Revealing direction of coupling between neuronal oscillators from time series: phase dynamics modeling versus partial directed coherence

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

Contributions to surface air temperature trends estimated from climate time series: Medium-term causalities

Contributions of various natural and anthropogenic factors to trends of surface air temperatures at different latitudes of the Northern and Southern hemispheres on various temporal horizons are estimated from climate data since the 19th century in empirical autoregressive models. Along with anthropogenic forcing, we assess the impact of several natural climate modes including Atlantic Multidecadal Oscillation, El-Nino/Southern Oscillation, Interdecadal Pacific Oscillation, Pacific Decadal Oscillation, and Antarctic Oscillation. On relatively short intervals of the length of two or three decades, contributions of climate variability modes are considerable and comparable to the contributions of greenhouse gases and even exceed the latter. On longer intervals of about half a century and greater, the contributions of greenhouse gases dominate at all latitudinal belts including polar, middle, and tropical ones.

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.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Subacute melamine exposure disrupts task-based hippocampal information flow via inhibiting the subunits 2 and 3 of AMPA glutamate receptors expression

Although melamine exposure induces cognitive deficits and dysfunctional neurotransmission in hippocampal Cornus Ammonis (CA) 1 region of rats, it is unclear whether the neural function, such as neural oscillations between hippocampal CA3-CA1 pathway and postsynaptic receptors involves in these effects. The levels of alpha-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptor (AMPAR) subunit glutamate receptor (GluR) 1 and GluR2/3 in CA1 region of melamine-treated rats, which were intragastric treated with 300 mg/kg/day for 4 weeks, were detected. Following systemic or intra-hippocampal CA1 injection with GluR2/3 agonist, spatial learning of melamine-treated rats was assessed in Morris water maze (MWM) task. Local field potentials were recorded in CA3-CA1 pathway before and during behavioral test. General Partial Directed Coherence approach was applied to determine directionality of neural information flow between CA3 and CA1 regions. Results showed that melamine exposure reduced GluR2/3 but not GluR1 level and systemic or intra-hippocampal CA1 injection with GluR2/3 agonist effectively mitigated the learning deficits. Phase synchronization between CA3 and CA1 regions were significantly diminished in delta, theta and alpha oscillations. Coupling directional index and strength of CA3 driving CA1 were marked reduced as well. Intra-hippocampal CA1 infusion with GluR2/3 agonist significantly enhanced the phase locked value and reversed the melamine-induced reduction in the neural information flow (NIF) from CA3 to CA1 region. These findings support that melamine exposure decrease the expression of GluR2/3 subunit involved in weakening directionality index of NIF, and thereby induced spatial learning deficits.

Transient and equilibrium causal effects in coupled oscillators

Two quite different types of causal effects are given by (i) changes in near future states of a driven system under changes in a current state of a driving system and (ii) changes in statistical characteristics of a driven system dynamics under changes in coupling parameters, e.g., under switching the coupling off. The former can be called transient causal effects and can be estimated from a time series within the well established framework of the Wiener-Granger causality, while the latter represent equilibrium (or stationary) causal effects which are often most interesting but generally inaccessible to estimation from an observed time series recorded at fixed coupling parameters. In this work, relationships between the two kinds of causal effects are found for unidirectionally coupled stochastic linear oscillators depending on their frequencies and damping factors. Approximate closed-form expressions for these relationships are derived. Their limitations and possible extensions are discussed, and their practical applicability to extracting equilibrium causal effects from time series is argued.

Reconstructing links in directed networks from noisy dynamics

We address the long-standing challenge of how to reconstruct links in directed networks from measurements, and present a general method that makes use of a noise-induced relation between network structure and both the time-lagged covariance of measurements taken at two different times and the covariance of measurements taken at the same time. When the coupling functions have certain additional properties, we can further reconstruct the weights of the links.

Reconstruction of a neural network from a time series of firing rates

Randomly coupled neural fields demonstrate irregular variation of firing rates, if the coupling is strong enough, as has been shown by Sompolinsky et al. [Phys. Rev. Lett. 61, 259 (1988)]PRLTAO0031-900710.1103/PhysRevLett.61.259. We present a method for reconstruction of the coupling matrix from a time series of irregular firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set and a small measurement noise, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function.

Assessing directed interactions from neurophysiological signals--an overview

The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.

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.

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.

Inferring directional interactions from transient signals with symbolic transfer entropy

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.

