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Different across-layer synchronization types of chimera states in multilayer networks have been discovered recently. We investigate possible relations between them, for example, if the onset of some synchronization type implies the onset of some other type. For this purpose, we use a two-layer network with multiplex inter-layer coupling. Each layer consists of a ring of non-locally coupled phase oscillators. While oscillators in each layer are identical, the layers are made non-identical by introducing mismatches in the oscillators’ mean frequencies and phase lag parameters of the intra-layer coupling. We use different metrics to quantify the degree of various across-layer synchronization types. These include phase-locking between individual interacting oscillators, amplitude and phase synchronization between the order parameters of each layer, generalized synchronization between the driver and response layer, and the alignment of the incoherent oscillator groups’ position on the two rings. For positive phase lag parameter mismatches, we get a cascaded onset of synchronization upon a gradual increase of the inter-layer coupling strength. For example, the two order parameters show phase synchronization before any of the interacting oscillator pairs does. For negative mismatches, most synchronization types have their onset in a narrow range of the coupling strength. Weaker couplings can destabilize chimera states in the response layer toward an almost fully coherent or fully incoherent motion. Finally, in the absence of a phase lag mismatch, sufficient coupling turns the response dynamics into a replica of the driver dynamics with the phases of all oscillators shifted by a constant lag. We acknowledge funding from the Spanish Ministry of Science and Innovation and the State Research Agency (Grant No. PID2020-118196GBI00/MICIN/AEI/10.13039/501100011033).

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.

Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map's complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 033125 (2020) and may be found at https://doi.org/10.1063/5.0002272., We numerically study a network of two identical populations of identical real-valued quadratic maps. Upon variation of the coupling strengths within and across populations, the network exhibits a rich variety of distinct dynamics. The maps in individual populations can be synchronized or desynchronized. Their temporal evolution can be periodic or aperiodic. Furthermore, one can find blends of synchronized with desynchronized states and periodic with aperiodic motions. We show symmetric patterns for which both populations have the same type of dynamics as well as chimera states of a broken symmetry. The network can furthermore show multistability by settling to distinct dynamics for different realizations of random initial conditions or by switching intermittently between distinct dynamics for the same realization. We conclude that our system of two populations of a particularly simple map is the most simple system that can show this highly diverse and complex behavior, which includes but is not limited to chimera states. As an outlook to future studies, we explore the stability of two populations of quadratic maps with a complex-valued control parameter. We show that bounded and diverging dynamics are separated by fractal boundaries in the complex plane of this control parameter.

Inferring the topology of a network using the knowledge of the signals of each of the interacting units is key to understanding real-world systems. One way to address this problem is using data- driven methods like cross-correlation or mutual information. However, these measures lack the ability to distinguish the direction of coupling. Here, we use a rank-based nonlinear interdependence measure originally developed for pairs of signals. This measure not only allows one to measure the strength but also the direction of the coupling. Our results for a system of coupled Lorenz dynamics show that we are able to consistently infer the underlying network for a subrange of the coupling strength and link density. Furthermore, we report that the addition of dynamical noise can benefit the reconstruction. Finally, we show an application to multichannel electroencephalographic recordings from an epilepsy patient. This work was funded by the EU via H2020 Marie SklodowskaCurie project COSMOS, grant no. 642563 (M.G.L., I.M., Z.L., and R.G.A). R.G.A. and C.G.B.M. acknowledge funding from the Spanish Ministry of Econ- omy and Competitiveness (Grant FIS2014-54177-R) and the CERCA Programme of the Generalitat de Catalunya. C.G.B.M. acknowledges the support by the Spanish Ministry of Economy and Competitiveness under the Maria de Maeztu Units of Excellence Programme (MDM-2015-0502). Z.L. acknowledges funding from the Slovenian Re- search Agency via program Complex Networks P1-0383 and project J5-8236.

This scientific commentary refers to ‘Crowdsourcing reproducible seizure forecasting in human and canine epilepsy’, by Brinkmann et al. (doi:10.1093/brain/aww045). Epilepsy is characterized by recurrent spontaneous seizures that seem to occur without warning, like a bolt from the blue. Most patients with epilepsy spend over 99.9% of their time in a perfectly normal brain state. But even though epileptic seizures constitute less than 0.1% of their lifetime, it is the apparent unpredictability of these events that puts an immense psychological burden on patients and substantially compromises their quality of life. A reliable method of forecasting seizure occurrence would thus greatly diminish psychological stress for patients. Moreover, it could lead to novel therapeutic options such as on-demand treatment and ultimately even closed-loop interventional devices to prevent impending seizures from occurring (Morrell and Halpern, 2016). In this issue of Brain , Brinkmann and co-workers have used the concept of crowdsourcing to tackle the problem of seizure prediction (Brinkmann et al. , 2016). They hosted epochs of continuous multi-day recordings of intracranial EEG (iEEG) activity from both humans and canines on a website (www.kaggle.com), and started a competition open to the public in which the aim was to correctly characterize these data clips as preictal (indicative of an impending seizure) or interictal (in this study, at least 1 week away from any seizure). Participating teams were provided with a set of training data consisting of clips that were labelled as being preictal or interictal. Using these data clips, they could train and optimize their algorithms before running them on the other clips provided—the testing data. For each of these remaining clips, the candidate algorithms had to estimate the probability that the clip was preictal. Receiver operating …

Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks. We acknowledge funding from the Volkswagen foundation, the Spanish Ministry of Economy and Competitiveness, Grant No. FIS2014-54177-R, the CERCA Programme of the Generalitat de Catalunya (R.G.A. and G.R.), and from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 642563 (R.G.A. and I.M.).

