Comparison of Bootstrap Methods for Estimating Causality in Linear Dynamic Systems: A Review.

In this study, we present a thorough comparison of the performance of four different bootstrap methods for assessing the significance of causal analysis in time series data. For this purpose, multivariate simulated data are generated by a linear feedback system. The methods investigated are uncorrelated Phase Randomization Bootstrap (uPRB), which generates surrogate data with no cross-correlation between variables by randomizing the phase in the frequency domain; Time Shift Bootstrap (TSB), which generates surrogate data by randomizing the phase in the time domain; Stationary Bootstrap (SB), which calculates standard errors and constructs confidence regions for weakly dependent stationary observations; and AR-Sieve Bootstrap (ARSB), a resampling method based on AutoRegressive (AR) models that approximates the underlying data-generating process. The uPRB method accurately identifies variable interactions but fails to detect self-feedback in some variables. The TSB method, despite performing worse than uPRB, is unable to detect feedback between certain variables. The SB method gives consistent causality results, although its ability to detect self-feedback decreases, as the mean block width increases. The ARSB method shows superior performance, accurately detecting both self-feedback and causality across all variables. Regarding the analysis of the Impulse Response Function (IRF), only the ARSB method succeeds in detecting both self-feedback and causality in all variables, aligning well with the connectivity diagram. Other methods, however, show considerable variations in detection performance, with some detecting false positives and others only detecting self-feedback.

State Space Modeling of Event Count Time Series.

This paper proposes a class of algorithms for analyzing event count time series, based on state space modeling and Kalman filtering. While the dynamics of the state space model is kept Gaussian and linear, a nonlinear observation function is chosen. In order to estimate the states, an iterated extended Kalman filter is employed. Positive definiteness of covariance matrices is preserved by a square-root filtering approach, based on singular value decomposition. Non-negativity of the count data is ensured, either by an exponential observation function, or by a newly introduced "affinely distorted hyperbolic" observation function. The resulting algorithm is applied to time series of the daily number of seizures of drug-resistant epilepsy patients. This number may depend on dosages of simultaneously administered anti-epileptic drugs, their superposition effects, delay effects, and unknown factors, making the objective analysis of seizure counts time series arduous. For the purpose of validation, a simulation study is performed. The results of the time series analysis by state space modeling, using the dosages of the anti-epileptic drugs as external control inputs, provide a decision on the effect of the drugs in a particular patient, with respect to reducing or increasing the number of seizures.

Decomposition of neurological multivariate time series by state space modelling.

Decomposition of multivariate time series data into independent source components forms an important part of preprocessing and analysis of time-resolved data in neuroscience. We briefly review the available tools for this purpose, such as Factor Analysis (FA) and Independent Component Analysis (ICA), then we show how linear state space modelling, a methodology from statistical time series analysis, can be employed for the same purpose. State space modelling, a generalization of classical ARMA modelling, is well suited for exploiting the dynamical information encoded in the temporal ordering of time series data, while this information remains inaccessible to FA and most ICA algorithms. As a result, much more detailed decompositions become possible, and both components with sharp power spectrum, such as alpha components, sinusoidal artifacts, or sleep spindles, and with broad power spectrum, such as FMRI scanner artifacts or epileptic spiking components, can be separated, even in the absence of prior information. In addition, three generalizations are discussed, the first relaxing the independence assumption, the second introducing non-stationarity of the covariance of the noise driving the dynamics, and the third allowing for non-Gaussianity of the data through a non-linear observation function. Three application examples are presented, one electrocardigram time series and two electroencephalogram (EEG) time series. The two EEG examples, both from epilepsy patients, demonstrate the separation and removal of various artifacts, including hum noise and FMRI scanner artifacts, and the identification of sleep spindles, epileptic foci, and spiking components. Decompositions obtained by two ICA algorithms are shown for comparison.

Source imaging of deep-brain activity using the regional spatiotemporal Kalman filter.

