Open Access
EPJ Nonlinear Biomed. Phys.
Volume 5, 2017
Article Number 2
Number of page(s) 16
Section Physics of Biological Systems and Their Interactions
Published online 30 June 2017

© K. Lehnertz et al., published by EDP Sciences, 2017

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Due to its complex structure and its immense functionality, the human brain is one of the most complex and fascinating systems in nature. The neocortex of human is a thin, extended, convoluted sheet of tissue with a surface area of approx. 2600 cm2 and thickness 3–4 mm [1,2]. It contains up to 1010 neurons (and about three times more glia cells [3]), which are connected with each other and with cells in other parts of the brain by about 1012 synapses [4]. The length of all connections amounts to 107–109 m. The highly interconnected networks in the brain, which are neither random nor entirely regular, span multiple spatial scales, from individual cells and synapses via cortical columns to (sub)cortical areas. These networks support a rich repertoire of behavioral and cognitive functions [517] that are – for the most part – shared among all individuals, despite enormous differences in morphology and connection structure. Moreover, in the case of brain pathologies, normal and abnormal functions and/or structures can coexist [1820].

In humans, macroscopic brain dynamics can be assessed non-invasively with three major recording techniques, namely electroencephalography (EEG) [21], magnetoencephalography (MEG) [22], and functional magnetic resonance imaging (fMRI) [23]. While EEG and MEG capture electric and magnetic correlates of gross neural activities outside the head, fMRI assesses these activities indirectly via associated changes in blood oxygenation. In some epilepsy patients that undergo presurgical evaluation [24], invasive intracranial EEG (iEEG) provides access to the meso- (≈105 neurons) and the micro-scale (single neurons) [2528] via electrodes implanted chronically on the surface of the brain and/or within deep brain structures (e.g., the hippocampus). Each of these techniques has its own temporal and spatial resolution. EEG/iEEG and MEG sample brain dynamics with a high time resolution (typically a few milliseconds), and spatial resolution is limited by sensor placement. fMRI allows for a higher spatial resolution but the temporal resolution is orders of magnitude lower than with EEG or MEG. At present, EEG/iEEG is the only technique that allows for the continuous multichannel recording of time series of brain dynamics over extended periods of time (days to weeks), such that a wide spectrum of physiological and pathophysiological activities can be captured.

Despite access to the various spatial scales, both the respective dynamics and particularly the mapping between them are not fully understood. EEG/iEEG time series exhibit oscillations at a variety of frequencies and these oscillations are thought to represent synchronized activity over a network of neurons [21]. Several of these oscillations have characteristic frequency ranges and spatial distributions and can be associated with different states of brain (dys)functioning (e.g., waking, various sleep stages, focal epileptic seizures). Recent investigations that simultaneously assess dynamics from the macro- (gross neural activities) and the micro-scale (single/multi-neuron activities) point to complex relationships between these scales, and combining insights from both these dynamics is a key challenge for the future [29].

Another characteristic of brain dynamics, which contrasts the aforementioned oscillation-based view to brain (dys)functioning, is that it appears to not contain a predominant temporal scale [30]; rather time series exhibit a 1/fα-like power spectrum (with ) at many spatial–temporal scales [12,3134]. Such a scale-free dynamics, coexisting with an oscillatory one, has long been considered as brain noise, but there is now increasing evidence that parameters of this dynamics are closely related to brain (dys)functioning [35,34].

These various, apparently irregular dynamics might point to nonlinear generating mechanisms. Indeed, in the brain, nonlinearity is already introduced on the cellular level – given that the dynamical behavior of individual neurons is governed by integration, threshold, and saturation phenomena [36]. It might, however, not be valid to expect that a huge network of such nonlinear elements also behaves in a nonlinear way [37,38]. Despite a number of approaches to test for nonlinearity [39,40], compelling support for nonlinearity in brain dynamics on the meso- and macro-scale as well as for any brain (dys)functioning has as yet not been provided. A prominent counterexample is epilepsy, for which abundant evidence for nonlinear brain dynamics not only during seizures but also during the seizure-free interval has been produced (see, e.g., [4155]).

Since the brain is a complex network of interacting nonstationary subsystems, its spatial-temporal dynamics is also influenced by properties of such interactions. Indeed, interactions mediated by synchronization phenomena on various scales were shown to correlate with perceptual binding, and specific synchronization patterns appear to be directly related to behavior and cognition [8,11]. Conversely, abnormal patterns of synchronization seem to be a key pathophysiological mechanism underlying various neurological and psychiatric disorders, such as epilepsy, schizophrenia, dementia, and Parkinson’s disease [1820]. It is, however, not yet clear which of the many forms of synchronization known from physics (for an overview, see, e.g., [5659]) best characterizes brain (dys)functioning.

A large number of time-series-analysis techniques – which are based on different mathematical and physical concepts – is available (see, e.g., [56,6076]), and these techniques are highly suitable to further improve the characterization of brain dynamics. Depending on whether the employed technique is used to characterize the dynamics sampled at a single recording site or relations between dynamics sampled simultaneously at two or more sites, it is referred to as a univariate, bivariate, or multivariate time-series-analysis technique, respectively. Such techniques provide various indices that allow characterization of different properties of some dynamics and of properties of interactions between two or more systems. Univariate time-series-analysis techniques that assess nonlinear properties of brain dynamics such as complexity or fractality were evaluated in a number of reviews (see, e.g., [7782]). Here, we discuss recent developments of (mostly bivariate and multivariate) time-series-analysis techniques that allow construction and characterization of weighted and directed, evolving large-scale functional brain networks, which can contribute to gain deeper insights into cooperative phenomena that are supposed to underly brain (dys)functioning.

We begin with a brief overview of commonly used approaches to tackle the problems one is confronted with when investigating inherently nonstationary brain dynamics on various spatial and temporal scales. We then discuss time-series-analysis techniques that allow a data-driven characterization of interactions between two or more simultaneously sampled brain regions and that form the basis for constructing evolving functional brain networks from multichannel recordings of brain dynamics. We proceed with concepts and indices from network/graph theory that are frequently used to characterize statistical and spectral properties of these networks. Given that the significance of indices for interaction and network properties can be affected by influencing factors from various sources, we then discuss approaches for surrogate testing that can help eliminating or at least minimizing those influences. Before we draw our conclusions, we illustrate recent efforts on characterizing aspects of human epileptic brain dynamics varying on time scales spanning up to seven orders of magnitude.

2 Tackling nonstationarity

The human brain is an open, dissipative, and adaptive dynamical system and thus inherently nonstationary. This property is usually regarded as an obstacle, although nonstationarity might actually represent an interesting aspect of the dynamics [8385], and the vast majority of time-series-analysis techniques require the system to be (at least approximately) stationary in order to allow for a robust and reliable characterization.

For long-lasting recordings of ongoing brain dynamics, the most common way of dealing with nonstationarity is to apply a sliding-window analysis (see Fig. 1A), thereby cutting the time series of an appropriate observable into successive (non)overlapping segments (or windows) during which dynamics of the system can be regarded as approximately stationary. A characterizing index is then calculated for each data window. Various methods have been proposed to assess the length of a window (see, e.g., [49,83,84,8693]) and findings range between a few hundreds of milliseconds to some tens of seconds. When using shorter windows, one needs to take into account that the decreased number of data points entering some algorithm may increase the uncertainty of a characterizing index. Thus, the choice of a window length is often a compromise between the required statistical accuracy for the estimation of indices and approximate stationarity within a window length.

Other techniques make use of multiple realizations of the corresponding short-lasting (typically a few 100 ms) transient dynamics (e.g., recordings of evoked or event-related potentials/fields [94]). By replacing the temporal average with an ensemble average, these techniques allow for a time-resolved analysis of the transient dynamics [9598].

thumbnail Fig. 1

Schematic of interaction-analysis-based approaches to characterize time-varying brain dynamics. (A) Sliding-window analysis: long-lasting multichannel recordings of brain dynamics (here: intracranial EEG) are segmented into successive (non)overlapping windows; Tn denotes the time point associated with the left boundary of the nth window (marked in gray). (B) Time-dependent sequence of interaction matrices: each matrix contains estimates of an interaction property (strength or direction) calculated from the brain dynamics within a given window for all pairs of N sampled brain regions. (C) A bivariate analysis approach renders a time-dependent sequence of an interaction property (here: strength of interaction) for each pair of sampled brain regions (upper three sequences are shifted upwards to enhance readability). (D) A network analysis approach renders a time-dependent sequence of a local or global network characteristic. Time-dependent sequences depicted in (C) and (D) are then subject to further analyses.

3 Characterizing interactions

As already mentioned above, a large repertoire of analysis techniques – derived, e.g., from statistics, nonlinear dynamics, synchronization theory, and statistical physics – is available that allow one to characterize interactions between two or more systems (here: brain regions). Here we consider some of the general aspects and recent developments for inferring interactions from multichannel time-series data.

An interaction between two observed, possibly nonlinear dynamical systems is usually characterized by its strength and its direction, and these properties can be assessed with analysis techniques based on different concepts, such as (statistical) correlation [99], predictability [100,101], information flow [102,103], phase synchronization [104,105], or generalized synchronization [51,106108], to name just a few. Since each concept relies on certain characteristic features of the dynamical systems under investigation and thus captures different aspects of an interaction, the use of the respective analysis techniques for an investigation of interaction properties depends on the specific problem. In the neurosciences, and particularly in context of functional neuroimaging, the characterization of interactions is often based on the so-called brain connectivity approach that relates structure to function (and vice versa) [109]. Structural connectivity describes physical (and chemical) connections between neuronal populations or individual neurons. Functional connectivity describes the (statistical) dependence (or similarity) between signals recorded from some neuronal units regardless of whether these units are connected by direct structural edges. Effective connectivity requires formulation of a mechanistic model of the causal effects upon which the data to be observed are based, and may be viewed as a union of the aforementioned concepts. Inferring causality – particularly from short and noisy signals – is notoriously problematic.

We note that many indices, particularly those that characterize the strength of an interaction, were shown to be somewhat correlated with each other in comparative studies using both model dynamical systems [110,111] and EEG/iEEG recordings [112114]. Nevertheless, since interaction properties may vary considerably over time [114,115] (see Fig. 2), it is advisable to employ analysis techniques from various concepts to allow for a comprehensive characterization.

Despite being based on a common concept, most techniques assess either exclusively the strength or the direction of an interaction, and in many cases estimators may depend differently on type and strength of coupling [115119]. As an example (see Fig. 3), in some driver–responder system, estimates for the strength of an interaction increase monotonically with an increasing coupling strength until they saturate in the limit of strong couplings. Estimates for the direction of an interaction also increase, but only up to some maximum at intermediate coupling strengths and eventually vanish in the limit of strong couplings. This reflects the fact that the dynamics of coupled systems become almost indistinguishable in the limit of strong couplings, but estimators for the direction of an interaction require some form of distinguishability. It has been suggested that further insights into how estimators depend on type and strength of coupling may be gained with data-driven methods that aim at extracting and reconstructing the coupling functions between interacting systems [120124].

If interactions are to be characterized between more than two systems (i.e., a network composed of multiple coupled systems), the aforementioned concepts and their analysis techniques can – in general – be applied to all pairs of systems, which leads to an interaction matrix I that represents a network (see Fig. 1B and next section). Despite being widely used in various scientific disciplines, with this approach one might be faced with the demanding problem to distinguish between direct and indirect interactions (mediated by another – even unobserved – system), which can lead to severe misinterpretations of possible causal relationships. In order to overcome this problem of transitivity, various analysis techniques based on partialization analysis have been proposed (see, e.g., [125144]). All these techniques involve calculating properties of an interaction between two systems, holding constant the external influences of a third. Nevertheless, one needs to take into consideration that volume conduction and asymmetric signal-to-noise ratios [145,146] as well as the number of interacting systems and connection density [147149] appear to severely limit the efficiency of many techniques. More recent approaches to distinguish between direct and indirect interactions include concepts from network theory as well as assumptions regarding the network topology and causality [150156], but their suitability for the analysis of brain dynamics remains to be shown.

