Complex brain networks: From topological communities to clustered dynamics

Recent research has revealed a rich and complicated network topology in the cortical connectivity of mammalian brains. A challenging task is to understand the implications of such network structures on the functional organisation of the brain activities. We investigate synchronisation dynamics on the corticocortical network of the cat by modelling each node of the network (cortical area) with a subnetwork of interacting excitable neurons. We find that this network of networks displays clustered synchronisation behaviour and the dynamical clusters closely coincide with the topological community structures observed in the anatomical network. The correlation between the firing rate of the areas and the areal intensity is additionally examined. Our results provide insights into the relationship between the global organisation and the functional specialisation of the brain cortex.      

Multivariate analysis of spatially heterogeneous phase synchronisation in complex systems: application to self-organised control of material flows in networks

Networks of interacting components are a class of complex systems that has attracted considerable interest over the last decades. In particular, if the dynamics of the autonomous components is characterised by an oscillatory behaviour, different types of synchronisation can be observed in dependence on the type and strength of interactions. In this contribution, we study the transition from non-synchronised to synchronised phase dynamics in complex networks. The most common approach to quantify the degree of phase synchronisation in such systems is the consideration of measures of phase coherence which are averaged over all pairs of interacting components. However, this approach implicitly assumes a spatially homogeneous synchronisation process, which is typically not present in complex networks. As a potential alternative, two novel methods of multivariate phase synchronisation analysis are considered: synchronisation cluster analysis (SCA) and the linear variance decay (LVD) dimension method. The strengths and weaknesses of the traditional as well as both new approaches are briefly illustrated for a Kuramoto model with long-range coupling. As a practical application, we study how spatial heterogeneity influences the transition to phase synchronisation in traffic networks where intersecting material flows are subjected to a self-organised decentralised control. We find that the network performance and the degree of phase synchronisation are closely related to each other and decrease significantly in the case of structural heterogeneities. The influences of the different parameters of our control approach on the synchronisation process are systematically studied, yielding a sequence of Arnold tongues which correspond to different locking modes.     

簇间连接方式不同的簇网络的同步过程研究

本文对簇间连接方式不同的三类簇网络的同步能力和同步过程进行研究. 构成簇网络的两个子网均为BA无标度网络, 当簇间连接方式是双向耦合时, 称其为TWD网络模型, 当簇间连接是大子网驱动小子网时, 称其为BDS网络模型, 当簇间连接是小子网驱动大子网时, 称其为SDB网络模型. 研究表明, 当小子网和大子网节点数目的比值大于某一临界值时, TWD网络模型的同步能力大于BDS网络模型的同步能力, 当该比值小于某一临界值时, TWD网络模型的同步能力小于BDS网络模型的同步能力, SDB网络模型的同步能力是三种网络结构中最差的. 对于簇间连接具有方向性的单向驱动网络, 簇网络的整体同步能力与被驱动子网的节点数和簇间连接数有关, 与驱动网络自身节点数无关. 增加簇间连接数在开始时会降低各子网的同步速度, 但最终各子网到达完全同步的时间减少, 网络的整体同步能力增强. 文中以Kuramoto相振子作为网络节点, 研究了不同情况下三种簇网络的同步过程, 证明了所得结论的正确性. 关键词： 簇网络 有向连接 同步能力 Kuramoto振子     

Synchronisation of fractional-order time delayed chaotic systems with ring connection

In this paper, synchronisation of fractional-order time delayed chaotic systems in ring networks is investigated. Based on Lyapunov stability theory, a new generic synchronisation criterion for N-coupled chaotic systems with time delay is proposed. The synchronisation scheme is applied to N-coupled fractional-order time delayed simplified Lorenz systems, and the Adomian decomposition method (ADM) is developed for solving these chaotic systems. Performance analysis of the synchronisation network is carried out. Numerical experiments demonstrate that synchronisation realises in both state variables and intermediate variables, which verifies the effectiveness of the proposed method.     

Optimization of synchronization in complex clustered networks

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.     

Synchronizing spiral waves in a coupled Rssler system

The synchronisation of spiral patterns in a drive-response Rssler system is studied.The existence of three types of synchronisation is revealed by inspecting the coupling parameter space.Two transient stages of phase synchronisation and partial synchronisation are observed in a comparatively weak feedback coupling parameter regime,whilst complete synchronisation of spirals is found with strong negative couplings.Detailed observations of the synchronous process,such as oscillatory frequencies,parameters mismatches and amplitude variations,etc,are investigated via numerical simulations.     