This study characterizes the spatial-temporal distribution of epileptic activity in mesial and neocortical temporal lobe epilepsy (TLE) assessed by subdural and depth electrodes during sleep. We determined in 13 patients the frequency, lateralization, and localization of interictal epileptic discharges (IEDs). As compared to the waking state, IEDs increased in light sleep (196%, p0.05) and in deep sleep (601%, p0.05) but did not change significantly in REM sleep (-8.33%, p = 0.94). From 11 patients with unilateral TLE, in all cases, IEDs lateralized to the seizure onset side during REM sleep and the waking state. In mesial TLE, IEDs tended to shift in an anterior-posterior axis and remained always localized in the amygdalo-hippocampal complex. By contrast, in neocortical TLE, the maximal activity moved in a mesial-lateral axis between neocortical and mesial structures. Twenty-six seizures were registered in 7 patients, 22 of which occurred in light sleep and 4 in wakefulness, but none occurred in deep or REM sleep.

Objective To show that time-irreversible EEG signals recorded with intracranial electrodes during seizures can serve as markers of the epileptogenic zone. Methods We use the recently developed method of mapping time series into directed horizontal graphs (dHVG). Each node of the dHVG represents a time point in the original intracranial EEG (iEEG) signal. Statistically significant differences between the distributions of the nodes’ number of input and output connections are used to detect time-irreversible iEEG signals. Results In 31 of 32 seizure recordings we found time-irreversible iEEG signals. The maximally time-irreversible signals always occurred during seizures, with highest probability in the middle of the first seizure half. These signals spanned a large range of frequencies and amplitudes but were all characterized by saw-tooth like shaped components. Brain regions removed from patients who became post-surgically seizure-free generated significantly larger time-irreversibilities than regions removed from patients who still had seizures after surgery. Conclusions Our results corroborate that ictal time-irreversible iEEG signals can indeed serve as markers of the epileptogenic zone and can be efficiently detected and quantified in a time-resolved manner by dHVG based methods. Significance Ictal time-irreversible EEG signals can help to improve pre-surgical evaluation in patients suffering from pharmaco-resistant epilepsies.

Time scale parametric spike train distances like the Victor and the van Rossum distances/nare often applied to study the neural code based on neural stimuli discrimination./nDifferent neural coding hypotheses, such as rate or coincidence coding,/ncan be assessed by combining a time scale parametric spike train distance with a/nclassifier in order to obtain the optimal discrimination performance. The time scale/nfor which the responses to different stimuli are distinguished best is assumed to be/nthe discriminative precision of the neural code. The relevance of temporal coding/nis evaluated by comparing the optimal discrimination performance with the one/nachieved when assuming a rate code./nWe here characterize the measures quantifying the discrimination performance,/nthe discriminative precision, and the relevance of temporal coding. Furthermore,/nwe evaluate the information these quantities provide about the neural code. We/nshow that the discriminative precision is too unspecific to be interpreted in terms/nof the time scales relevant for encoding. Accordingly, the time scale parametric/nnature of the distances is mainly an advantage because it allows maximizing the/ndiscrimination performance across a whole set of measures with different sensitivities/ndetermined by the time scale parameter, but not due to the possibility to/nexamine the temporal properties of the neural code. DC is supported by the grant 2010FI-B2 00079 of the ”Comissionat per a Universitats/ni Recerca del Departament d’Innovació, Universitats i Empresa de la Generalitat/nde Catalunya i del Fons Social Europeu” and grant 2008BE1 00166 of the ”Comissionat per a Universitats i Recerca del Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya”. RGA acknowledges grant BFU2007-61710 of the Spanish Ministry of Education and Science.

The sudden and apparently unpredictable nature of seizures is one of the most disabling aspects of the disease epilepsy. A method capable of predicting the occurrence of seizures from the electroencephalogram (EEG) of epilepsy patients would open new therapeutic possibilities. Since the 1970s investigations on the predictability of seizures have advanced from preliminary descriptions of seizure precursors to controlled studies applying prediction algorithms to continuous multi-day EEG recordings. While most of the studies published in the 1990s and around the turn of the millennium yielded rather promising results, more recent evaluations could not reproduce these optimistic findings, thus raising a debate about the validity and reliability of previous investigations. In this review, we will critically discuss the literature on seizure prediction and address some of the problems and pitfalls involved in the designing and testing of seizure-prediction algorithms. We will give an account of the current state of this research field, point towards possible future developments and propose methodological guidelines for future studies on seizure prediction.

