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Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This—together with its conceptual simplicity and robustness against measurement noise—makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks. [ABSTRACT FROM AUTHOR]

The collective dynamics of complex networks of FitzHugh–Nagumo units exhibits rare and recurrent events of high amplitude (extreme events) that are preceded by so-called proto-events during which a certain fraction of the units become excited. Although it is well known that a sufficiently large fraction of excited units is required to turn a proto-event into an extreme event, it is not yet clear how the other units are being recruited into the final generation of an extreme event. Addressing this question and mimicking typical experimental situations, we investigate the centrality of edges in time-dependent interaction networks. We derived these networks from time series of the units' dynamics employing a widely used bivariate analysis technique. Using our recently proposed edge-centrality concepts together with an edge-based network decomposition technique, we observe that the recruitment is primarily facilitated by sets of certain edges that have no equivalent in the underlying topology. Our finding might aid to improve the understanding of generation of extreme events in natural networked dynamical systems. [ABSTRACT FROM AUTHOR]

Epilepsy is now conceptualized as a network disease. The epileptic brain network comprises structurally and functionally connected cortical and subcortical brain regions – spanning lobes and hemispheres –, whose connections and dynamics evolve in time. With this concept, focal and generalized seizures as well as other related pathophysiological phenomena are thought to emerge from, spread via, and be terminated by network vertices and edges that also generate and sustain normal, physiological brain dynamics. Research over the last years has advanced concepts and techniques to identify and characterize the evolving epileptic brain network and its constituents on various spatial and temporal scales. Network-based approaches further our understanding of how seizures emerge from the evolving epileptic brain network, and they provide both novel insights into pre-seizure dynamics and important clues for success or failure of measures for network-based seizure control and prevention. In this review, we summarize the current state of knowledge and address several important challenges that would need to be addressed to move network-based prediction and control of seizures closer to clinical translation. [ABSTRACT FROM AUTHOR]

Nodes in large-scale epileptic networks that are crucial for seizure facilitation and termination can be regarded as potential targets for individualized focal therapies. Graph-theoretical approaches based on centrality concepts can help to identify such important nodes, however, they may be influenced by the way networks are derived from empirical data. Here we investigate evolving functional epileptic brain networks during 82 focal seizures with different anatomical onset locations that we derive from multichannel intracranial electroencephalographic recordings from 51 patients. We demonstrate how the various methodological steps (from the recording montage via node and link inference to the assessment of node centralities) affect importance estimation and discuss their impact on the interpretability of findings in the context of pathophysiological aspects of seizure dynamics. [ABSTRACT FROM AUTHOR]

The human brain is a complex network of interacting nonstationary subsystems, whose complicated spatial-temporal dynamics is still poorly understood. Deeper insights can be gained from recent improvements of time-series-analysis techniques to assess strength and direction of interactions together with methodologies for deriving and characterizing evolving networks from empirical time series. Wehere review these developments, and by taking the example of evolving epileptic brain networks, we discuss the progress that has beenmade in capturing and understanding brain dynamics that varies on time scales ranging from seconds to years. [ABSTRACT FROM AUTHOR]

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫ 10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach. [ABSTRACT FROM AUTHOR]

We investigate the relative merit of phase-based methods--mean phase coherence, unweighted and weighted phase lag index--for estimating the strength of interactions between dynamical systems from empirical time series which are affected by common sources and noise. By numerically analyzing the interaction dynamics of coupled model systems, we compare these methods to each other with respect to their ability to distinguish between different levels of coupling for various simulated experimental situations. We complement our numerical studies by investigating consistency and temporal variations of the strength of interactions within and between brain regions using intracranial electroencephalographic recordings from an epilepsy patient. Our findings indicate that the unweighted and weighted phase lag index are less prone to the influence of common sources but that this advantage may lead to constrictions limiting the applicability of these methods. [ABSTRACT FROM AUTHOR]

Abstract: Network theory provides novel concepts that promise an improved characterization of interacting dynamical systems. Within this framework, evolving networks can be considered as being composed of nodes, representing systems, and of time-varying edges, representing interactions between these systems. This approach is highly attractive to further our understanding of the physiological and pathophysiological dynamics in human brain networks. Indeed, there is growing evidence that the epileptic process can be regarded as a large-scale network phenomenon. We here review methodologies for inferring networks from empirical time series and for a characterization of these evolving networks. We summarize recent findings derived from studies that investigate human epileptic brain networks evolving on timescales ranging from few seconds to weeks. We point to possible pitfalls and open issues, and discuss future perspectives. [Copyright &y& Elsevier]

We investigate interaction networks that we derive from multivariate time series with methods frequently employed in diverse scientific fields such as biology, quantitative finance, physics, earth and climate sciences, and the neurosciences. Mimicking experimental situations, we generate time series with finite length and varying frequency content but from independent stochastic processes. Using the correlation coefficient and the maximum cross-correlation, we estimate interdependencies between these time series. With clustering coefficient and average shortest path length, we observe unweighted interaction networks, derived via thresholding the values of interdependence, to possess non-trivial topologies as compared to Erdös-Rényi networks, which would indicate small-world characteristics. These topologies reflect the mostly unavoidable finiteness of the data, which limits the reliability of typically used estimators of signal interdependence. We propose random networks that are tailored to the way interaction networks are derived from empirical data. Through an exemplary investigation of multichannel electroencephalographic recordings of epileptic seizures - known for their complex spatial and temporal dynamics - we show that such random networks help to distinguish network properties of interdependence structures related to seizure dynamics from those spuriously induced by the applied methods of analysis. [ABSTRACT FROM AUTHOR]