BACKGROUND AND OBJECTIVE: The human brain displays rich and complex patterns of interaction within and among brain networks that involve both cortical and subcortical brain regions. Due to the limited spatial resolution of surface electroencephalography (EEG), EEG source imaging is used to reconstruct brain sources and investigate their spatial and temporal dynamics. The majority of EEG source imaging methods fail to detect activity from subcortical brain structures. The reconstruction of subcortical sources is a challenging task because the signal from these sources is weakened and mixed with artifacts and other signals from cortical sources. In this proof-of-principle study we present a novel EEG source imaging method, the regional spatiotemporal Kalman filter (RSTKF), that can detect deep brain activity. METHODS: The regional spatiotemporal Kalman filter (RSTKF) is a generalization of the spatiotemporal Kalman filter (STKF), which allows for the characterization of different regional dynamics in the brain. It is based on state-space modeling with spatially heterogeneous dynamical noise variances, since models with spatial and temporal homogeneity fail to describe the dynamical complexity of brain activity. First, RSTKF is tested using simulated EEG data from sources in the frontal lobe, putamen, and thalamus. After that, it is applied to non-averaged interictal epileptic spikes from a presurgical epilepsy patient with focal epileptic activity in the amygdalo-hippocampal complex. The results of RSTKF are compared to those of low-resolution brain electromagnetic tomography (LORETA) and of standard STKF. RESULTS: Only RSTKF is successful in consistently and accurately localizing the sources in deep brain regions. Additionally, RSTKF shows improved spatial resolution compared to LORETA and STKF. CONCLUSIONS: RSTKF is a generalization of STKF that allows for accurate, focal, and consistent localization of sources, especially in the deeper brain areas. In contrast to standard source imaging methods, RSTKF may find application in the localization of the epileptogenic zone in deeper brain structures, such as mesial frontal and temporal lobe epilepsies, especially in EEG recordings for which no reliable averaged spike shape can be obtained due to lack of the necessary number of spikes required to reach a certain signal-to-noise ratio level after averaging.

Optimal HRF and smoothing parameters for fMRI time series within an autoregressive modeling framework.

The analysis of time series obtained by functional magnetic resonance imaging (fMRI) may be approached by fitting predictive parametric models, such as nearest-neighbor autoregressive models with exogeneous input (NNARX). As a part of the modeling procedure, it is possible to apply instantaneous linear transformations to the data. Spatial smoothing, a common preprocessing step, may be interpreted as such a transformation. The autoregressive parameters may be constrained, such that they provide a response behavior that corresponds to the canonical haemodynamic response function (HRF). We present an algorithm for estimating the parameters of the linear transformations and of the HRF within a rigorous maximum-likelihood framework. Using this approach, an optimal amount of both the spatial smoothing and the HRF can be estimated simultaneously for a given fMRI data set. An example from a motor-task experiment is discussed. It is found that, for this data set, weak, but non-zero, spatial smoothing is optimal. Furthermore, it is demonstrated that activated regions can be estimated within the maximum-likelihood framework.

SVD Square-root Iterated Extended Kalman Filter for Modeling of Epileptic Seizure Count Time Series with External Inputs.

In this paper a nonlinear filtering algorithm for count time series is developed that takes the non-negativity of the data into account and preserves positive definiteness of the covariance matrices of the model. For this purpose, a recently proposed variant of Kalman Filtering based on Singular Value Decomposition is incorporated into Iterative Extended Kalman Filtering, in order to estimate the states of a nonlinear state space model. The resulting algorithm is applied to the evaluation and design of therapies for patients suffering from Myoclonic Astatic Epilepsy, employing time series of daily seizure rate. The analysis provides a decision whether for a specific patient a particular anti-epileptic drug is increasing or reducing the seizure rate. Through a simulation study the proposed algorithm is validated. Additionally, for clinical data results obtained by the proposed algorithm are compared with the results from a Cox-Stuart trend test as well as with the visual assessment of experienced pediatric epileptologists.

Magnetic particle mapping using magnetoelectric sensors as an imaging modality.

Magnetic nanoparticles (MNPs) are a hot topic in the field of medical life sciences, as they are highly relevant in diagnostic applications. In this regard, a large variety of novel imaging methods for MNP in biological systems have been invented. In this proof-of-concept study, a new and novel technique is explored, called Magnetic Particle Mapping (MPM), using resonant magnetoelectric (ME) sensors for the detection of MNPs that could prove to be a cheap and efficient way to localize the magnetic nanoparticles. The simple and straightforward setup and measurement procedure includes the detection of higher harmonic excitations of MNP ensembles. We show the feasibility of this approach by building a measurement setup particularly suited to exploit the inherent sensor properties. We measure the magnetic response from 2D MNP distributions and reconstruct the distribution by solving the inverse problem. Furthermore, biological samples with magnetically labeled cells were measured and reconstruction of the distribution was compared with light microscope images. Measurement results suggest that the approach presented here is promising for MNP localization.

