Characterizing direction of coupling from experimental observations

Phase-dynamic causalities within dynamical effects framework

This work investigates numerics of several widely known phase-dynamic quantifiers of directional (causal) couplings between oscillatory systems: transfer entropy (TE), differential quantifier, and squared-coefficients quantifier based on an evolution map. The study is performed on the system of two stochastic Kuramoto oscillators within the framework of dynamical causal effects. The quantifiers are related to each other and to an asymptotic effect of the coupling on phase diffusion. Several novel findings are listed as follows: (i) for a non-synchronous regime and high enough noise levels, the TE rate multiplied by a certain characteristic time (called here reduced TE) equals twice an asymptotic effect of a directional coupling on phase diffusion; (ii) "information flow" expressed by the TE rate unboundedly rises with the coupling coefficient even in the domain of effective synchronization; (iii) in any effective synchronization regime, the reduced TE is equal to 1/8 n.u. in each direction for equal coupling coefficients and equal noise intensities, and it is in general a simple function of the ratio of noise intensities and the ratio of coupling coefficients.

Coupling strength versus coupling impact in nonidentical bidirectionally coupled dynamics

The understanding of interacting dynamics is important for the characterization of real-world networks. In general, real-world networks are heterogeneous in the sense that each node of the network is a dynamics with different properties. For coupled nonidentical dynamics symmetric interactions are not straightforwardly defined from the coupling strength values. Thus, a challenging issue is whether we can define a symmetric interaction in this asymmetric setting. To address this problem we introduce the notion of the coupling impact. The coupling impact considers not only the coupling strength but also the energy of the individual dynamics, which is conveyed via the coupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs of coupled model dynamics using two different connectivity measures. We find that the coupling impact, but not the coupling strength, correctly detects a symmetric interaction between pairs of coupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows us to reveal the real impact that one dynamics has on the other and hence to define symmetric interactions in pairs of nonidentical dynamics.

Quantification of causal couplings via dynamical effects: a unifying perspective

Quantitative characterization of causal couplings from time series is crucial in studies of complex systems of different origin. Various statistical tools for that exist and new ones are still being developed with a tendency to creating a single, universal, model-free quantifier of coupling strength. However, a clear and generally applicable way of interpreting such universal characteristics is lacking. This work suggests a general conceptual framework for causal coupling quantification, which is based on state space models and extends the concepts of virtual interventions and dynamical causal effects. Namely, two basic kinds of interventions (state space and parametric) and effects (orbital or transient and stationary or limit) are introduced, giving four families of coupling characteristics. The framework provides a unifying view of apparently different well-established measures and allows us to introduce new characteristics, always with a definite "intervention-effect" interpretation. It is shown that diverse characteristics cannot be reduced to any single coupling strength quantifier and their interpretation is inevitably model based. The proposed set of dynamical causal effect measures quantifies different aspects of "how the coupling manifests itself in the dynamics," reformulating the very question about the "causal coupling strength."

Neural mechanisms of object-based attention

How we attend to objects and their features that cannot be separated by location is not understood. We presented two temporally and spatially overlapping streams of objects, faces versus houses, and used magnetoencephalography and functional magnetic resonance imaging to separate neuronal responses to attended and unattended objects. Attention to faces versus houses enhanced the sensory responses in the fusiform face area (FFA) and parahippocampal place area (PPA), respectively. The increases in sensory responses were accompanied by induced gamma synchrony between the inferior frontal junction, IFJ, and either FFA or PPA, depending on which object was attended. The IFJ appeared to be the driver of the synchrony, as gamma phases were advanced by 20 ms in IFJ compared to FFA or PPA. Thus, the IFJ may direct the flow of visual processing during object-based attention, at least in part through coupled oscillations with specialized areas such as FFA and PPA.

Detecting effective connectivity in networks of coupled neuronal oscillators

The application of data-driven time series analysis techniques such as Granger causality, partial directed coherence and phase dynamics modeling to estimate effective connectivity in brain networks has recently gained significant prominence in the neuroscience community. While these techniques have been useful in determining causal interactions among different regions of brain networks, a thorough analysis of the comparative accuracy and robustness of these methods in identifying patterns of effective connectivity among brain networks is still lacking. In this paper, we systematically address this issue within the context of simple networks of coupled spiking neurons. Specifically, we develop a method to assess the ability of various effective connectivity measures to accurately determine the true effective connectivity of a given neuronal network. Our method is based on decision tree classifiers which are trained using several time series features that can be observed solely from experimentally recorded data. We show that the classifiers constructed in this work provide a general framework for determining whether a particular effective connectivity measure is likely to produce incorrect results when applied to a dataset.

Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators

Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time.

