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.

Impact of biological rhythms on the importance hierarchy of constituents in time-dependent functional brain networks.

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 century. Yet, only little is known on how biological rhythms influence the properties of EEG-derived evolving functional brain networks. Here, we derive such networks from multiday, multichannel EEG recordings and use different centrality concepts to assess the time-varying importance hierarchy of the networks' vertices and edges as well as the various aspects of their structural integration in the network. We observe strong circadian and ultradian influences that highlight distinct subnetworks in the evolving functional brain networks. Our findings indicate the existence of a vital and fundamental subnetwork that is rather generally involved in ongoing brain activities during wakefulness and sleep.

Modifying functional brain networks in focal epilepsy by manual visceral-osteopathic stimulation of the vagus nerve at the abdomen.

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 abdomen (voVNS). We explore the impact of short-term voVNS on various local and global characteristics of EEG-derived, large-scale evolving functional brain networks from a group of 20 subjects with and without epilepsy. We observe differential voVNS-mediated alterations of these characteristics that can be interpreted as a reconfiguration and modification of networks and their stability and robustness properties. Clearly, future studies are necessary to assess the impact of such a non-pharmaceutical intervention on clinical decision-making in the treatment of epilepsy. However, our findings may add to the current discussion on the importance of the gut-brain axis in health and disease. Clinical Trial Registration: https://drks.de/search/en/trial/DRKS00029914, identifier DRKS00029914.

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.

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 spatiotemporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni- and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments, which would be necessary to advance characterization of evolving functional brain networks.

Epileptic-network-based prediction and control of seizures in humans.

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

Seizure forecasting: Where do we stand?

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 recently from wearable and implantable devices that record not only electroencephalography, but also data on motor behavior, acoustic signals, and various signals of the autonomic nervous system. This multimodal monitoring can be performed for ultralong timescales covering months or years. Accordingly, features and metrics extracted from these data now assess seizure dynamics with a greater degree of completeness. Most prominently, this has allowed the confirmation of the long-suspected cyclical nature of interictal epileptiform activity, seizure risk, and seizures. The timescales cover daily, multi-day, and yearly cycles. Progress has also been fueled by approaches originating from the interdisciplinary field of network science. Considering epilepsy as a large-scale network disorder yielded novel perspectives on the pre-ictal dynamics of the evolving epileptic brain. In addition to discrete predictions that a seizure will take place in a specified prediction horizon, the community broadened the scope to probabilistic forecasts of a seizure risk evolving continuously in time. This shift of gears triggered the incorporation of additional metrics to quantify the performance of forecasting algorithms, which should be compared to the chance performance of constrained stochastic null models. An imminent task of utmost importance is to find optimal ways to communicate the output of seizure-forecasting algorithms to patients, caretakers, and clinicians, so that they can have socioeconomic impact and improve patients' well-being.

The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives.

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 continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.

Stimulation-related modifications of evolving functional brain networks in unresponsive wakefulness.

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, basal stimulation, and daily behavioral assessment as applied routinely during complex early rehabilitation treatment. We investigated global characteristics of EEG-derived evolving functional brain networks during the repeated (3-6 weeks apart) evaluation of brain dynamics at rest as well as during and after multisensory stimulation in ten patients who were diagnosed with an unresponsive wakefulness syndrome (UWS). The age-corrected average clustering coefficient C* allowed to discriminate between individual patients at first (three patients) and second assessment (all patients). Clinically, only two patients changed from UWS to minimally conscious state. Of note, most patients presented with significant changes of C* due to stimulations, along with treatment, and with an increasing temporal distance to injury. These changes tended towards the levels of nine healthy controls. Our approach allowed to monitor both, short-term effects of individual therapy sessions and possibly long-term recovery. Future studies will need to assess its full potential for disease monitoring and control of individualized treatment decisions.

Transcutaneous Auricular Vagus Nerve Stimulation Differently Modifies Functional Brain Networks of Subjects With Different Epilepsy Types.

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 of EEG-derived functional brain networks in different types of epilepsy. Thirty subjects (nine with focal epilepsy, 11 with generalized epilepsy, and 10 without epilepsy or seizures) underwent a 3-h continuous EEG-recording (1 h pre-stimulation, 1 h taVNS stimulation, 1 h post-stimulation) from which we derived evolving functional brain networks. We assessed-in a time-resolved manner-important global (topological, robustness, and stability properties) and local (centralities of vertices and edges) network characteristics. Compared to the subjects with focal epilepsies and without epilepsy, those with generalized epilepsies clearly presented with different topological properties of their functional brain network already at rest. Furthermore, subjects with focal and generalized epilepsies reacted differently to the stimulation, expressed as different taVNS-induced immediate and enduring reorganization of global network characteristics. On the local network scale, no discernible spatial pattern could be detected, which points to a rather unspecific and generalized modification of brain activity. Assessing functional brain network characteristics can provide additional information for differentiating between focal and generalized epilepsy. TaVNS-related modifications of global network characteristics clearly differ between epilepsy types. Impact of such a non-pharmaceutical intervention on clinical decision-making in the treatment of different epilepsy types needs to be assessed in future studies.

Network structure from a characterization of interactions in complex systems.