Synchronized Current Oscillations of Formic Acid Electro-oxidation in a Microchip-based Dual-Electrode Flow Cell

We investigate the oscillatory electro-oxidation of formic acid on platinum in a microchip-based dual-electrode cell with microfluidic flow control. The main dynamical features of current oscillations on single Pt electrode that had been observed in macro-cells are reproduced in the microfabricated electrochemical cell. In dual-electrode configuration nearly in-phase synchronized current oscillations occur when the reference/counter electrodes are placed far away from the microelectrodes. The synchronization disappears with close reference/counter electrode placements. We show that the cause for synchronization is weak albeit important, bidirectional electrical coupling between the electrodes; therefore the unidirectional mass transfer interactions are negligible. The experimental design enables the investigation of the dynamical behavior in micro-electrode arrays with well-defined control of flow of the electrolyte in a manner where the size and spacing of the electrodes can be easily varied.

Inferring direct directed-information flow from multivariate nonlinear time series

Estimating the functional topology of a network from multivariate observations is an important task in nonlinear dynamics. We introduce the nonparametric partial directed coherence that allows disentanglement of direct and indirect connections and their directions. We illustrate the performance of the nonparametric partial directed coherence by means of a simulation with data from synchronized nonlinear oscillators and apply it to real-world data from a patient suffering from essential tremor.

Sleep-dependent directional coupling between human neocortex and hippocampus

Complex interactions between neocortex and hippocampus are the neural basis of memory formation. Two-step theories of memory formation suggest that initial encoding of novel information depends on the induction of rapid plasticity within the hippocampus, and is followed by a second sleep-dependent step of memory consolidation. These theories predict information flow from the neocortex into the hippocampus during waking state and in the reverse direction during sleep. However, experimental evidence that interactions between hippocampus and neocortex have a predominant direction which reverses during sleep rely on cross-correlation analysis of data from animal experiments and yielded inconsistent results. Here, we investigated directional coupling in intracranial EEG data from human subjects using a phase-modeling approach which is well suited to reveal functional interdependencies in oscillatory data. In general, we observed that the anterior hippocampus predominantly drives nearby and remote brain regions. Surprisingly, however, the influence of neocortical regions on the hippocampus significantly increased during sleep as compared to waking state. These results question the standard model of hippocampal-neocortical interactions and suggest that sleep-dependent consolidation is accomplished by an active retrieval of hippocampal information by the neocortex.

Detection of couplings in ensembles of stochastic oscillators

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Interactions between medial temporal lobe, prefrontal cortex, and inferior temporal regions during visual working memory: a combined intracranial EEG and functional magnetic resonance imaging study

It is a fundamental question whether the medial temporal lobe (MTL) supports only long-term memory encoding, or contributes to working memory (WM) processes as well. Recent data suggest that the MTL is activated whenever multiple items or item features are being maintained in WM. This may rely on interactions between the MTL or the prefrontal cortex (PFC) and content-specific areas in the inferior temporal (IT) cortex. Here, we investigated the neural mechanism through which the MTL, PFC, and IT cortex interact during WM maintenance. First, we quantified phase synchronization of intracranial EEG data in epilepsy patients with electrodes in both regions. Second, we used directional coupling analysis to study whether oscillatory activity in the IT cortex drives the MTL or vice versa. Finally, we investigated functional connectivity in functional magnetic resonance imaging data of healthy subjects with seeds in the MTL and PFC. With increasing load, EEG phase synchronization between the IT cortex and anterior parahippocampal gyrus and within the MTL increased. Coupling was bidirectional in all load conditions, but changed toward an increased top-down (anterior parahippocampal gyrus --> IT) coupling in the high gamma range (51-75 Hz) with increasing load. Functional connectivity between the MTL seed and the visual association cortex increased with load, but activity within the MTL and the PFC correlated with fewer voxels, suggesting that more specific neural networks were engaged. These data indicate that WM for multiple items depends on an increased strength of top-down control of activity within the IT cortex by the MTL.

Measuring interdependences in dissipative dynamical systems with estimated Fokker-Planck coefficients

We propose a data-driven approach to measure interdependences between dissipative dynamical systems under the influence of noise. We estimate drift and diffusion coefficients of a Fokker-Planck equation and derive measures that allow one to quantify the asymmetry in coupling in a fully automated and computationally inexpensive and simple way. Our approach makes it possible to discriminate between interdependences in the deterministic and stochastic parts of the dynamics. We report results of numerical studies of exemplary time series from coupled stochastic and deterministic model systems and of an application to electroencephalographic recordings from epilepsy patients.

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.