Another and closely related issue are spurious indications of strength and direction of interactions, which can be caused by sampling the same subsystem, i.e., a common source, and that can also lead to severe misinterpretations. Such an instantaneous mixture of activities from multiple sources may be caused by, e.g., a too close spatial sampling of some brain region or – as in case of EEG recordings – due to the unavoidable referential recording. Extensions to and modifications of particularly phase-based approaches [157160] have been proposed that appear to be much less affected by the influence of common sources, their suitability for analyses of brain dynamics, however, continues to be matter of debate [161165].

Before closing this section, we mention other developments that are of importance to further improve characterization of interacting brain systems. Given that axonal delays between brain areas can amount to several tens of milliseconds, recently proposed analysis techniques to detect and to characterize time-delayed directional interactions [166170] can help to better interpret the direction of interactions between brain regions. Last but not least, scale-bridging analysis techniques for characterizing couplings between point processes and flows [171] can help to better understand interactions between the single-neuron level and the neuron-network level.

thumbnail Fig. 2

Exemplary time-dependent sequences of mean interaction strength and various network characteristics. Sequences derived from multichannel (N = 90), long-term (14.7 days) iEEG recordings from a patient with a focal epilepsy undergoing presurgical evaluation (the patient signed informed consent that the clinical data might be used and published for research purposes, and the study protocol had previously been approved the by ethics committee of the University of Bonn). From top to bottom: mean interaction strength (average over all estimated strength values – calculated using the mean phase coherence R [105] – from all pairs of sampled brain regions), assortativity a, clustering coefficient c, synchronizability s, betweenness centrality of an exemplary brain region near the seizure-generating area, and betweenness centrality of a homologous region in the opposite brain hemisphere. Left column shows full sequences (smoothed with a moving average over 40 windows (13.65 min) to improve legibility, and tics on x-axis denote midnight); right column shows zooms in (1 h segment from day 1) of the unsmoothed sequences. Grey bars mark discontinuities in the sequences due to recording gaps.

thumbnail Fig. 3

Schematic dependence of strength and direction of an interaction on coupling strength. Estimates for the strength (red, solid line) and direction of an interaction (blue, dashed line) for some driver-responder system evaluated for increasing coupling strengths.

4 From pairwise interactions to functional brain networks

As already mentioned in the previous section, the aforementioned concepts and their analysis techniques can be applied to characterize interactions between multiple coupled systems, from which an interaction matrix I – that represents a network – can be derived (see Fig. 1B). Indeed, research over the last decade indicates that with this ansatz so called functional brain networks can be derived from multichannel recordings of neuronal activities (see, e.g., [172174] for an overview). Brain regions (nodes) are usually associated with sensors that are placed to sufficiently capture the node dynamics, and edges represent interactions between pairs of nodes. These nodes and edges constitute a functional brain network. This initial definition of network constituents, however, can have profound consequences for how functional brain networks are configured and interpreted [175182]. Moreover, it remains unclear which analysis techniques best represents the underlying neurobiological reality.

The interaction matrix , where N denotes the number of nodes, usually serves as starting point for the construction of binary or weighted and undirected or directed networks. An undirected binary network characterizes interacting brain regions in terms of connected or disconnected and can be described by a symmetric adjacency matrix A ∈ {0, 1}N×N. An entry Akl in this matrix is 1 if there is an edge between nodes k and l, and 0 otherwise. Typically, two nodes are assumed to be connected by an edge, if an estimator for the strength of interaction exceeds some threshold Θ. As of now, there are no commonly accepted criteria for the choice of Θ [176,183187]. Alternatively, one can derive a minimum spanning tree from the interaction matrix and use it as adjacency matrix [188191].

It is often of interest how strongly nodes interact with each other. This can be characterized by an undirected weighted network, which can be described by a symmetric weight matrix . While it is possible to again define some threshold to exclude edges with non-significant interaction strengths, in most cases all edges are considered to exist in such a complete network. The most direct way to determine the weight matrix W is to assume identity between a weight and the estimated strength of an interaction, i.e., Wkl = Ikl. Albeit most estimators for the strength of an interaction are normalized, in general, the weight matrix is not; a suitable normalization of this matrix is thus advisable. Eventually, dominant influences of the distribution of estimated strengths of interaction on network properties of interest [192] can be minimized by assigning weights to the edges using ranks of the entries of W [193].

Directed functional brain networks have been investigated only rarely (see, e.g., [194196]). Expanding a binary network to a directed binary network appears intuitive. Borrowing from the construction of the undirected case, an estimator for the direction of interaction can be thresholded to exclude non-significant directionality indications and to derive a directed adjacency matrix D ∈ {0, 1}N×N. In this asymmetric matrix an entry Dkl is 1, if there is a directed edge from node k to l, and 0 otherwise. If the entry of the inverse direction Dlk is also 1 (Dkl = Dlk = 1), the edge is called bidirectional. One should keep in mind though, that there are no commonly accepted criteria for the choice of the threshold. Assuming identity between a directed edge and the estimated direction of an interaction, Dkl = Ikl is problematic given the lack of physical interpretability of an estimator’s modulus; in many cases only the sign indicates the direction.

Merging both interaction properties – strength and direction – into a weighted and directed network is more complex and, as yet, not solved in a conclusive manner. For such a network, it is necessary to remember that strength and direction are different but not unrelated properties of interactions [115]. Quite often has the modulus of an estimator for the direction of an interaction been interpreted as the strength of an interaction, which might not generally be valid and particularly for uncoupled and strongly coupled systems can lead to severe misinterpretations [115,116,118]. It is therefore advisable to estimate both interaction properties separately but using methods that are based on the same concept (e.g., synchronization theory or information theory), as it is unclear how different concepts translate to each other. Moreover, it is necessary to define how the weights should be allocated to forward and backward direction of the edges, as strength of interaction is invariant under exchange of nodes, while direction of interaction is not.

With a chosen network construction approach and using a sliding-window analysis, a sequence of functional brain networks – a so called evolving brain network with fixed nodes and time-varying edges – is usually derived, whose further characterization is discussed in the next section.

5 Characterizing evolving brain networks

Characteristics of networks are assessed with approaches from graph theory (see [58,173,197200] for an overview). In the following, we concentrate on those characteristics that are frequently determined when investigating an undirected binary or weighted evolving brain network (extensions to directed networks are subject of current research). Characteristics can be divided into local ones, which describe properties of parts of the network, e.g., individual nodes or edges and into global ones, which assess properties of a network as a whole. Having determined the temporal sequence of some network characteristic (see Fig. 1D), this sequence is then subject to further analyses (e.g., using linear or nonlinear time-series-analysis techniques).

In a binary network, the degree of a node is the number of edges incident to it, while in a weighted network the strength – the sum over the weights of all edges incident to that node – is used instead. Degree and strength are often used as indicators for the connectedness of nodes or to assess the importance of a node (see below).

In a binary network, the shortest path between two nodes l and k is the smallest number of edges one has to traverse to reach node l from node k. The diameter of a network is the longest of all shortest paths. In a weighted network, defining the length of a single path between two nodes is not straightforward, but quite often is the inverse of the edge weight used as length since the ratio between the weights of two edges is the same as the ratio between their lengths. The mean shortest path is the mean over all shortest paths in a given network. For an unconnected network (no path exists between some nodes k and l) the mean shortest path will diverge. The easiest and most often applied ansatz to circumvent this problem is to ensure that the network is always connected (e.g., by choosing an appropriate threshold for network construction). Recent studies indicate that the shortest path is also affected by other influencing factors such as common sources and indirect interactions [184,201]. The mean shortest path is often used to assess the efficiency of information or mass transport on a network.

In a binary network, the (local) clustering coefficient of a node is the ratio of that node’s neighbors that are connected to each other. Taking the mean over all nodes gives an estimate for the (global) clustering coefficient of that network. The sometimes employed concept of transitivity, defined as the rate at which nodes with a common neighbor are connected, is not the same as the clustering coefficient, though similar. Several suggestions have been made to extend the concept of a clustering coefficient to weighted networks [202], but their suitability for the analysis of evolving brain networks remains to be shown. Similar to the mean shortest path, the clustering coefficient is prone to influences resulting from oversampling and common sources [184,201]. The clustering coefficient is often used to assess the robustness of a network to deletion of individual nodes.

Clustering coefficient and mean shortest path are often employed to determine whether a network is a small-world network or not [203]. This property has repeatedly been reported for both structural and functional, physiological and pathophysiological brain networks [172,204,205]. With the identification of factors that influence clustering coefficient and mean shortest path, however, these findings continue to be matter of considerable debate [175,177,182,206].

The tendency of nodes to connect to other nodes with similar properties is called assortativity. In the context of functional brain networks, most often the preference to connect to nodes with similar degrees is investigated. Such degree-degree correlations have far-reaching consequences for network resilience (assortative networks are less vulnerable to attacks [207]), for the ability of a network to globally synchronize (assortative networks are harder to synchronize), and the tendency of a network to separate into distinct groups (assortative networks have a stronger tendency to do so) [208].

Synchronizability describes the stability of the global synchronization state, i.e., the propensity of a network of coupled dynamical systems for synchronization. This property can be derived from the eigenvalue spectrum of the graph Laplacian via the ratio of the largest and the second smallest eigenvalue (for an overview, see, e.g., [209,210]).

The concept of centrality assigns an influence or importance to some node. Various centrality indices had been proposed and each of these indices capture different aspects of importance within a network [211,212]. Nodes with high degree (or strength) centrality (see above) are often called hubs. Closeness centrality is the average of the distances (i.e., shortest paths) of a node to all other nodes. Degree and closeness centrality thus consider nodes which are better connected to the rest of the network. In contrast, betweenness centrality is defined as the ratio of all shortest paths (between all nodes) passing through a given node. A node with a high betweenness centrality is important for connecting different regions of the network by acting as a bridge. Eigenvector centrality for node i is defined as the ith element of the eigenvector to the largest eigenvalue of the adjacency matrix A. Through this recursive definition, it has high values for nodes that are connected to other nodes with a high eigenvector centrality.

We note that the aforementioned centrality indices can be defined analogously for edges. Moreover, the characterization of networks based on minimum spanning trees may require additional indices.

6 Surrogates, null models, and network comparison

Characteristics of time series and evolving networks can be affected by a number of influencing factors, which can lead to severe misinterpretations. Such factors may arise from specifics of the applied recording technique, from specifics and uncertainties of the various analysis methods, from the way networks are derived from multichannel time-series data, or may be due to unavoidable noise contaminations. An elimination or at least a minimization of those influences can be achieved through surrogate testing. This statistical approach requires the formulation of a null hypothesis [213], which specifies properties of influencing factors that might lead to the results of an analysis. An ensemble of surrogate data – Monte Carlo-simulated [214] instances of an appropriate null model – is then generated, which preserves all important statistical properties of the original data but not the property which is tested for. The null hypothesis will be accepted if some discriminating statistics for the original data falls within the expectation range for the surrogate ensemble.

Surrogate-generating concepts and algorithms are available that can be applied on the level of time series or on the network level. These are briefly described in the following.

A large number of techniques is available to generate surrogates for multichannel time-series data (see, e.g., [39,40,215] for an overview). When investigating oscillatory brain dynamics, particularly Fourier-based surrogate techniques are prone to false rejections of the null hypothesis due to quasi-periodicities or coherency, and various extensions have been proposed that aim at minimizing the risk for false rejections [216218]. Alternatives to such surrogate techniques are based on the wavelet transform [219221] or on recurrence plots [222]. Techniques that take into account long-term fluctuations [223,224] or other forms of nonstationarities [225,226] may prove helpful when analyzing interactions in multichannel recordings that extend over longer time periods (hours to days to weeks).

