Inferring directed networks using a rank-based connectivity measure

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.

Phase irregularity: A conceptually simple and efficient approach to characterize electroencephalographic recordings from epilepsy patients

The severe neurological disorder epilepsy affects almost 1% of the world population. For patients who suffer from pharmacoresistant focal-onset epilepsy, electroencephalographic (EEG) recordings are essential for the localization of the brain area where seizures start. Apart from the visual inspection of the recordings, quantitative EEG signal analysis techniques proved to be useful for this purpose. Among other features, regularity versus irregularity and phase coherence versus phase independence allowed characterizing brain dynamics from the measured EEG signals. Can phase irregularities also characterize brain dynamics? To address this question, we use the univariate coefficient of phase velocity variation, defined as the ratio of phase velocity standard deviation and the mean phase velocity. Beyond that, as a bivariate measure we use the classical mean phase coherence to quantify the degree of phase locking. All phase-based measures are combined with surrogates to test null hypotheses about the dynamics underlying the signals. In the first part of our analysis, we use the Rössler model system to study our approach under controlled conditions. In the second part, we use the Bern-Barcelona EEG database which consists of focal and nonfocal signals extracted from seizure-free recordings. Focal signals are recorded from brain areas where the first seizure EEG signal changes can be detected, and nonfocal signals are recorded from areas that are not involved in the seizure at its onset. Our results show that focal signals have less phase variability and more phase coherence than nonfocal signals. Once combined with surrogates, the mean phase velocity proved to have the highest discriminative power between focal and nonfocal signals. In conclusion, conceptually simple and easy to compute phase-based measures can help to detect features induced by epilepsy from EEG signals. This holds not only for the classical mean phase coherence but even more so for univariate measures of phase irregularity.

Reconstructing network structures from partial measurements

The dynamics of systems of interacting agents is determined by the structure of their coupling network. The knowledge of the latter is, therefore, highly desirable, for instance, to develop efficient control schemes, to accurately predict the dynamics, or to better understand inter-agent processes. In many important and interesting situations, the network structure is not known, however, and previous investigations have shown how it may be inferred from complete measurement time series on each and every agent. These methods implicitly presuppose that, even though the network is not known, all its nodes are. Here, we investigate the different problem of inferring network structures within the observed/measured agents. For symmetrically coupled dynamical systems close to a stable equilibrium, we establish analytically and illustrate numerically that velocity signal correlators encode not only direct couplings, but also geodesic distances in the coupling network within the subset of measurable agents. When dynamical data are accessible for all agents, our method is furthermore algorithmically more efficient than the traditional ones because it does not rely on matrix inversion.

Impact of local network characteristics on network reconstruction

When a network is inferred from data, two types of errors can occur: false positive and false negative conclusions about the presence of links. We focus on the influence of local network characteristics on the probability α of false positive conclusions, and on the probability β of false negative conclusions, in the case of networks of coupled oscillators. We demonstrate that false conclusion probabilities are influenced by local connectivity measures such as the shortest path length and the detour degree, which can also be estimated from the inferred network when the true underlying network is not known a priori. These measures can then be used for quantification of the confidence level of link conclusions, and for improving the network reconstruction via advanced concepts of link weights thresholding.

Seizure Onset Zone Lateralization Using a Non-linear Analysis of Micro vs. Macro Electroencephalographic Recordings During Seizure-Free Stages of the Sleep-Wake Cycle From Epilepsy Patients

The application of non-linear signal analysis techniques to biomedical data is key to improve our knowledge about complex physiological and pathological processes. In particular, the use of non-linear techniques to study electroencephalographic (EEG) recordings can provide an advanced characterization of brain dynamics. In epilepsy these dynamics are altered at different spatial scales of neuronal organization. We therefore apply non-linear signal analysis to EEG recordings from epilepsy patients derived with intracranial hybrid electrodes, which are composed of classical macro contacts and micro wires. Thereby, these electrodes record EEG at two different spatial scales. Our aim is to test the degree to which the analysis of the EEG recorded at these different scales allows us to characterize the neuronal dynamics affected by epilepsy. For this purpose, we retrospectively analyzed long-term recordings performed during five nights in three patients during which no seizures took place. As a benchmark we used the accuracy with which this analysis allows determining the hemisphere that contains the seizure onset zone, which is the brain area where clinical seizures originate. We applied the surrogate-corrected non-linear predictability score (ψ), a non-linear signal analysis technique which was shown previously to be useful for the lateralization of the seizure onset zone from classical intracranial EEG macro contact recordings. Higher values of ψ were found predominantly for signals recorded from the hemisphere containing the seizure onset zone as compared to signals recorded from the opposite hemisphere. These differences were found not only for the EEG signals recorded with macro contacts, but also for those recorded with micro wires. In conclusion, the information obtained from the analysis of classical macro EEG contacts can be complemented by the one of micro wire EEG recordings. This combined approach may therefore help to further improve the degree to which quantitative EEG analysis can contribute to the diagnostics in epilepsy patients.

Using machine learning to assess short term causal dependence and infer network links

We introduce and test a general machine-learning-based technique for the inference of short term causal dependence between state variables of an unknown dynamical system from time-series measurements of its state variables. Our technique leverages the results of a machine learning process for short time prediction to achieve our goal. The basic idea is to use the machine learning to estimate the elements of the Jacobian matrix of the dynamical flow along an orbit. The type of machine learning that we employ is reservoir computing. We present numerical tests on link inference of a network of interacting dynamical nodes. It is seen that dynamical noise can greatly enhance the effectiveness of our technique, while observational noise degrades the effectiveness. We believe that the competition between these two opposing types of noise will be the key factor determining the success of causal inference in many of the most important application situations.

Reconstructing dynamical networks via feature ranking

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the structure of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the properties and dynamics of the scrutinized network. Here, we use the insights from machine learning to design a new method of network reconstruction that essentially makes no such assumptions. Specifically, we interpret the available trajectories (data) as "features" and use two independent feature ranking approaches-Random Forest and RReliefF-to rank the importance of each node for predicting the value of each other node, which yields the reconstructed adjacency matrix. We show that our method is fairly robust to coupling strength, system size, trajectory length, and noise. We also find that the reconstruction quality strongly depends on the dynamical regime.