A subharmonic dynamical bifurcation during in vitro epileptiform activity

Epileptic seizures are considered to result from a sudden change in the synchronization of firing neurons in brain neural networks. We have used an in vitro model of status epilepticus (SE) to characterize dynamical regimes underlying the observed seizure-like activity. Time intervals between spikes or bursts were used as the variable to construct first-return interpeak or interburst interval plots, for studying neuronal population activity during the transition to seizure, as well as within seizures. Return maps constructed for a brief epoch before seizures were used for approximating the local system dynamics during that time window. Analysis of the first-return maps suggests that intermittency is a dynamical regime underlying the observed epileptic activity. This type of analysis may be useful for understanding the collective dynamics of neuronal populations in the normal and pathological brain.     

基于Kendall改进的同步算法癫痫脑网络分析

提出了一种基于Kendall等级相关改进的同步算法IRC(inverse rank correlation).Kendall等级相关是非线性动力学分析的一般化算法,可有效地度量变量间的非线性相关性.复杂网络的研究已逐渐深入到社会科学的各个领域,脑网络的研究已经成为当今脑功能研究的热点.利用改进的IRC算法,基于脑电EEG(electroencephalogram)数据来构建大脑功能性网络.对构建的脑功能网络的度指标进行了分析,以调查癫痫脑功能网络是否异于正常人.结果显示:使用该改进的算法能够对癫痫和正常脑功能网络显著区分,且只需要记录很短的脑电数据.实验结果数据表明,该方法适用于区分癫痫和正常脑组织网络度指标,它可有助于进一步地加深对大脑的神经动力学行为的研究,并为临床诊断提供有效工具.     

Synchronization of multi-phase oscillators: an Axelrod-inspired model

Inspired by Axelrod’s model of culture dissemination, we introduce and analyze a model for a population of coupled oscillators where different levels of synchronization can be assimilated to different degrees of cultural organization. The state of each oscillator is represented by a set of phases, and the interaction – which occurs between homologous phases – is weighted by a decreasing function of the distance between individual states. Both ordered arrays and random networks are considered. We find that the transition between synchronization and incoherent behaviour is mediated by a clustering regime with rich organizational structure, where any two oscillators can be synchronized in some of their phases, while their remain unsynchronized in the others.     

Explosive synchronization: From synthetic to real-world networks

Synchronization is a widespread phenomenon in both synthetic and real-world networks. This collective behavior of simple and complex systems has been attracting much research during the last decades. Two different routes to synchrony are defined in networks; first-order, characterized as explosive, and second-order, characterized as continuous transition. Although pioneer researches explained that the transition type is a generic feature in the networks, recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization. The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions. Despite different theoretical analyses about the appearance of the first-order transition, studies are limited to the mean-field theory, which cannot be generalized to all networks. There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization, e.g., the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks. In this review article, explosive synchronization is discussed from two main aspects. First, pioneer articles are categorized from the dynamical-structural framework point of view. Then, articles that considered different oscillators in the explosive synchronization frameworks are studied. In this article, the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators. Also, efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.     

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.     

Ictal imaging using functional magnetic resonance

Magnetic resonance imaging (MRI) can now provide maps of human brain function with high spatial and temporal resolution. This noninvasive technique can also map the coritical activation that occurs during focal seizures, as demonstrated here by the results obtained using a conventional 1.5 T clinical MRI system for the investigation of a 4-year-old boy suffering from frequent partial motor seizures of his right side. FLASH images (TE = 60 ms) were acquired every 10 s over a period of 25 min, and activation images derived by subtracting baseline images from images obtained during clinical seizures. Functional MRI revealed sequential activation associated with specific gyri within the left hemisphere with each of five consecutive clinical seizures, and also during a period that was not associated with a detectable clinical seizure. The activated regions included gyri that were structurally abnormal. These results demonstrate (a) that functional MRI can potentially provide new insights into the dynamic events that occur in the epileptic brain and their relationship to brain structure; and (b) that there is the possibility of obtaining similar information in the absence of clinical seizures, suggesting the potential for studies in patients with interictal electrical disturbances.     

Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control

In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fractional order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more practical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective.     

