A copula approach to assessing Granger causality

Causalized convergent cross-mapping and its approximate equivalence with directed information in causality analysis

Convergent cross-mapping (CCM) has attracted increased attention recently due to its capability to detect causality in nonseparable systems under deterministic settings, which may not be covered by the traditional Granger causality. From an information-theoretic perspective, causality is often characterized as the directed information (DI) flowing from one side to the other. As information is essentially nondeterministic, a natural question is: does CCM measure DI flow? Here, we first causalize CCM so that it aligns with the presumption in causality analysis-the future values of one process cannot influence the past of the other, and then establish and validate the approximate equivalence of causalized CCM (cCCM) and DI under Gaussian variables through both theoretical derivations and fMRI-based brain network causality analysis. Our simulation result indicates that, in general, cCCM tends to be more robust than DI in causality detection. The underlying argument is that DI relies heavily on probability estimation, which is sensitive to data size as well as digitization procedures; cCCM, on the other hand, gets around this problem through geometric cross-mapping between the manifolds involved. Overall, our analysis demonstrates that cross-mapping provides an alternative way to evaluate DI and is potentially an effective technique for identifying both linear and nonlinear causal coupling in brain neural networks and other settings, either random or deterministic, or both.

Parallel factor analysis for multidimensional decomposition of functional near-infrared spectroscopy data

SIGNIFICANCE

Current techniques for data analysis in functional near-infrared spectroscopy (fNIRS), such as artifact correction, do not allow to integrate the information originating from both wavelengths, considering only temporal and spatial dimensions of the signal's structure. Parallel factor analysis (PARAFAC) has previously been validated as a multidimensional decomposition technique in other neuroimaging fields.

AIM

We aimed to introduce and validate the use of PARAFAC for the analysis of fNIRS data, which is inherently multidimensional (time, space, and wavelength).

APPROACH

We used data acquired in 17 healthy adults during a verbal fluency task to compare the efficacy of PARAFAC for motion artifact correction to traditional two-dimensional decomposition techniques, i.e., target principal (tPCA) and independent component analysis (ICA). Correction performance was further evaluated under controlled conditions with simulated artifacts and hemodynamic response functions.

RESULTS

PARAFAC achieved significantly higher improvement in data quality as compared to tPCA and ICA. Correction in several simulated signals further validated its use and promoted it as a robust method independent of the artifact's characteristics.

CONCLUSIONS

This study describes the first implementation of PARAFAC in fNIRS and provides validation for its use to correct artifacts. PARAFAC is a promising data-driven alternative for multidimensional data analyses in fNIRS and this study paves the way for further applications.

Intermuscular coupling network analysis of upper limbs based on R-vine copula transfer entropy

In the field of neuroscience, it is very important to evaluate the causal coupling characteristics between bioelectrical signals accurately and effectively. Transfer entropy is commonly used to analyze complex data, especially the causal relationship between data with non-linear, multidimensional characteristics. However, traditional transfer entropy needs to estimate the probability density function of the variable, which is computationally complex and unstable. In this paper, a new and effective method for entropy transfer is proposed, by means of applying R-vine copula function estimation. The effectiveness of R-vine copula transfer entropy is first verified on several simulations, and then applied to intermuscular coupling analysis to explore the characteristics of the intermuscular coupling network of muscles in non-fatigue and fatigue conditions. The experiment results show that as the muscle group enters the fatigue state, the community structure can be adjusted and the muscle nodes participating in the exercise are fully activated, enabling the two-way interaction between different communities. Finally, it comes to the conclusion that the proposed method can make accurate inferences about complex causal coupling. Moreover, the characteristics of the intermuscular coupling network in both non-fatigue and fatigue states can provide a new theoretical perspective for the diagnosis of neuromuscular fatigue and sports rehabilitation, which has good application value.