The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a ‘controlled’ setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others. c 2006 Elsevier B.V. All rights reserved.

Conceptually and structurally simple mathematical models of coupled oscillator networks can show a rich variety of complex dynamics, providing fundamental insights into many real-world phenomena.A recent and not yet fully understood example is the collapse of coexisting synchronous and synchronous oscillations into a globally synchronous motion found in networks of identical oscillators. Here we show that this sudden collapse is promoted by a further decrease of synchronization, rather than by critically high synchronization. This strikingly counterintuitive mechanism can be found also in nature, as we demonstrate on epileptic seizures in humans. Analyzing spatiotemporal correlation profiles derived from intracranial electroencephalographic recordings (EEG) of seizures in epilepsy patients, we found a pronounced decrease of correlation at the seizure onsets. Applying our findings in a closed-loop control scheme to models of coupled oscillators in chimera states, we succeed in both provoking and preventing outbreaks of global synchronization. Our findings not only advance the understanding of networks of coupled dynamics but can open new ways to control them, thus offering a vast range of potential new applications. The authors are grateful to Ernest Montbrió and Arkady Pikovsky for useful discussion. R.G.A. and F.M. acknowledge funding from the Volkswagen foundation. R.G.A was supported by the Spanish Ministry of Economy and Competitiveness (Grant FIS2014-54177-R). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642563 (R.G.A.). F.M. acknowledges support from the German Research Council (DFG, MO 930/4-1 and SFB 1089) and the European Commission (FP7 602102 EPITARGET). K.S. and C.R. are grateful for support by the Swiss National Science Foundation (Projects No. SNF 320030_122010, No. 33CM30-140332, No. 32003B_155950).

Objective: An important issue in epileptology is the question whether information extracted from the EEG of epilepsy patients can be used for the prediction of seizures. Several studies have claimed evidence for the existence of a pre-seizure state that can be detected using different characterizing measures. In this paper, we evaluate the predictability of seizures by comparing the predictive performance of a variety of univariate and bivariate measures comprising both linear and non-linear approaches. Methods: We compared 30 measures in terms of their ability to distinguish between the interictal period and the pre-seizure period. After completely analyzing continuous inctracranial multi-channel recordings from five patients lasting over days, we used ROC curves to distinguish between the amplitude distributions of interictal and preictal time profiles calculated for the respective measures. We compared different evaluation schemes including channelwise and seizurewise analysis plus constant and adaptive reference levels. Particular emphasis was placed on statistical validity and significance. Results: Univariate measures showed statistically significant performance only in a channelwise, seizurewise analysis using an adaptive baseline. Preictal changes for these measures occurred 5‐30 min before seizures. Bivariate measures exhibited high performance values reaching statistical significance for a channelwise analysis using a constant baseline. Preictal changes were found at least 240 min before seizures. Linear measures were found to perform similar or better than non-linear measures. Conclusions: Results provide statistically significant evidence for the existence of a preictal state. Based on our findings, the most promising approach for prospective seizure anticipation could be a combination of bivariate and univariate measures. Significance: Many measures reported capable of seizure prediction in earlier studies are found to be insignificant in performance, which underlines the need for statistical validation in this field. q 2004 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.

A number of recent studies indicate that nonlinear electroencephalogram (EEG) analyses allow one to define a state predictive of an impending epileptic seizure. In this paper, we combine a method for detecting nonlinear determinism with a novel test for stationarity to characterize EEG recordings from both the seizure-free interval and the preseizure phase. We discuss differences between these periods, particularly an increased occurrence of stationary, nonlinear segments prior to seizures. These differences seem most prominent for recording sites within the seizure-generating area and for EEG segments less than one minute's length.

The exact mechanisms leading to the occurrence of epileptic seizures in humans are still poorly understood. It is widely accepted, however, that the process of seizure generation is closely associated with an abnormal synchronization of neurons. In order to investigate this process, we here measure phase synchronization between different regions of the brain using intracranial EEG recordings. Based on our preliminary finding of a preictal drop in synchronization, we investigate whether this phenomenon can be used as a sensitive and specific criterion to characterize a preseizure state and to distinguish this state from the interictal interval. Applying an automated technique for detecting decreased synchronization to EEG recordings from a group of 18 patients with focal epilepsy comprising a total of 117 h, we observe a characteristic decrease in synchronization prior to 26 out of 32 analyzed seizures at a very high specificity as tested on interictal recordings. The duration of this preictal state is found to range from several minutes up to a few hours. Investigation of the spatial distribution of preictal desynchronization indicates that the process of seizure generation in focal epilepsy is not necessarily confined to the focus itself but may instead involve more distant, even contralateral areas of the brain. Finally, we demonstrate an intrahemispheric asymmetry in the spatial dynamics of preictal desynchronization that is found in the majority of seizures and appears to be an immanent part of the mechanisms underlying the initiation of seizures in humans.

