Andreas Galka

A solution to the dynamical inverse problem of EEG generation using spatiotemporal Kalman filtering

We present a new approach for estimating solutions of the dynamical inverse problem of EEG generation. In contrast to previous approaches, we reinterpret this problem as a filtering problem in a state space framework; for the purpose of its solution, we propose a new extension of Kalman filtering to the case of spatiotemporal dynamics. The temporal evolution of the distributed generators of the EEG can be reconstructed at each voxel of a discretisation of the gray matter of brain. By fitting linear autoregressive models with neighbourhood interactions to EEG time series, new classes of inverse solutions with improved resolution and localisation ability can be explored. For the purposes of model comparison and parameter estimation from given data, we employ a likelihood maximisation approach. Both for instantaneous and dynamical inverse solutions, we derive estimators of the time-dependent estimation error at each voxel. The performance of the algorithm is demonstrated by application to simulated and clinical EEG recordings. It is shown that by choosing appropriate dynamical models, it becomes possible to obtain inverse solutions of considerably improved quality, as compared to the usual instantaneous inverse solutions.

Recursive penalized least squares solution for dynamical inverse problems of EEG generation

In the dynamical inverse problem of electroencephalogram (EEG) generation where a specific dynamics for the electrical current distribution is assumed, we can impose general spatiotemporal constraints onto the solution by casting the problem into a state space representation and assuming a specific class of parametric models for the dynamics. The Akaike Bayesian Information Criterion (ABIC), which is based on the Type II likelihood, was used to estimate the parameters and evaluate the model. In addition, dynamic low‐resolution brain electromagnetic tomography (LORETA), a new approach for estimating the current distribution is introduced. A recursive penalized least squares (RPLS) step forms the main element of our implementation. To obtain improved inverse solutions, dynamic LORETA exploits both spatial and temporal information, whereas LORETA uses only spatial information. A considerable improvement in performance compared to LORETA was found when dynamic LORETA was applied to simulated EEG data, and the new method was applied also to clinical EEG data. Hum. Brain Mapp. 21:221–235, 2004.

Evaluating the Performance of Kalman-Filter-Based EEG Source Localization

Electroencephalographic (EEG) source localization is an important tool for noninvasive study of brain dynamics, due to its ability to probe neural activity more directly, with better temporal resolution than other imaging modalities. One promising technique for solving the EEG inverse problem is Kalman filtering, because it provides a natural framework for incorporating dynamic EEG generation models in source localization. Here, a recently developed inverse solution is introduced, which uses spatiotemporal Kalman filtering tuned through likelihood maximization. Standard diagnostic tests for objectively evaluating Kalman filter performance are then described and applied to inverse solutions for simulated and clinical EEG data. These tests, employed for the first time in Kalman-filter-based source localization, check the statistical properties of the innovation and validate the use of likelihood maximization for filter tuning. However, this analysis also reveals that the filters existing space- and time-invariant process model, which contains a single fixed-frequency resonance, is unable to completely model the complex spatiotemporal dynamics of EEG data. This finding indicates that the algorithm could be improved by allowing the process model parameters to vary in space.

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.

Estimating the dimension of high-dimensional attractors: A comparison between two algorithms

Abstract We compare two algorithms for the numerical estimation of the correlation dimension from a finite set of vectors: the “classical” algorithm of Grassberger and Procaccia (GPA) and the recently proposed algorithm of Judd (JA). Data set size requirements and their relations to systematic and statistical errors of the estimates are investigated. It is demonstrated that correlation dimensions of the order of 6 can correctly be resolved on the basis of the about 100000 data points in the case of a continuous trajectory on a strange attractor; the minimum data set size is, however, noticeably dependent on the geometrical structure of the system from which the vectors were sampled.