Sparse representations of high dimensional neural data

Conventional Vector Autoregressive (VAR) modelling methods applied to high dimensional neural time series data result in noisy solutions that are dense or have a large number of spurious coefficients. This reduces the speed and accuracy of auxiliary computations downstream and inflates the time required to compute functional connectivity networks by a factor that is at least inversely proportional to the true network density. As these noisy solutions have distorted coefficients, thresholding them as per some criterion, statistical or otherwise, does not alleviate the problem. Thus obtaining a sparse representation of such data is important since it provides an efficient representation of the data and facilitates its further analysis. We propose a fast Sparse Vector Autoregressive Greedy Search (SVARGS) method that works well for high dimensional data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments, our methods show high accuracy in recovering the true sparse model. The relative absence of spurious coefficients permits accurate, stable and fast evaluation of derived quantities such as power spectrum, coherence and Granger causality. Consequently, sparse functional connectivity networks can be computed, in a reasonable time, from data comprising tens of thousands of channels/voxels. This enables a much higher resolution analysis of functional connectivity patterns and community structures in such large networks than is possible using existing time series methods. We apply our method to EEG data where computed network measures and community structures are used to distinguish emotional states as well as to ADHD fMRI data where it is used to distinguish children with ADHD from typically developing children.

Piecewise Causality Study between Power Load and Vibration in Hydro-Turbine Generator Unit for a Low-Carbon Era

With the rapid development of wind and photovoltaic power generation, hydro-turbine generator units have to operate in a challenging way, resulting in obvious vibration problems. Because of the significant impact of vibration on safety and economical operation, it is of great significance to study the causal relationship between vibration and other variables. The complexity of the hydro-turbine generator unit makes it difficult to analyze the causality of the mechanism. This paper studied the correlation based on a data-driven method, then transformed the correlation into causality based on the mechanism. In terms of correlation, traditional research only judges whether there is a correlation between all data. When the data with correlation are interfered with by the data without correlation, the traditional methods cannot accurately identify the correlation. A piecewise correlation method based on change point detection was proposed to fill this research gap. The proposed method segmented time series pairs, then analyzed the correlation between subsequences. The causality between power load and vibration of a hydro-turbine generator unit was further analyzed. It indicated that when the power load is less than 200 MW, the causality is weak, and when the power load is greater than 375 MW, the causality is strong. The results show that the causality between vibration and power load is not fixed but piecewise. Furthermore, the piecewise correlation method compensated for the limitation of high variance of the maximum information coefficient.

Analysis of corticomuscular connectivity during walking using vine copula

Corticomuscular connectivity plays an important role in the neural control of human motion. This study recorded electroencephalography (EEG) and surface electromyography (sEMG) signals from subjects performing specific tasks (walking on level ground and on stairs) based on metronome instructions. This study presents a novel method based on vine copula to jointly model EEG and sEMG signals. The advantage of vine copula is its applicability in the construction of dependency structures to describe the connectivity between the cortex and muscles during different movements. A corticomuscular function network was also constructed by analyzing the dependence of each channel sample. The successfully constructed network shows information transmission between different divisions of the cortex, between muscles, and between the cortex and muscles when the body performs lower limb movements. Additionally, it highlights the potential of the vine copula concept used in this study, indicating that significant changes in the corticomuscular network under lower limb movements can be quantified by effective connectivity values.

Characteristics of the neural coding of causality

While causality processing is an essential cognitive capacity of the neural system, a systematic understanding of the neural coding of causality is still elusive. We propose a physically fundamental analysis of this issue and demonstrate that the neural dynamics encodes the original causality between external events near homomorphically. The causality coding is memory robust for the amount of historical information and features high precision but low recall. This coding process creates a sparser representation for the external causality. Finally, we propose a statistic characterization for the neural coding mapping from the original causality to the coded causality in neural dynamics.

Network Analysis of Multivariate Transfer Entropy of Cryptocurrencies in Times of Turbulence

We investigate the effects of the recent financial turbulence of 2020 on the market of cryptocurrencies taking into account the hourly price and volume of transactions from December 2019 to April 2020. The data were subdivided into time frames and analyzed the directed network generated by the estimation of the multivariate transfer entropy. The approach followed here is based on a greedy algorithm and multiple hypothesis testing. Then, we explored the clustering coefficient and the degree distributions of nodes for each subperiod. It is found the clustering coefficient increases dramatically in March and coincides with the most severe fall of the recent worldwide stock markets crash. Further, the log-likelihood in all cases bent over a power law distribution, with a higher estimated power during the period of major financial contraction. Our results suggest the financial turbulence induce a higher flow of information on the cryptocurrency market in the sense of a higher clustering coefficient and complexity of the network. Hence, the complex properties of the multivariate transfer entropy network may provide early warning signals of increasing systematic risk in turbulence times of the cryptocurrency markets.