Topologies of real-world complex networks are rarely accessible, but can often be reconstructed from experimentally obtained time series via suitable network reconstruction methods. Extending our earlier work on methods based on statistics of derivative-variable correlations, we here present a new method built on integrating an evolutionary optimization algorithm into the derivative-variable correlation method. Results obtained from our modi cation of the method in general outperform the original results, demonstrating the suitability of evolutionary optimization logic in network reconstruction problems. We show the method's usefulness in realistic scenarios where the reconstruction precision can be limited by the nature of the time series. We also discuss important limitations coming from various dynamical regimes that time series can belong to. This work was founded by the EU via H2020 Marie SklodowskaCurie project COSMOS, grant no. 642563. R G A acknowledges funding from the Volkswagen foundation, the Spanish Ministry of Economy and Competitiveness (Grant FIS2014-54177-R) and the CERCA Programme of the Generalitat de Catalunya. Z L acknowledges funding from the Slovenian Research Agency via program Complex Networks P1-0383 and project J5- 8236.

The rank-based nonlinear predictability score was recently introduced as a test for determinism in point processes. We here adapt this measure to time series sampled from time-continuous flows. We use noisy Lorenz signals to compare this approach against a classical amplitude-based nonlinear prediction error. Both measures show an almost identical robustness against Gaussian white noise. In contrast, when the amplitude distribution of the noise has a narrower central peak and heavier tails than the normal distribution, the rank-based nonlinear predictability score outperforms the amplitude-based nonlinear prediction error. For this type of noise, the nonlinear predictability score has a higher sensitivity for deterministic structure in noisy signals. It also yields a higher statistical power in a surrogate test of the null hypothesis of linear stochastic correlated signals. We show the high relevance of this improved performance in an application to electroencephalographic (EEG) recordings from epilepsy patients. Here the nonlinear predictability score again appears of higher sensitivity to nonrandomness. Importantly, it yields an improved contrast between signals recorded from brain areas where the first ictal EEG signal changes were detected (focal EEG signals) versus signals recorded from brain areas that were not involved at seizure onset (nonfocal EEG signals). R.G.A. acknowledges Grant No. FIS-2010-18204 of the Spanish Ministry of Education and Science and funding from the Volkswagen Foundation. K.S. and C.R. are grateful for support by the Swiss National Science Foundation (Projects No. SNF 320030-122010 and No. 33CM30-124089).

The detection of a nonrandom structure from experimental data can be crucial for the classification, understanding, and interpretation of the generating process. We here introduce a rank-based nonlinear predictability score to detect determinism from point process data. Thanks to its modular nature, this approach can be adapted to whatever signature in the data one considers indicative of deterministic structure. After validating our approach using point process signals from deterministic and stochastic model dynamics, we show an application to neuronal spike trains recorded in the brain of an epilepsy patient. While we illustrate our approach in the context of temporal point processes, it can be readily applied to spatial point processes as well. R.G.A. acknowledges Grant No. FIS-2010-18204 of the Spanish Ministry of Education and Science. R.G.A. and F.M. acknowledge funding from the Volkswagen Foundation. T.K. acknowledges funding from the European Commission through the Marie Curie Initial Training Network ‘Neural Engineering Transformative Technologies (NETT)’, Project No. 289146.

In the analysis of physiological time series it is important to investigate the inter-dependence structure among the observed variables of the system. For this, a number of measures of the so-called Granger causality have been developed, and among them information measures have gained much attention. However, information measures have to deal with the estimation of probability distributions of high-dimensional vector variables, typically formed through uniform delay embedding of the time series. Here, the focus is on measures derived after nonuniform embedding. Particularly, two schemes are considered, one based on conditional mutual information and the other based on conditional entropy. We evaluate the two measures on simulated nonlinear coupled dynamical systems and apply them to signals of heart rate variability, respiration, and oxygen concentration in the blood.

The theory of deterministic chaos addresses simple deterministic dynamics in which nonlinearity gives rise to complex temporal behavior. Although biological neuronal networks such as the brain are highly complicated, a number of studies provide growing evidence that nonlinear time series analysis of brain electrical activity in patients with epilepsy is capable of providing potentially useful diagnostic information. In the present study, this analysis framework was extended by introducing a new measure ξ , designed to discriminate between nonlinear deterministic and linear stochastic dynamics. For the evaluation of its discriminative power, ξ was extracted from intracranial multi-channel EEGs recorded during the interictal state in 25 patients with unilateral mesial temporal lobe epilepsy. Strong indications of nonlinear determinism were found in recordings from within the epileptogenic zone, while EEG signals from other sites mainly resembled linear stochastic dynamics. In all investigated cases, this differentiation allowed to retrospectively determine the side of the epileptogenic zone in full agreement with results of the presurgical workup.

