CR.

Conclusions

In a neural network, the evolutionarily selected CR (10-30%) optimizes the occurrence of SBA, and APFL serves a pivotal network motif required to maximize the occurrence of SBA.

Synchronization in small networks of time-delay coupled chaotic diode lasers

Topologies of two, three and four time-delay-coupled chaotic semiconductor lasers are experimentally and theoretically found to show new types of synchronization. Generalized zero-lag synchronization is observed for two lasers separated by long distances even when their self-feedback delays are not equal. Generalized sub-lattice synchronization is observed for quadrilateral geometries while the equilateral triangle is zero-lag synchronized. Generalized zero-lag synchronization, without the limitation of precisely matched delays, opens possibilities for advanced multi-user communication protocols.

Effect of duration of synaptic activity on spike rate of a Hodgkin-Huxley neuron with delayed feedback

A recurrent loop consisting of a single Hodgkin-Huxley neuron influenced by a chemical excitatory delayed synaptic feedback is considered. We show that the behavior of the system depends on the duration of the activity of the synapse, which is determined by the activation and deactivation time constants of the synapse. For the fast synapses, those for which the effect of the synaptic activity is small compared to the period of firing, depending on the delay time, spiking with single and multiple interspike intervals is possible and the average firing rate can be smaller or larger than that of the open loop neuron. For slow synapses for which the synaptic time constants are of order of the period of the firing, the self-excitation increases the firing rate for all values of the delay time. We also show that for a chain consisting of few similar oscillators, if the synapses are chosen from different time constants, the system will follow the dynamics imposed by the fastest element, which is the oscillator that receives excitations via a slow synapse. The generalization of the results to other types of relaxation oscillators is discussed and the results are compared to those of the loops with inhibitory synapses as well as with gap junctions.