Time-Frequency-Domain Copula-Based Granger Causality and Application to Corticomuscular Coupling in Stroke

The corticomuscular coupling (CMC) characterization between the motor cortex and muscles during motion control is a valid biomarker of motor system function after stroke, which can improve clinical decision-making. However, traditional CMC analysis is mainly based on the coherence method that can’t determine the coupling direction, whereas Granger Causality (GC) is limited in identifying linear cause–effect relationship. In this paper, a time-frequency domain copula-based GC (copula-GC) method is proposed to assess CMC characteristic. The 32-channel electroencephalogram (EEG) signals over brain scalp and electromyography (EMG) signals from upper limb were recorded during controlling and maintaining steady-state force output for five stroke patients and five healthy controls. Then, the time-frequency copula-GC analysis was applied to evaluate the CMC strength in both directions. Experimental results show that the CMC strength of descending direction is greater than that of ascending direction in the time domain for healthy controls. With the increase of grip strength, the bi-directional CMC strength has an increasing trend. Meanwhile, the bi-directional CMC strength of right hand is larger than that of left hand. In addition, the bi-directional CMC strength of stroke patients is lower than that of healthy controls. In the frequency domain, the strongest CMC is observed at the beta frequency band. Additionally, the CMC strength of descending direction is slightly larger than that of ascending direction in healthy controls, while the CMC strength of descending direction is lower than that of ascending direction in stroke patients. We suggest that the proposed time-frequency domain analysis approach based on copula-GC can effectively detect complex functional coupling between cortical oscillations and muscle activities, and provide a potential quantitative analysis measure for motion control and rehabilitation evaluation.

A study of dependency features of spike trains through copulas

Despite the progresses of statistical and machine learning techniques, simultaneous recordings from many neurons hide important information and the connections characterizing the network remain generally undiscovered. Discerning the presence of direct links between neurons from data is still a not completely solved problem. We propose the use of copulas, to enlarge the number of tools for detecting the network structure, pursuing on a research direction we started in Sacerdote et al. (2012). Here, our aim is to distinguish different types of connections on a very simple network. Our proposal consists in choosing suitable random intervals in pairs of spike trains determining the shapes of their copulas. We show that this approach allows to detect different types of dependencies. We illustrate the features of the proposed method on synthetic data from suitably connected networks of two or three formal neurons directly connected or influenced by the surrounding network. We show how a smart choice of pairs of random times together with the use of empirical copulas allows to discern between direct and indirect interactions.

Spike-field Granger causality for hybrid neural data analysis

Neurotechnological innovations allow for simultaneous recording at various scales, ranging from spiking activity of individual neurons to large neural populations’ local field potentials (LFPs). This capability necessitates developing multiscale analysis of spike-field activity. A joint analysis of the hybrid neural data is crucial for bridging the scales between single neurons and local networks. Granger causality is a fundamental measure to evaluate directional influences among neural signals. However, it is mainly limited to inferring causal influence between the same type of signals—either LFPs or spike trains—and not well developed between two different signal types. Here we propose a model-free, nonparametric spike-field Granger causality measure for hybrid data analysis. Our measure is distinct from existing methods in that we use “binless” spikes (precise spike timing) rather than “binned” spikes (spike counts within small consecutive time windows). The latter clearly distort the information in the mixed analysis of spikes and LFP. Therefore, our spectral estimate of spike trains is directly applied to the neural point process itself, i.e., sequences of spike times rather than spike counts. Our measure is validated by an extensive set of simulated data. When the measure is applied to LFPs and spiking activity simultaneously recorded from visual areas V1 and V4 of monkeys performing a contour detection task, we are able to confirm computationally the long-standing experimental finding of the input-output relationship between LFPs and spikes. Importantly, we demonstrate that spike-field Granger causality can be used to reveal the modulatory effects that are inaccessible by traditional methods, such that spike→LFP Granger causality is modulated by the behavioral task, whereas LFP→spike Granger causality is mainly related to the average synaptic input.