On the network level, one may want to test first how the way functional networks are derived from time-series data affect findings. To this end, the results of the whole analysis procedure can be compared to those for surrogate time series [177,201]. Characteristics of the derived networks can then be compared to those that can be expected for null model or surrogate networks that constrain specific properties [173]. Eventually, an appropriate surrogate normalization of the specific network characteristics can aid with their interpretation, and thus can prevent deriving inappropriate conclusions (see, e.g., [192] and references therein).

When investigating time-dependent networks, it is often of interest, how networks change from time step to time step, which requires some form of comparative analysis between networks [176]. Such an analysis, however, is highly nontrivial, given that topological properties of a given network are necessarily dependent on the number of edges and the number of nodes, each of which may change over time, rendering an unbiased comparison between networks difficult. An improved testing for differences between time-evolving networks may be achieved with distance metrics [227230] or with other recently proposed methods for generating surrogates for time-evolving networks [231233].

7 Capturing time-varying dynamics in the human epileptic brain

Electroencephalographic recordings (either non-invasive or invasive) from epilepsy patients are particularly suitable for an investigation of time-varying brain dynamics on various spatial and temporal scales. During clinical monitoring for epilepsy surgery such recordings are usually performed over extended periods of time (days to weeks), with high spatial and temporal resolution. Moreover, the recent development of an ambulatory seizure advisory system [234] allowed to record invasively electrical activity of the brains from epilepsy patients for a period of several years. Long-term recordings capture a wide spectrum of physiological and pathophysiological activities as well as interactions between these activities. In the following, we briefly illustrate some recent findings that have been achieved by analyzing such recordings.

A large number of studies investigating long-term recordings from epilepsy patients is concerned with the question whether information predictive of an impending seizure can be extracted from ongoing EEG/iEEG signals using univariate, bivariate, or multivariate analysis techniques (see, e.g., [235238] for an overview). Findings achieved so far indicate that, at least in some patients, pre-seizure states can indeed be identified with various techniques well ahead of seizure occurrence. Pre-seizure changes in the time-dependent sequence of some characteristics, however, may also be identifiable – but through less pronounced changes – during the seizure-free interval. It thus remains unclear whether identification performance suffices for clinical applications of prediction-based seizure-prevention techniques [239]. Given the heterogeneity of seizures and epilepsies, successful state identification probably requires a combination of various univariate and bivariate, both linear and nonlinear, analysis techniques. In a recent seizure-forecasting contest [240], a large number of spectral EEG properties together with various machine-learning algorithms were reported to perform better than some random predictor on a large iEEG data-base from canines and humans with epilepsy. Although such a contest allows to directly test and compare various seizure-forecasting strategies, the multiple testing [241] of a plethora of algorithms requires a reasonable control of the type-I error. In addition, findings unfortunately add little – if at all – to our understanding of possible neurophysiological mechanisms associated with pre-seizure changes.

Other studies aim at pinpointing factors that might influence the identification of pre-seizure states, such as seizure occurrences, changes in antiepileptic medication, the circadian rhythm, to name just a few. Inter-seizure intervals of more than 10 000 seizures (derived from long-term (up to 2 yrs) ambulatory intracranial EEG recordings) were reported [242] to follow a power-law distribution [243] with an exponent indicative of long-range memory in human seizure frequency. This observation was also supported by estimated Hurst exponents H in the range 0.5 < H < 1, and the observed memory spanned time scales from 30 min to 40 d. This finding is in stark contrast to the notion that timing and occurrence of seizures can best be described by a Poisson process [244,245].

Changes in cortical excitability are believed to play a role in normal conditions during the sleep-wake cycle, while pathological changes are closely related to initiation and spread of seizure activity. Cortical excitability – characterized from the ongoing EEG/iEEG with bivariate indices for synchronization – was reported [246] to correlate well with the load of antiepileptic drugs (used to control the degree of cortical excitability) and to exhibit a characteristic modulation over 24 h, progressively increasing during wakefulness and rebalanced during sleep. Together with an earlier report on a brain-region-dependent impact of the circadian rhythm [247], these findings indicate that EEG-based indices for synchronization can probably regarded a reliable characteristic for excitability.

There is increasing evidence for epilepsy to be a network disease (see, e.g., [72,173,248,249] for an overview), with relevant interactions extending well beyond the seizure-generating brain structure (epileptic focus or epileptogenic zone). With this concept, seizures (even focal ones) and other related pathophysiological dynamics are regarded as emerging from, spreading via, and being terminated by network constituents that generate and sustain normal, physiological brain dynamics during the seizure-free interval. Deeper insights into epileptic networks and their dynamics can be achieved by investigating – at various scales – characteristics of functional brain networks derived from epilepsy patients. A number of studies of functional brain networks during seizures reported various local and global network characteristics to provide important clues about seizure dynamics, how they spread and terminate (see, e.g., [208,250260]). These findings – exhibiting a high similarity of topological evolution across different types of epilepsies, seizures, medication, age, gender, and other clinical features – are of high relevance for further improving existing seizure-prevention techniques or for developing new ones, based, e.g., on behaviorally induced modifications of epileptic networks [238,261,262]. As regards the seizure-free interval, most studies investigated functional brain networks that were derived from selected EEG/iEEG epochs that ranged from some seconds to a few minutes only (see [263] for an overview), and only rarely [256,264267] had the evolution of an epileptic network been monitored over longer periods (days to weeks; for an example, see Fig. 2). These studies reported on characteristic modulations of local and global network characteristics over 24 h and also at the subharmonics at about 12 and 8 h, which points to a strong influence of the circadian rhythm. There are further indications for other influences from infra- and ultradian rhythms and possibly from changes of the anticonvulsant medication. On the other hand, important aspects of the epileptic process that typically act on time scales from seconds up to a few hours were reported to contribute to a lesser extent to the temporal variability of the network characteristics. These findings are of importance for future seizure-prediction studies based on network characteristics as well as for studies that aim at shedding further light into the complex interactions between physiological and pathophysiological activities in large-scale brain networks.

8 Conclusions and outlook

We summarized developments of bivariate time-series-analyses techniques that aim at assessing strength and direction of interactions as well as methodologies for deriving and characterizing evolving networks from empirical time series. At the example of evolving epileptic brain networks, we discussed the progress that has been made in capturing and improving our understanding of brain dynamics that varies on time scales ranging from seconds to years. In a work of this scope, it is inevitable that some contributions may be over- or underemphasized, depending upon the points to be made in the text.

It is quite difficult to decide which of all the approaches presented here is the best, given problems for which there are by now no satisfactory and commonly accepted solutions. It is therefore advisable to consider the different characteristics as tentative indices of interacting brain systems and to carefully apply various methodologies that are sensitive to different aspects of time-varying brain dynamics.

Capturing time-varying brain dynamics remains a demanding issue, given the difficulties associated with unequivocally deriving causal relationships from empirical data together with the challenges associated with mapping between multichannel time-series data and complex networks as well as their robust classification. We are confident though that further developments will allow for an improved inference and characterization of time-varying physiological and pathophysiological brain dynamics.

Conflicts of interest

The authors declare that they have no conflicts of interest in relation to this article.

Authors' contribution

All authors have written the paper and have given final approval for its publication.


The authors would like to thank G. Ansmann and J. Brose for critical comments on earlier versions of the manuscript. This work was supported by W Science Laboratories AG.