噪声环境下时滞耦合网络的广义投影滞后同步

时滞和噪声在复杂网络中普遍存在,而含有耦合时滞和噪声摄动的耦合网络同步的研究工作却极其稀少. 本文针对噪声环境下具有不同节点动力学、不同拓扑结构及不同节点数目的耦合时滞网络,提出了两个网络之间的广义投影滞后同步. 首先,构建了更加贴近现实的驱动-响应网络同步的理论框架;其次,基于随机时滞微分方程LaSalle不变性原理,严格证明了在合理的控制器作用下,驱动网络和响应网络在几乎必然渐近稳定性意义下能够取得广义投影滞后同步;最后,借助于计算机仿真,通过具体的网络模型验证了理论推理的有效性. 数值模拟结果表明,驱动网络与响应网络不但能够达到广义投影滞后同步,而且同步效果不依赖于耦合时滞和比例因子的选取,同时也揭示了更新增益和耦合时滞对同步收敛速度的显著性影响. 关键词： 复杂网络 广义投影滞后同步 随机噪声 时滞     

非一致通信时滞动力学网络上的接连滞后同步

在动力学网络中,节点与节点之间的通信通常存在时滞,并且不同节点之间的通信时滞往往是不同的(即非一致通信时滞),研究非一致通信时滞动力学网络上的接连滞后同步,更具现实意义.为此,本文首先构建含有非一致通信时滞的动力学网络模型.其次分别设计线性反馈控制和自适应反馈控制,利用Lyapunov函数方法,重点分析了该网络的接连滞后同步的稳定性,得到了同步稳定的充分条件.最后,选取蔡氏电路作为局部动力学,又分别选取了链式网络和星型网络这两种拓扑结构来验证理论结果的正确性和有效性.     

Testing statistical significance of multivariate time series analysis techniques for epileptic seizure prediction

Nonlinear time series analysis techniques have been proposed to detect changes in the electroencephalography dynamics prior to epileptic seizures. Their applicability in practice to predict seizure onsets is hampered by the present lack of generally accepted standards to assess their performance. We propose an analytic approach to judge the prediction performance of multivariate seizure prediction methods. Statistical tests are introduced to assess patient individual results, taking into account that prediction methods are applied to multiple time series and several seizures. Their performance is illustrated utilizing a bivariate seizure prediction method based on synchronization theory.     

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.     

Clinical applications: MRI, SPECT, and PET

MRI, PET, and SPECT are all used to image abnormalities in the epileptic brain. Comparison of the techniques is difficult because they measure different aspects of the epileptic process—structure, metabolism, and perfusion. SPECT is the only one that can be systematically applied during seizures, while all three are used to image interictal abnormalities. Literature review suggests that of interictal techniques, PET has the highest diagnostic sensitivity in temporal lobe epilepsy (TLE) (84% vs. 66% for SPECT, 55% for qualitative MRI, 71% for quantitative MRI) while SPECT has the highest sensitivity in extratemporal epilepsy (ETE) (60% vs. 43% for MRI and 33% for PET). The highest diagnostic sensitivity and specificity were achieved by ictal imaging with SPECT (90% in TLE, 81% in ETE). The techniques, however, were not always redundant. One reason for the wide discrepancy of results in TLE and ETE might be the differing pathologic substrates. A literature review of imaging findings associated with mesial temporal sclerosis (MTS), developmental lesion or tumor as the underlying abnormality associated with epilepsy supports this explantion. PET and MRI are much more sensitive to MTS than SPECT (100%, 95% vs. 70%). On the other hand, in developmental lesions the three techniques are equally sensitive (88–92%) and in tumors, MRI was most sensitive (96%) and SPECT least (82%). A study at NIH explains the differing sensitivities: using PET to measure both blood flow and metabolism revealed discrepant findings in the same patients. Preliminary evidence also indicates that the distribution of hyperperfusion on ictal SPECT can differentiate subtypes of TLE. Combining the results of refined imaging techniques holds great promise in epilepsy localization and diagnosis.     

Symbolic synchronization and the detection of global properties of coupled dynamics from local information

We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the homogeneous unsynchronized phase, where the coupled dynamics is in a chaotic unsynchronized state, but exhibits qualitative similar symbolic dynamics at all the nodes in the network. We refer to this dynamical behavior as symbolic synchronization. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behavior of the largest Lyapunov exponent and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local chaotic dynamical function. We present results for two model dynamics, iterations of the one-dimensional logistic map and the two-dimensional Henon map, as local dynamical function.     

The influence of hubs in the structure of a neuronal network during an epileptic seizure

In this work, we propose changes in the structure of a neuronal network with the intention to provoke strong synchronization to simulate episodes of epileptic seizure. Starting with a network of Izhikevich neurons we slowly increase the number of connections in selected nodes in a controlled way, to produce (or not) hubs. We study how these structures alter the synchronization on the spike firings interval, on individual neurons as well as on mean values, as a function of the concentration of connections for random and non-random (hubs) distribution. We also analyze how the post-ictal signal varies for the different distributions. We conclude that a network with hubs is more appropriate to represent an epileptic state.     