NEW & NOTEWORTHY It is a pressing question to study the directional interactions between local field potential (LFP) and spiking activity. In this report, we propose a model-free, nonparametric spike-field Granger causality measure that can be used to reveal directional influences between spikes and LFPs. This new measure is crucial for bridging the scales between single neurons and neural networks; hence it represents an important step to explicate how the brain orchestrates information processing.

Causal Discovery with Attention-Based Convolutional Neural Networks

Having insight into the causal associations in a complex system facilitates decision making, e.g., for medical treatments, urban infrastructure improvements or financial investments. The amount of observational data grows, which enables the discovery of causal relationships between variables from observation of their behaviour in time. Existing methods for causal discovery from time series data do not yet exploit the representational power of deep learning. We therefore present the Temporal Causal Discovery Framework (TCDF), a deep learning framework that learns a causal graph structure by discovering causal relationships in observational time series data. TCDF uses attention-based convolutional neural networks combined with a causal validation step. By interpreting the internal parameters of the convolutional networks, TCDF can also discover the time delay between a cause and the occurrence of its effect. Our framework learns temporal causal graphs, which can include confounders and instantaneous effects. Experiments on financial and neuroscientific benchmarks show state-of-the-art performance of TCDF on discovering causal relationships in continuous time series data. Furthermore, we show that TCDF can circumstantially discover the presence of hidden confounders. Our broadly applicable framework can be used to gain novel insights into the causal dependencies in a complex system, which is important for reliable predictions, knowledge discovery and data-driven decision making.

Copula directional dependence for inference and statistical analysis of whole-brain connectivity from fMRI data

INTRODUCTION

Inferring connectivity between brain regions has been raising a lot of attention in recent decades. Copula directional dependence (CDD) is a statistical measure of directed connectivity, which does not require strict assumptions on probability distributions and linearity.

METHODS

In this work, CDDs between pairs of local brain areas were estimated based on the fMRI responses of human participants watching a Pixar animation movie. A directed connectivity map of fourteen predefined local areas was obtained for each participant, where the network structure was determined by the strengths of the CDDs. A resampling technique was further applied to determine the statistical significance of the connectivity directions in the networks. In order to demonstrate the effectiveness of the suggested method using CDDs, statistical group analysis was conducted based on graph theoretic measures of the inferred directed networks and CDD intensities. When the 129 fMRI participants were grouped by their age (3-5 year-old, 7-12 year-old, adult) and gender (F, M), nonparametric two-way analysis of variance (ANOVA) results could identify which cortical regions and connectivity structures correlated with the two physiological factors.

RESULTS

Especially, we could identify that (a) graph centrality measures of the frontal eye fields (FEF), the inferior temporal gyrus (ITG), and the temporopolar area (TP) were significantly affected by aging, (b) CDD intensities between FEF and the primary motor cortex (M1) and between ITG and TP were highly significantly affected by aging, and (c) CDDs between M1 and the anterior prefrontal cortex (aPFC) were highly significantly affected by gender.

SOFTWARE

The R source code for fMRI data preprocessing, estimation of directional dependences, network visualization, and statistical analyses are available at https://github.com/namgillee/CDDforFMRI.

Estimation of Vector Autoregressive Parameters and Granger Causality From Noisy Multichannel Data

OBJECTIVE

The objective of this paper is to estimate the parameters of a multivariate autoregressive process from noisy multichannel data.

METHODS

Using a multivariate generalization of the Cadzow method, we propose a new method for estimating autoregressive parameters from noisy data: the nonlinear Cadzow method.

RESULTS

We show that our method outperforms existing multivariate methods such as higher order Yule-Walker method and Kalman EM method on simulated data. We apply our method to estimation of Granger causality from noisy data and again obtain superior results compared to previous methods. Finally, when applied to experimental local field potential data from monkey somatosensory and motor cortical areas, our method produces results consistent with cortical physiology.

CONCLUSION

The proposed nonlinear Cadzow method outperforms existing methods in obtaining denoised estimates of multivariate autoregressive parameters.

SIGNIFICANCE

Since multichannel recordings have become commonplace in biomedical applications ranging from discovering functional connectivity in the brain to speech data analysis and these recordings are inevitably contaminated by measurement noise, we believe our method has the potential for significant impact.