The framework of the theory of nonlinear dynamics provides powerful concepts and algorithms to study complicated dynamics such as brain electrical activity (electroencephalogram, EEG). Although different influencing factors render the use of nonlinear measures in a strict sense problematic, converging evidence from various investigations now indicates that nonlinear EEG analysis provides a means to reliably characterize different states of physiological and pathophysiological brain function. We here focus on applications of nonlinear EEG analysis in epileptology. Epilepsy affects more than 50 million individuals worldwide – approximately 1% of the world's population. The disease is characterized by a recurrent and sudden malfunction of the brain that is termed seizure. Nonlinear EEG analysis techniques allow to reliably identify the seizure generating structure (epileptic focus) in different areas of the brain even during seizure-free intervals, to disentangle complex spatio-temporal interactions between the epileptic focus and other areas of the brain, and to define a specific state predictive of an impending seizure. Nonlinear EEG analysis provides supplementary information about the epileptogenic process in humans, contributes to an improvement of the presurgical evaluation of epilepsy patients, and offers a basis for the development of new therapy concepts for seizure prevention. [ABSTRACT FROM AUTHOR]

We investigate the relative merit of different linear and nonlinear synchronization measures for a characterization of the spatio-temporal dynamics of the epileptic process. Analyzing long-lasting multichannel electroencephalographic recordings from more than 20 epilepsy patients we show that all measures are able to identify brain regions of pathological synchronization associated with epilepsy, even during the seizure-free interval, and are able to detect a long-lasting transitional preseizure state. These findings render synchronization measures attractive for future prospective studies on seizure prediction. [ABSTRACT FROM AUTHOR]

Measuring the directionality of coupling between dynamical systems is one of the challenging problems in nonlinear time series analysis. We investigate the relative merit of two approaches to assess directionality, one based on phase dynamics modeling and one based on state space topography. We analyze unidirectionally coupled model systems to investigate the ability of the two approaches to detect driver-responder relationships and discuss certain problems and pitfalls. In addition we apply both approaches to the intracranial electroencephalogram (EEG) recorded from one epilepsy patient during the seizure-free interval to demonstrate the general suitability of directionality measures to reflect the pathological interaction of the epileptic focus with other brain areas. [ABSTRACT FROM AUTHOR]

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data. [ABSTRACT FROM AUTHOR]

Stochastic approaches to complex dynamical systems have recently provided broader insights into spatial-temporal aspects of epileptic brain dynamics. Stochastic qualifiers based on higher-order Kramers-Moyal coefficients derived directly from time series data indicate improved differentiability between physiological and pathophysiological brain dynamics. It remains unclear, however, to what extent stochastic qualifiers of brain dynamics are affected by other endogenous and/or exogenous influencing factors. Addressing this issue, we investigate multi-day, multi-channel electroencephalographic recordings from a subject with epilepsy. We apply a recently proposed criterion to differentiate between Langevin-type and jump-diffusion processes and observe the type of process most qualified to describe brain dynamics to change with time. Stochastic qualifiers of brain dynamics are strongly affected by endogenous and exogenous rhythms acting on various time scales—ranging from hours to days. Such influences would need to be taken into account when constructing evolution equations for the epileptic brain or other complex dynamical systems subject to external forcings. [ABSTRACT FROM AUTHOR]

We introduce the bivariate jump-diffusion process, consisting of two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. We present a data-driven, nonparametric estimation procedure of higher-order (up to 8) Kramers-Moyal coefficients that allows one to reconstruct relevant aspects of the underlying jump-diffusion processes and to recover the underlying parameters. The procedure is validated with numerically integrated data using synthetic bivariate time series from continuous and discontinuous processes. We further evaluate the possibility of estimating the parameters of the jump-diffusion model via data-driven analyses of the higher-order Kramers-Moyal coefficients, and the limitations arising from the scarcity of points in the data or disproportionate parameters in the system. [ABSTRACT FROM AUTHOR]

We extend the concept of symbolic transfer entropy to enable the time-resolved investigation of directional relationships between coupled dynamical systems from short and transient noisy time series. For our approach, we consider an observed ensemble of a sufficiently large number of time series as multiple realizations of a process. We derive an index that quantifies the preferred direction of transient interactions and assess its significance using a surrogate-based testing scheme. Analyzing time series from noisy chaotic systems, we demonstrate numerically the applicability and limitations of our approach. Our findings obtained from an analysis of event-related brain activities underline the importance of our method to improve understanding of gross neural interactions underlying cognitive processes. [ABSTRACT FROM AUTHOR]

Abstract: We present a client–server application for the distributed multivariate analysis of time series using standard PCs. We here concentrate on analyses of multichannel EEG/MEG data, but our method can easily be adapted to other time series. Due to the rapid development of new analysis techniques, the focus in the design of our application was not only on computational performance, but also on high flexibility and expandability of both the client and the server programs. For this purpose, the communication between the server and the clients as well as the building of the computational tasks has been realized via the Extensible Markup Language (XML). Running our newly developed method in an asynchronous distributed environment with random availability of remote and heterogeneous resources, we tested the system''s performance for a number of different univariate and bivariate analysis techniques. Results indicate that for most of the currently available analysis techniques, calculations can be performed in real time, which, in principle, allows on-line analyses at relatively low cost. [Copyright &y& Elsevier]