arXiv subjects

Klaus Lehnertz

Publications and source records attributed to Klaus Lehnertz.

At least 19 recordsLinked to original sources

arXiv2026

Classifying absence seizure generation mechanisms: A critical transitions framework

Understanding how the brain switches from normal activity to an epileptic seizure is essential for improving seizure therapy, yet the underlying seizure generation mechanisms remain largely unknown. In particular, while seizure onset has been described as a critical transition (CT), there is no consensus on whether (i) bifurcation-induced, (ii) noise-induced, or (iii) bifurcation/noise-induced CTs are responsible. To clarify this, we develop a versatile CT-classification framework that can be applied to seizures in both animals and humans. First, we identify a canonical mathematical model which displays CTs that closely resemble voltage recordings of real seizures and can be of the three types mentioned above. We then identify distinctive properties of each CT-type in the model's output and use them to train a machine learning CT-type classifier. Finally, we apply the model-trained classifier to voltage recordings from epileptic rodents which consist of thousands of real absence seizures. We find that the largest proportion of analysed seizures are classified as noise-induced CTs. In other words, our results on absence seizures in rodents are in contrast to the conventional view that seizures are predominantly bifurcation-induced, and indicate that different CT mechanisms may dominate different seizure types.

math.DS↗

arXiv2026

Detecting seizure onset and offset times using human intelligence: A critical-transitions-based approach

Most existing seizure detection algorithms require extensive pre-processing of the data and rely on heuristic or currently unexplainable machine learning approaches. These approaches often struggle with balancing detection sensitivity and specificity in the presence of variable seizure morphologies, interictal epileptiform discharges, and artefacts. Here, we consider an alternative approach: our seizure detection algorithm, which is based on the concept of critical transitions and overcomes the aforementioned limitations. Specifically, we perform a receiver-operating-characteristic analysis to quantify the performance of our algorithm in terms of its agreement with expert annotations of seizure onset and offset times in the voltage recordings of seizure activity in epileptic rodents with different seizure morphologies. We demonstrate how performance depends on algorithm parameters and varies across different rodent recording sessions. We determine the optimal set of algorithm parameters for each recording session, with near expert-level performance achieved in most cases. Finally, we derive a single general set of algorithm parameters applicable across all recording sessions. The algorithm maintains its high performance in this general setting, demonstrating its versatility, robustness across varying seizure morphologies, and potential to complement machine learning algorithms.

math.DS↗

arXiv2026

Complex network topological and spectral determinants of extreme events

We study the impact of the coupling topology on the ability of various networked dynamical systems to generate extreme events. By determining the coupling strength that is necessary to generate an extreme event in the collective dynamics of a given system, we observe a power-law-like relationship between this coupling threshold and both topological (edge density) and spectral (algebraic connectivity) properties of various coupling topologies. Interestingly, this relationship appears to be largely independent of both the investigated system and the underlying mechanism to generate extreme events. This may indicate that the observed relationship is primarily mediated by aspects of the coupling topology.

nlin.CD↗

arXiv2025

Transcript-based estimators for characterizing interactions

The concept of transcripts was introduced in 2009 as a means to characterize various aspects of the functional relationship between time series of interacting systems. Based on this concept that utilizes algebraic relations between ordinal patterns derived from time series, estimators for the strength, direction, and complexity of interactions have been introduced. These estimators, however, have not yet found widespread application in studies of interactions between real-world systems. Here, we revisit the concept of transcripts and showcase the usage of transcript-based estimators for a time-series-based investigation of interactions between coupled paradigmatic dynamical systems of varying complexity. At the example of a time-resolved analysis of multichannel and multiday recordings of ongoing human brain dynamics, we demonstrate the potential of the methods to provide novel insights into the intricate spatial-temporal interactions in the human brain underlying different vigilance states.

nlin.CD↗

arXiv2025

Quantifying the irregularity of a time series

We introduce circulance, a scalar measure for classifying time series of dynamical systems. Circulance captures the extent of temporal regularity or irregularity that is encoded in the topology of a directed ordinal pattern transition network derived from a time series. We demonstrate numerically that circulance sensitively and robustly positions time series of canonical model systems, representative of preset dynamical regimes, along a continuous spectrum from regularity to randomness. Analyzing empirical data from long-term observations of high-dimensional, complex systems -- human brain and the Sun -- reveals that circulance aids in elucidating different dynamical regimes.