Modulation of brain functional connectivity with manual acupuncture in healthy subjects: An electroencephalograph case studying

Manual acupuncture is widely used for pain treatment and stress control. Previous studies on acupuncture have shown its modulatory effects on functional connectivity associated with one or a few preselected brain regions. To investigate how manual acupuncture modulates the organization of functional networks at a whole-brain level, we acupuncture at ST36 of right leg to obtain electroencephalograph (EEG) signals. By coherence estimation, we determine the synchronizations between all pairwise combinations of EEG channels in three acupuncture states. The resulting synchronization matrices are converted into functional networks by applying a threshold, and clustering coefficients and path lengths are computed as a function of threshold. The results show that acupuncture can increase functional connections and synchronizations between different brain areas. For a wide range of threshold, the clustering coefficient during acupuncture and post-acupuncture period is higher than that during the pre-acupuncture control period, whereas characteristic path length is shorter. We provide further support for the presence of "small-world" network characteristics in functional networks by acupuncture. These preliminary results highlight the beneficial modulations of functional connectivity by manual acupuncture, which could contribute to the understanding of acupuncture effects on the entire brain, as well as the neurophysiological mechanisms underlying acupuncture. Moreover, the proposed method may be a useful approach to the further investigation of the complexity of patterns of interrelations between EEG channels.     

The modulation of brain functional connectivity with manual acupuncture in healthy subjects:An electroencephalograph case study

Manual acupuncture is widely used for pain relief and stress control.Previous studies on acupuncture have shown its modulatory effects on the functional connectivity associated with one or a few preselected brain regions.To investigate how manual acupuncture modulates the organization of functional networks at a whole-brain level,we acupuncture at ST36 of a right leg to obtain electroencephalograph(EEG) signals.By coherence estimation,we determine the synchronizations between all pairwise combinations of EEG channels in three acupuncture states.The resulting synchronization matrices are converted into functional networks by applying a threshold,and the clustering coefficients and path lengths are computed as a function of threshold.The results show that acupuncture can increase functional connections and synchronizations between different brain areas.For a wide range of thresholds,the clustering coefficient during acupuncture and postacupuncture period is higher than that during the pre-acupuncture control period,whereas the characteristic path length is shorter.We provide further support for the presence of "small-world" network characteristics in functional networks by using acupuncture.These preliminary results highlight the beneficial modulations of functional connectivity by manual acupuncture,which could contribute to the understanding of the effects of acupuncture on the entire brain,as well as the neurophysiological mechanisms underlying acupuncture.Moreover,the proposed method may be a useful approach to the further investigation of the complexity of patterns of interrelations between EEG channels.     

Synchronization between different motifs

In this paper,we study the synchronization between different motifs.First,the synchronization between two networks with different topology structures and different dynamical behaviours is studied.With the open-plus-closed-loop(OPCL) method,conditions for two different networks to realize synchronization are given.Then based on the theoretical results achieved,the synchronization between different motifs is studied,which verifies the effectiveness and feasibility of the synchronization scheme.     

基于脑电图的大脑功能性网络分析

利用脑电图数据建立了大脑功能性网络.分析了该网络的复杂网络统计特征，发现它的聚类系数远大于相应随机网络，明显具有小世界网络的特征，其度分布也接近于无标度网络.进一步验证了大脑功能性网络的复杂网络特性，发现患者的各项复杂网络特征指数与正常人相比有明显不同.定义了大脑神经网络信息熵及神经网络标准信息熵的概念，发现脑病患者的大脑神经网络信息熵明显小于正常人.从一个全新的角度量度了大脑的复杂网络特征，并提示了临床脑病诊疗的判断依据. 关键词： 脑电图 大脑功能性网络 复杂网络统计特征 信息熵     

Comparison of Synchronization Ability of Four Types of Regular Coupled Networks

We investigate the synchronization ability of four types of regular coupled networks. By introducing the proper error variables and Lyapunov functions, we turn the stability of synchronization manifold into that of null solution of error equations, further, into the negative definiteness of some symmetric matrices, thus we get the sufficient synchronization stability conditions. To test the valid of the results, we take the Chua's circuit as an example. Although the theoretical synchronization thresholds appear to be very conservative, they provide new insights about the influence of topology and scale of networks on synchronization, and that the theoretical results and our numerical simulations are consistent.