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.

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.

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.

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.

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.

Impact of Transcutaneous Auricular Vagus Nerve Stimulation on Large-Scale Functional Brain Networks: From Local to Global.

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 examination schedule to probe for immediate taVNS-induced modifications of large-scale epileptic brain networks. With this schedule, we observed short-term taVNS to have a topology-modifying, robustness- and stability-enhancing immediate effect on large-scale functional brain networks from subjects with focal epilepsies. We here expand on this study and investigate the impact of short-term taVNS on various local and global characteristics of large-scale evolving functional brain networks from a group of 30 subjects with and without central nervous system diseases. Our findings point to differential, at first glance counterintuitive, taVNS-mediated alterations of local and global topological network characteristics that result in a reconfiguration of networks and a modification of their stability and robustness properties. We propose a model of a stimulation-related stretching and compression of evolving functional brain networks that may help to better understand the mode of action of taVNS.

Testing Jump-Diffusion in Epileptic Brain Dynamics: Impact of Daily Rhythms.

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 dynamics. It remains unclear, however, to what extent stochastic qualifiers of brain dynamics are affected by other endogenous and/or exogenous influencing factors. Addressing this issue, we investigate multi-day, multi-channel electroencephalographic recordings from a subject with epilepsy. We apply a recently proposed criterion to differentiate between Langevin-type and jump-diffusion processes and observe the type of process most qualified to describe brain dynamics to change with time. Stochastic qualifiers of brain dynamics are strongly affected by endogenous and exogenous rhythms acting on various time scales-ranging from hours to days. Such influences would need to be taken into account when constructing evolution equations for the epileptic brain or other complex dynamical systems subject to external forcings.

Transcutaneous auricular vagus nerve stimulation induces stabilizing modifications in large-scale functional brain networks: towards understanding the effects of taVNS in subjects with epilepsy.

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 modifications of large-scale epileptic brain networks and accompanying changes of cognition and behaviour. In this prospective trial, we applied short-term (1 h) taVNS to 14 subjects with epilepsy during a continuous 3-h EEG recording which was embedded in two standardized neuropsychological assessments. From these EEG, we derived evolving epileptic brain networks and tracked important topological, robustness, and stability properties of networks over time. In the majority of investigated subjects, taVNS induced measurable and persisting modifications in network properties that point to a more resilient epileptic brain network without negatively impacting cognition, behaviour, or mood. The stimulation was well tolerated and the usability of the device was rated good. Short-term taVNS has a topology-modifying, robustness- and stability-enhancing immediate effect on large-scale epileptic brain networks. It has no detrimental effects on cognition and behaviour. Translation into clinical practice requires further studies to detail knowledge about the exact mechanisms by which taVNS prevents or inhibits seizures.

Time in Brain: How Biological Rhythms Impact on EEG Signals and on EEG-Derived Brain Networks.

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 technique that enables the continuous recording of brain dynamics over periods of time that range from a few seconds to hours and days and beyond. When taking long-term recordings, various endogenous and exogenous biological rhythms may impinge on characteristics of EEG signals. While the impact of the circadian rhythm and of ultradian rhythms on spectral characteristics of EEG signals has been investigated for more than half a century, only little is known on how biological rhythms influence characteristics of brain dynamics assessed with modern EEG analysis techniques. At the example of multiday, multichannel non-invasive and invasive EEG recordings, we here discuss the impact of biological rhythms on temporal changes of various characteristics of human brain dynamics: higher-order statistical moments and interaction properties of multichannel EEG signals as well as local and global characteristics of EEG-derived evolving functional brain networks. Our findings emphasize the need to take into account the impact of biological rhythms in order to avoid erroneous statements about brain dynamics and about evolving functional brain networks.

Reconfiguration of human evolving large-scale epileptic brain networks prior to seizures: an evaluation with node centralities.