  1. V. Braitenberg, A. Schütz, Anatomy of the cortex – statistics and geometry (Springer, Berlin, 1991) [Google Scholar]
  2. J.M.J. Murre, D.P.F. Sturdy, The connectivity of the brain: multi-level quantitative analysis, Biol. Cybern. 73, 529 (1995) [CrossRef] [Google Scholar]
  3. A. Volterra, J. Meldolesi, Astrocytes, from brain glue to communication elements: the revolution continues, Nat. Rev. Neurosci. 6, 626 (2005) [Google Scholar]
  4. V.B. Mountcastle, The columnar organization of the neocortex, Brain 120, 701 (1997) [CrossRef] [Google Scholar]
  5. B. Hutcheon, Y. Yarom, Resonance, oscillation and the intrinsic frequency preferences of neurons, Trends Neurosci. 23, 216 (2000) [CrossRef] [PubMed] [Google Scholar]
  6. A.K. Engel, P. Fries, W. Singer, Dynamic predictions: oscillations and synchrony in top-down processing, Nat. Rev. Neurosci. 2, 704 (2001) [CrossRef] [PubMed] [Google Scholar]
  7. E. Salinas, T.J. Sejnowski, Correlated neuronal activity and the flow of neural information, Nat. Rev. Neurosci. 2, 539 (2001) [CrossRef] [PubMed] [Google Scholar]
  8. F.J. Varela, J.P. Lachaux, E. Rodriguez, J. Martinerie, The brain web: Phase synchronization and large-scale integration, Nat. Rev. Neurosci. 2, 229 (2001) [CrossRef] [PubMed] [Google Scholar]
  9. S. Makeig, S. Debener, J. Onton, A. Delorme, Mining event-related brain dynamics, Trends Cogn. Sci. 8, 204 (2004) [CrossRef] [Google Scholar]
  10. S.L. Bressler, V. Menon, Large-scale brain networks in cognition: emerging methods and principles, Trends Cogn. Sci. 14, 277 (2010) [CrossRef] [Google Scholar]
  11. J. Fell, N. Axmacher, The role of phase synchronization in memory processes, Nat. Rev. Neurosci. 12, 105 (2011) [CrossRef] [PubMed] [Google Scholar]
  12. G. Buzsáki, C.A. Anastassiou, C. Koch, The origin of extracellular fields and currents–EEG, ECoG, LFP and spikes, Nat. Rev. Neurosci. 13, 407 (2012) [CrossRef] [PubMed] [Google Scholar]
  13. W. Freeman, Neurodynamics: an exploration in mesoscopic brain dynamics (Springer, London, UK, 2012) [Google Scholar]
  14. M.I. Rabinovich, K.J. Friston, P. Varona, eds., Principles of brain dynamics: global state interactions (MIT Press, Cambridge, MA, 2012) [Google Scholar]
  15. M. Siegel, T.H. Donner, A.K. Engel, Spectral fingerprints of large-scale neuronal interactions, Nat. Rev. Neurosci. 13, 121 (2012) [Google Scholar]
  16. A.K. Engel, C. Gerloff, C.C. Hilgetag, G. Nolte, Intrinsic coupling modes: multiscale interactions in ongoing brain activity, Neuron 80, 867 (2013) [CrossRef] [Google Scholar]
  17. H.S. Lee, A. Ghetti, A. Pinto-Duarte, X. Wang, G. Dziewczapolski, F. Galimi, S. Huitron-Resendiz, J.C. Pina-Crespo, A.J. Roberts, I.M. Verma, T.J. Sejnowski, S.F. Heinemann, Astrocytes contribute to gamma oscillations and recognition memory, Proc. Natl. Acad. Sci. USA 111, 3343 (2014) [CrossRef] [Google Scholar]
  18. A. Schnitzler, J. Gross, Normal and pathological oscillatory communication in the brain, Nat. Rev. Neurosci. 6, 285 (2005) [CrossRef] [PubMed] [Google Scholar]
  19. P.J. Uhlhaas, W. Singer, Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology, Neuron 52, 155 (2006) [CrossRef] [PubMed] [Google Scholar]
  20. N.D. Schiff, T. Nauvel, J.D. Victor, Large-scale brain dynamics in disorders of consciousness, Curr. Opin. Neurobiol. 25, 7 (2014) [CrossRef] [Google Scholar]
  21. E. Niedermeyer, F. Lopes da Silva, Electroencephalography: basic principles, clinical applications, and related fields (Lippincott Williams and Williams, Philadelphia, 2005) [Google Scholar]
  22. M. Hämäläinen, R. Hari, R.J. Ilmoniemi, J. Knuutila, O.V. Lounasmaa, Magnetoencephalography – theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys. 65, 413 (1993) [CrossRef] [Google Scholar]
  23. S.A. Huettel, A.W. Song, G. McCarthy, Functional magnetic resonance imaging, 3rd edn (Sinauer Associates Sunderland, Sunderland, MA, USA, 2004) [Google Scholar]
  24. F. Rosenow, H. Lüders, Presurgical evaluation of epilepsy, Brain 124, 1683 (2001) [CrossRef] [PubMed] [Google Scholar]
  25. A.K. Engel, C.K.E. Moll, I. Fried, G.A. Ojemann, Invasive recordings from the human brain: clinical insights and beyond, Nat. Rev. Neurosci. 6, 35 (2005) [CrossRef] [PubMed] [Google Scholar]
  26. S.S. Cash, L.R. Hochberg, The emergence of single neurons in clinical neurology, Neuron 86, 79 (2015) [CrossRef] [Google Scholar]
  27. E.F. Chang, Towards large-scale, human-based, mesoscopic neurotechnologies, Neuron 86, 68 (2015) [CrossRef] [Google Scholar]
  28. J. Niediek, J. Boström, C.E. Elger, F. Mormann, Reliable analysis of single-unit recordings from the human brain under noisy conditions: tracking neurons over hours, PLOS ONE 11, 0166598 (2016) [CrossRef] [Google Scholar]
  29. S. Panzeri, J.H. Macke, J. Gross, C. Kayser, Neural population coding: combining insights from microscopic and mass signals, Trends Cogn. Sci. 19, 162 (2015) [CrossRef] [Google Scholar]
  30. S. Marom, Neural timescales or lack thereof, Prog. Neurobiol. 90, 16 (2010) [CrossRef] [PubMed] [Google Scholar]
  31. T. Gisiger, Scale invariance in biology: coincidence or footprint of a universal mechanism? Biol. Rev. 76, 161 (2001) [CrossRef] [Google Scholar]
  32. C. Bédard, H. Kröger, A. Destexhe, Does the 1∕f frequency scaling of brain signals reflect self-organized critical states? Phys. Rev. Lett. 97, 118102 (2006) [CrossRef] [PubMed] [Google Scholar]
  33. G. Werner, Fractals in the nervous system: conceptual implications for theoretical neuroscience, Front. Physiol. 1, 15 (2010) [Google Scholar]
  34. B.J. He, Scale-free brain activity: past, present, and future, Trends Cogn. Sci. 18, 480 (2014) [CrossRef] [Google Scholar]
  35. C. Meisel, C. Kuehn, Scaling effects and spatio-temporal multilevel dynamics in epileptic seizures, PLOS ONE 7, 30371 (2012) [CrossRef] [Google Scholar]
  36. P.L. Nunez, R. Srinivasan, Electric fields of the brain: the neurophysics of EEG (Oxford University Press, Oxford, UK, 2006), 2nd edn [CrossRef] [Google Scholar]
  37. K. Lehnertz, J. Arnhold, P. Grassberger, C.E. Elger, Chaos in brain? (World Scientific, Singapore, 2000) [CrossRef] [Google Scholar]
  38. M. Breakspear, Dynamic models of large-scale brain activity, Nat. Neurosci. 20, 340 (2017) [CrossRef] [Google Scholar]
  39. T. Schreiber, A. Schmitz, Surrogate time series, Physica D 142, 346 (2000) [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  40. M. Paluš, From nonlinearity to causality: statistical testing and inference of physical mechanisms underlying complex dynamics, Contemp. Phys. 48, 307 (2007) [CrossRef] [Google Scholar]
  41. M.J. Van der Heyden, C. Diks, J.P.M. Pijn, D.N. Velis, Time reversibility of intracranial human EEG recordings in mesial temporal lobe epilepsy, Phys. Lett. A 216, 283 (1996) [CrossRef] [Google Scholar]
  42. M.C. Casdagli, L.D. Iasemidis, R.S. Savit, R.L. Gilmore, S. Roper, J.C. Sackellares, Non-linearity in invasive EEG recordings from patients with temporal lobe epilepsy, Electroencephalogr. Clin. Neurophysiol. 102, 98 (1997) [CrossRef] [Google Scholar]
  43. J.P. Pijn, D.N. Velis, M.J. van der Heyden, J. DeGoede, C.W.M. van Veelen, Lopes da Silva, F.H.: Nonlinear dynamics of epileptic seizures on basis of intracranial EEG recordings, Brain Topogr. 9, 249 (1997) [CrossRef] [Google Scholar]
  44. M. Feucht, U. Möller, H. Witte, K. Schmidt, M. Arnold, F. Benninger, K. Steinberger, M.H. Friedrich, Nonlinear dynamics of 3 Hz spike-and-wave discharges recorded during typical absence seizures in children, Cereb. Cortex 8, 524 (1998) [CrossRef] [Google Scholar]
  45. J.L. Perez Velazquez, H. Khosravani, A. Lozano, B.L. Bardakijan, P.L. Carlen, R. Wennberg, Type III intermittency in human partial epilepsy, Eur. J. Neurosci. 11, 2571 (1999) [CrossRef] [Google Scholar]
  46. R.G. Andrzejak, G. Widman, K. Lehnertz, P. David, C.E. Elger, The epileptic process as nonlinear deterministic dynamics in a stochastic environment: An evaluation on mesial temporal lobe epilepsy, Epilepsy Res. 44, 129 (2001) [CrossRef] [Google Scholar]
  47. R.G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David, C.E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state, Phys. Rev. E 64, 061907 (2001) [CrossRef] [Google Scholar]
  48. T. Gautama, D.P. Mandic, M.M. Van Hulle, Indications of nonlinear structures in brain electrical activity, Phys. Rev. E 67, 046204 (2003) [CrossRef] [Google Scholar]
  49. C. Rieke, F. Mormann, R.G. Andrzejak, T. Kreuz, P. David, C.E. Elger, K. Lehnertz, Discerning nonstationarity from nonlinearity in seizure-free and preseizure EEG recordings from epilepsy patients, IEEE Trans. Biomed. Eng. 50, 634 (2003) [CrossRef] [Google Scholar]
  50. R.G. Andrzejak, F. Mormann, G. Widmann, T. Kreuz, C.E. Elger, K. Lehnertz, Improved spatial characterization of the epileptic brain by focusing on nonlinearity, Epilepsy Res. 69, 30 (2006) [CrossRef] [Google Scholar]
  51. R.G. Andrzejak, K. Schindler, C. Rummel, Nonrandomness, nonlinear dependence, and nonstationarity of electroencephalographic recordings from epilepsy patients, Phys. Rev. E 86, 046206 (2012) [CrossRef] [Google Scholar]
  52. J.F. Donges, R.V. Donner, J. Kurths, Testing time series irreversibility using complex network methods, Europhys. Lett. 102, 10004 (2013) [Google Scholar]
  53. M. Anvari, M.R.R. Tabar, J. Peinke, K. Lehnertz, Disentangling the stochastic behavior of complex time series, Sci. Rep. 6, 35435 (2016) [CrossRef] [Google Scholar]
  54. K. Schindler, C. Rummel, R.G. Andrzejak, M. Goodfellow, F. Zubler, E. Abela, R. Wiest, C. Pollo, A. Steimer, H. Gast, Ictal time-irreversible intracranial EEG signals as markers of the epileptogenic zone, Clin. Neurophysiol. 127, 3051 (2016) [CrossRef] [Google Scholar]
  55. M. Rizzi, I. Weissberg, D.Z. Milikovsky, A. Friedman, Following a potential epileptogenic insult, prolonged high rates of nonlinear dynamical regimes of intermittency type is the hallmark of epileptogenesis, Sci. Rep. 6, 35510 (2016) [CrossRef] [Google Scholar]
  56. A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization: a universal concept in nonlinear sciences (Cambridge University Press, Cambridge, UK, 2001) [Google Scholar]
  57. S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, The synchronization of chaotic systems, Phys. Rep. 366, 1 (2002) [CrossRef] [MathSciNet] [Google Scholar]
  58. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008) [CrossRef] [MathSciNet] [Google Scholar]
  59. Y. Tang, F. Qian, H. Gao, J. Kurths, Synchronization in complex networks and its application – a survey of recent advances and challenges, Annu. Rev. Control 38, 184 (2014) [CrossRef] [Google Scholar]
  60. G.E.P. Box, G.M. Jenkins, Time series analysis: forecasting and control, revised ed (Holden-Day, San Francisco, 1993) [Google Scholar]
  61. D. Brillinger, Time series: data analysis and theory (Holden-Day, San Francisco, USA, 1981) [Google Scholar]
  62. M.B. Priestley, Nonlinear and non-stationary time series analysis (Academic Press, London, 1988) [Google Scholar]
  63. B. Boashash, Time frequency signal analysis: methods and applications (Longman Cheshire, Melbourne, 1992) [Google Scholar]
  64. H.D.I. Abarbanel, Analysis of observed chaotic data (Springer, New York, 1996) [Google Scholar]
  65. J.S. Bendat, A.G. Piersol, Random data analysis and measurement procedure (Wiley, New York, 2000) [Google Scholar]
  66. H. Kantz, T. Schreiber, Nonlinear time series analysis (Cambridge University Press, Cambridge, UK, 2003), 2nd edn [Google Scholar]
  67. E. Pereda, R. Quian Quiroga, J. Bhattacharya, Nonlinear multivariate analysis of neurophysiological signals, Prog. Neurobiol. 77, 1 (2005) [CrossRef] [PubMed] [Google Scholar]
  68. C.J. Stam, Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field, Clin. Neurophysiol. 116, 2266 (2005) [CrossRef] [PubMed] [Google Scholar]
  69. K. Hlaváčková-Schindler, M. Paluš, M. Vejmelka, J. Bhattacharya, Causality detection based on information-theoretic approaches in time series analysis, Phys. Rep. 441, 1 (2007) [CrossRef] [Google Scholar]
  70. N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Recurrence plots for the analysis of complex systems, Phys. Rep. 438, 237 (2007) [Google Scholar]
  71. H. Osterhage, K. Lehnertz, Nonlinear time series analysis in epilepsy, Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 3305 (2007) [CrossRef] [Google Scholar]
  72. K. Lehnertz, S. Bialonski, M.-T. Horstmann, D. Krug, A. Rothkegel, M. Staniek, T. Wagner, Synchronization phenomena in human epileptic brain networks, J. Neurosci. Methods 183, 42 (2009) [CrossRef] [PubMed] [Google Scholar]
  73. R. Friedrich, J. Peinke, M. Sahimi, M.R.R. Tabar, Approaching complexity by stochastic methods: from biological systems to turbulence, Phys. Rep. 506, 87 (2011) [CrossRef] [Google Scholar]
