Klaus Lehnertz

University of Bonn | Uni Bonn · Klinik und Poliklinik für Epileptologie

Electroencephalography (EEG) is a widely employed tool for exploring brain dynamics and is used extensively in various domains, ranging from clinical diagnosis via neuroscience, cognitive science, cognitive psychology, psychophysiology, neuromarketing, neurolinguistics, and pharmacology to research on brain computer interfaces. EEG is the only tech...

Vertex degree-the number of edges that are incident to a vertex-is a fundamental concept in network theory. It is the historically first and conceptually simplest centrality concept to rate the importance of a vertex for a network's structure and dynamics. Unlike many other centrality concepts, for which joint metrics have been proposed for both ve...

Estimating resilience of adaptive, networked dynamical systems remains a challenge. Resilience refers to a system's capacity "to absorb exogenous and/or endogenous disturbances and to reorganize while undergoing change so as to still retain essentially the same functioning, structure, and feedbacks." The majority of approaches to estimate resilienc...

Transcutaneous auricular vagus nerve stimulation (taVNS) is a novel non-invasive treatment option for different diseases and symptoms, such as epilepsy or depression. Its mechanism of action, however, is still not fully understood. We investigated short-term taVNS-induced changes of local and global properties of EEG-derived, evolving functional br...

Epilepsy types differ by pathophysiology and prognosis. Transcutaneous auricular vagus nerve stimulation (taVNS) is a non-invasive treatment option in epilepsy. Nevertheless, its mode of action and impact on different types of epilepsy are still unknown. We investigated whether short-term taVNS differently affects local and global characteristics o...

Electrophysiological power spectra typically consist of two components: An aperiodic part usually following an 1/f power law $$P\propto 1/{f}^{\beta }$$ P ∝ 1 / f β and periodic components appearing as spectral peaks. While the investigation of the periodic parts, commonly referred to as neural oscillations, has received considerable attention, the...

Epilepsy is a neurological disorder affecting approximately 70 million people worldwide. It is characterized by seizures that are complex aberrant dynamical events typically treated with drugs and surgery. Unfortunately, not all patients become seizure-free, and there is an opportunity for novel approaches to treat epilepsy using a network view of...

There is evidence that biofeedback of electrodermal activity (EDA) can reduce seizure frequency in people with epilepsy. Prior studies have linked EDA biofeedback to a diffuse brain activation as a potential functional mechanism. Here, we investigated whether short-term EDA biofeedback alters EEG-derived large-scale functional brain networks in peo...

Electrophysiological power spectra typically consist of two components: An aperiodic part usually following an 1/f power law P ∝1/ f β and periodic components appearing as spectral peaks. While the investigation of the periodic parts, commonly referred to as neural oscillations, has received considerable attention, the study of the aperiodic part h...

Transcutaneous auricular vagus nerve stimulation (taVNS) is a novel non-invasive brain stimulation technique considered as a potential supplementary treatment option for a wide range of diseases. Although first promising findings were obtained so far, the exact mode of action of taVNS is not fully understood yet. We recently developed an examinatio...

Lehnertz, KlausEpilepsyEpilepsy is a malfunction of the brain that affects about 0.8% of the world population. Epileptic seizures are the cardinal symptom of the disease and are usually related to an overly synchronized firing of neuronsNeurons, as reflected by oscillationsOscillation on the electroencephalogram (EEG)Electroencephalogram. Seizures...

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. W...

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. W...

Transcutaneous auricular vagus nerve stimulation (taVNS) is a novel non-invasive brain stimulation technique considered as a potential supplementary treatment option for subjects with refractory epilepsy. Its exact mechanism of action is not yet fully understood. We developed an examination schedule to probe for immediate taVNS-induced modification...

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 dyn...

We evaluate the significance of a recently proposed bivariate jump- diffusion model for a data-driven characterization of interactions between complex dynamical systems. For various coupled and non-coupled jump-diffusion processes, we find that the inevitably finite sampling interval of time-series data negatively affects the reconstruction accurac...

We evaluate the significance of a recently proposed bivariate jump-diffusion model for a data-driven characterization of interactions between complex dynamical systems. For various coupled and non-coupled jump-diffusion processes, we find that the inevitably finite sampling interval of time-series data negatively affects the reconstruction accuracy...

The field of Network Physiology aims to advance our understanding of how physiological systems and subsystems 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...

We study patterns of partial synchronization in a network of FitzHugh–Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay b...

Previous research has indicated that temporal changes of centrality of specific nodes in human evolving large-scale epileptic brain networks carry information predictive of impending seizures. Centrality is a fundamental network-theoretical concept that allows one to assess the role a node plays in a network. This concept allows for various interpr...

The human brain is a complex network of interacting nonstationary subsystems (network of networks), and its complicated spatio-temporal dynamics remain poorly understood. Recent developments in the fi eld of time-series analysis and complex network theory promise new and improved insights into the dynamics of brain networks on various spatio-tempor...