nlin.CD↗

arXiv2025

Discontinuous stochastic forcing in Greenland ice core data

Paleoclimate proxy records from Greenland ice cores, archiving e.g. $δ^{18}$O as a proxy for surface temperature, show that sudden climatic shifts called Dansgaard-Oeschger events (DO) occurred repeatedly during the last glacial interval. They comprised substantial warming of the Arctic region from cold to milder conditions. Concomitant abrupt changes in the dust concentrations of the same ice cores suggest that sudden reorganisations of the hemispheric-scale atmospheric circulation have accompanied the warming events. Genuine bistability of the North Atlantic climate system is commonly hypothesised to explain the existence of stadial (cold) and interstadial (milder) periods in Greenland. However, the physical mechanisms that drove abrupt transitions from the stadial to the interstadial state, and more gradual yet still abrupt reverse transitions, remain debated. Here, we conduct a one-dimensional data-driven analysis of the Greenland temperature and atmospheric circulation proxies under the purview of stochastic processes. We take the Kramers-Moyal equation to estimate each proxy's drift and diffusion terms within a Markovian model framework. We then assess noise contributions beyond Gaussian white noise. The resulting stochastic differential equation (SDE) models feature a monostable drift for the Greenland temperature proxy and a bistable one for the atmospheric circulation proxy. Indicators of discontinuity in stochastic processes suggest to include higher-order terms of the Kramers-Moyal equation when modelling the Greenland temperature proxy's evolution. This constitutes a qualitative difference in the characteristics of the two time series, which should be further investigated from the standpoint of climate dynamics.

physics.ao-ph↗

arXiv2025

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 \textquotedblleft follows\textquotedblright\ 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 \textquotedblleft noiseless scenario\textquotedblright\ just described when noise is present in the driver, a more realistic situation that we call the \textquotedblleft noisy scenario\textquotedblright . 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 \textquotedblleft generic\textquotedblright\ 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.

nlin.CD↗

arXiv2024

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

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' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

physics.data-an↗

arXiv2024

Synchronization dynamics of phase oscillators on power grid models

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 assess the synchronization dynamics and observe the order parameter to exhibit either constant or periodic or non-periodic, possibly chaotic temporal evolutions for a given coupling strength but depending on initial conditions and the systems' disorder. Interestingly, both topological and spectral characteristics of the power grids point to a diminished capability of these networks to support a temporarily stable synchronization dynamics. We find non-trivial commonalities between the synchronization dynamics of oscillators on seemingly opposing topologies.

physics.soc-ph↗

arXiv2023

Data-Driven Reconstruction of Stochastic Dynamical Equations based on Statistical Moments

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) equation, a partial differential equation that involves an infinite number of coefficients, which depend on the state variable. The KM coefficients must be evaluated based on measured time series for a data-driven reconstruction of the governing equations for the stochastic dynamics. We present an accurate method of computing the KM coefficients, which relies on computing the coefficients' conditional moments based on the statistical moments of the time series. The method's advantages over state-of-the-art approaches are demonstrated by investigating prototypical stochastic processes with well-known properties.

cond-mat.stat-mech↗

arXiv2023

A perturbation-based approach to identifying potentially superfluous network constituents

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 ranging from global to local scale. We derive a perturbation-based method to identify potentially superfluous network constituents that makes use of vertex and edge centrality concepts. We investigate the suitability of our approach through analyses of weighted small-world, scale-free, random, and complete networks.

physics.soc-ph↗

arXiv2023

Ordinal methods for a characterization of evolving functional brain networks

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 spatial-temporal 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.

q-bio.NC↗

arXiv2021

Arbitrary-order finite-time corrections for the Kramers-Moyal operator

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.

nlin.AO↗

arXiv2021

Enhancing the accuracy of a data-driven reconstruction of bivariate jump-diffusion models with corrections for higher orders of the sampling interval

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 of higher-order conditional moments that are required to reconstruct the underlying jump-diffusion equations. We derive correction terms for conditional moments in higher orders of the sampling interval and demonstrate their suitability to strongly enhance the data-driven reconstruction accuracy.

physics.data-an↗

arXiv2020

Identifying edges that facilitate the generation of extreme events in networked dynamical systems

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.

nlin.AO↗

arXiv2019

Analysis and data-driven reconstruction of bivariate jump-diffusion processes

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 (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.

nlin.AO↗

arXiv2019

Centrality-based identification of important edges in complex networks

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 network are important between other pairs of vertices. Focusing on the importance of edges, we propose an edge-centrality-based network decomposition technique to identify a hierarchy of sets of edges, where each set is associated with a different level of importance. We evaluate the efficiency of our methods using various paradigmatic network models and apply the novel concepts to identify important edges and important sets of edges in a commonly used benchmark model in social network analysis as well as in evolving epileptic brain networks.

physics.soc-ph↗

arXiv2019

No evidence for critical slowing down prior to human epileptic seizures

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 international effort has successfully identified predictors of imminent seizures. However, the suitability of typically applied early warning indicators for critical slowing down, namely variance and lag-1 autocorrelation, for indexing seizure susceptibility is still controversially discussed. Here, we investigated long-term, multichannel recordings of brain dynamics from 28 subjects with epilepsy. Using a surrogate-based evaluation procedure of sensitivity and specificity of time-resolved estimates of early warning indicators, we found no evidence for critical slowing down prior to 105 epileptic seizures.

physics.med-ph↗