Cross-approximate entropy of cortical local field potentials quantifies effects of anesthesia - a pilot study in rats

Background

Anesthetics dose-dependently shift electroencephalographic (EEG) activity towards high-amplitude, slow rhythms, indicative of a synchronization of neuronal activity in thalamocortical networks. Additionally, they uncouple brain areas in higher (gamma) frequency ranges possibly underlying conscious perception. It is currently thought that both effects may impair brain function by impeding proper information exchange between cortical areas. But what happens at the local network level? Local networks with strong excitatory interconnections may be more resilient towards global changes in brain rhythms, but depend heavily on locally projecting, inhibitory interneurons. As anesthetics bias cortical networks towards inhibition, we hypothesized that they may cause excessive synchrony and compromise information processing already on a small spatial scale. Using a recently introduced measure of signal independence, cross-approximate entropy (XApEn), we investigated to what degree anesthetics synchronized local cortical network activity. We recorded local field potentials (LFP) from the somatosensory cortex of three rats chronically implanted with multielectrode arrays and compared activity patterns under control (awake state) with those at increasing concentrations of isoflurane, enflurane and halothane.

Results

Cortical LFP signals were more synchronous, as expressed by XApEn, in the presence of anesthetics. Specifically, XApEn was a monotonously declining function of anesthetic concentration. Isoflurane and enflurane were indistinguishable; at a concentration of 1 MAC (the minimum alveolar concentration required to suppress movement in response to noxious stimuli in 50% of subjects) both volatile agents reduced XApEn by about 70%, whereas halothane was less potent (50% reduction).

Conclusions

The results suggest that anesthetics strongly diminish the independence of operation of local cortical neuronal populations, and that the quantification of these effects in terms of XApEn has a similar discriminatory power as changes of spontaneous action potential rates. Thus, XApEn of field potentials recorded from local cortical networks provides valuable information on the anesthetic state of the brain.

     

Hierarchical organization unveiled by functional connectivity in complex brain networks

How do diverse dynamical patterns arise from the topology of complex networks? We study synchronization dynamics in the cortical brain network of the cat, which displays a hierarchically clustered organization, by modeling each node (cortical area) with a subnetwork of interacting excitable neurons. We find that in the biologically plausible regime the dynamics exhibits a hierarchical modular organization, in particular, revealing functional clusters coinciding with the anatomical communities at different scales. Our results provide insights into the relationship between network topology and functional organization of complex brain networks.     

Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction

It is shown that the nonlinear dynamics of chaotic time-delay systems can be reconstructed using a new type of neural network with two modules: one for nonfeedback part with input data delayed by the embedding time, and a second one for the feedback part with input data delayed by the feedback time. The method is applied to both simulated and experimental data from an electronic analog circuit of the Mackey–Glass system. Better results are obtained for the modular than for feedforward neural networks for the same number of parameters. It is found that the complexity of the neural network model required to reconstruct nonlinear dynamics does not increase with the delay time. Synchronization between the data and the model with diffusive coupling is also achieved. We have also shown by iterating the model from the present point that the dynamics can be predicted with a forecast horizon larger than the feedback delay time.     

Synchronisation of chaos and its applications

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronisation, a classical paradigm of order. We survey the main theory behind complete, generalised and phase synchronisation phenomena in simple as well as complex networks and discuss applications to secure communications, parameter estimation and the anticipation of chaos.     

Generalised synchronisation of spatiotemporal chaos using feedback control method and phase compression

Coupled map lattices are taken as examples to study the synchronisation of spatiotemporal chaotic systems. First, a generalised synchronisation of two coupled map lattices is realised through selecting an appropriate feedback function and appropriate range of feedback parameter. Based on this method we use the phase compression method to extend the range of the parameter. So, we integrate the feedback control method with the phase compression method to implement the generalised synchronisation and obtain an exact range of feedback parameter. This technique is simple to implement in practice. Numerical simulations show the effectiveness and the feasibility of the proposed program.     

Projective synchronisation with non-delayed and delayed coupling in complex networks consisting of identical nodes and different nodes

This paper first investigates the projective synchronisation problem with non-delayed and delayed coupling between drive-response dynamical networks consisting of identical nodes and different nodes.Based on Lyapunov stability theory,several nonlinear controllers are applied to achieve the projective synchronisation between the drive-response dynamical networks;simultaneously the topological structure of the drive dynamical complex networks can be exactly identified.Moreover,numerical examples are presented to verify the feasibility and effectiveness of the theorems.     

Novel pinning control strategies for synchronisation of complex networks with nonlinear coupling dynamics

This paper considers the global stability of controlling an uncertain complex network to a homogeneous trajectory of the uncoupled system by a local pinning control strategy. Several sufficient conditions are derived to guarantee the network synchronisation by investigating the relationship among pinning synchronisation, network topology, and coupling strength. Also, some fundamental and yet challenging problems in the pinning control of complex networks are discussed: (1) what nodes should be selected as pinned candidates? (2) How many nodes are needed to be pinned for a fixed coupling strength? Furthermore, an adaptive pinning control scheme is developed. In order to achieve synchronisation of an uncertain complex network, the adaptive tuning strategy of either the coupling strength or the control gain is utilised. As an illustrative example, a network with the Lorenz system as node self-dynamics is simulated to verify the efficacy of theoretical results.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Plasticity-induced characteristic changes of pattern dynamics and the related phase transitions in small-world neuronal networks