We study patterns of partial synchronization in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay b...

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-...

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-...

Debates on six controversial topics on the network theory of epilepsy were held during two debate sessions, as part of the International Conference for Technology and Analysis of Seizures, 2019 (ICTALS 2019) convened at the University of Exeter, UK, September 2-5 2019. The debate topics were (1) From pathologic to physiologic: is the epileptic netw...

Complex systems are omnipresent and play a vital role in in our every-day lives. Adverse behavior of such systems has generated considerable interest in being able to control complex systems modeled as networks. Here, we propose a topology-dynamics-based approach for controlling complex systems modeled as networks of coupled multi-dimensional dynam...

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 p...

Cross‐frequency phase‐phase coupling (PPC) has been suggested to play a role in cognitive processing and, in particular, in memory consolidation during sleep. Controversial results have been reported regarding the existence of spontaneous phase‐phase coupling in the hippocampus. Here, we investigated this phenomenon in intracranial EEG recordings f...

There is an ongoing debate whether generic early warning signals for critical transitions exist that can be applied across diverse systems. The human epileptic brain is often considered as a prototypical system, given the devastating and, at times, even life-threatening nature of the extreme event epileptic seizure. More than three decades of inter...

Dynamical aspects of spatially extended complex systems are captured in edge properties, such as their weight. Hence, understanding "importance" as a characteristic of edges may lead to a better comprehension, control, or manipulation of spatially extended complex systems such as the brain. Analyzing evolving epileptic networks derived from the lon...

Centrality is one of the most fundamental metrics in network science. Despite an abundance of methods for measuring centrality of individual vertices, there are by now only a few metrics to measure centrality of individual edges. We modify various, widely used centrality concepts for vertices to those for edges, in order to find which edges in a ne...

There is a ongoing debate whether generic early warning signals for critical transitions exist that can be applied across diverse systems. The human epileptic brain is often considered as a prototypical system, given the devastating and, at times, even life-threatening nature of the extreme event epileptic seizure. More than three decades of intern...

We present novel centrality concepts as well as a network decomposition method to identify edges/groups of edges that are important between other pairs/groups of vertices in complex networks. At the example of prototypical networks, we address the questions as to whether there is a relationship between importance of edges and nodes and whether imp...

Knowing when, where, and how seizures are initiated in large-scale epileptic brain networks remains a widely unsolved problem. Seizure precursors – changes in brain dynamics predictive of an impending seizure – can now be identified well ahead of clinical manifestations, but either the seizure onset zone or remote brain areas are reported as networ...

We introduce the bivariate jump-diffusion process, comprising two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. We present a data-driven, non-parametric estimation procedure of higher-order Kramers--Moyal coefficients that allows one to reconstruct relevant aspects of the underlying jump-diffusion processes an...

We report on self-induced switchings between multiple distinct space-time patterns in the dynamics of a spatially extended excitable system. These switchings between low-amplitude oscillations, nonlinear waves, and extreme events strongly resemble a random process, although the system is deterministic. We show that a chaotic saddle-which contains a...

Systems of FitzHugh-Nagumo units with different coupling topologies are capable of self-generating and-terminating strong deviations from their regular dynamics that can be regarded as extreme events due to their rareness and recurrent occurrence. Here we demonstrate the crucial role of an interior crisis in the emergence of extreme events. In para...

Purpose: Research into epileptic networks has recently allowed deeper insights into the epileptic process. Here we investigated the importance of individual network nodes for seizure dynamics. Methods: We analysed intracranial electroencephalographic recordings of 86 focal seizures with different anatomical onset locations. With time-resolved corre...

We propose a Markov chain method to efficiently generate surrogate networks that are random under the constraint of given vertex strengths. With these strength-preserving surrogates and with edge-weight-preserving surrogates we investigate the clustering coefficient and the average shortest path length of functional networks of the human brain as w...

We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Emplo...

Graph-theoretical analyses of complex brain networks is a rapidly evolving field with a strong impact for neuroscientific and related clinical research. Due to a number of confounding variables, however, a reliable and meaningful characterization of particularly functional brain networks is a major challenge. Addressing this problem, we present an...

We investigate the suitability of selected measures of complexity based on recurrence quantification analysis and recurrence networks for an identification of pre-seizure states in multi-day, multi-channel, invasive electroencephalo-graphic recordings from five epilepsy patients. We employ several statistical techniques to avoid spurious findings d...

We propose a straightforward extension of symbolic transfer entropy to enable the investigation of delayed directional relationships between coupled dynamical systems from time series. Analyzing time series from chaotic model systems, we demonstrate the applicability and limitations of our approach. Our findings obtained from applying our method to...

We propose an analytical approach to study non-Markov random walks by employing an exact enumeration method. Using the method, we derive an exact expansion for the first-passage time (FPT) distribution for any continuous, differentiable non-Markov random walk with Gaussian or non-Gaussian multivariate distribution. As an example, we study the FPT d...