  74. K. Lehnertz, Assessing directed interactions from neurophysiological signals – an overview, Physiol. Meas. 32, 1715 (2011) [CrossRef] [PubMed] [Google Scholar]
  75. P. Clemson, G. Lancaster, A. Stefanovska, Reconstructing time-dependent dynamics, Proc. IEEE 104, 223 (2016) [CrossRef] [Google Scholar]
  76. A. Porta, L. Faes, Wiener-Granger causality in network physiology with applications to cardiovascular control and neuroscience, Proc. IEEE 104, 282 (2016) [CrossRef] [Google Scholar]
  77. W.S. Pritchard, D.S. Duke, Measuring chaos in the brain: a tutorial review of nonlinear dynamical EEG analysis, Int. J. Neurosci. 67, 31 (1992) [CrossRef] [Google Scholar]
  78. K. Lehnertz, R.G. Andrzejak, J. Arnhold, T. Kreuz, F. Mormann, C. Rieke, G. Widman, C.E. Elger, Nonlinear EEG analysis in epilepsy: its possible use for interictal focus localization, seizure anticipation, and prevention, J. Clin. Neurophysiol. 18, 209 (2001) [CrossRef] [Google Scholar]
  79. A. Eke, P. Herman, L. Kocsis, L.R. Kozak, Fractal characterization of complexity in temporal physiological signals, Physiol. Meas. 23, 1 (2002) [CrossRef] [Google Scholar]
  80. J. Kwapień, S. Drożdż, Physical approach to complex systems, Phys. Rep. 515, 115 (2012) [CrossRef] [Google Scholar]
  81. A. Di Ieva, F. Grizzi, H. Jelinek, A.J. Pellionisz, G.A. Losa, Fractals in the neurosciences, part I: general principles and basic neurosciences, Neuroscientist 20, 403 (2014) [CrossRef] [Google Scholar]
  82. A. Di Ieva, F.J. Esteban, F. Grizzi, W. Klonowski, M. Martín-Landrove, Fractals in the neurosciences, part II: clinical applications and future perspectives, Neuroscientist 21, 30 (2015) [CrossRef] [Google Scholar]
  83. R. Hegger, H. Kantz, L. Matassini, T. Schreiber, Coping with non-stationarity by overembedding, Phys. Rev. Lett. 84, 4092 (2000) [CrossRef] [Google Scholar]
  84. C. Rieke, K. Sternickel, R.G. Andrzejak, C.E. Elger, P. David, K. Lehnertz, Measuring nonstationarity by analyzing the loss of recurrence in dynamical systems, Phys. Rev. Lett. 88, 244102 (2002) [CrossRef] [Google Scholar]
  85. E.C.A. Hansen, D. Battaglia, A. Spiegler, G. Deco, V.K. Jirsa, Functional connectivity dynamics: Modeling the switching behavior of the resting state, NeuroImage 105, 525 (2015) [CrossRef] [Google Scholar]
  86. J.S. Barlow, Methods of analysis of nonstationary EEGs with emphasis on segmentation techniques: a comparative review, J. Clin. Neurophysiol. 2, 267 (1985) [CrossRef] [Google Scholar]
  87. S. Blanco, H. Garcia, R. Quian Quiroga, L. Romanelli, O.A. Rosso, Stationarity of the EEG series, IEEE Eng. Med. Biol. 4, 395 (1995) [CrossRef] [Google Scholar]
  88. T. Schreiber, Detecting and analysing nonstationarity in a time series using nonlinear cross predictions, Phys. Rev. Lett. 78, 843 (1997) [CrossRef] [Google Scholar]
  89. A. Witt, J. Kurths, A. Pikovsky, Testing stationarity in time series, Phys. Rev. E 58, 1800 (1998) [CrossRef] [Google Scholar]
  90. C. Rieke, R.G. Andrzejak, F. Mormann, K. Lehnertz, Improved statistical test for nonstationarity using recurrence time statistics, Phys. Rev. E 69, 046111 (2004) [CrossRef] [Google Scholar]
  91. T. Dikanev, D. Smirnov, R. Wennberg, J. Velazquez, B. Bezruchko, EEG nonstationarity during intracranially recorded seizures: statistical and dynamical analysis, Clin. Neurophysiol. 116, 1796 (2005) [CrossRef] [Google Scholar]
  92. A.Y. Kaplan, A.A. Fingelkurts, A.A. Fingelkurts, S.V. Borisov, B.S. Darkhovsky, Nonstationary nature of the brain activity as revealed by EEG/MEG: methodological, practical and conceptual challenges, Signal Proc. 85, 2190 (2005) [CrossRef] [Google Scholar]
  93. S. Tong, Z. Li, Y. Zhu, N.V. Thakor, Describing the nonstationarity level of neurological signals based on quantifications of time-frequency representation, IEEE Trans. Biomed. Eng. 54, 1780 (2007) [CrossRef] [Google Scholar]
  94. S.J. Luck, An introduction to the event-related potential technique (MIT Press, Cambridge, MA, 2005) [Google Scholar]
  95. R.G. Andrzejak, A. Ledberg, G. Deco, Detecting event-related time-dependent directional couplings, New J. Phys. 8, 6 (2006) [CrossRef] [Google Scholar]
  96. S. Łeski, D.K. Wójcik, Inferring coupling strength from event-related dynamics, Phys. Rev. E 78, 41918 (2008) [CrossRef] [Google Scholar]
  97. M. Martini, T.A. Kranz, T. Wagner, K. Lehnertz, Inferring directional interactions from transient signals with symbolic transfer entropy, Phys. Rev. E 83, 011919 (2011) [CrossRef] [MathSciNet] [Google Scholar]
  98. P. Wollstadt, M. Martinez-Zarzuela, R. Vicente, F.J. Diaz-Pernas, M. Wibral, Efficient transfer entropy analysis of non-stationary neural time series, PLOS ONE 9, 1 (2014) [Google Scholar]
  99. J.L. Rodgers, W.A. Nicewander, Thirteen ways to look at the correlation coefficient, Am. Stat. 42, 59 (1988) [CrossRef] [Google Scholar]
  100. C.W.J. Granger, Investigating causal relations by econometric models and cross–spectral methods, Econometrica 37, 424 (1969) [Google Scholar]
  101. M. Eichler, A graphical approach for evaluating effective connectivity in neural systems, Philos. Trans. R. Soc. Lond. B: Biol. Sci. 360, 953 (2005) [CrossRef] [Google Scholar]
  102. T. Schreiber, Measuring information transfer, Phys. Rev. Lett. 85, 461 (2000) [CrossRef] [PubMed] [Google Scholar]
  103. Z. Liu, Measuring the degree of synchronization from time series data, Europhys. Lett. 68, 19 (2004) [CrossRef] [EDP Sciences] [Google Scholar]
  104. M.G. Rosenblum, A.S. Pikovsky, Detecting direction of coupling in interacting oscillators, Phys. Rev. E 64, 045202 (2001) [CrossRef] [Google Scholar]
  105. F. Mormann, K. Lehnertz, P. David, C.E. Elger, Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients, Physica D 144, 358 (2000) [CrossRef] [Google Scholar]
  106. J. Arnhold, P. Grassberger, K. Lehnertz, C.E. Elger, A robust method for detecting interdependences: application to intracranially recorded EEG, Physica D 134, 419 (1999) [CrossRef] [Google Scholar]
  107. R. Quian Quiroga, J. Arnhold, P. Grassberger, Learning driver-response relationships from synchronization patterns, Phys. Rev. E 61, 5142 (2000) [CrossRef] [Google Scholar]
  108. R.G. Andrzejak, D. Chicharro, K. Lehnertz, F. Mormann, Using bivariate signal analysis to characterize the epileptic focus: the benefit of surrogates, Phys. Rev. E 83, 046203 (2011) [CrossRef] [Google Scholar]
  109. K.J. Friston, Functional and effective connectivity: a review, Brain Connect. 1, 13 (2011) [Google Scholar]
  110. O. David, D. Cosmelli, K.J. Friston, Evaluation of different measures of functional connectivity using a neural mass model, NeuroImage 21, 659 (2004) [CrossRef] [Google Scholar]
  111. T. Kreuz, F. Mormann, R.G. Andrzejak, A. Kraskov, K. Lehnertz, P. Grassberger, Measuring synchronization in coupled model systems: a comparison of different approaches, Physica D 225, 29 (2007) [CrossRef] [MathSciNet] [Google Scholar]
  112. R. Quian Quiroga, A. Kraskov, T. Kreuz, P. Grassberger, Performance of different synchronization measures in real data: a case study on electroencephalographic signals, Phys. Rev. E 65, 041903 (2002) [CrossRef] [Google Scholar]
  113. H. Osterhage, F. Mormann, M. Staniek, K. Lehnertz, Measuring synchronization in the epileptic brain: a comparison of different approaches, Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 3539 (2007) [CrossRef] [Google Scholar]
  114. H. Dickten, S. Porz, C.E. Elger, K. Lehnertz, Weighted and directed interactions in evolving large-scale epileptic brain networks, Sci. Rep. 6, 34824 (2016) [CrossRef] [Google Scholar]
  115. K. Lehnertz, H. Dickten, Assessing directionality and strength of coupling through symbolic analysis: an application to epilepsy patients, Philos. Trans. R. Soc. A 373, 20140094 (2015) [Google Scholar]
  116. M. Paluš, M. Vejmelka, Directionality of coupling from bivariate time series: how to avoid false causalities and missed connections, Phys. Rev. E 75, 056211 (2007) [CrossRef] [MathSciNet] [Google Scholar]
  117. H. Osterhage, F. Mormann, T. Wagner, K. Lehnertz, Measuring the directionality of coupling: phase versus state space dynamics and application to EEG time series, Int. J. Neural. Syst. 17, 139 (2007) [CrossRef] [PubMed] [Google Scholar]
  118. H. Osterhage, F. Mormann, T. Wagner, K. Lehnertz, Detecting directional coupling in the human epileptic brain: Limitations and potential pitfalls, Phys. Rev. E 77, 011914 (2008) [CrossRef] [Google Scholar]
  119. M. Staniek, K. Lehnertz, Symbolic transfer entropy, Phys. Rev. Lett. 100, 158101 (2008) [CrossRef] [PubMed] [Google Scholar]
  120. K.A. Blaha, A. Pikovsky, M. Rosenblum, M.T. Clark, C.G. Rusin, J.L. Hudson, Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators, Phys. Rev. E 84, 046201 (2011) [CrossRef] [Google Scholar]
  121. T. Stankovski, A. Duggento, P.V.E. McClintock, A. Stefanovska, Inference of time-evolving coupled dynamical systems in the presence of noise, Phys. Rev. Lett. 109, 024101 (2012) [CrossRef] [PubMed] [Google Scholar]
  122. B. Kralemann, M. Frühwirth, A. Pikovsky, M. Rosenblum, T. Kenner, J. Schaefer, M. Moser, In vivo cardiac phase response curve elucidates human respiratory heart rate variability, Nat. Commun. 4, 2418 (2013) [CrossRef] [PubMed] [Google Scholar]
  123. T. Stankovski, V. Ticcinelli, P.V.E. McClintock, A. Stefanovska, Coupling functions in networks of oscillators, New J. Phys. 17, 035002 (2015) [CrossRef] [Google Scholar]
  124. J. Wilting, K. Lehnertz, Bayesian inference of interaction properties of noisy dynamical systems with time-varying coupling: capabilities and limitations, Eur. Phys. J. B 88, 193 (2015) [CrossRef] [EDP Sciences] [Google Scholar]
  125. M. Eichler, R. Dahlhaus, J. Sandkühler, Partial correlation analysis for the identification of synaptic connections, Biol. Cybern. 89, 289 (2003) [CrossRef] [Google Scholar]
  126. Y. Chen, G. Rangarajan, J. Feng, M. Ding, Analyzing multiple nonlinear time series with extended Granger causality, Phys. Lett. A 324, 26 (2004) [CrossRef] [Google Scholar]
  127. B. Schelter, M. Winterhalder, R. Dahlhaus, J. Kurths, J. Timmer, Partial phase synchronization for multivariate synchronizing systems, Phys. Rev. Lett. 96, 208103 (2006) [CrossRef] [PubMed] [Google Scholar]
  128. B. Schelter, M. Winterhalder, M. Eichler, M. Peifer, B. Hellwig, B. Guschlbauer, C.H. Lücking, R. Dahlhaus, J. Timmer, Testing for directed influences among neural signals using partial directed coherence, J. Neurosci. Methods 152, 210 (2006) [CrossRef] [Google Scholar]
  129. S. Frenzel, B. Pompe, Partial mutual information for coupling analysis of multivariate time series, Phys. Rev. Lett. 99, 204101 (2007) [CrossRef] [Google Scholar]
  130. D.A. Smirnov, B.P. Bezruchko, Detection of couplings in ensembles of stochastic oscillators, Phys. Rev. E 79, 046204 (2009) [CrossRef] [MathSciNet] [Google Scholar]
  131. V.A. Vakorin, O.A. Krakovska, A.R. McIntosh, Confounding effects of indirect connections on causality estimation, J. Neurosci. Methods 184, 152 (2009) [CrossRef] [PubMed] [Google Scholar]
  132. J. Nawrath, M.C. Romano, M. Thiel, I.Z. Kiss, M. Wickramasinghe, J. Timmer, J. Kurths, B. Schelter, Distinguishing direct from indirect interactions in oscillatory networks with multiple time scales, Phys. Rev. Lett. 104, 038701 (2010) [CrossRef] [PubMed] [Google Scholar]
  133. M. Jalili, M.G. Knyazeva, Constructing brain functional networks from EEG: partial and unpartial correlations, J. Integr. Neurosci. 10, 213 (2011) [CrossRef] [Google Scholar]
  134. Y. Zou, M.C. Romano, M. Thiel, N. Marwan, J. Kurths, Inferring indirect coupling by means of recurrences, Int. J. Bifurc. Chaos Appl. Sci. Eng. 21, 1099 (2011) [CrossRef] [Google Scholar]
  135. J. Runge, J. Heitzig, V. Petoukhov, J. Kurths, Escaping the curse of dimensionality in estimating multivariate transfer entropy, Phys. Rev. Lett. 108, 258701 (2012) [CrossRef] [Google Scholar]
  136. S. Stramaglia, G.-R. Wu, M. Pellicoro, D. Marinazzo, Expanding the transfer entropy to identify information circuits in complex systems, Phys. Rev. E 86, 066211 (2012) [CrossRef] [Google Scholar]
  137. D. Kugiumtzis, Partial transfer entropy on rank vectors, Eur. Phys. J. Special Topics 222, 401 (2013) [CrossRef] [EDP Sciences] [Google Scholar]
  138. L. Leistritz, B. Pester, A. Doering, K. Schiecke, F. Babiloni, L. Astolfi, H. Witte, Time-variant partial directed coherence for analysing connectivity: a methodological study, Philos. Trans. R. Soc. A 371, 20110616 (2013) [Google Scholar]