Analysis of the effects of medication for the treatment of epilepsy by ensemble Iterative Extended Kalman filtering.

This paper proposes an objective methodology for the analysis of epileptic seizure count time series by developing a non-linear state space model. An iterative extended Kalman filter (IEKF) is employed for the estimation of the states of the non-linear state space model. In order to improve convergence of the IEKF, the recently proposed Levenberg-Marquardt variant of the IEKF is explored. As external inputs time-dependent dosages of several simultaneously administered anticonvulsants are included. The aim of the analysis is to decide whether each anticonvulsant decreases or increases the number of seizures per day. The performance of the analysis is analyzed for simulated data, as well as for real data from a patient suffering from myoclonic-astatic epilepsy.

Pipeline for Forward Modeling and Source Imaging of Magnetocardiographic Recordings via Spatiotemporal Kalman Filtering.

the aim of this proof-of-concept work was to apply the spatiotemporal Kalman filter (STKF) algorithm to magnetocardiographic (MCG) recordings of the heart. Due to the lack of standardized software and pipelines for MCG source imaging, we needed to construct a pipeline for MCG forward modeling before we could apply the STKF method. In the forward module, the finite element method (FEM) solvers in SimBio software are used to solve the MCG forward problem. In the inverse module, STKF and Low Resolution Brain Electromagnetic Tomography (LORETA) algorithms are applied. The work was conducted using two simulated datasets contaminated with different levels of additive white Gaussian noise (AWGN). Then the inverse problem was solved using both LORETA and STKF. The results indicate that STKF outperformed LORETA for MCG datasets with low signal-to-noise ratio (SNR). In the future clinical MCG recordings and more sophisticated simulations will be used to evaluate the accuracy of MCG source imaging via STKF.

Multifocal epilepsy in children is associated with increased long-distance functional connectivity: An explorative EEG-fMRI study.

OBJECTIVE: Multifocal epileptic activity is an unfavourable feature of a number of epileptic syndromes (Lennox-Gastaut syndrome, West syndrome, severe focal epilepsies) which suggests an overall vulnerability of the brain to pathological synchronization. However, the mechanisms of multifocal activity are insufficiently understood. This explorative study investigates whether pathological connectivity within brain areas of the default mode network as well as thalamus, brainstem and retrosplenial cortex may predispose individuals to multifocal epileptic activity. METHODS: 33 children suffering from multifocal and monofocal (control group) epilepsies were investigated using EEG-fMRI recordings during sleep. The blood oxygenated level dependent (BOLD) signal of 15 regions of interest was extracted and temporally correlated (resting-state functional connectivity). RESULTS: Patients with monofocal epilepsies were characterized by strong correlations between the corresponding interhemispheric homotopic regions. This pattern of correlations with pronounced short-distance and weak long-distance functional connectivity resembles the connectivity pattern described for healthy children. Patients with multifocal epileptic activity, however, demonstrated significantly stronger correlations between a large number of regions of the default mode network as well as thalamus and brainstem, with a significant increase in long-distance connectivity compared to children with monofocal epileptic activity. In the group of patients with multifocal epilepsies there were no differences in functional connectivity between patients with or without Lennox-Gastaut syndrome. CONCLUSION: This explorative study shows that multifocal activity is associated with generally increased long-distance functional connectivity in the brain. It can be suggested that this pronounced connectivity may represent either a risk to pathological over-synchronization or a consequence of the multifocal epileptic activity.

Spatial projection as a preprocessing step for EEG source reconstruction using spatiotemporal Kalman filtering.

The reconstruction of brain sources from non-invasive electroencephalography (EEG) or magnetoencephalography (MEG) via source imaging can be distorted by information redundancy in case of high-resolution recordings. Dimensionality reduction approaches such as spatial projection may be used to alleviate this problem. In this proof-of-principle paper we apply spatial projection to solve the problem of information redundancy in case of source reconstruction via spatiotemporal Kalman filtering (STKF), which is based on state-space modeling. We compare two approaches for incorporating spatial projection into the STKF algorithm and select the best approach based on its performance in source localization with respect to accurate estimation of source location, lack of spurious sources, computational speed and small number of required optimization steps in state-space model parameter estimation. We use state-of-the-art simulated EEG data based on neuronal population models, for which the number and location of sources is known, to validate the source reconstruction results of the STKF. The incorporation of spatial projection into the STKF algorithm solved the problem of information redundancy, resulting in correct source localization with no spurious sources, and decreased the overall computational time in STKF analysis. The results help make STKF analyses of high-density EEG, MEG or simultaneous MEG-EEG data more feasible.