A Copula-Based Granger Causality Measure for the Analysis of Neural Spike Train Data

In systems neuroscience, it is becoming increasingly common to record the activity of hundreds of neurons simultaneously via electrode arrays. The ability to accurately measure the causal interactions among multiple neurons in the brain is crucial to understanding how neurons work in concert to generate specific brain functions. The development of new statistical methods for assessing causal influence between spike trains is still an active field of neuroscience research. Here, we suggest a copula-based Granger causality measure for the analysis of neural spike train data. This method is built upon our recent work on copula Granger causality for the analysis of continuous-valued time series by extending it to point-process neural spike train data. The proposed method is therefore able to reveal nonlinear and high-order causality in the spike trains while retaining all the computational advantages such as model-free, efficient estimation, and variability assessment of Granger causality. The performance of our algorithm can be further boosted with time-reversed data. Our method performed well on extensive simulations, and was then demonstrated on neural activity simultaneously recorded from primary visual cortex of a monkey performing a contour detection task.

Enhanced Brain Network Activity in Complex Movement Rhythms: A Simultaneous Functional Magnetic Resonance Imaging and Electroencephalography Study

Generating movement rhythms is known to involve a network of distributed brain regions associated with motor planning, control, execution, and perception of timing for the repertoire of motor actions. What brain areas are bound in the network and how the network activity is modulated by rhythmic complexity have not been completely explored. To contribute to answering these questions, we designed a study in which nine healthy participants performed simple to complex rhythmic finger movement tasks while undergoing simultaneous functional magnetic resonance imaging and electroencephalography (fMRI-EEG) recordings of their brain activity during the tasks and rest. From fMRI blood oxygenation-level-dependent (BOLD) measurements, we found that the complexity of rhythms was associated with brain activations in the primary motor cortex (PMC), supplementary motor area (SMA), and cerebellum (Cb), and with network interactions from these cortical regions to the cerebellum. The spectral analysis of single-trial EEG source waveforms at the cortical regions further showed that there were bidirectional interactions between PMC and SMA, and the complexity of rhythms was associated with power spectra and Granger causality spectra in the beta (13-30 Hz) frequency band, not in the alpha (8-12 Hz) and gamma (30-58 Hz) bands. These results provide us new insights into the mechanisms for movement rhythm complexity.

A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula

We begin by reviewing the statistical framework of information theory as applicable to neuroimaging data analysis. A major factor hindering wider adoption of this framework in neuroimaging is the difficulty of estimating information theoretic quantities in practice. We present a novel estimation technique that combines the statistical theory of copulas with the closed form solution for the entropy of Gaussian variables. This results in a general, computationally efficient, flexible, and robust multivariate statistical framework that provides effect sizes on a common meaningful scale, allows for unified treatment of discrete, continuous, unidimensional and multidimensional variables, and enables direct comparisons of representations from behavioral and brain responses across any recording modality. We validate the use of this estimate as a statistical test within a neuroimaging context, considering both discrete stimulus classes and continuous stimulus features. We also present examples of analyses facilitated by these developments, including application of multivariate analyses to MEG planar magnetic field gradients, and pairwise temporal interactions in evoked EEG responses. We show the benefit of considering the instantaneous temporal derivative together with the raw values of M/EEG signals as a multivariate response, how we can separately quantify modulations of amplitude and direction for vector quantities, and how we can measure the emergence of novel information over time in evoked responses. Open-source Matlab and Python code implementing the new methods accompanies this article. Hum Brain Mapp 38:1541-1573, 2017. © 2016 Wiley Periodicals, Inc.

Structure-Function Network Mapping and Its Assessment via Persistent Homology

Understanding the relationship between brain structure and function is a fundamental problem in network neuroscience. This work deals with the general method of structure-function mapping at the whole-brain level. We formulate the problem as a topological mapping of structure-function connectivity via matrix function, and find a stable solution by exploiting a regularization procedure to cope with large matrices. We introduce a novel measure of network similarity based on persistent homology for assessing the quality of the network mapping, which enables a detailed comparison of network topological changes across all possible thresholds, rather than just at a single, arbitrary threshold that may not be optimal. We demonstrate that our approach can uncover the direct and indirect structural paths for predicting functional connectivity, and our network similarity measure outperforms other currently available methods. We systematically validate our approach with (1) a comparison of regularized vs. non-regularized procedures, (2) a null model of the degree-preserving random rewired structural matrix, (3) different network types (binary vs. weighted matrices), and (4) different brain parcellation schemes (low vs. high resolutions). Finally, we evaluate the scalability of our method with relatively large matrices (2514x2514) of structural and functional connectivity obtained from 12 healthy human subjects measured non-invasively while at rest. Our results reveal a nonlinear structure-function relationship, suggesting that the resting-state functional connectivity depends on direct structural connections, as well as relatively parsimonious indirect connections via polysynaptic pathways.