  139. R. Ramb, M. Eichler, A. Ing, M. Thiel, C. Weiller, C. Grebogi, C. Schwarzbauer, J. Timmer, B. Schelter, The impact of latent confounders in directed network analysis in neuroscience, Philos. Trans. R. Soc. A 371, 20110612 (2013) [CrossRef] [Google Scholar]
  140. B. Kralemann, A. Pikovsky, M. Rosenblum, Reconstructing effective phase connectivity of oscillator networks from observations, New J. Phys. 16, 085013 (2014) [CrossRef] [Google Scholar]
  141. H. Elsegai, H. Shiells, M. Thiel, B. Schelter, Network inference in the presence of latent confounders: the role of instantaneous causalities, J. Neurosci. Methods 245, 91 (2015) [CrossRef] [Google Scholar]
  142. L. Faes, D. Kugiumtzis, G. Nollo, F. Jurysta, D. Marinazzo, Estimating the decomposition of predictive information in multivariate systems, Phys. Rev. E 91, 032904 (2015) [CrossRef] [Google Scholar]
  143. W. Mader, M. Mader, J. Timmer, M. Thiel, B. Schelter, Networks: On the relation of bi-and multivariate measures, Sci. Rep. 5, 10805 (2015) [CrossRef] [Google Scholar]
  144. J. Zhao, Y. Zhou, X. Zhang, L. Chen, Part mutual information for quantifying direct associations in networks, Proc. Natl. Acad. Sci. USA 113, 5130 (2016) [CrossRef] [Google Scholar]
  145. Z. Albo, G.V. Di Prisco, Y. Chen, G. Rangarajan, W. Truccolo, J. Feng, R.P. Vertes, M. Ding, Is partial coherence a viable technique for identifying generators of neural oscillations? Biol. Cybern. 90, 318 (2004) [CrossRef] [Google Scholar]
  146. G. Nolte, O. Bai, L. Wheaton, Z. Mari, S. Vorbach, M. Hallett, Identifying true brain interaction from EEG data using the imaginary part of coherency, Clin. Neurophysiol. 115, 2292 (2004) [Google Scholar]
  147. T. Zerenner, P. Friederichs, K. Lehnertz, A. Hense, A Gaussian graphical model approach to climate networks, Chaos 24, 023103 (2014) [CrossRef] [Google Scholar]
  148. N. Rubido, A.C. Marti, E. Bianco-Martinez, C. Grebogi, M.S. Baptista, C. Masoller, Exact detection of direct links in networks of interacting dynamical units, New J. Phys. 16, 093010 (2014) [CrossRef] [Google Scholar]
  149. T. Rings, K. Lehnertz, Distinguishing between direct and indirect directional couplings in large oscillator networks: partial or non-partial phase analyses? Chaos 26, 093106 (2016) [CrossRef] [Google Scholar]
  150. R. Guimerà, M. Sales-Pardo, Missing and spurious interactions and the reconstruction of complex networks, Proc. Natl. Acad. Sci. USA 106, 22073 (2009) [CrossRef] [Google Scholar]
  151. B. Barzel, A.-L. Barabasi, Network link prediction by global silencing of indirect correlations, Nat. Biotechnol. 31, 720 (2013) [CrossRef] [PubMed] [Google Scholar]
  152. V. Pernice, S. Rotter, Reconstruction of sparse connectivity in neural networks from spike train covariances, J. Stat. Mech. Theor. Exp. 2013, 03008 (2013) [CrossRef] [Google Scholar]
  153. Z. Shen, W.-X. Wang, Y. Fan, Z. Di, Y.-C. Lai, Reconstructing propagation networks with natural diversity and identifying hidden sources, Nat. Commun. 5, 4323 (2014) [Google Scholar]
  154. J. Runge, Quantifying information transfer and mediation along causal pathways in complex systems, Phys. Rev. E 92, 062829 (2015) [CrossRef] [Google Scholar]
  155. Y.V. Zaytsev, A. Morrison, M. Deger, Reconstruction of recurrent synaptic connectivity of thousands of neurons from simulated spiking activity, J. Comput. Neurosci. 39, 77 (2015) [CrossRef] [Google Scholar]
  156. L. Pan, T. Zhou, L. Lü, C.-K. Hu, Predicting missing links and identifying spurious links via likelihood analysis, Sci. Rep. 6, 22955 (2016) [CrossRef] [Google Scholar]
  157. C.J. Stam, G. Nolte, A. Daffertshofer, Phase lag index: assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources, Hum. Brain Mapp. 28, 1178 (2007) [CrossRef] [PubMed] [Google Scholar]
  158. M. Vinck, R. Oostenveld, M. van Wingerden, F. Battaglia, C.M.A. Pennartz, An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias, NeuroImage 55, 1548 (2011) [CrossRef] [Google Scholar]
  159. C.J. Stam, E.C.W. van Straaten, Go with the flow: use of a directed phase lag index (dPLI) to characterize patterns of phase relations in a large-scale model of brain dynamics, NeuroImage 62, 1415 (2012) [CrossRef] [Google Scholar]
  160. M. Hardmeier, F. Hatz, H. Bousleiman, C. Schindler, C.J. Stam, P. Fuhr, Reproducibility of functional connectivity and graph measures based on the Phase Lag Index (PLI) and Weighted Phase Lag Index (wPLI) derived from high resolution EEG, PLOS ONE 9, 108648 (2014) [CrossRef] [Google Scholar]
  161. D. Yu, S. Boccaletti, Real-time estimation of interaction delays, Phys. Rev. E 80, 036203 (2009) [CrossRef] [Google Scholar]
  162. L.R. Peraza, A.U.R. Asghar, G. Green, D.M. Halliday, Volume conduction effects in brain network inference from electroencephalographic recordings using phase lag index, J. Neurosci. Methods 207, 189 (2012) [CrossRef] [Google Scholar]
  163. S.M. Gordon, P.J. Franaszczuk, W.D. Hairston, M. Vindiola, K. McDowell, Comparing parametric and nonparametric methods for detecting phase synchronization in EEG, J. Neurosci. Methods 212, 247 (2013) [CrossRef] [Google Scholar]
  164. S. Porz, M. Kiel, K. Lehnertz, Can spurious indications for phase synchronization due to superimposed signals be avoided? Chaos 24, 033112 (2014) [CrossRef] [PubMed] [Google Scholar]
  165. G.L. Colclough, M.W. Woolrich, P.K. Tewarie, M.J. Brookes, A.J. Quinn, S.M. Smith, How reliable are MEG resting-state connectivity metrics? NeuroImage 138, 284 (2016) [CrossRef] [Google Scholar]
  166. L. Cimponeriu, M. Rosenblum, A. Pikovsky, Estimation of delay in coupling from time series, Phys. Rev. E 70, 046213 (2004) [CrossRef] [Google Scholar]
  167. N. Wessel, A. Suhrbier, M. Riedl, N. Marwan, H. Malberg, G. Bretthauer, T. Penzel, J. Kurths, Detection of time-delayed interactions in biosignals using symbolic coupling traces, Europhys. Lett. 87, 10004 (2009) [CrossRef] [EDP Sciences] [Google Scholar]
  168. A.N. Silchenko, I. Adamchic, N. Pawelczyk, C. Hauptmann, M. Maarouf, V. Sturm, P.A. Tass, Data-driven approach to the estimation of connectivity and time delays in the coupling of interacting neuronal subsystems, J. Neurosci. Methods 191, 32 (2010) [CrossRef] [Google Scholar]
  169. H. Dickten, K. Lehnertz, Identifying delayed directional couplings with symbolic transfer entropy, Phys. Rev. E 90, 062706 (2014) [CrossRef] [Google Scholar]
  170. H. Ye, E.R. Deyle, L.J. Gilarranz, G. Sugihara, Distinguishing time-delayed causal interactions using convergent cross mapping, Sci. Rep. 5, 14750 (2015) [CrossRef] [Google Scholar]
  171. R.G. Andrzejak, T. Kreuz, Characterizing unidirectional couplings between point processes and flows, Europhys. Lett. 96, 50012 (2011) [CrossRef] [EDP Sciences] [Google Scholar]
  172. E. Bullmore, O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems, Nat. Rev. Neurosci. 10, 186 (2009) [CrossRef] [PubMed] [Google Scholar]
  173. K. Lehnertz, G. Ansmann, S. Bialonski, H. Dickten, C. Geier, S. Porz, Evolving networks in the human epileptic brain, Physica D 267, 7 (2014) [CrossRef] [Google Scholar]
  174. D. Papo, M. Zanin, J.A. Pineda-Pardo, S. Boccaletti, J.M. Buldú, Functional brain networks: great expectations, hard times and the big leap forward, Philos. Trans. R. Soc. B 369, 20130525 (2014) [CrossRef] [Google Scholar]
  175. S. Bialonski, M.-T. Horstmann, K. Lehnertz, From brain to earth and climate systems: Small-world interaction networks or not? Chaos 20, 013134 (2010) [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  176. B.C.M. van Wijk, C.J. Stam, A. Daffertshofer, Comparing brain networks of different size and connectivity density using graph theory, PLOS ONE 5, 13701 (2010) [CrossRef] [PubMed] [Google Scholar]
  177. J. Hlinka, D. Hartman, M. Paluš, Small-world topology of functional connectivity in randomly connected dynamical systems, Chaos 22, 033107 (2012) [CrossRef] [Google Scholar]
  178. A. Joudaki, N. Salehi, M. Jalili, M.G. Knyazeva, EEG-based functional brain networks: does the network size matter? PLOS ONE 7, 35673 (2012) [CrossRef] [Google Scholar]
  179. A. Fornito, A. Zalesky, M. Breakspear, Graph analysis of the human connectome: promise, progress, and pitfalls, NeuroImage 80, 426 (2013) [CrossRef] [Google Scholar]
  180. M.L. Stanley, M.N. Moussa, B. Paolini, R.G. Lyday, J.H. Burdette, P.J. Laurienti, Defining nodes in complex brain networks, Front. Comput. Neurosci. 7, 169 (2013) [CrossRef] [Google Scholar]
  181. V. Wens, Investigating complex networks with inverse models: analytical aspects of spatial leakage and connectivity estimation, Phys. Rev. E 91, 012823 (2015) [CrossRef] [Google Scholar]
  182. D. Papo, M. Zanin, J.H. Martínez, J.M. Buldú, Beware of the small-world neuroscientist! Front. Hum. Neurosci. 10, 96 (2016) [Google Scholar]
  183. B.S. Anderson, C. Butts, K. Carley, The interaction of size and density with graph-level indices, Soc. Netw. 21, 239 (1999) [CrossRef] [Google Scholar]
  184. A.A. Ioannides, Dynamic functional connectivity, Curr. Opin. Neurobiol. 17, 161 (2007) [CrossRef] [Google Scholar]
  185. M.A. Kramer, U.T. Eden, S.S. Cash, E.D. Kolaczyk, Network inference with confidence from multivariate time series, Phys. Rev. E 79, 061916 (2009) [CrossRef] [MathSciNet] [Google Scholar]
  186. M. Rubinov, O. Sporns, Complex network measures of brain connectivity: Uses and interpretations, NeuroImage 52, 1059 (2010) [Google Scholar]
  187. M. Zanin, P. Sousa, D. Papo, R. Bajo, J. Garcia-Prieto, F. del Pozo, E. Menasalvas, S. Boccaletti, Optimizing functional network representation of multivariate time series, Sci. Rep. 2, 630 (2012) [CrossRef] [PubMed] [Google Scholar]
  188. R. Rammal, G. Toulouse, M.A. Virasoro, Ultrametricity for physicists, Rev. Mod. Phys. 58, 765 (1986) [CrossRef] [Google Scholar]
  189. U. Lee, S. Kim, K.-Y. Jung, Classification of epilepsy types through global network analysis of scalp electroencephalograms, Phys. Rev. E 73, 041920 (2006) [CrossRef] [Google Scholar]
  190. C.J. Ortega, R.G. Sola, J. Pastor, Complex network analysis of human ECoG data, Neurosci. Lett. 447, 129 (2008) [CrossRef] [Google Scholar]
  191. C. Stam, P. Tewarie, E. Van Dellen, E. Van Straaten, A. Hillebrand, P. Van Mieghem, The trees and the forest: characterization of complex brain networks with minimum spanning trees, Int. J. Psychophysiol. 92, 129 (2014) [CrossRef] [Google Scholar]
  192. G. Ansmann, K. Lehnertz, Surrogate-assisted analysis of weighted functional brain networks, J. Neurosci. Methods 208, 165 (2012) [CrossRef] [Google Scholar]
  193. P. Macdonald, E. Almaas, A.-L. Barabási, Minimum spanning trees of weighted scale-free networks, Europhys. Lett. 72, 308 (2005) [CrossRef] [EDP Sciences] [Google Scholar]
  194. F. De Vico Fallani, L. Astolfi, F. Cincotti, D. Mattia, A. Tocci, S. Salinari, M. Marciani, H. Witte, A. Colosimo, F. Babiloni, Brain network analysis from high-resolution EEG recordings by the application of theoretical graph indexes, IEEE Trans. Neural Syst. Rehab. Eng. 16, 442 (2008) [CrossRef] [Google Scholar]
  195. W.J. Marshall, C.L. Lackner, P. Marriott, D.L. Santesso, S.J. Segalowitz, Using phase shift Granger causality to measure directed connectivity in EEG recordings, Brain Connect. 4, 826 (2014) [CrossRef] [Google Scholar]
  196. M.H.I. Shovon, N. Nandagopal, R. Vijayalakshmi, J.T. Du, B. Cocks, Directed connectivity analysis of functional brain networks during cognitive activity using transfer entropy, Neural Process. Lett. 45, 807 (2016) [CrossRef] [Google Scholar]
  197. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Complex networks: structure and dynamics, Phys. Rep. 424, 175 (2006) [Google Scholar]
  198. S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010) [Google Scholar]
  199. M. Barthélemy, Spatial networks, Phys. Rep. 499, 1 (2011) [CrossRef] [MathSciNet] [Google Scholar]
  200. M.E.J. Newman, Communities, modules and large-scale structure in networks, Nat. Phys. 8, 25 (2012) [CrossRef] [Google Scholar]
  201. S. Bialonski, M. Wendler, K. Lehnertz, Unraveling spurious properties of interaction networks with tailored random networks, PLOS ONE 6, 22826 (2011) [CrossRef] [Google Scholar]
  202. J. Saramäki, M. Kivelä, J.P. Onnela, K. Kaski, J. Kertész, Generalizations of the clustering coefficient to weighted complex networks, Phys. Rev. E 75, 027105 (2007) [CrossRef] [Google Scholar]
  203. D.S. Bassett, E. Bullmore, Small-world brain networks, Neuroscientist 12, 512 (2006) [CrossRef] [PubMed] [Google Scholar]
  204. J.C. Reijneveld, S.C. Ponten, H.W. Berendse, C.J. Stam, The application of graph theoretical analysis to complex networks in the brain, Clin. Neurophysiol. 118, 2317 (2007) [CrossRef] [Google Scholar]
  205. C.J. Stam, Modern network science of neurological disorders, Nat. Rev. Neurosci. 15, 683 (2014) [CrossRef] [Google Scholar]
