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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Complex systems are omnipresent and play a vital role in in our every-day lives. Adverse behavior of such systems has generated considerable interest in being able to control complex systems modeled as networks. Here, we propose a topology-dynamics-based approach for controlling complex systems modeled as networks of coupled multi-dimensional dynamical entities. For given dynamics and topology, we introduce an efficient scheme to identify in polynomial time a finite set of driver nodes, which – when endowed with the control function – steer the network to the desired behavior. We demonstrate the high suitability of our approach by controlling various networked multi-dimensional dynamics, coupled onto different topologies. [ABSTRACT FROM AUTHOR]

We study scale-free networks of FitzHugh-Nagumo units whose collective dynamics exhibits rare and recurrent events of high amplitude (extreme events). We investigate which units play a crucial role in the generation of these events. Unexpectedly, we observe that extreme events originate from low-degree units, while high-degree units (hubs) appear to be crucial for the event to engulf the whole network. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate a recently published analysis framework based on Bayesian inference for the time-resolved characterization of interaction properties of noisy, coupled dynamical systems. It promises wide applicability and a better time resolution than well-established methods. At the example of representative model systems, we show that the analysis framework has the same weaknesses as previous methods, particularly when investigating interacting, structurally different non-linear oscillators. We also inspect the tracking of time-varying interaction properties and propose a further modification of the algorithm, which improves the reliability of obtained results. We exemplarily investigate the suitability of this algorithm to infer strength and direction of interactions between various regions of the human brain during an epileptic seizure. Within the limitations of the applicability of this analysis tool, we show that the modified algorithm indeed allows a better time resolution through Bayesian inference when compared to previous methods based on least square fits. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems' dynamics that may qualify as precursors to extreme events. We investigate these features and elucidate mechanisms that may be responsible for the generation of the extreme events. [ABSTRACT FROM AUTHOR]

Systems of FitzHugh-Nagumo units with different coupling topologies are capable of self-generating and -terminating strong deviations from their regular dynamics that can be regarded as extreme events due to their rareness and recurrent occurrence. Here we demonstrate the crucial role of an interior crisis in the emergence of extreme events. In parameter space we identify this interior crisis as the organizing center of the dynamics by employing concepts of mixed-mode oscillations and of leaking chaotic systems. We find that extreme events occur in certain regions in parameter space, and we show the robustness of this phenomenon with respect to the system size. [ABSTRACT FROM AUTHOR]

We extend the concept of symbolic transfer entropy to enable the time-resolved investigation of directional relationships between coupled dynamical systems from short and transient noisy time series. For our approach, we consider an observed ensemble of a sufficiently large number of time series as multiple realizations of a process. We derive an index that quantifies the preferred direction of transient interactions and assess its significance using a surrogate-based testing scheme. Analyzing time series from noisy chaotic systems, we demonstrate numerically the applicability and limitations of our approach. Our findings obtained from an analysis of event-related brain activities underline the importance of our method to improve understanding of gross neural interactions underlying cognitive processes. [ABSTRACT FROM AUTHOR]