Phase transitions widely exist in nature and occur when some control parameters are changed. In neural systems, their macroscopic states are represented by the activity states of neuron populations, and phase transitions between different activity states are closely related to corresponding functions in the brain. In particular, phase transitions to some rhythmic synchronous firing states play significant roles on diverse brain functions and disfunctions, such as encoding rhythmical external stimuli, epileptic seizure, etc. However, in previous studies, phase transitions in neuronal networks are almost driven by network parameters （e.g., external stimuli）, and there has been no investigation about the transitions between typical activity states of neuronal networks in a self-organized way by applying plastic connection weights. In this paper, we discuss phase transitions in electrically coupled and lattice-based small-world neuronal networks （LBSW networks） under spike-timing-dependent plasticity （STDP）. By applying STDP on all electrical synapses, various known and novel phase transitions could emerge in LBSW networks, particularly, the phenomenon of self-organized phase transitions （SOPTs）： repeated transitions between synchronous and asynchronous firing states. We further explore the mechanics generating SOPTs on the basis of synaptic weight dynamics.     

Bursting synchronization in clustered neuronal networks

Neuronal networks in the brain exhibit the modular (clustered) property, i.e., they are composed of certain subnetworks with differential internal and external connectivity. We investigate bursting synchronization in a clustered neuronal network. A transition to mutual-phase synchronization takes place on the bursting time scale of coupled neurons, while on the spiking time scale, they behave asynchronously. This synchronization transition can be induced by the variations of inter- and intra- coupling strengths, as well as the probability of random links between different subnetworks. Considering that some pathological conditions are related with the synchronization of bursting neurons in the brain, we analyze the control of bursting synchronization by using a time-periodic external signal in the clustered neuronal network. Simulation results show a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even in the presence of external driving. Hence, effective synchronization suppression can be realized with the driving parameters outside the frequency locking region.     

Phase transitions for information diffusion in random clustered networks

We study the conditions for the phase transitions of information diffusion in complexnetworks. Using the random clustered network model, a generalisation of the Chung-Lurandom network model incorporating clustering, we examine the effect of clustering underthe Susceptible-Infected-Recovered (SIR) epidemic diffusion model with heterogeneouscontact rates. For this purpose, we exploit the branching process to analyse informationdiffusion in random unclustered networks with arbitrary contact rates, and provide noveliterative algorithms for estimating the conditions and sizes of global cascades,respectively. Showing that a random clustered network can be mapped into a factor graph,which is a locally tree-like structure, we successfully extend our analysis to randomclustered networks with heterogeneous contact rates. We then identify the conditions forphase transitions of information diffusion using our method. Interestingly, for variouscontact rates, we prove that random clustered networks with higher clustering coefficientshave strictly lower phase transition points for any given degree sequence. Finally, weconfirm our analytical results with numerical simulations of both synthetically-generatedand real-world networks.     

Decentralised control of material or traffic flows in networks using phase-synchronisation

We present a self-organising, decentralised control method for material flows in networks. The concept applies to networks where time sharing mechanisms between conflicting flows in nodes are required and where a coordination of these local switches on a system-wide level can improve the performance. We show that, under certain assumptions, the control of nodes can be mapped to a network of phase-oscillators.By synchronising these oscillators, the desired global coordination is achieved. We illustrate the method in the example of traffic signal control for road networks. The proposed concept is flexible, adaptive, robust and decentralised. It can be transferred to other queuing networks such as production systems. Our control approach makes use of simple synchronisation principles found in various biological systems in order to obtain collective behaviour from local interactions.     

不同耦合下混沌神经元网络的同步

采用Hindmarsh-Rose神经元动力学模型, 对二维点阵上的神经元网络的同步进行了研究. 为了解不同耦合对网络同步的影响, 提出了一般反馈耦合、分层反馈耦合和分层局域平均场反馈耦合三种方案.研究表明:在耦合强度较小的近邻耦合下, 一般反馈耦合不能使网络达到完全同步, 而分层反馈耦合和分层局域平均场反馈耦合可以使网络出现局部同步和全局同步. 不同形式的耦合会导致网络出现不同的斑图, 随着耦合强度的增大, 网络从不同步到同步的过程也不相同, 一般反馈耦合和分层反馈耦合网络是突然出现全局同步, 同步之前网络出现非周期性的相干斑图; 对于分层局域平均场反馈耦合网络, 同层神经元之间先出现从簇放电同步到同步的转变, 形成靶波, 然后同步区由中心向外逐渐扩大, 最终达到网络的全局同步. 这些结果表明, 只有适当的耦合才能实现信号的无损耗的传递. 此外我们发现分层局域平均场反馈耦合可以促进网络的同步.