Summary:Several recent studies emphasize the high value of nonlinear EEG analysis particularly for improved characterization of epileptic brain states. In this review the authors report their work to increase insight into the spatial and temporal dynamics of the epileptogenic process. Specifically

To derive tests for randomness, nonlinear-independence, and stationarity, we combine surrogates with a nonlinear prediction error, a nonlinear interdependence measure, and linear variability measures, respectively. We apply these tests to intracranial electroencephalographic recordings (EEG) from patients suffering from pharmacoresistant focal-onset epilepsy. These recordings had been performed prior to and independent from our study as part of the epilepsy diagnostics. The clinical purpose of these recordings was to delineate the brain areas to be surgically removed in each individual patient in order to achieve seizure control. This allowed us to define two distinct sets of signals: One set of signals recorded from brain areas where the first ictal EEG signal changes were detected as judged by expert visual inspection (“focal signals”) and one set of signals recorded from brain areas that were not involved at seizure onset (“nonfocal signals”). We find more rejections for both the randomness and the nonlinear-independence test for focal versus nonfocal signals. In contrast more rejections of the stationarity test are found for nonfocal signals. Furthermore, while for nonfocal signals the rejection of the stationarity test increases the rejection probability of the randomness and nonlinear-independence test substantially, we find a much weaker influence for the focal signals. In consequence, the contrast between the focal and nonfocal signals obtained from the randomness and nonlinear-independence test is further enhanced when we exclude signals for which the stationarity test is rejected. To study the dependence between the randomness and nonlinear-independence test we include only focal signals for which the stationarity test is not rejected. We show that the rejection of these two tests correlates across signals. The rejection of either test is, however, neither necessary nor sufficient for the rejection of the other test. Thus, our results suggest that EEG signals from epileptogenic brain areas are less random, more nonlinear-dependent, and more stationary compared to signals recorded from nonepileptogenic brain areas. We provide the data, source code, and detailed results in the public domain. R.G.A. acknowledges grant FIS-2010-18204 of the Spanish Ministry of Education and Science. K.S. and C.R. are grateful for support by the Swiss National Science Foundation (projects nos. SNF 320030-122010 and 33CM30- 124089).

Recently, the SPIKE-distance has been proposed as a parameter-free and time-scale independent measure of spike train synchrony. This measure is time-resolved since it relies on instantaneous estimates of spike train dissimilarity. However, its original definition led to spuriously high instantaneous values for event-like firing patterns. Here we present a substantial improvement of this measure which eliminates this shortcoming. The reliability gained allows us to track changes in instantaneous clustering, i.e., time-localized patterns of (dis)similarity among multiple spike trains. Additional new features include selective and triggered temporal averaging as well as the instantaneous comparison of spike train groups. In a second step, a causal SPIKE-distance is defined such that the instantaneous values of dissimilarity rely on past information only so that time-resolved spike train synchrony can be estimated in real-time. We demonstrate that these methods are capable of extracting valuable information from field data by monitoring the synchrony between neuronal spike trains during an epileptic seizure. Finally, the applicability of both the regular and the real-time SPIKE-distance to continuous data is illustrated on model electroencephalographic (EEG) recordings., Comment: 16 pages, 10 figures, 35 references; 1 supplementary figure, 1 supplementary movie (see author's webpage http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html)

Comunicació presentada a ACM Multimedia, International Workshop on Machine Learning and Music (MML), celebrat del 25 al 29 d'octubre de 2010 a Florència, Itàlia. Current systems for cover song detection are based on a model-free approach: they basically search for similarities in descriptor time series reflecting the evolution of tonal information in a musical piece. In this contribution we propose the use of a model-based approach. In particular, we explore threshold autoregressive models and the concept of cross-prediction error, i.e. a measure of to which extent a model trained on one song's descriptor time series is able to predict the covers'. Results indicate that the considered approach can provide competitive accuracies while being considerably fast and with potentially less storage requirements. Furthermore, the approach is parameter-free from the user's perspective, what provides a robust and straightforward application of it. We thank Emilia G´omez for her review on a previous version of this article. J.S. has been partially funded by the A/09/96235 grant from the Deutscher Akademisch Austausch Dienst and by the Music 3.0 (TSI-070100- 2008-318) and Buscamedia (CEN-20091026) projects. R.G.A. has been funded by the BFU2007-61710 grant of the Spanish Ministry of Education and Science.