  206. C.C. Hilgetag, A. Goulas, Is the brain really a small-world network? Brain Struct. Funct. 221, 2361 (2016) [CrossRef] [Google Scholar]
  207. M.E.J. Newman, Assortative mixing in networks, Phys. Rev. Lett. 89, 208701 (2002) [CrossRef] [PubMed] [Google Scholar]
  208. S. Bialonski, K. Lehnertz, Assortative mixing in functional brain networks during epileptic seizures, Chaos 23, 033139 (2013) [CrossRef] [PubMed] [Google Scholar]
  209. F.M. Atay, T. Bıyıkoğlu, J. Jost, Network synchronization: Spectral versus statistical properties, Physica D 224, 35 (2006) [CrossRef] [MathSciNet] [Google Scholar]
  210. F. Comellas, S. Gago, Synchronizability of complex networks, J. Phys. A 40, 4483 (2007) [CrossRef] [Google Scholar]
  211. D. Koschützki, K. Lehmann, L. Peeters, S. Richter, D. Tenfelde-Podehl, O. Zlotowski, Centrality indices, in Network analysis. Lecture Notes in Computer Science, edited by U. Brandes, T. Erlebach (Springer, Berlin, Heidelberg, 2005), Vol. 3418, p. 16 [CrossRef] [Google Scholar]
  212. M.-T. Kuhnert, C. Geier, C.E. Elger, K. Lehnertz, Identifying important nodes in weighted functional brain networks: a comparison of different centrality approaches, Chaos 22, 023142 (2012) [CrossRef] [Google Scholar]
  213. B. Efron, Large-scale simultaneous hypothesis testing: the choice of a null hypothesis, JASA 99, 465 (2004) [Google Scholar]
  214. B. Efron, The jackknife, the bootstrap and other resampling plans (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1982) [Google Scholar]
  215. R.G. Andrzejak, A. Kraskov, H. Stögbauer, F. Mormann, T. Kreuz, Bivariate surrogate techniques: necessity, strengths, and caveats, Phys. Rev. E 68, 066202 (2003) [CrossRef] [Google Scholar]
  216. M. Small, D. Yu, R.G. Harrison, Surrogate test for pseudoperiodic time series data, Phys. Rev. Lett. 87, 188101 (2001) [CrossRef] [Google Scholar]
  217. K.T. Dolan, A. Neiman, Surrogate analysis of coherent multichannel data, Phys. Rev. E 65, 026108 (2002) [CrossRef] [Google Scholar]
  218. L. Faes, G.D. Pinna, A. Porta, R. Maestri, G. Nollo, Surrogate data analysis for assessing the significance of the coherence function, IEEE Trans. Biomed. Eng. 51, 1156 (2004) [CrossRef] [Google Scholar]
  219. M. Breakspear, M. Brammer, P.A. Robinson, Construction of multivariate surrogate sets from nonlinear data using the wavelet transform, Physica D 182, 1 (2003) [CrossRef] [Google Scholar]
  220. C.J. Keylock, A wavelet-based method for surrogate data generation, Physica D 225, 219 (2007) [CrossRef] [Google Scholar]
  221. M. Paluš, Bootstrapping multifractals: Surrogate data from random cascades on wavelet dyadic trees, Phys. Rev. Lett. 101, 134101 (2008) [CrossRef] [Google Scholar]
  222. M.C. Romano, M. Thiel, J. Kurths, K. Mergenthaler, R. Engbert, Hypothesis test for synchronization: twin surrogates revisited, Chaos 19, 015108 (2009) [CrossRef] [Google Scholar]
  223. T. Nakamura, M. Small, Y. Hirata, Testing for nonlinearity in irregular fluctuations with long-term trends, Phys. Rev. E 74, 026205 (2006) [CrossRef] [Google Scholar]
  224. T. Nakamura, T. Tanizawa, M. Small, Constructing networks from a dynamical system perspective for multivariate nonlinear time series, Phys. Rev. E 93, 032323 (2016) [CrossRef] [Google Scholar]
  225. T. Suzuki, T. Ikeguchi, M. Suzuki, Effects of data windows on the methods of surrogate data, Phys. Rev. E 71, 056708 (2005) [CrossRef] [Google Scholar]
  226. J. Lucio, R. Valdés, L. Rodríguez, Improvements to surrogate data methods for nonstationary time series, Phys. Rev. E 85, 056202 (2012) [CrossRef] [Google Scholar]
  227. A.M. Bronstein, M.M. Bronstein, R. Kimmel, Efficient computation of isometry-invariant distances between surfaces, SIAM J. Sci. Comput. 28, 1812 (2006) [CrossRef] [Google Scholar]
  228. M. Muskulus, S. Houweling, S. Verduyn-Lunel, A. Daffertshofer, Functional similarities and distance properties, J. Neurosci. Methods 183, 31 (2009) [CrossRef] [Google Scholar]
  229. F. Mémoli, Gromov-Wasserstein distances and the metric approach to object matching, Found. Comput. Math. 11, 417 (2011) [CrossRef] [MathSciNet] [Google Scholar]
  230. H. Lee, M.K. Chung, H. Kang, B.-N. Kim, D.S. Lee, Computing the shape of brain networks using graph filtration and Gromov-Hausdorff metric, in International Conference on Medical Image Computing and Computer-Assisted Intervention (Springer, 2011), pp. 302–309 [Google Scholar]
  231. D.S. Bassett, M.A. Porter, N.F. Wymbs, S.T. Grafton, J.M. Carlson, P.J. Mucha, Robust detection of dynamic community structure in networks, Chaos 23, 013142 (2013) [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  232. Y. Hulovatyy, H. Chen, T. Milenković, Exploring the structure and function of temporal networks with dynamic graphlets, Bioinformatics 31, 171 (2015) [CrossRef] [PubMed] [Google Scholar]
  233. N. Dianati, Unwinding the hairball graph: pruning algorithms for weighted complex networks, Phys. Rev. E 93, 012304 (2016) [CrossRef] [Google Scholar]
  234. M.J. Cook, T.J. O'Brien, S.F. Berkovic, M. Murphy, A. Morokoff, G. Fabinyi, W. D'Souza, R. Yerra, J. Archer, L. Litewka, S. Hosking, P. Lightfoot, V. Ruedebusch, W.D. Sheffield, D. Snyder, K. Leyde, D. Himes, Prediction of seizure likelihood with a long-term, implanted seizure advisory system in patients with drug-resistant epilepsy: a first-in-man study, Lancet Neurol. 12, 563 (2013) [CrossRef] [Google Scholar]
  235. F. Mormann, R. Andrzejak, C.E. Elger, K. Lehnertz, Seizure prediction: the long and winding road, Brain 130, 314 (2007) [CrossRef] [PubMed] [Google Scholar]
  236. S. Ramgopal, S. Thome-Souza, M. Jackson, N.E. Kadish, I.S. Fernández, J. Klehm, W. Bosl, C. Reinsberger, S. Schachter, T. Loddenkemper, Seizure detection, seizure prediction, and closed-loop warning systems in epilepsy, Epilepsy Behav. 37, 291 (2014) [CrossRef] [Google Scholar]
  237. K. Gadhoumi, J.-M. Lina, F. Mormann, J. Gotman, Seizure prediction for therapeutic devices: a review, J. Neurosci. Methods 260, 270 (2016) [CrossRef] [Google Scholar]
  238. K. Lehnertz, H. Dickten, S. Porz, C. Helmstaedter, C.E. Elger, Predictability of uncontrollable multifocal seizures – towards new treatment options, Sci. Rep. 6, 24584 (2016) [CrossRef] [Google Scholar]
  239. R.S. Fisher, Therapeutic devices for epilepsy, Ann. Neurol. 71, 157 (2012) [CrossRef] [Google Scholar]
  240. B.H. Brinkmann, J. Wagenaar, D. Abbot, P. Adkins, S.C. Bosshard, M. Chen, Q.M. Tieng, J. He, F.J. Muñoz-Almaraz, P. Botella-Rocamora, J. Pardo, F. Zamora-Martinez, M. Hills, W. Wu, I. Korshunova, W. Cukierski, C. Vite, E.E. Patterson, B. Litt, G.A. Worrell, Crowdsourcing reproducible seizure forecasting in human and canine epilepsy, Brain 139, 1713 (2016) [CrossRef] [Google Scholar]
  241. H. Feldwisch-Drentrup, M. Staniek, A. Schulze-Bonhage, J. Timmer, H. Dickten, C.E. Elger, B. Schelter, K. Lehnertz, Identification of preseizure states in epilepsy: a data-driven approach for multichannel EEG recordings, Front. Comput. Neurosci. 5, 32 (2011) [CrossRef] [Google Scholar]
  242. M.J. Cook, A. Varsavsky, D. Himes, K. Leyde, S.F. Berkovic, T. O’Brien, I. Mareels, Long memory processes are revealed in the dynamics of the epileptic brain, Front. Neurol. 5, 217 (2014) [Google Scholar]
  243. A. Clauset, C.R. Shalizi, M.E.J. Newman, Power-law distributions in empirical data, SIAM Rev. 51, 661 (2009) [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  244. E. Taubøll, A. Lundervold, L. Gjerstada, Temporal distribution of seizures in epilepsy, Epilepsy Res. 8, 153 (1991) [CrossRef] [Google Scholar]
  245. P. Suffczynski, S. Kalitzin, F.H.L. da Silva, Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network, Neuroscience 126, 467 (2004) [CrossRef] [Google Scholar]
  246. C. Meisel, A. Schulze-Bonhage, D. Freestone, M.J. Cook, P. Achermann, D. Plenz, Intrinsic excitability measures track antiepileptic drug action and uncover increasing/decreasing excitability over the wake/sleep cycle, Proc. Natl. Acad. Sci. USA 112, 14694 (2015) [CrossRef] [Google Scholar]
  247. T. Kreuz, R.G. Andrzejak, F. Mormann, A. Kraskov, H. Stögbauer, C.E. Elger, K. Lehnertz, P. Grassberger, Measure profile surrogates: a method to validate the performance of epileptic seizure prediction algorithms, Phys. Rev. E 69, 061915 (2004) [CrossRef] [Google Scholar]
  248. M.A. Kramer, S.S. Cash, Epilepsy as a disorder of cortical network organization, Neuroscientist 18, 360 (2012) [CrossRef] [Google Scholar]
  249. M.P. Richardson, Large scale brain models of epilepsy: dynamics meets connectomics, J. Neurol. Neurosurg. Psychiatry 83, 1238 (2012) [CrossRef] [PubMed] [Google Scholar]
  250. K. Schindler, H. Leung, C.E. Elger, K. Lehnertz, Assessing seizure dynamics by analysing the correlation structure of multichannel intracranial EEG, Brain 130, 65 (2007) [Google Scholar]
  251. K. Schindler, C.E. Elger, K. Lehnertz, Increasing synchronization may promote seizure termination: Evidence from status epilepticus, Clin. Neurophysiol. 118, 1955 (2007) [CrossRef] [PubMed] [Google Scholar]
  252. K. Schindler, S. Bialonski, M.-T. Horstmann, C.E. Elger, K. Lehnertz, Evolving functional network properties and synchronizability during human epileptic seizures, Chaos 18, 033119 (2008) [CrossRef] [PubMed] [Google Scholar]
  253. M.A. Kramer, E.D. Kolaczyk, H.E. Kirsch, Emergent network topology at seizure onset in humans, Epilepsy Res. 79, 173 (2008) [CrossRef] [PubMed] [Google Scholar]
  254. S.C. Ponten, L. Douw, F. Bartolomei, J.C. Reijneveld, C.J. Stam, Indications for network regularization during absence seizures: weighted and unweighted graph theoretical analysis, Exp. Neurol. 217, 197 (2009) [CrossRef] [PubMed] [Google Scholar]
  255. M.A. Kramer, W. Truccolo, U.T. Eden, K.Q. Lepage, L.R. Hochberg, E.N. Eskandar, J.R. Madsen, J.W. Lee, A. Maheshwari, E. Halgren, C.J. Chu, S.S. Cash, Human seizures self-terminate across spatial scales via a critical transition, Proc. Natl. Acad. Sci. USA 109, 21116 (2012) [CrossRef] [Google Scholar]
  256. S.P. Burns, S. Santaniello, R.B. Yaffe, C.C. Jouny, N.E. Crone, G.K. Bergey, W.S. Anderson, S.V. Sarma, Network dynamics of the brain and influence of the epileptic seizure onset zone, Proc. Natl. Acad. Sci. USA 111, 5321 (2014) [CrossRef] [Google Scholar]
  257. C. Geier, S. Bialonski, C.E. Elger, K. Lehnertz, How important is the seizure onset zone for seizure dynamics? Seizure 25, 160 (2015) [CrossRef] [PubMed] [Google Scholar]
  258. A.N. Khambhati, K.A. Davis, B.S. Oommen, S.H. Chen, T.H. Lucas, B. Litt, D.S. Bassett, Dynamic network drivers of seizure generation, propagation and termination in human neocortical epilepsy, PLoS Comput. Biol. 11, 1 (2015) [CrossRef] [Google Scholar]
  259. F. Zubler, H. Gast, E. Abela, C. Rummel, M. Hauf, R. Wiest, C. Pollo, K. Schindler, Detecting functional hubs of ictogenic networks, Brain Topogr. 28, 305 (2015) [CrossRef] [Google Scholar]
  260. M. Goodfellow, C. Rummel, E. Abela, M.P. Richardson, K. Schindler, J.R. Terry, Estimation of brain network ictogenicity predicts outcome from epilepsy surgery, Sci. Rep. 6, 29215 (2016) [CrossRef] [Google Scholar]
  261. M.-T. Horstmann, S. Bialonski, N. Noennig, H. Mai, J. Prusseit, J. Wellmer, H. Hinrichs, K. Lehnertz, State dependent properties of epileptic brain networks: comparative graph-theoretical analyses of simultaneously recorded EEG and MEG, Clin. Neurophysiol. 121, 172 (2010) [CrossRef] [Google Scholar]
  262. M.-T. Kuhnert, S. Bialonski, N. Noennig, H. Mai, H. Hinrichs, C. Helmstaedter, K. Lehnertz, Incidental and intentional learning of verbal episodic material differentially modifies functional brain networks, PLOS ONE 8, 80273 (2013) [CrossRef] [Google Scholar]
  263. E. van Diessen, W.J.E.M. Zweiphenning, F.E. Jansen, C.J. Stam, K.P.J. Braun, W.M. Otte, Brain network organization in focal epilepsy: a systematic review and meta-analysis, PLOS ONE 9, 114606 (2014) [CrossRef] [Google Scholar]
  264. M.-T. Kuhnert, C.E. Elger, K. Lehnertz, Long-term variability of global statistical properties of epileptic brain networks, Chaos 20, 043126 (2010) [CrossRef] [Google Scholar]
  265. C. Geier, K. Lehnertz, S. Bialonski, Time-dependent degree-degree correlations in epileptic brain networks: from assortative to dissortative mixing, Front. Hum. Neurosci. 9, 462 (2015) [CrossRef] [Google Scholar]
  266. M.A. Kramer, U.T. Eden, K.Q. Lepage, E.D. Kolaczyk, M.T. Bianchi, S.S. Cash, Emergence of persistent networks in long-term intracranial EEG recordings, J. Neurosci. 31, 15757 (2011) [CrossRef] [Google Scholar]
  267. C. Geier, K. Lehnertz, Long-term variability of importance of brain regions in evolving epileptic brain networks, Chaos 27, 043112 (2017) [CrossRef] [Google Scholar]