Source reconstruction via the spatiotemporal Kalman filter and LORETA from EEG time series with 32 or fewer electrodes.

The clinical routine of non-invasive electroencephalography (EEG) is usually performed with 8-40 electrodes, especially in long-term monitoring, infants or emergency care. There is a need in clinical and scientific brain imaging to develop inverse solution methods that can reconstruct brain sources from these low-density EEG recordings. In this proof-of-principle paper we investigate the performance of the spatiotemporal Kalman filter (STKF) in EEG source reconstruction with 9-, 19- and 32- electrodes. We used simulated EEG data of epileptic spikes generated from lateral frontal and lateral temporal brain sources using state-of-the-art neuronal population models. For validation of source reconstruction, we compared STKF results to the location of the simulated source and to the results of low-resolution brain electromagnetic tomography (LORETA) standard inverse solution. STKF consistently showed less localization bias compared to LORETA, especially when the number of electrodes was decreased. The results encourage further research into the application of the STKF in source reconstruction of brain activity from low-density EEG recordings.

Localized short-range correlations in the spectrum of the equal-time correlation matrix.

We suggest a procedure to identify those parts of the spectrum of the equal-time correlation matrix C where relevant information about correlations of a multivariate time series is induced. Using an ensemble average over each of the distances between eigenvalues, all nearest-neighbor distributions can be calculated individually. We present numerical examples, where (a) information about cross correlations is found in the so-called "bulk" of eigenvalues (which generally is thought to contain only random correlations) and where (b) the information extracted from the lower edge of the spectrum of C is statistically more significant than that extracted from the upper edge. We apply the analysis to electroencephalographic recordings with epileptic events.

Modelling non-stationary variance in EEG time series by state space GARCH model.

We present a new approach to modelling non-stationarity in EEG time series by a generalized state space approach. A given time series can be decomposed into a set of noise-driven processes, each corresponding to a different frequency band. Non-stationarity is modelled by allowing the variances of the driving noises to change with time, depending on the state prediction error within the state space model. The method is illustrated by an application to EEG data recorded during the onset of anaesthesia.

Detection and characterization of changes of the correlation structure in multivariate time series.

We propose a method based on the equal-time correlation matrix as a sensitive detector for phase-shape correlations in multivariate data sets. The key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix. Consequently, detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors. The performance of the technique is demonstrated by application to N(f)-tori, autoregressive models, and coupled chaotic systems. The high sensitivity, the comparatively small computational effort, and the excellent time resolution of the method recommend it for application to the analysis of complex, spatially extended, nonstationary systems.

The performance of the spatiotemporal Kalman filter and LORETA in seizure onset localization.

The assumption of spatial-smoothness is often used to solve the bioelectric inverse problem during electroencephalographic (EEG) source imaging, e.g., in low resolution electromagnetic tomography (LORETA). Since the EEG data show a temporal structure, the combination of the temporal-smoothness and the spatial-smoothness constraints may improve the solution of the EEG inverse problem. This study investigates the performance of the spatiotemporal Kalman filter (STKF) method, which is based on spatial and temporal smoothness, in the localization of a focal seizure's onset and compares its results to those of LORETA. The main finding of the study was that the STKF with an autoregressive model of order two significantly outperformed LORETA in the accuracy and consistency of the localization, provided that the source space consists of a whole-brain volumetric grid. In the future, these promising results will be confirmed using data from more patients and performing statistical analyses on the results. Furthermore, the effects of the temporal smoothness constraint will be studied using different types of focal seizures.

The choice of the source space and the Laplacian matrix in LORETA and the spatio-temporal Kalman filter EEG inverse methods.