Since the introduction of a family of cost-based distancesbetween spike trains by Victor and Purpura [1,2], spiketrain distances are widely used to study the precision ofthe temporal coding of neurons (see [3] for a review).Subsequently, other distances that act on convolutedspike trains have been proposed [4,5]. All of these dis-tances involve the choice of a parameter determining thetime scale at which the spike trains are compared. In gen-eral, the extent of the stimulus-specific clustering manifestin the distances between spike trains of a neuron firing inresponse to a set of trials of different stimuli is used toquantify the mutual information between the stimuli andthe spike trains in dependence of the respective parame-ter. The time scale for which the maximum mutual infor-mation is obtained serves as an indicator about theprecision of the temporal coding.Here we discuss limitations of the use of spike train dis-tances as well as principal shortcomings of the mutualinformation maximization procedure for the study ofneural coding. The implementation of this whole proce-dure relies on some initial implicit assumptions about theactual neural code: the single neuron is assumed as thecoding unit, the stimuli are conjectured to be the onesencoded by this neuron and the existence of an optimumtemporal precision is taken for granted. However, sincethe calculation of the spike train distances and the maxi-mization of mutual information are always possible andwill always provide an optimal time scale even if theseassumptions are incompatible with the actual code, naiveapplications can lead to wrong conclusions.To demonstrate the dangers of such implicit assumptions,we simulate examples in which the relation between stim-uli and spike trains is known and study the performanceof the techniques recovering the actual coding when theassumptions above are not justifiable. In particular, weconsider the non-existence of a unique time scale simulat-ing spike trains with bimodal or non-stationary firingrates. For this case we also compare the use of the timescale parametric distances with a scale adaptive distance[6]. Furthermore we show that misinterpretations can becaused by concluding a precision for a temporal codefrom neurons which in fact participate in a populationcoding. We also show that false conclusions can be drawndue to an incorrect identification of the stimulus relevantfor the neural code.

Measures of multiple spike train synchrony are essential in order to study issues such as spike timing reliability, network synchronization, and neuronal coding. These measures can broadly be divided in multivariate measures and averages over bivariate measures. One of the most recent bivariate approaches, the ISI-distance, employs the ratio of instantaneous interspike intervals. In this study we propose two extensions of the ISI-distance, the straightforward averaged bivariate ISI-distance and the multivariate ISI-diversity based on the coefficient of variation. Like the original measure these extensions combine many properties desirable in applications to real data. In particular, they are parameter free, time scale independent, and easy to visualize in a time-resolved manner, as we illustrate with in vitro recordings from a cortical neuron. Using a simulated network of Hindemarsh-Rose neurons as a controlled configuration we compare the performance of our methods in distinguishing different levels of multi-neuron spike train synchrony to the performance of six other previously published measures. We show and explain why the averaged bivariate measures perform better than the multivariate ones and why the multivariate ISI-diversity is the best performer among the multivariate methods. Finally, in a comparison against standard methods that rely on moving window estimates, we use single-unit monkey data to demonstrate the advantages of the instantaneous nature of our methods., Comment: 15 pages, 17 figures, 30 references Changes: Abstract corrected, one Figure and one Section in Appendix added, plus some minor corrections (Final Version)

We introduce a new technique designed to quantify directional couplings between nonlinear dynamics and we evaluate its application to the study of neuronal data. Similar to existing approaches [1,2] (and references therein) this technique quantifies the degree to which similar states of the one dynamics are mapped to similar states of the other dynamics based on reconstructed state-spaces. It differs from these existing techniques in that it uses rank statistics instead of distance statistics.

We present a client-server application for the distributed multivariate analysis of time series using standard PCs. We here concentrate on analyses of multichannel EEG/MEG data, but our method can easily be adapted to other time series. Due to the rapid development of new analysis techniques, the focus in the design of our application was not only on computational performance, but also on high flexibility and expandability of both the client and the server programs. For this purpose, the communication between the server and the clients as well as the building of the computational tasks has been realized via the Extensible Markup Language (XML). Running our newly developed method in an asynchronous distributed environment with random availability of remote and heterogeneous resources, we tested the system's performance for a number of different univariate and bivariate analysis techniques. Results indicate that for most of the currently available analysis techniques, calculations can be performed in real time, which, in principle, allows on-line analyses at relatively low cost.

Obtaining the most independent components from a mixture (under a chosen model) is only the first part of an ICA analysis. After that, it is necessary to measure the actual dependency between the components and the reliability of the decomposition. We have to identify one- and multidimensional components (i.e., clusters of mutually dependent components) or channels which are too close to Gaussians to be reliably separated. For the determination of the dependencies we use a new highly accurate mutual information (MI) estimator. The variability of the MI under remixing provides us a measure for the stability. A rapid growth of the MI under mixing identifies stable components. On the other hand a low variability identifies unreliable components. The method is illustrated on artificial datasets. The usefulness in real-world data is shown on biomedical data.

In this overview we present our work investigating the spatio-temporal dynamics of the epileptogenic process using time series analysis techniques derived from the theory of nonlinear dynamics. Apart from a localization of epileptic foci in different anatomical locations during the seizure-free interval we discuss possibilities for seizure prediction, a field that represents one of the greatest challenges in epileptology. The unequivocal definition of a pre-seizure state of a sufficient length would enable investigation of basic mechanisms leading to seizure initiation in humans and provide a basis for the development of adequate seizure prevention strategies.

Attempting to increase insight into the spatio-temporal dynamics of the epileptogenic process to address one of the greatest challenges in epileptology. In the field of EEG analysis the search for the hidden information predictive of an impending seizure has a long history. The nonlinear EEG analysis techniques we have used in principle allow one to define a pre-ictal state and to characterize different temporal and spatial aspects of this state. The results obtained so far emphasize the high value of nonlinear EEG analysis techniques for the detection of a long-lasting pre-ictal state. Once given a sufficient sensitivity and specificity of seizure prediction techniques, more extensive clinical studies on a larger population of patients, either at home or in a clinical setting, can be envisaged.