Cite this article as: Klaus Lehnertz, Christian Geier, Thorsten Rings, Kirsten Stahn, Capturing time-varying brain dynamics, EPJ Nonlinear Biomed. Phys. 5, 2 (2017)

All Figures

thumbnail Fig. 1

Schematic of interaction-analysis-based approaches to characterize time-varying brain dynamics. (A) Sliding-window analysis: long-lasting multichannel recordings of brain dynamics (here: intracranial EEG) are segmented into successive (non)overlapping windows; Tn denotes the time point associated with the left boundary of the nth window (marked in gray). (B) Time-dependent sequence of interaction matrices: each matrix contains estimates of an interaction property (strength or direction) calculated from the brain dynamics within a given window for all pairs of N sampled brain regions. (C) A bivariate analysis approach renders a time-dependent sequence of an interaction property (here: strength of interaction) for each pair of sampled brain regions (upper three sequences are shifted upwards to enhance readability). (D) A network analysis approach renders a time-dependent sequence of a local or global network characteristic. Time-dependent sequences depicted in (C) and (D) are then subject to further analyses.

In the text
thumbnail Fig. 2

Exemplary time-dependent sequences of mean interaction strength and various network characteristics. Sequences derived from multichannel (N = 90), long-term (14.7 days) iEEG recordings from a patient with a focal epilepsy undergoing presurgical evaluation (the patient signed informed consent that the clinical data might be used and published for research purposes, and the study protocol had previously been approved the by ethics committee of the University of Bonn). From top to bottom: mean interaction strength (average over all estimated strength values – calculated using the mean phase coherence R [105] – from all pairs of sampled brain regions), assortativity a, clustering coefficient c, synchronizability s, betweenness centrality of an exemplary brain region near the seizure-generating area, and betweenness centrality of a homologous region in the opposite brain hemisphere. Left column shows full sequences (smoothed with a moving average over 40 windows (13.65 min) to improve legibility, and tics on x-axis denote midnight); right column shows zooms in (1 h segment from day 1) of the unsmoothed sequences. Grey bars mark discontinuities in the sequences due to recording gaps.

In the text
thumbnail Fig. 3

Schematic dependence of strength and direction of an interaction on coupling strength. Estimates for the strength (red, solid line) and direction of an interaction (blue, dashed line) for some driver-responder system evaluated for increasing coupling strengths.

In the text