Temporal Information of Directed Causal Connectivity in Multi-Trial ERP Data using Partial Granger Causality

Partial Granger causality (PGC) has been applied to analyse causal functional neural connectivity after effectively mitigating confounding influences caused by endogenous latent variables and exogenous environmental inputs. However, it is not known how this connectivity obtained from PGC evolves over time. Furthermore, PGC has yet to be tested on realistic nonlinear neural circuit models and multi-trial event-related potentials (ERPs) data. In this work, we first applied a time-domain PGC technique to evaluate simulated neural circuit models, and demonstrated that the PGC measure is more accurate and robust in detecting connectivity patterns as compared to conditional Granger causality and partial directed coherence, especially when the circuit is intrinsically nonlinear. Moreover, the connectivity in PGC settles faster into a stable and correct configuration over time. After method verification, we applied PGC to reveal the causal connections of ERP trials of a mismatch negativity auditory oddball paradigm. The PGC analysis revealed a significant bilateral but asymmetrical localised activity in the temporal lobe close to the auditory cortex, and causal influences in the frontal, parietal and cingulate cortical areas, consistent with previous studies. Interestingly, the time to reach a stable connectivity configuration (~250–300 ms) coincides with the deviation of ensemble ERPs of oddball from standard tones. Finally, using a sliding time window, we showed higher resolution dynamics of causal connectivity within an ERP trial. In summary, time-domain PGC is promising in deciphering directed functional connectivity in nonlinear and ERP trials accurately, and at a sufficiently early stage. This data-driven approach can reduce computational time, and determine the key architecture for neural circuit modeling.

Causal Inference Based on the Analysis of Events of Relations for Non-stationary Variables

The main concept behind causality involves both statistical conditions and temporal relations. However, current approaches to causal inference, focusing on the probability vs. conditional probability contrast, are based on model functions or parametric estimation. These approaches are not appropriate when addressing non-stationary variables. In this work, we propose a causal inference approach based on the analysis of Events of Relations (CER). CER focuses on the temporal delay relation between cause and effect, and a binomial test is established to determine whether an "event of relation" with a non-zero delay is significantly different from one with zero delay. Because CER avoids parameter estimation of non-stationary variables per se, the method can be applied to both stationary and non-stationary signals.

Data on copula modeling of mixed discrete and continuous neural time series

Copula is an important tool for modeling neural dependence. Recent work on copula has been expanded to jointly model mixed time series in neuroscience (“Hu et al., 2016, Joint Analysis of Spikes and Local Field Potentials using Copula” [1]). Here we present further data for joint analysis of spike and local field potential (LFP) with copula modeling. In particular, the details of different model orders and the influence of possible spike contamination in LFP data from the same and different electrode recordings are presented. To further facilitate the use of our copula model for the analysis of mixed data, we provide the Matlab codes, together with example data.

Inference of biological networks using Bi-directional Random Forest Granger causality

The standard ordinary least squares based Granger causality is one of the widely used methods for detecting causal interactions between time series data. However, recent developments in technology limit the utilization of some existing implementations due to the availability of high dimensional data. In this paper, we are proposing a technique called Bi-directional Random Forest Granger causality. This technique uses the random forest regularization together with the idea of reusing the time series data by reversing the time stamp to extract more causal information. We have demonstrated the effectiveness of our proposed method by applying it to simulated data and then applied it to two real biological datasets, i.e., fMRI and HeLa cell. fMRI data was used to map brain network involved in deductive reasoning while HeLa cell dataset was used to map gene network involved in cancer.