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, it is an unsolved issue if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We address this issue by deriving functional networks from characterizing interactions in paradigmatic oscillator networks with widely-used time-series-analysis techniques for various factors that alter the collective network dynamics. Surprisingly, we find that particularly key constituents of functional networks-as identified with betweenness and eigenvector centrality-coincide with ground truth to a high degree, while global topological and spectral properties-clustering coefficient, average shortest path length, assortativity, and synchronizability-clearly deviate. We obtain similar concurrences for an empirical network. Our findings are of relevance for various scientific fields and call for conceptual and methodological refinements to further our understanding of the relationship between structure and function of complex dynamical systems.

Modifications of Functional Human Brain Networks by Transcutaneous Auricular Vagus Nerve Stimulation: Impact of Time of Day.

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 brain networks from eighteen subjects who underwent two 1 h stimulation phases (morning and afternoon) during continuous EEG-recording. In the majority of subjects, taVNS induced measurable modifications of network properties. Network alterations induced by stimulation in the afternoon were clearly more pronounced than those induced by stimulation in the morning. Alterations mostly affected the networks' topology and stability properties. On the local network scale, no clear-cut spatial stimulation-related patterns could be discerned. Our findings indicate that the possible impact of diurnal influences on taVNS-induced network modifications would need to be considered for future research and clinical studies of this non-pharmaceutical intervention approach.

Separating Neural Oscillations from Aperiodic 1/f Activity: Challenges and Recommendations.

Electrophysiological power spectra typically consist of two components: An aperiodic part usually following an 1/f power law [Formula: see text] 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 has only recently gained more interest. The periodic part is usually quantified by center frequencies, powers, and bandwidths, while the aperiodic part is parameterized by the y-intercept and the 1/f exponent [Formula: see text]. For investigation of either part, however, it is essential to separate the two components. In this article, we scrutinize two frequently used methods, FOOOF (Fitting Oscillations & One-Over-F) and IRASA (Irregular Resampling Auto-Spectral Analysis), that are commonly used to separate the periodic from the aperiodic component. We evaluate these methods using diverse spectra obtained with electroencephalography (EEG), magnetoencephalography (MEG), and local field potential (LFP) recordings relating to three independent research datasets. Each method and each dataset poses distinct challenges for the extraction of both spectral parts. The specific spectral features hindering the periodic and aperiodic separation are highlighted by simulations of power spectra emphasizing these features. Through comparison with the simulation parameters defined a priori, the parameterization error of each method is quantified. Based on the real and simulated power spectra, we evaluate the advantages of both methods, discuss common challenges, note which spectral features impede the separation, assess the computational costs, and propose recommendations on how to use them.

A straightforward edge centrality concept derived from generalizing degree and strength.

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 vertices and edges, by now there is no concept for an edge centrality analogous to vertex degree. Here, we propose such a concept-termed nearest-neighbor edge centrality-and demonstrate its suitability for a non-redundant identification of central edges in paradigmatic network models as well as in real-world networks from various scientific domains.

Electrodermal Activity Biofeedback Alters Evolving Functional Brain Networks in People With Epilepsy, but in a Non-specific Manner.

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 people with epilepsy. In this prospective controlled trial, thirty participants were quasi-randomly assigned to one of three biofeedback conditions (arousal, sham, or relaxation) and performed a single, 30-min biofeedback training while undergoing continuous EEG recordings. Based on the EEG, we derived evolving functional brain networks and examined their topological, robustness, and stability properties over time. Potential effects on attentional-executive functions and mood were monitored via a neuropsychological assessment and subjective self-ratings. Participants assigned to the relaxation group seemed to be most successful in meeting the task requirements for this specific control condition (i.e., decreasing EDA). Participants in the sham group were more successful in increasing EDA than participants in the arousal group. However, only the arousal biofeedback training was associated with a prolonged robustness-enhancing effect on networks. Effects on other network properties were mostly unspecific for the different groups. None of the biofeedback conditions affected attentional-executive functions or subjective behavioral measures. Our results suggest that global characteristics of evolving functional brain networks are modified by EDA biofeedback. Some alterations persisted after the single training session; however, the effects were largely unspecific across the different biofeedback protocols. Further research should address changes of local network characteristics and whether multiple training sessions will result in more specific network modifications.

What Models and Tools can Contribute to a Better Understanding of Brain Activity?

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 mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.

Towards a Data-Driven Estimation of Resilience in Networked Dynamical Systems: Designing a Versatile Testbed.

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 resilience requires exact knowledge of the underlying equations of motion; the few data-driven approaches so far either lack appropriate strategies to verify their suitability or remain subject of considerable debate. We develop a testbed that allows one to modify resilience of a multistable networked dynamical system in a controlled manner. The testbed also enables generation of multivariate time series of system observables to evaluate the suitability of data-driven estimators of resilience. We report first findings for such an estimator.

Perspectives on Understanding Aberrant Brain Networks in Epilepsy.

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 the brain. The traditional seizure focus theory presumed that seizures originated within a discrete cortical area with subsequent recruitment of adjacent cortices with seizure progression. However, a more recent view challenges this concept, suggesting that epilepsy is a network disease, and both focal and generalized seizures arise from aberrant activity in a distributed network. Changes in the anatomical configuration or widespread neural activities spanning lobes and hemispheres could make the brain more susceptible to seizures. In this perspective paper, we summarize the current state of knowledge, address several important challenges that could further improve our understanding of the human brain in epilepsy, and invite novel studies addressing these challenges.