# Klaus LehnertzUniversity of Bonn | Uni Bonn · Klinik und Poliklinik für Epileptologie

Klaus Lehnertz

Prof. Dr. rer. nat.

## About

382

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Introduction

Working at the interface between physics and medicine my research aims at improving our understanding of normal brain functions and epileptic processes, particularly transitions into and out of the extreme event - epileptic seizure.
To this end, we improve existing and develop new methods and time series analysis techniques from statistical physics, dynamical systems theory, synchronization and graph theory and model the multiscale brain dynamics on complex networks.

## Publications

Publications (382)

Many natural and man-made complex dynamical systems can be represented by networks with vertices representing system units and edges the coupling between vertices. If edges of such a structural network are inaccessible, a widely used approach is to identify them with interactions between vertices, thereby setting up a functional network. However, i...

Stochastic processes are encountered in many contexts, ranging from generation sizes of bacterial colonies and service times in a queueing system to displacements of Brownian particles and frequency fluctuations in an electrical power grid.
If such processes are Markov, then their probability distribution is governed by the Kramers-Moyal (KM) equa...

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus-a discrete cortical area from which seizures originate-to a widespread epileptic network-spanning lobes and hemispheres-considerably advanced our understanding of epilepsy and continue...

Natural and manmade complex systems are comprised of different elementary units, being either system components or diverse subsystems as in the case of networked systems. These units interact with each other in a possibly nonlinear way, which results in a complex dynamics that is generally dissipative and nonstationary. One of the challenges in the...

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems’ in...

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' in...

Time series often exhibit a combination of long-range drift and short-term stochastic fluctuations. Traditional methods for analyzing such series involve fitting regression models to capture the drift component and using the residuals to estimate the random component. We demonstrate, however, that estimating the drift in a real-time (time-resolved)...

The human brain is a complex system whose temporally and spatially multiscale
structure gives rise to an immense functionality, which can coexist with
pathophysiologic functioning in case of diseased states. Many important advances
in various fields of research critically continue to improve our understanding of the
structure and (dys-)function of...

We investigate topological and spectral properties of models of European and US-American power grids and of paradigmatic network models as well as their implications for the synchronization dynamics of phase oscillators with heterogeneous natural frequencies. We employ the complex-valued order parameter-a widely used indicator for phase ordering-to...

Biological rhythms are natural, endogenous cycles with period lengths ranging from less than 24 h (ultradian rhythms) to more than 24 h (infradian rhythms). The impact of the circadian rhythm (approximately 24 h) and ultradian rhythms on spectral characteristics of electroencephalographic (EEG) signals has been investigated for more than half a cen...

Non-invasive transcutaneous vagus nerve stimulation elicits similar therapeutic effects as invasive vagus nerve stimulation, offering a potential treatment alternative for a wide range of diseases, including epilepsy. Here, we present a novel, non-invasive stimulation of the vagus nerve, which is performed manually
viscero-osteopathically on the ab...

Constructing networks from empirical time-series data is often faced with the as yet unsolved issue of how to avoid potentially superfluous network constituents. Such constituents can result, e.g., from spatial and temporal oversampling of the system's dynamics, and neglecting them can lead to severe misinterpretations of network characteristics ra...

Constructing networks from empirical time series data is often faced with the as yet unsolved issue of how to avoid potentially superfluous network constituents. Such constituents can result, e.g., from spatial and temporal oversampling of the system's dynamics, and neglecting them can lead to severe misinterpretations of network characteristics ra...

Stochastic processes are encountered in many contexts, ranging from generation sizes of bacterial colonies and service times in a queueing system to displacements of Brownian particles and frequency fluctuations in an electrical power grid. If such processes are Markov, then their probability distribution is governed by the Kramers-Moyal (KM) equat...

A lot of mileage has been made recently on the long and winding road toward seizure forecasting. Here we briefly review some selected milestones passed along the way, which were discussed at the International Conference for Technology and Analysis of Seizures—ICTALS 2022—convened at the University of Bern, Switzerland. Major impetus was gained rece...

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 phe...

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...

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...

KEYWORDS brain network, brain dynamics, brain pathologies, biological rhythms, sleep, network models, neuron models, synchronization Editorial on the Research Topic Network physiology, insights into the brain system: 2021

Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathemati...

Recent advances in neurophysiological brain network analysis have demonstrated novel potential for diagnosis and prognosis of disorders of consciousness. While most progress has been achieved on the population-sample level, time-economic and easy-to-apply personalized solutions are missing. This prospective controlled study combined EEG recordings,...

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...

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...

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...

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...

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...

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...

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...

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 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...

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...

The Open Access license, which previously read:

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...