Joint analysis of spikes and local field potentials using copula

Recent technological advances, which allow for simultaneous recording of spikes and local field potentials (LFPs) at multiple sites in a given cortical area or across different areas, have greatly increased our understanding of signal processing in brain circuits. Joint analysis of simultaneously collected spike and LFP signals is an important step to explicate how the brain orchestrates information processing. In this contribution, we present a novel statistical framework based on Gaussian copula to jointly model spikes and LFP. In our approach, we use copula to link separate, marginal regression models to construct a joint regression model, in which the binary-valued spike train data are modeled using generalized linear model (GLM) and the continuous-valued LFP data are modeled using linear regression. Model parameters can be efficiently estimated via maximum-likelihood. In particular, we show that our model offers a means to statistically detect directional influence between spikes and LFP, akin to Granger causality measure, and that we are able to assess its statistical significance by conducting a Wald test. Through extensive simulations, we also show that our method is able to reliably recover the true model used to generate the data. To demonstrate the effectiveness of our approach in real setting, we further apply the method to a mixed neural dataset, consisting of spikes and LFP simultaneously recorded from the visual cortex of a monkey performing a contour detection task.

Top-Down Control of Visual Alpha Oscillations: Sources of Control Signals and Their Mechanisms of Action

Alpha oscillations (8-12 Hz) are thought to inversely correlate with cortical excitability. Goal-oriented modulation of alpha has been studied extensively. In visual spatial attention, alpha over the region of visual cortex corresponding to the attended location decreases, signifying increased excitability to facilitate the processing of impending stimuli. In contrast, in retention of verbal working memory, alpha over visual cortex increases, signifying decreased excitability to gate out stimulus input to protect the information held online from sensory interference. According to the prevailing model, this goal-oriented biasing of sensory cortex is effected by top-down control signals from frontal and parietal cortices. The present study tests and substantiates this hypothesis by (a) identifying the signals that mediate the top-down biasing influence, (b) examining whether the cortical areas issuing these signals are task-specific or task-independent, and (c) establishing the possible mechanism of the biasing action. High-density human EEG data were recorded in two experimental paradigms: a trial-by-trial cued visual spatial attention task and a modified Sternberg working memory task. Applying Granger causality to both sensor-level and source-level data we report the following findings. In covert visual spatial attention, the regions exerting top-down control over visual activity are lateralized to the right hemisphere, with the dipoles located at the right frontal eye field (FEF) and the right inferior frontal gyrus (IFG) being the main sources of top-down influences. During retention of verbal working memory, the regions exerting top-down control over visual activity are lateralized to the left hemisphere, with the dipoles located at the left middle frontal gyrus (MFG) being the main source of top-down influences. In both experiments, top-down influences are mediated by alpha oscillations, and the biasing effect is likely achieved via an inhibition-disinhibition mechanism.

Causality Analysis of fMRI Data Based on the Directed Information Theory Framework

This paper aims to conduct fMRI-based causality analysis in brain connectivity by exploiting the directed information (DI) theory framework. Unlike the well-known Granger causality (GC) analysis, which relies on the linear prediction technique, the DI theory framework does not have any modeling constraints on the sequences to be evaluated and ensures estimation convergence. Moreover, it can be used to generate the GC graphs. In this paper, first, we introduce the core concepts in the DI framework. Second, we present how to conduct causality analysis using DI measures between two time series. We provide the detailed procedure on how to calculate the DI for two finite-time series. The two major steps involved here are optimal bin size selection for data digitization and probability estimation. Finally, we demonstrate the applicability of DI-based causality analysis using both the simulated data and experimental fMRI data, and compare the results with that of the GC analysis. Our analysis indicates that GC analysis is effective in detecting linear or nearly linear causal relationship, but may have difficulty in capturing nonlinear causal relationships. On the other hand, DI-based causality analysis is more effective in capturing both linear and nonlinear causal relationships. Moreover, it is observed that brain connectivity among different regions generally involves dynamic two-way information transmissions between them. Our results show that when bidirectional information flow is present, DI is more effective than GC to quantify the overall causal relationship.