Reconstructing phase dynamics of oscillator networks

We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling.

Reducing the bias of causality measures

Measures of the direction and strength of the interdependence between two time series are evaluated and modified to reduce the bias in the estimation of the measures, so that they give zero values when there is no causal effect. For this, point shuffling is employed as used in the frame of surrogate data. This correction is not specific to a particular measure and it is implemented here on measures based on state space reconstruction and information measures. The performance of the causality measures and their modifications is evaluated on simulated uncoupled and coupled dynamical systems and for different settings of embedding dimension, time series length, and noise level. The corrected measures, and particularly the suggested corrected transfer entropy, turn out to stabilize at the zero level in the absence of a causal effect and detect correctly the direction of information flow when it is present. The measures are also evaluated on electroencephalograms (EEG) for the detection of the information flow in the brain of an epileptic patient. The performance of the measures on EEG is interpreted in view of the results from the simulation study.

Order parameter expansion and finite-size scaling study of coherent dynamics induced by quenched noise in the active rotator model

We use a recently developed order parameter expansion method to study the transition to synchronous firing occurring in a system of coupled active rotators under the exclusive presence of quenched noise. The method predicts correctly the existence of a transition from a rest state to a regime of synchronous firing and another transition out of it as the intensity of the quenched noise increases and leads to analytical expressions for the critical noise intensities in the large coupling regime. It also predicts the order of the transitions for different probability distribution functions of the quenched variables. Using numerical simulations and finite-size scaling theory to estimate the critical exponents of the transitions, we found values which are consistent with those reported in other scalar systems in the exclusive presence of additive static disorder.

Detecting couplings between interacting oscillators with time-varying basic frequencies: instantaneous wavelet bispectrum and information theoretic approach

In the natural world, the properties of interacting oscillatory systems are not constant, but evolve or fluctuating continuously in time. Thus, the basic frequencies of the interacting oscillators are time varying, which makes the system analysis complex. For studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory. We show how these methods uncover the interacting properties and reveal the nature, strength, and direction of coupling. Wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver-response relationships in complex systems. We generate bivariate time-series numerically to mimic typical situations that occur in real measured data, apply both methods to the same time-series and discuss the results. The approach is applicable quite generally to any system of coupled nonlinear oscillators.

From Granger causality to long-term causality: application to climatic data

Quantitative characterization of interaction between processes from time series is often required in different fields of natural science including geophysics and biophysics. Typically, one estimates "short-term" influences, e.g., the widely used Granger causality is defined via one-step-ahead predictions. Such an approach does not reveal how strongly the "long-term" behavior of one process under study is affected by the others. To overcome this problem, we introduce the concept of long-term causality, which extends the concept of Granger causality. The long-term causality is estimated from data via empirical modeling and analysis of model dynamics under different conditions. Apart from mathematical examples, we apply both approaches to find out how strongly the global surface temperature (GST) is affected by variations in carbon dioxide atmospheric content, solar activity, and volcanic activity during the last 150 years. Influences of all the three factors on GST are detected with the Granger causality. However, the long-term causality shows that the rise in GST during the last decades can be explained only if the anthropogenic factor (CO2) is taken into account in a model.

Phase dynamics of coupled oscillators reconstructed from data

We systematically develop a technique for reconstructing the phase dynamics equations for coupled oscillators from data. For autonomous oscillators and for two interacting oscillators we demonstrate how phase estimates obtained from general scalar observables can be transformed to genuine phases. This allows us to obtain an invariant description of the phase dynamics in terms of the genuine, observable-independent phases. We discuss the importance of this transformation for characterization of strength and directionality of interaction from bivariate data. Moreover, we demonstrate that natural (autonomous) frequencies of oscillators can be recovered if several observations of coupled systems at different, yet unknown coupling strengths are available. We illustrate our method by several numerical examples and apply it to a human electrocardiogram and to a physical experiment with coupled metronomes.

Detecting directional coupling in the human epileptic brain: limitations and potential pitfalls

We study directional relationships-in the driver-responder sense-in networks of coupled nonlinear oscillators using a phase modeling approach. Specifically, we focus on the identification of drivers in clusters with varying levels of synchrony, mimicking dynamical interactions between the seizure generating region (epileptic focus) and other brain structures. We demonstrate numerically that such an identification is not always possible in a reliable manner. Using the same analysis techniques as in model systems, we study multichannel electroencephalographic recordings from two patients suffering from focal epilepsy. Our findings demonstrate that--depending on the degree of intracluster synchrony--certain subsystems can spuriously appear to be driving others, which should be taken into account when analyzing field data with unknown underlying dynamics.