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 interpretations, which is reflected in a number of centrality indices. Here we aim to achieve a more general understanding of local and global network reconfigurations during the pre-seizure period as indicated by changes of different node centrality indices. To this end, we investigate-in a time-resolved manner-evolving large-scale epileptic brain networks that we derived from multi-day, multi-electrode intracranial electroencephalograpic recordings from a large but inhomogeneous group of subjects with pharmacoresistant epilepsies with different anatomical origins. We estimate multiple centrality indices to assess the various roles the nodes play while the networks transit from the seizure-free to the pre-seizure period. Our findings allow us to formulate several major scenarios for the reconfiguration of an evolving epileptic brain network prior to seizures, which indicate that there is likely not a single network mechanism underlying seizure generation. Rather, local and global aspects of the pre-seizure network reconfiguration affect virtually all network constituents, from the various brain regions to the functional connections between them.

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

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

No evidence for spontaneous cross-frequency phase-phase coupling in the human hippocampus.

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 from the human hippocampus acquired during waking state and different sleep stages. We estimated the strength of interactions between different pairs of frequency bands and evaluated the statistical significance of findings using surrogates that build on different null hypotheses. Indications for spontaneous phase-phase coupling were only observed when testing with less rigorous surrogates. When requiring that all four surrogate tests be passed, however, there were no significant indications for phase-phase coupling. In conclusion, we did not detect evidence for spontaneous cross-frequency phase-phase coupling in the human hippocampus.

No evidence for critical slowing down prior to human epileptic seizures.

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 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.

Precursors of seizures due to specific spatial-temporal modifications of evolving large-scale epileptic brain networks.

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 network nodes from which seizure precursors emerge. We aimed to shed more light on the role of constituents of evolving epileptic networks that recurrently transit into and out of seizures. We constructed such networks from more than 3200 hours of continuous intracranial electroencephalograms recorded in 38 patients with medication refractory epilepsy. We succeeded in singling out predictive edges and predictive nodes. Their particular characteristics, namely edge weight respectively node centrality (a fundamental concept of network theory), from the pre-ictal periods of 78 out of 97 seizures differed significantly from the characteristics seen during inter-ictal periods. The vast majority of predictive nodes were connected by most of the predictive edges, but these nodes never played a central role in the evolving epileptic networks. Interestingly, predictive nodes were entirely associated with brain regions deemed unaffected by the focal epileptic process. We propose a network mechanism for a transition into the pre-seizure state, which puts into perspective the role of the seizure onset zone in this transition and highlights the necessity to reassess current concepts for seizure generation and seizure prevention.

Traceability and dynamical resistance of precursor of extreme events.

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, knowledge about dynamical resistance of a system prior to an extreme event is of utmost importance. Here we introduce a novel time-series-based and non-perturbative approach to efficiently monitor dynamical resistance and apply it to high-resolution observations of brain activities from 43 subjects with uncontrollable epileptic seizures. We gain surprising insights into pre-seizure dynamical resistance of brains that also provide important clues for success or failure of measures for seizure prevention. The novel resistance monitoring perspective advances our understanding of precursor dynamics in complex spatio-temporal systems with potential applications in refining control strategies.

Impact of type of intracranial EEG sensors on link strengths of evolving functional brain networks.

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 cortex. It is still unclear, however, whether the use of different sensors for an identification of network nodes affects properties of functional brain networks. We address this question by investigating properties of links of such networks that we characterize by assessing interactions in multi-sensor, multi-day EEG data recorded from 49 epilepsy patients during presurgical evaluation. These data allow us to study the impact of different types of sensors together with the impact of various physiologic and pathophysiologic activities on the properties of links. MAIN RESULTS: We observe that different types of sensors differently impact on spatial means and temporal fluctuations of link strengths. Moreover, the impact depends on the relative anatomical location of sensors with respect to location and extent of sources of the prevailing activities. SIGNIFICANCE: Type and location of sensors should be considered when constructing networks.