The discretization of the brain and the definition of the Laplacian matrix influence the results of methods based on spatial and spatio-temporal smoothness, since the Laplacian operator is used to define the smoothness based on the neighborhood of each grid point. In this paper, the results of low resolution electromagnetic tomography (LORETA) and the spatiotemporal Kalman filter (STKF) are computed using, first, a greymatter source space with the standard definition of the Laplacian matrix and, second, using a whole-brain source space and a modified definition of the Laplacian matrix. Electroencephalographic (EEG) source imaging results of five inter-ictal spikes from a pre-surgical patient with epilepsy are used to validate the two aforementioned approaches. The results using the whole-brain source space and the modified definition of the Laplacian matrix were concentrated in a single source activation, stable, and concordant with the location of the focal cortical dysplasia (FCD) in the patient's brain compared with the results which use a grey-matter grid and the classical definition of the Laplacian matrix. This proof-of-concept study demonstrates a substantial improvement of source localization with both LORETA and STKF and constitutes a basis for further research in a large population of patients with epilepsy.

Analysis of epileptic seizure count time series by ensemble state space modelling.

We propose an approach for the analysis of epileptic seizure count time series within a state space framework. Time-dependent dosages of several simultaneously administered anticonvulsants are included as external inputs. The method aims at distinguishing which temporal correlations in the data are due to the medications, and which correspond to an unrelated background signal. Through this method it becomes possible to disentagle the effects of the individual anticonvulsants, i.e., to decide which anticonvulsant in a particular patient decreases or rather increases the number of seizures.

Impulse response function based on multivariate AR model can differentiate focal hemisphere in temporal lobe epilepsy.

The purpose of this study is to propose and investigate a new approach for discriminating between focal and non-focal hemispheres in intractable temporal lobe epilepsy, based on applying multivariate time series analysis to the discharge-free background brain activity observed in nocturnal electrocorticogram (ECoG) time series. Five unilateral focal patients and one bilateral focal patient were studied. In order to detect the location of epileptic foci, linear multivariate autoregressive (MAR) models were fitted to the ECoG data; as a new approach for the purpose of summarizing these models in a single relevant parameter, the behavior of the corresponding impulse response functions was studied and described by an attenuation coefficient. In the majority of unilateral focal patients, the averaged attenuation coefficient was found to be almost always significantly larger in the focal hemisphere, as compared to the non-focal hemisphere. Also the amplitude of the fluctuations of the attenuation coefficient was significantly larger in the focal hemisphere. Moreover, in one patient showing a typical regular sleep cycle, the attenuation coefficient in the focal hemisphere tended to be larger during REM sleep and smaller during Non-REM sleep. In the bilateral focal patient, no statistically significant distinction between the hemispheres was found. This study provides encouraging results for new investigations of brain dynamics by multivariate parametric modeling. It opens up the possibility of relating diseases like epilepsy to the properties of inconspicuous background brain dynamics, without the need to record and analyze epileptic seizures or other evidently pathological waveforms.

Combining EEG and MEG for the reconstruction of epileptic activity using a calibrated realistic volume conductor model.

To increase the reliability for the non-invasive determination of the irritative zone in presurgical epilepsy diagnosis, we introduce here a new experimental and methodological source analysis pipeline that combines the complementary information in EEG and MEG, and apply it to data from a patient, suffering from refractory focal epilepsy. Skull conductivity parameters in a six compartment finite element head model with brain anisotropy, constructed from individual MRI data, are estimated in a calibration procedure using somatosensory evoked potential (SEP) and field (SEF) data. These data are measured in a single run before acquisition of further runs of spontaneous epileptic activity. Our results show that even for single interictal spikes, volume conduction effects dominate over noise and need to be taken into account for accurate source analysis. While cerebrospinal fluid and brain anisotropy influence both modalities, only EEG is sensitive to skull conductivity and conductivity calibration significantly reduces the difference in especially depth localization of both modalities, emphasizing its importance for combining EEG and MEG source analysis. On the other hand, localization differences which are due to the distinct sensitivity profiles of EEG and MEG persist. In case of a moderate error in skull conductivity, combined source analysis results can still profit from the different sensitivity profiles of EEG and MEG to accurately determine location, orientation and strength of the underlying sources. On the other side, significant errors in skull modeling are reflected in EEG reconstruction errors and could reduce the goodness of fit to combined datasets. For combined EEG and MEG source analysis, we therefore recommend calibrating skull conductivity using additionally acquired SEP/SEF data.