Detecting causality between different frequencies

Biological systems are usually non-linear and, as a result, the driving signal frequency (say, MHz) is in general not identical with the output frequency (say, N Hz). Coherence and causality analysis have been well-developed to measure the (directional) correlation between input and output signals with identical frequencies (N=M), but they are not applicable to the cases with different frequencies (N not equal M). In this paper, we propose a novel method called frequency-modified causality (coherence) analysis to resolve the issue. The input or output signal is first modulated by up-sampling or down-sampling, coherence and causality analysis are then applied to the frequency modulated and filtered signals. An optimal coherence and causality is found, revealing the true input-output relationship between signals. The method is successfully tested on data generated from a toy model, the van der Pol oscillator and then employed to analyze data recorded from Parkinson's disease (PD) patients.

Measuring direction in the coupling of biological oscillators: a case study for electroreceptors of paddlefish

Recently developed methods for estimating directionality in the coupling between oscillators were tested on experimental time series data from electroreceptors of paddlefish, because each electroreceptor contains two distinct types of noisy oscillators. One type of oscillator is in the sensory epithelia, and another type is in the terminals of afferent neurons. Based on morphological organization and our previous work, we expected unidirectional coupling, whereby epithelial oscillations synaptically influence the spiking oscillators of afferent neurons. Using directionality analysis we confirmed unidirectional coupling of oscillators embedded in electroreceptors. We studied the performance of directionality algorithms for decreasing length of data. Also, we experimentally varied the strength of oscillator coupling, to test the effect of coupling strength on directionality algorithms.

Estimation of coupling between oscillators from short time series via phase dynamics modeling: limitations and application to EEG data

We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D. A. Smirnov and B. P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and nonvanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization.

Detection of weak directional coupling: phase-dynamics approach versus state-space approach

We compare two conceptually different approaches to the detection of weak directional couplings between two oscillatory systems from bivariate time series. The first approach is based on the analysis of the systems' phase dynamics, whereas the other one tests for interdependencies in the reconstructed state spaces of the systems. We analyze the sensitivity of both techniques to weak couplings in numerical experiments by considering couplings between almost identical as well as between significantly different nonlinear systems. We study different degrees of phase diffusion, test the robustness of the two techniques against observational noise, and investigate the influence of the time series length. Our results show that none of the two approaches is generally superior to the other, and we conclude that it is probably the combination of both techniques that would allow the most comprehensive and reliable characterization of coupled systems.

Estimation of delay in coupling from time series

We demonstrate that a time delay in weak coupling between two self-sustained oscillators can be estimated from the observed time series data. We present two methods which are based on the analysis of interrelations between the phases of the signals. We show analytically and numerically that irregularity of the phase dynamics (due to the intrinsic noise or chaos) is essential for determination of the delay. We compare and contrast both methods to the standard cross-correlation analysis.

Directionality of coupling of physiological subsystems: age-related changes of cardiorespiratory interaction during different sleep stages in babies

Activity of many physiological subsystems has a well-expressed rhythmic character. Often, a dependency between physiological rhythms is established due to interaction between the corresponding subsystems. Traditional methods of data analysis allow one to quantify the strength of interaction but not the causal interrelation that is indispensable for understanding the mechanisms of interaction. Here we present a recently developed method for quantification of coupling direction and apply it to an important problem. Namely, we study the mutual influence of respiratory and cardiovascular rhythms in healthy newborns within the first 6 mo of life in quiet and active sleep. We find an age-related change of the coupling direction: the interaction is nearly symmetric during the first days and becomes practically unidirectional (from respiration to heart rhythm) at the age of 6 mo. Next, we show that the direction of interaction is mainly determined by respiratory frequency. If the latter is less than approximately 0.6 Hz, the interaction occurs dominantly from respiration to heart. With higher respiratory frequencies that only occur at very young ages, the dominating direction is less pronounced or even abolished. The observed dependencies are not related to sleep stage, suggesting that the coupling direction is determined by system-inherent dynamical processes, rather than by functional modulations. The directional analysis may be applied to other interacting narrow band oscillatory systems, e.g., in the central nervous system. Thus it is an important step forward in revealing and understanding causal mechanisms of interactions.

Estimation of interaction strength and direction from short and noisy time series

A technique for determination of character and intensity of interaction between the elements of complex systems based on reconstruction of model equations for phase dynamics is extended to the case of short and noisy time series. Corrections, which eliminate systematic errors of the estimates, and expressions for confidence intervals are derived. Analytic results are presented for a particular case of linear uncoupled systems, and their validity for a much wider range of situations is demonstrated with numerical examples. The technique should be useful for the analysis of nonstationary processes in real time, including the situations of significant noise and restrictions on the observation time.