Synthetic neuronal datasets for benchmarking directed functional connectivity metrics

A novel method for efficient estimation of brain effective connectivity in EEG

BACKGROUND AND OBJECTIVE

Brain connectivity plays a pivotal role in understanding the brain's information processing functions by providing various details including magnitude, direction, and temporal dynamics of inter-neuron connections. While the connectivity may be classified as structural, functional and causal, a complete in-vivo directional analysis is guaranteed by the latter and is referred to as Effective Connectivity (EC). Two most widely used EC techniques are Directed Transfer Function (DTF) and Partial Directed Coherence (PDC) which are based on multivariate autoregressive models. The drawbacks of these techniques include poor frequency resolution and the requirement for experimental approach to determine signal normalization and thresholding techniques in identifying significant connectivities between multivariate sources.

METHODS

In this study, the drawbacks of DTF and PDC are addressed by proposing a novel technique, termed as Efficient Effective Connectivity (EEC), for the estimation of EC between multivariate sources using AR spectral estimation and Granger causality principle. In EEC, a linear predictive filter with AR coefficients obtained via multivariate EEG is used for signal prediction. This leads to the estimation of full-length signals which are then transformed into frequency domain by using Burg spectral estimation method. Furthermore, the newly proposed normalization method addressed the effect on each source in EEC using the sum of maximum connectivity values over the entire frequency range. Lastly, the proposed dynamic thresholding works by subtracting the first moment of causal effects of all the sources on one source from individual connections present for that source.

RESULTS

The proposed method is evaluated using synthetic and real resting-state EEG of 46 healthy controls. A 3D-Convolutional Neural Network is trained and tested using the PDC and EEC samples. The result indicates that compared to PDC, EEC improves the EEG eye-state classification accuracy, sensitivity and specificity by 5.57%, 3.15% and 8.74%, respectively.

CONCLUSION

Correct identification of all connections in synthetic data and improved resting-state classification performance using EEC proved that EEC gives better estimation of directed causality and indicates that it can be used for reliable understanding of brain mechanisms. Conclusively, the proposed technique may open up new research dimensions for clinical diagnosis of mental disorders.

Estimation of Granger causality through Artificial Neural Networks: applications to physiological systems and chaotic electronic oscillators

One of the most challenging problems in the study of complex dynamical systems is to find the statistical interdependencies among the system components. Granger causality (GC) represents one of the most employed approaches, based on modeling the system dynamics with a linear vector autoregressive (VAR) model and on evaluating the information flow between two processes in terms of prediction error variances. In its most advanced setting, GC analysis is performed through a state-space (SS) representation of the VAR model that allows to compute both conditional and unconditional forms of GC by solving only one regression problem. While this problem is typically solved through Ordinary Least Square (OLS) estimation, a viable alternative is to use Artificial Neural Networks (ANNs) implemented in a simple structure with one input and one output layer and trained in a way such that the weights matrix corresponds to the matrix of VAR parameters. In this work, we introduce an ANN combined with SS models for the computation of GC. The ANN is trained through the Stochastic Gradient Descent L1 (SGD-L1) algorithm, and a cumulative penalty inspired from penalized regression is applied to the network weights to encourage sparsity. Simulating networks of coupled Gaussian systems, we show how the combination of ANNs and SGD-L1 allows to mitigate the strong reduction in accuracy of OLS identification in settings of low ratio between number of time series points and of VAR parameters. We also report how the performances in GC estimation are influenced by the number of iterations of gradient descent and by the learning rate used for training the ANN. We recommend using some specific combinations for these parameters to optimize the performance of GC estimation. Then, the performances of ANN and OLS are compared in terms of GC magnitude and statistical significance to highlight the potential of the new approach to reconstruct causal coupling strength and network topology even in challenging conditions of data paucity. The results highlight the importance of of a proper selection of regularization parameter which determines the degree of sparsity in the estimated network. Furthermore, we apply the two approaches to real data scenarios, to study the physiological network of brain and peripheral interactions in humans under different conditions of rest and mental stress, and the effects of the newly emerged concept of remote synchronization on the information exchanged in a ring of electronic oscillators. The results highlight how ANNs provide a mesoscopic description of the information exchanged in networks of multiple interacting physiological systems, preserving the most active causal interactions between cardiovascular, respiratory and brain systems. Moreover, ANNs can reconstruct the flow of directed information in a ring of oscillators whose statistical properties can be related to those of physiological networks.

Modeling time-varying brain networks with a self-tuning optimized Kalman filter

Brain networks are complex dynamical systems in which directed interactions between different areas evolve at the sub-second scale of sensory, cognitive and motor processes. Due to the highly non-stationary nature of neural signals and their unknown noise components, however, modeling dynamic brain networks has remained one of the major challenges in contemporary neuroscience. Here, we present a new algorithm based on an innovative formulation of the Kalman filter that is optimized for tracking rapidly evolving patterns of directed functional connectivity under unknown noise conditions. The Self-Tuning Optimized Kalman filter (STOK) is a novel adaptive filter that embeds a self-tuning memory decay and a recursive regularization to guarantee high network tracking accuracy, temporal precision and robustness to noise. To validate the proposed algorithm, we performed an extensive comparison against the classical Kalman filter, in both realistic surrogate networks and real electroencephalography (EEG) data. In both simulations and real data, we show that the STOK filter estimates time-frequency patterns of directed connectivity with significantly superior performance. The advantages of the STOK filter were even clearer in real EEG data, where the algorithm recovered latent structures of dynamic connectivity from epicranial EEG recordings in rats and human visual evoked potentials, in excellent agreement with known physiology. These results establish the STOK filter as a powerful tool for modeling dynamic network structures in biological systems, with the potential to yield new insights into the rapid evolution of network states from which brain functions emerge.

