Detecting directional coupling in the human epileptic brain: limitations and potential pitfalls

Generalized synchronization in the presence of dynamical noise and its detection via recurrent neural networks

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder "follows" the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the "noiseless scenario" just described when noise is present in the driver, a more realistic situation that we call the "noisy scenario." Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for "generic" driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real-world data demonstrate the capability of our method.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Coupling strength versus coupling impact in nonidentical bidirectionally coupled dynamics

The understanding of interacting dynamics is important for the characterization of real-world networks. In general, real-world networks are heterogeneous in the sense that each node of the network is a dynamics with different properties. For coupled nonidentical dynamics symmetric interactions are not straightforwardly defined from the coupling strength values. Thus, a challenging issue is whether we can define a symmetric interaction in this asymmetric setting. To address this problem we introduce the notion of the coupling impact. The coupling impact considers not only the coupling strength but also the energy of the individual dynamics, which is conveyed via the coupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs of coupled model dynamics using two different connectivity measures. We find that the coupling impact, but not the coupling strength, correctly detects a symmetric interaction between pairs of coupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows us to reveal the real impact that one dynamics has on the other and hence to define symmetric interactions in pairs of nonidentical dynamics.

Weighted and directed interactions in evolving large-scale epileptic brain networks

Epilepsy can be regarded as a network phenomenon with functionally and/or structurally aberrant connections in the brain. Over the past years, concepts and methods from network theory substantially contributed to improve the characterization of structure and function of these epileptic networks and thus to advance understanding of the dynamical disease epilepsy. We extend this promising line of research and assess-with high spatial and temporal resolution and using complementary analysis approaches that capture different characteristics of the complex dynamics-both strength and direction of interactions in evolving large-scale epileptic brain networks of 35 patients that suffered from drug-resistant focal seizures with different anatomical onset locations. Despite this heterogeneity, we find that even during the seizure-free interval the seizure onset zone is a brain region that, when averaged over time, exerts strongest directed influences over other brain regions being part of a large-scale network. This crucial role, however, manifested by averaging on the population-sample level only - in more than one third of patients, strongest directed interactions can be observed between brain regions far off the seizure onset zone. This may guide new developments for individualized diagnosis, treatment and control.

Distinguishing between direct and indirect directional couplings in large oscillator networks: Partial or non-partial phase analyses?

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.

Direction of information flow in large-scale resting-state networks is frequency-dependent

Normal brain function requires interactions between spatially separated, and functionally specialized, macroscopic regions, yet the directionality of these interactions in large-scale functional networks is unknown. Magnetoencephalography was used to determine the directionality of these interactions, where directionality was inferred from time series of beamformer-reconstructed estimates of neuronal activation, using a recently proposed measure of phase transfer entropy. We observed well-organized posterior-to-anterior patterns of information flow in the higher-frequency bands (alpha1, alpha2, and beta band), dominated by regions in the visual cortex and posterior default mode network. Opposite patterns of anterior-to-posterior flow were found in the theta band, involving mainly regions in the frontal lobe that were sending information to a more distributed network. Many strong information senders in the theta band were also frequent receivers in the alpha2 band, and vice versa. Our results provide evidence that large-scale resting-state patterns of information flow in the human brain form frequency-dependent reentry loops that are dominated by flow from parieto-occipital cortex to integrative frontal areas in the higher-frequency bands, which is mirrored by a theta band anterior-to-posterior flow.

Assessing directionality and strength of coupling through symbolic analysis: an application to epilepsy patients

Inferring strength and direction of interactions from electroencephalographic (EEG) recordings is of crucial importance to improve our understanding of dynamical interdependencies underlying various physiological and pathophysiological conditions in the human epileptic brain. We here use approaches from symbolic analysis to investigate--in a time-resolved manner--weighted and directed, short- to long-ranged interactions between various brain regions constituting the epileptic network. Our observations point to complex spatial-temporal interdependencies underlying the epileptic process and their role in the generation of epileptic seizures, despite the massive reduction of the complex information content of multi-day, multi-channel EEG recordings through symbolization. We discuss limitations and potential future improvements of this approach.