A data-driven model of the generation of human EEG based on a spatially distributed stochastic wave equation

We discuss a model for the dynamics of the primary current density vector field within the grey matter of human brain. The model is based on a linear damped wave equation, driven by a stochastic term. By employing a realistically shaped average brain model and an estimate of the matrix which maps the primary currents distributed over grey matter to the electric potentials at the surface of the head, the model can be put into relation with recordings of the electroencephalogram (EEG). Through this step it becomes possible to employ EEG recordings for the purpose of estimating the primary current density vector field, i.e. finding a solution of the inverse problem of EEG generation. As a technique for inferring the unobserved high-dimensional primary current density field from EEG data of much lower dimension, a linear state space modelling approach is suggested, based on a generalisation of Kalman filtering, in combination with maximum-likelihood parameter estimation. The resulting algorithm for estimating dynamical solutions of the EEG inverse problem is applied to the task of localising the source of an epileptic spike from a clinical EEG data set; for comparison, we apply to the same task also a non-dynamical standard algorithm.

Measuring the Complexity of Currency Markets by Fractal Dimension Analysis

We use the theory of nonlinear dynamical systems to measure the complexity of currency markets by estimating the correlation dimension of the returns of the Dollar/Pound and Dollar/Yen daily exchange rates (the spot rates). We test the significance of the results by comparing them to correlation dimension estimates for surrogate time series, i.e. stochastic linear time series with the same power spectrum and amplitude distribution as given by the original data. We find discernible nonlinear structure in the returns of the Dollar/Pound daily rate.

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.

Effective Connectivity of Cortical Sensorimotor Networks During Finger Movement Tasks: A Simultaneous fNIRS, fMRI, EEG Study

Recently, interest has been growing to understand the underlying dynamic directional relationship between simultaneously activated regions of the brain during motor task performance. Such directionality analysis (or effective connectivity analysis), based on non-invasive electrophysiological (electroencephalography—EEG) and hemodynamic (functional near infrared spectroscopy—fNIRS; and functional magnetic resonance imaging—fMRI) neuroimaging modalities can provide an estimate of the motor task-related information flow from one brain region to another. Since EEG, fNIRS and fMRI modalities achieve different spatial and temporal resolutions of motor-task related activation in the brain, the aim of this study was to determine the effective connectivity of cortico-cortical sensorimotor networks during finger movement tasks measured by each neuroimaging modality. Nine healthy subjects performed right hand finger movement tasks of different complexity (simple finger tapping-FT, simple finger sequence-SFS, and complex finger sequence-CFS). We focused our observations on three cortical regions of interest (ROIs), namely the contralateral sensorimotor cortex (SMC), the contralateral premotor cortex (PMC) and the contralateral dorsolateral prefrontal cortex (DLPFC). We estimated the effective connectivity between these ROIs using conditional Granger causality (GC) analysis determined from the time series signals measured by fMRI (blood oxygenation level-dependent-BOLD), fNIRS (oxygenated-O2Hb and deoxygenated-HHb hemoglobin), and EEG (scalp and source level analysis) neuroimaging modalities. The effective connectivity analysis showed significant bi-directional information flow between the SMC, PMC, and DLPFC as determined by the EEG (scalp and source), fMRI (BOLD) and fNIRS (O2Hb and HHb) modalities for all three motor tasks. However the source level EEG GC values were significantly greater than the other modalities. In addition, only the source level EEG showed a significantly greater forward than backward information flow between the ROIs. This simultaneous fMRI, fNIRS and EEG study has shown through independent GC analysis of the respective time series that a bi-directional effective connectivity occurs within a cortico-cortical sensorimotor network (SMC, PMC and DLPFC) during finger movement tasks.

Testing for nonlinearity in high-dimensional time series from continuous dynamics

Abstract We address the issue of testing for nonlinearity in time series from continuous dynamics and propose a quantitative measure for nonlinearity which is based on discrete parametric modelling. The well-known problems of modelling continuous dynamical systems by discrete models are addressed by a subsampling approach. This measure should preferably be combined with conventional surrogate data testing. The performance of the test is demonstrated by application to simulated, heavily noise-contaminated time series from high-dimensional Lorenz systems, and to experimental time series from a high-dimensional mode of Taylor–Couette flow. We also discuss the discrimination power of the test under surrogate data testing, when compared with other well-tried test statistics.