During normal behavior, brains transition between functional network states several times per second. This allows humans to quickly read a sentence, and a frog to catch a fly. Understanding these fast network dynamics is fundamental to understanding how brains work, but up to now it has proven very difficult to model fast brain dynamics for various methodological reasons. To overcome these difficulties, we designed a new Kalman filter (STOK) by innovating on previous solutions from control theory and state-space modelling. We show that STOK accurately models fast network changes in simulations and real neural data, making it an essential new tool for modelling fast brain networks in the time and frequency domain.

A regularized and smoothed General Linear Kalman Filter for more accurate estimation of time-varying directed connectivity

Adaptive algorithms based on the Kalman filter are valuable tools to model the dynamic and directed Granger causal interactions between neurophysiological signals simultaneously recorded from multiple cortical regions. Among these algorithms, the General Linear Kalman Filter (GLKF) has proven to be the most accurate and reliable. Here we propose a regularized and smoothed GLKF (spsm-GLKF) with ℓ1 norm penalties based on lasso or group lasso and a fixedinterval smoother. We show that the group lasso penalty promotes sparse solutions by shrinking spurious connections to zero, while the smoothing increases the robustness of the estimates. Overall, our results demonstrate that spsm-GLKF outperforms the original GLKF, and represents a more accurate tool for the characterization of dynamical and sparse functional brain networks.

Assessing the performance of Granger-Geweke causality: Benchmark dataset and simulation framework

Nonparametric methods based on spectral factorization offer well validated tools for estimating spectral measures of causality, called Granger–Geweke Causality (GGC). In Pagnotta et al. (2018) [1] we benchmarked nonparametric GGC methods using EEG data recorded during unilateral whisker stimulations in ten rats; here, we include detailed information about the benchmark dataset. In addition, we provide codes for estimating nonparametric GGC and a simulation framework to evaluate the effects on GGC analyses of potential problems, such as the common reference problem, signal-to-noise ratio (SNR) differences between channels, and the presence of additive noise. We focus on nonparametric methods here, but these issues also affect parametric methods, which can be tested in our framework as well. Our examples allow showing that time reversal testing for GGC (tr-GGC) mitigates the detrimental effects due to SNR imbalance and presence of mixed additive noise, and illustrate that, when using a common reference, tr-GGC unambiguously detects the causal influence׳s dominant spectral component, irrespective of the characteristics of the common reference signal. Finally, one of our simulations provides an example that nonparametric methods can overcome a pitfall associated with the implementation of conditional GGC in traditional parametric methods.

Consistency of EEG source localization and connectivity estimates

As the EEG inverse problem does not have a unique solution, the sources reconstructed from EEG and their connectivity properties depend on forward and inverse modeling parameters such as the choice of an anatomical template and electrical model, prior assumptions on the sources, and further implementational details. In order to use source connectivity analysis as a reliable research tool, there is a need for stability across a wider range of standard estimation routines. Using resting state EEG recordings of N=65 participants acquired within two studies, we present the first comprehensive assessment of the consistency of EEG source localization and functional/effective connectivity metrics across two anatomical templates (ICBM152 and Colin27), three electrical models (BEM, FEM and spherical harmonics expansions), three inverse methods (WMNE, eLORETA and LCMV), and three software implementations (Brainstorm, Fieldtrip and our own toolbox). Source localizations were found to be more stable across reconstruction pipelines than subsequent estimations of functional connectivity, while effective connectivity estimates where the least consistent. All results were relatively unaffected by the choice of the electrical head model, while the choice of the inverse method and source imaging package induced a considerable variability. In particular, a relatively strong difference was found between LCMV beamformer solutions on one hand and eLORETA/WMNE distributed inverse solutions on the other hand. We also observed a gradual decrease of consistency when results are compared between studies, within individual participants, and between individual participants. In order to provide reliable findings in the face of the observed variability, additional simulations involving interacting brain sources are required. Meanwhile, we encourage verification of the obtained results using more than one source imaging procedure.

A Simulation Framework for Benchmarking EEG-Based Brain Connectivity Estimation Methodologies

Due to its high temporal resolution, electroencephalography (EEG) is widely used to study functional and effective brain connectivity. Yet, there is currently a mismatch between the vastness of studies conducted and the degree to which the employed analyses are theoretically understood and empirically validated. We here provide a simulation framework that enables researchers to test their analysis pipelines on realistic pseudo-EEG data. We construct a minimal example of brain interaction, which we propose as a benchmark for assessing a methodology's general eligibility for EEG-based connectivity estimation. We envision that this benchmark be extended in a collaborative effort to validate methods in more complex scenarios. Quantitative metrics are defined to assess a method's performance in terms of source localization, connectivity detection and directionality estimation. All data and code needed for generating pseudo-EEG data, conducting source reconstruction and connectivity estimation using baseline methods from the literature, evaluating performance metrics, as well as plotting results, are made publicly available. While this article covers only EEG modeling, we will also provide a magnetoencephalography version of our framework online.