A rapidly growing number of studies deals with the prediction of epileptic seizures. For this purpose, various techniques derived from linear and nonlinear time series analysis have been applied to the electroencephalogram of epilepsy patients. In none of these works, however, the performance of the seizure prediction statistics is tested against a null hypothesis, an otherwise ubiquitous concept in science. In consequence, the evaluation of the reported performance values is problematic. Here, we propose the technique of seizure time surrogates based on a Monte Carlo simulation to remedy this deficit. C.E.E., T.K., K.L., F.M., and C.R. acknowledge support from the Deutsche Forschungsgemeinschaft.

Deterministic chaos offers a striking explanation for apparently irregular behavior of the brain that is evidenced in the EEG. Recent developments in the physical-mathematical framework of the theory of nonlinear dynamics (colloquially often termed chaos theory) provide new concepts and powerful algorithms to analyze such time series. Because of its high versatility, nonlinear time series analysis has already gone beyond the physical sciences and, at present, is being successfully applied in a variety of disciplines, including cardiology, neurology, psychiatry, and epileptology. However, it is well known that different influencing factors limit the use of nonlinear measures to characterize EEG dynamics in a strict sense. Nevertheless, when interpreted with care, relative estimates of, e.g., the correlation dimension or the Lyapunov exponents, can reliably characterize different states of normal and pathologic brain function. In epileptology, extraction of nonlinear measures from the intracranially recorded EEG promises to be important for clinical practice. In addition to an immense reduction of information content of long-lasting EEG recordings, previous studies have shown that these measures enable (a) localization of the primary epileptogenic area in different cerebral regions during the interictal state, (b) investigations of antiepileptic drug effects, (c) analyses of spatio-temporal interactions between the epileptogenic zone and other brain areas, and (d) detection of features predictive of imminent seizure activity. Nonlinear time series analysis provides new and supplementary information about the epileptogenic process and thus contributes to an improvement in presurgical evaluation.

Nonlinear interdependence measures can be used to detect directional couplings between stationary dynamical systems from a pair of signals measured from them. For many dynamics, however, intermittent directional couplings arise in causal relation to distinct events on timescales that are often too short to be resolved by nonlinear interdependence measures. On the other hand, in many experimental settings signals are measured for multiple instances of such events. We demonstrate how these multiple realizations can be exploited to reliably detect event-related time-dependent directional couplings. For this purpose, we propose the general concept of time-resolved causal statistics derived from embeddings across multiple realizations of time-dependent dynamics. Surrogates constructed by permuting the order of realizations can be used to test specified null hypotheses. We adapt a conventional nonlinear interdependence measure to serve as a time-resolved causal statistic and apply it to exemplary coupled Lorenz dynamics. This approach allows detecting event-related time-dependent directional couplings based on only a few tens of realizations. Changes of the coupling direction can be detected within one oscillation of the dynamics. Beyond this particular application, any metric bivariate or univariate measure can be adapted to serve as time-resolved causal statistics to characterize various aspects of event-related time-dependent dynamics.

The understanding of interacting dynamics is important for the characterization of real-world/nnetworks. In general real-world networks are heterogeneous in the sense that each node of the/nnetwork is a dynamics with di erent properties. For coupled non-identical dynamics symmetric/ninteractions are not straightforwardly de ned from the coupling strength values. Thus, a challenging/nissue is whether we can de ne a symmetric interaction in this asymmetric setting. To address this/nproblem we introduce the notion of the coupling impact. The coupling impact considers not only/nthe coupling strength but also the energy of the individual dynamics which is conveyed via the/ncoupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs/nof coupled model dynamics using two di erent connectivity measures. We nd that the coupling/nimpact, but not the coupling strength, correctly detects a symmetric interaction between pairs of/ncoupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows to reveal/nthe real impact that one dynamics has on the other and hence to de ne symmetric interactions in/npairs of non-identical dynamics. They acknowledge funding from the Volkswagen foundation and support by the Spanish Ministry of Economy and/nCompetitiveness (Grant FIS2014-54177-R). This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642563 (R.G.A.) and/nCERCA Programme/Generalitat de Catalunya (R.G.A).