Copula regression analysis of simultaneously recorded frontal eye field and inferotemporal spiking activity during object-based working memory

Inferotemporal (IT) neurons are known to exhibit persistent, stimulus-selective activity during the delay period of object-based working memory tasks. Frontal eye field (FEF) neurons show robust, spatially selective delay period activity during memory-guided saccade tasks. We present a copula regression paradigm to examine neural interaction of these two types of signals between areas IT and FEF of the monkey during a working memory task. This paradigm is based on copula models that can account for both marginal distribution over spiking activity of individual neurons within each area and joint distribution over ensemble activity of neurons between areas. Considering the popular GLMs as marginal models, we developed a general and flexible likelihood framework that uses the copula to integrate separate GLMs into a joint regression analysis. Such joint analysis essentially leads to a multivariate analog of the marginal GLM theory and hence efficient model estimation. In addition, we show that Granger causality between spike trains can be readily assessed via the likelihood ratio statistic. The performance of this method is validated by extensive simulations, and compared favorably to the widely used GLMs. When applied to spiking activity of simultaneously recorded FEF and IT neurons during working memory task, we observed significant Granger causality influence from FEF to IT, but not in the opposite direction, suggesting the role of the FEF in the selection and retention of visual information during working memory. The copula model has the potential to provide unique neurophysiological insights about network properties of the brain.

Brain effective connectivity during motor-imagery and execution following stroke and rehabilitation

Brain areas within the motor system interact directly or indirectly during motor-imagery and motor-execution tasks. These interactions and their functionality can change following stroke and recovery. How brain network interactions reorganize and recover their functionality during recovery and treatment following stroke are not well understood. To contribute to answering these questions, we recorded blood oxygenation-level dependent (BOLD) functional magnetic resonance imaging (fMRI) signals from 10 stroke survivors and evaluated dynamical causal modeling (DCM)-based effective connectivity among three motor areas: primary motor cortex (M1), pre-motor cortex (PMC) and supplementary motor area (SMA), during motor-imagery and motor-execution tasks. We compared the connectivity between affected and unaffected hemispheres before and after mental practice and combined mental practice and physical therapy as treatments. The treatment (intervention) period varied in length between 14 to 51 days but all patients received the same dose of 60 h of treatment. Using Bayesian model selection (BMS) approach in the DCM approach, we found that, after intervention, the same network dominated during motor-imagery and motor-execution tasks but modulatory parameters suggested a suppressive influence of SM A on M1 during the motor-imagery task whereas the influence of SM A on M1 was unrestricted during the motor-execution task. We found that the intervention caused a reorganization of the network during both tasks for unaffected as well as for the affected hemisphere. Using Bayesian model averaging (BMA) approach, we found that the intervention improved the regional connectivity among the motor areas during both the tasks. The connectivity between PMC and M1 was stronger in motor-imagery tasks whereas the connectivity from PMC to M1, SM A to M1 dominated in motor-execution tasks. There was significant behavioral improvement (p = 0.001) in sensation and motor movements because of the intervention as reflected by behavioral Fugl-Meyer (FMA) measures, which were significantly correlated (p = 0.05) with a subset of connectivity. These findings suggest that PMC and M1 play a crucial role during motor-imagery as well as during motor-execution task. In addition, M1 causes more exchange of causal information among motor areas during a motor-execution task than during a motor-imagery task due to its interaction with SM A. This study expands our understanding of motor network involved during two different tasks, which are commonly used during rehabilitation following stroke. A clear understanding of the effective connectivity networks leads to a better treatment in helping stroke survivors regain motor ability.

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Brain motor effective connectivity can change due to stroke and during recovery.

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Rehabilitative treatments caused significant changes in motor and sensation scores.

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Behavioral improvements were accompanied by specific changes in brain connectivity.

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SMA exerted a suppressive driving to M1 during motor imagery.

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SMA-to-M1connectivity was positively modulated during actual motor execution.

Effect of measurement noise on Granger causality

Most of the signals recorded in experiments are inevitably contaminated by measurement noise. Hence, it is important to understand the effect of such noise on estimating causal relations between such signals. A primary tool for estimating causality is Granger causality. Granger causality can be computed by modeling the signal using a bivariate autoregressive (AR) process. In this paper, we greatly extend the previous analysis of the effect of noise by considering a bivariate AR process of general order p. From this analysis, we analytically obtain the dependence of Granger causality on various noise-dependent system parameters. In particular, we show that measurement noise can lead to spurious Granger causality and can suppress true Granger causality. These results are verified numerically. Finally, we show how true causality can be recovered numerically using the Kalman expectation maximization algorithm.