Identification of the Epileptogenic Zone from Stereo-EEG Signals: A Connectivity-Graph Theory Approach

In the context of focal drug-resistant epilepsies, the surgical resection of the epileptogenic zone (EZ), the cortical region responsible for the onset, early seizures organization, and propagation, may be the only therapeutic option for reducing or suppressing seizures. The rather high rate of failure in epilepsy surgery of extra-temporal epilepsies highlights that the precise identification of the EZ, mandatory objective to achieve seizure freedom, is still an unsolved problem that requires more sophisticated methods of investigation. Despite the wide range of non-invasive investigations, intracranial stereo-EEG (SEEG) recordings still represent, in many patients, the gold standard for the EZ identification. In this contest, the EZ localization is still based on visual analysis of SEEG, inevitably affected by the drawback of subjectivity and strongly time-consuming. Over the last years, considerable efforts have been made to develop advanced signal analysis techniques able to improve the identification of the EZ. Particular attention has been paid to those methods aimed at quantifying and characterizing the interactions and causal relationships between neuronal populations, since is nowadays well assumed that epileptic phenomena are associated with abnormal changes in brain synchronization mechanisms, and initial evidence has shown the suitability of this approach for the EZ localization. The aim of this review is to provide an overview of the different EEG signal processing methods applied to study connectivity between distinct brain cortical regions, namely in focal epilepsies. In addition, with the aim of localizing the EZ, the approach based on graph theory will be described, since the study of the topological properties of the networks has strongly improved the study of brain connectivity mechanisms.

Inferring functional neural connectivity with phase synchronization analysis: a review of methodology

Functional neural connectivity is drawing increasing attention in neuroscience research. To infer functional connectivity from observed neural signals, various methods have been proposed. Among them, phase synchronization analysis is an important and effective one which examines the relationship of instantaneous phase between neural signals but neglecting the influence of their amplitudes. In this paper, we review the advances in methodologies of phase synchronization analysis. In particular, we discuss the definitions of instantaneous phase, the indexes of phase synchronization and their significance test, the issues that may affect the detection of phase synchronization and the extensions of phase synchronization analysis. In practice, phase synchronization analysis may be affected by observational noise, insufficient samples of the signals, volume conduction, and reference in recording neural signals. We make comments and suggestions on these issues so as to better apply phase synchronization analysis to inferring functional connectivity from neural signals.

Assessing directed interactions from neurophysiological signals--an overview

The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.

Reconstructing phase dynamics of oscillator networks

We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling.

Inferring directional interactions from transient signals with symbolic transfer entropy

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.

Symbolic transfer entropy: inferring directionality in biosignals

Inferring directional interactions from biosignals is of crucial importance to improve understanding of dynamical interdependences underlying various physiological and pathophysiological conditions. We here present symbolic transfer entropy as a robust measure to infer the direction of interactions between multidimensional dynamical systems. We demonstrate its performance in quantifying driver-responder relationships in a network of coupled nonlinear oscillators and in the human epileptic brain.

Effective phase dynamics of noise-induced oscillations in excitable systems

We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.

Sleep-dependent directional coupling between human neocortex and hippocampus

Complex interactions between neocortex and hippocampus are the neural basis of memory formation. Two-step theories of memory formation suggest that initial encoding of novel information depends on the induction of rapid plasticity within the hippocampus, and is followed by a second sleep-dependent step of memory consolidation. These theories predict information flow from the neocortex into the hippocampus during waking state and in the reverse direction during sleep. However, experimental evidence that interactions between hippocampus and neocortex have a predominant direction which reverses during sleep rely on cross-correlation analysis of data from animal experiments and yielded inconsistent results. Here, we investigated directional coupling in intracranial EEG data from human subjects using a phase-modeling approach which is well suited to reveal functional interdependencies in oscillatory data. In general, we observed that the anterior hippocampus predominantly drives nearby and remote brain regions. Surprisingly, however, the influence of neocortical regions on the hippocampus significantly increased during sleep as compared to waking state. These results question the standard model of hippocampal-neocortical interactions and suggest that sleep-dependent consolidation is accomplished by an active retrieval of hippocampal information by the neocortex.

Detection of couplings in ensembles of stochastic oscillators

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Epilepsy and nonlinear dynamics

This overview summarizes findings obtained from analyzing electroencephalographic (EEG) recordings from epilepsy patients with methods from the theory of nonlinear dynamical systems. The last two decades have shown that nonlinear time series analysis techniques allow an improved characterization of epileptic brain states and help to gain deeper insights into the spatial and temporal dynamics of the epileptic process. Nonlinear EEG analyses can help to improve the evaluation of patients prior to neurosurgery, and with an unequivocal identification of precursors of seizures, they can be of great value in the development of seizure warning and prevention techniques.

Measuring interdependences in dissipative dynamical systems with estimated Fokker-Planck coefficients

We propose a data-driven approach to measure interdependences between dissipative dynamical systems under the influence of noise. We estimate drift and diffusion coefficients of a Fokker-Planck equation and derive measures that allow one to quantify the asymmetry in coupling in a fully automated and computationally inexpensive and simple way. Our approach makes it possible to discriminate between interdependences in the deterministic and stochastic parts of the dynamics. We report results of numerical studies of exemplary time series from coupled stochastic and deterministic model systems and of an application to electroencephalographic recordings from epilepsy patients.