The idea of inferring causal interactions from a variable X to another variable Y from the reduction of the prediction error of Y when including the past of X was formulated by Wiener [1] and formalized by Granger [2]. While Granger provided a measure to study causal interactions for Gaussian linear processes, the underlying concept (Granger causality) can be used to derive a general criterion to test for causality [3]. In particular, in the framework of information theory, transfer entropy [4] extends the measure proposed by Granger so that it is applicable to stationary and non-stationary finite-order Markov processes [5]. We here show that the concept of Granger causality can be used to reliably determine between which nodes in a network causal interactions exist, but in contrast to commonly held beliefs, it is not adequate to quantify the strength of these causal interactions. Instead we propose a two-step procedure to infer and quantify causality. First, a Granger causality based measure, like transfer entropy, is applied to build a directed graph indicating causal interactions between the nodes. Second, for nodes fulfilling some conditions we describe, an alternative measure is used to quantify the strength of the causal interactions. This procedure is exemplified with bivariate stochastic processes for which the information theoretic measures can be calculated analytically. This avoids dealing with the problem of the identification of the underlying processes from the recorded time series, and the problem of estimating the probability distributions which further difficult the analysis of causal interactions from experimental data. These simple examples allow us to illustrate some drawbacks of using the measures based on the concept of Granger causality in the way they are commonly applied for the study of neural causal interactions [e. g. 6-8].

We study two-layer networks of identical phase oscillators. Each individual layer is a ring network for which a non-local intra-layer coupling leads to the formation of a chimera state. The number of oscillators and their natural frequencies is in general different across the layers. We couple the phases of individual oscillators in one layer to the phase of the mean field of the other layer. This coupling from the mean field to individual oscillators is done in both directions. For a sufficient strength of this interlayer coupling, the phases of the mean fields lock across the two layers. In contrast, both layers continue to exhibit chimera states with no locking between the phases of individual oscillators across layers, and the two mean field amplitudes remain uncorrelated. Hence, the networks’ mean fields show phase synchronization which is analogous to the one between low-dimensional chaotic oscillators. The required coupling strength to achieve this mean field phase synchronization increases with the mismatches in the network sizes and the oscillators’ natural frequencies. We acknowledge funding from the Spanish Ministry of Economy and Competitiveness, Grant FIS2014-54177-R (R.G.A and G.R.), the CERCA Programme of the Generalitat de Catalunya (R.G.A), the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642563 (R.G.A and I.M.) and by the DFG in the framework of SFB 910 (E.S., A.Z. and G.R.).

Intuitively, music has both predictable and unpredictable components. In this work we assess this qualitative statement in a quantitative way using common time series models fitted to state-of-the-art music descriptors. These descriptors cover different musical facets and are extracted from a large collection of real audio recordings comprising a variety of musical genres. Our findings show that music descriptor time series exhibit a certain predictability not only for short time intervals, but also for mid-term and relatively long intervals. This fact is observed independently of the descriptor, musical facet and time series model we consider. Moreover, we show that our findings are not only of theoretical relevance /nbut can also have practical impact. To this end we demonstrate that music predictability at relatively long time intervals can be exploited in a real-world application, namely the automatic identification of cover songs (i.e. different renditions or versions of the same musical piece). Importantly, this prediction strategy yields a parameter-free approach for cover song identification that is substantially faster, allows for reduced computational storage and still maintains highly competitive accuracies when compared to state-of-the-art systems. This work has been partially funded by the Deutscher Akademischer Austausch Dienst (A/09/96235), by the projects Classical Planet (MITYC: TSI-070100- 2009-407) and DRIMS (MICINN: TIN2009-14247-C02-01), by the Spanish Ministry of Education /nand Science (BFU2007-61710) and by the Max Planck Institute for the Physics of Complex Systems.

The concept of surrogates allows testing results from time series analysis against specified null hypotheses. In application to bivariate model dynamics we here compare different types of surrogates, each designed to test against a different null hypothesis, e.g., an underlying bivariate linear stochastic process. Two measures that aim at a characterization of interdependence between nonlinear deterministic dynamics were used as discriminating statistics. We analyze eight different stochastic and deterministic models not only to demonstrate the power of the surrogates, but also to reveal some pitfalls and limitations. T.K. and F.M. acknowledge support from the Deutsche Forschungsgemeinschaft.

The question whether information extracted from the electroencephalogram ~EEG! of epilepsy patients can be used for the prediction of seizures has recently attracted much attention. Several studies have reported evidence for the existence of a preseizure state that can be detected using different measures derived from the theory of dynamical systems. Most of these studies, however, have neglected to sufficiently investigate the specificity of the observed effects or suffer from other methodological shortcomings. In this paper we present an automated technique for the detection of a preseizure state from EEG recordings using two different measures for synchronization between recording sites, namely, the mean phase coherence as a measure for phase synchronization and the maximum linear cross correlation as a measure for lag synchronization. Based on the observation of characteristic drops in synchronization prior to seizure onset, we used this phenomenon for the characterization of a preseizure state and its distinction from the remaining seizure-free interval. After optimizing our technique on a group of 10 patients with temporal lobe epilepsy we obtained a successful detection of a preseizure state prior to 12 out of 14 analyzed seizures for both measures at a very high specificity as tested on recordings from the seizure-free interval. After checking for in-sample overtraining via cross validation, we applied a surrogate test to validate the observed predictability. Based on our results, we discuss the differences of the two synchronization measures in terms of the dynamics underlying seizure generation in focal epilepsies. This work was supported by the Deutsche Forschungsgemeinschaft.