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.

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.

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.

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.

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.

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.

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.

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.

FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena.

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 between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability. Among these networks, a small-world network with intermediate rewiring probability best mimics the findings achieved with the simulations using the empirical structural connectivity. For the other network topologies, either no spontaneously occurring epileptic-seizure-related synchronization phenomena can be observed in the simulated dynamics, or the overall degree of synchronization remains high throughout the simulation. This indicates that a topology with some balance between regularity and randomness favors the self-initiation and self-termination of episodes of seizure-like strong synchronization.

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.

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.

Complexity and irreducibility of dynamics on networks of networks.

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 a region in parameter space in which the interplay of within- and between-network couplings allows for a richer dynamical behavior than can be observed for a single sub-network. Adjoining this atypical region, our network of networks exhibits transitions to multistability. We elucidate bifurcations governing the transitions between the various dynamics when crossing this region and discuss how varying the couplings affects the effective structure of our network of networks. Our findings indicate that reducing a network of networks to a single (but bigger) network might not be accurate enough to properly understand the complexity of its dynamics.

Long-term variability of importance of brain regions in evolving epileptic brain networks.

We investigate the temporal and spatial variability of the importance of brain regions in evolving epileptic brain networks. We construct these networks from multiday, multichannel electroencephalographic data recorded from 17 epilepsy patients and use centrality indices to assess the importance of brain regions. Time-resolved indications of highest importance fluctuate over time to a greater or lesser extent, however, with some periodic temporal structure that can mostly be attributed to phenomena unrelated to the disease. In contrast, relevant aspects of the epileptic process contribute only marginally. Indications of highest importance also exhibit pronounced alternations between various brain regions that are of relevance for studies aiming at an improved understanding of the epileptic process with graph-theoretical approaches. Nonetheless, these findings may guide new developments for individualized diagnosis, treatment, and control.

Surrogate-assisted identification of influences of network construction on evolving weighted functional networks.

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 functional brain networks before, during, and after epileptic seizures that we derive from multichannel electroencephalographic data recorded from epilepsy patients. For these evolving networks, we estimate clustering coefficient and average shortest path length in a time-resolved manner. Lastly, we make use of surrogate concepts that we apply at various levels of the chain of analysis to assess to what extent network characteristics are dominated by properties of the electroencephalographic recordings and/or the evolving weighted networks, which may be accessible more easily. We observe that characteristics are differently affected by the unavoidable referencing of the electroencephalographic recording, by the time-series-analysis technique used to derive the properties of network links, and whether or not networks were normalized. Importantly, for the majority of analysis settings, we observe temporal evolutions of network characteristics to merely reflect the temporal evolutions of mean interaction strengths. Such a property of the data may be accessible more easily, which would render the weighted network approach-as used here-as an overly complicated description of simple aspects of the data.

Distinguishing between direct and indirect directional couplings in large oscillator networks: Partial or non-partial phase analyses?

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach.

Assessing directionality and strength of coupling through symbolic analysis: an application to epilepsy patients.

Inferring strength and direction of interactions from electroencephalographic (EEG) recordings is of crucial importance to improve our understanding of dynamical interdependencies underlying various physiological and pathophysiological conditions in the human epileptic brain. We here use approaches from symbolic analysis to investigate--in a time-resolved manner--weighted and directed, short- to long-ranged interactions between various brain regions constituting the epileptic network. Our observations point to complex spatial-temporal interdependencies underlying the epileptic process and their role in the generation of epileptic seizures, despite the massive reduction of the complex information content of multi-day, multi-channel EEG recordings through symbolization. We discuss limitations and potential future improvements of this approach.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges.

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.

Can spurious indications for phase synchronization due to superimposed signals be avoided?

We investigate the relative merit of phase-based methods-mean phase coherence, unweighted and weighted phase lag index-for estimating the strength of interactions between dynamical systems from empirical time series which are affected by common sources and noise. By numerically analyzing the interaction dynamics of coupled model systems, we compare these methods to each other with respect to their ability to distinguish between different levels of coupling for various simulated experimental situations. We complement our numerical studies by investigating consistency and temporal variations of the strength of interactions within and between brain regions using intracranial electroencephalographic recordings from an epilepsy patient. Our findings indicate that the unweighted and weighted phase lag index are less prone to the influence of common sources but that this advantage may lead to constrictions limiting the applicability of these methods.

A Gaussian graphical model approach to climate networks.

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyze networks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations. Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.