We propose an analytical approach to study non-Markov random walks by employing an exact enumeration method. Using the method, we derive an exact expansion for the first-passage time (FPT) distribution for any continuous, differentiable non-Markov random walk with Gaussian or non-Gaussian multivariate distribution. As an example, we study the FPT d...

We propose an analytical approach to study non-Markov random walks by employing an exact enumeration method. Using the method, we derive an exact expansion for the first-passage time (FPT) distribution of any continuous differentiable non-Markov random walk with Gaussian or non-Gaussian multivariate distribution. As an example, we study the FPT dis...

We present edge centrality concepts as well as a decomposition method to identify edges/groups of edges that are important between other pairs/groups of vertices in complex networks. At the example of prototypical networks, we address the questions as to whether there is a relationship between importance of edges and nodes and whether important ed...

Centrality is one of the most fundamental metrics in network science. Despite an abundance of methods for measuring centrality of individual vertices, there are by now only a few metrics to measure centrality of individual edges. We modify various, widely used centrality concepts for vertices to those for edges, in order to find which edges in a ne...

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Extreme events occur in a variety of natural, technical, and societal systems and often have catastrophic consequences. Their low-probability, high-impact nature has recently triggered research into improving our understanding of generating mechanisms, providing early warnings as well as developing control strategies. For the latter to be effective...

Data sampled at discrete times appears as a succession of discontinuous jumps, even if the underlying trajectory is continuous. We analytically derive a criterion that allows one to check whether for a given, even noisy time series the underlying process has a continuous (diffusion) trajectory or has jump discontinuities. This enables one to detect...

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh–Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identify...

Pharmaco-electroencephalography (pharmaco-EEG) has never gained great popularity in epilepsy research. Nevertheless, the electroencephalogram (EEG) is the most important neurological examination technique in this patient population. Development and investigation of antiepileptic drugs (AEDs) involves EEG for diagnosis and outcome evaluation. In con...

Epilepsy is a common disorder characterized by recurrent seizures. An overwhelming majority of people with epilepsy regard the unpredictability of seizures as a major issue. More than 30 years of international effort have been devoted to the prediction of seizures, aiming to remove the burden of unpredictability and to couple novel, time-specific t...

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh--Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identif...

Objective and Approach: Investigating properties of evolving functional brain networks has become a valuable tool to characterize the complex dynamics of the epileptic brain. Such networks are usually derived from electroencephalograms (EEG) recorded with sensors implanted chronically into deeper structures of the brain and/or placed onto the cort...

We aim at identifying factors that may affect the characteristics of evolving weighted networks derived from empirical observations. To this end, we employ various chains of analysis that are often used in field studies for a data-driven derivation and characterization of such networks. As an example, we consider fully connected, weighted functiona...

We study scale-free networks of FitzHugh-Nagumo units whose collective dynamics exhibits rare and recurrent events of high amplitude (extreme events). We investigate which units play a crucial role in the generation of these events. Unexpectedly, we observe that extreme events originate from low-degree units, while high-degree units (hubs) appear t...

We investigate the temporal and spatial variability of the importance of brain regions in evolving epileptic brain networks. We construct these networks from multiday, multichannel electroencephalographic data recorded from 17 epilepsy patients and use centrality indices to assess the importance of brain regions. Time-resolved indications of highes...

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 e...

Scientific Reports 6 : Article number: 34824 10.1038/srep34824 ; published online: 06 October 2016 ; updated: 09 December 2016 The Acknowledgments section in this Article is incomplete: “We are grateful to Gerrit Ansmann and Christian Geier for interesting discussions and for critical comments on earlier versions of the manuscript and by the Verein...

Complex systems involving a large number of degrees of freedom, generally exhibit non-stationary dynamics, which can result in either continuous or discontinuous sample paths of the corresponding time series. The latter sample paths may be caused by discontinuous events – or jumps – with some distributed amplitudes, and disentangling effects caused...

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 dis...

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 pr...

Drug-resistant, multifocal, non-resectable epilepsies are among the most difficult epileptic disorders to manage. An approach to control previously uncontrollable seizures in epilepsy patients would consist of identifying seizure precursors in critical brain areas combined with delivering a counteracting influence to prevent seizure generation. Pre...

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...

We report on self-induced switchings between multiple distinct space-time patterns in the dynamics of a spatially extended excitable system. These switchings between low-amplitude oscillations, nonlinear waves, and extreme events strongly resemble a random process, although the system is deterministic. We show that a chaotic saddle—which contains a...

We investigate the suitability of selected measures of complexity based on recurrence quantification analysis and recurrence networks for an identification of pre-seizure states in multi-day, multi-channel, invasive electroencephalographic recordings from five epilepsy patients. We employ several statistical techniques to avoid spurious findings du...