Multivariate approaches to extract neural interrelations between EEG channels Amir Omidvarnia 22 Oct. 2010 Outline Introduction to multivariate AR models Multivariate connectivity based on time-invariant methods Non-parametric approaches Parametric approaches Multivariate connectivity based on time-varying methods Non-iterative approaches Iterative approaches Introduction Methods based on the estimation of coherence/cross- correlation functions are widely used to extract mutual and synchronized activities between EEG channels. Most of these methods use multivariate AR models to define proper criteria. Detecting the direction of the information flow between EEG channel pairs is one of most important objectives of the newly suggested methods. As the EEG signal is non-stationary, time-varying MVAR based solutions should be taken into consideration. Multivariate AR models The MVAR model with N variables is defined by the equations : Multivariate AR models x1(n), . . ., xN(n) are the current values of each time series. a11(i) . . . aNN(i) are predictor coefficients at delay i. M is the model order, indicating the number of previous data points used for modelling. e1(n) . . . eN(n) are one-step prediction errors  Multivariate connectivity based on time-invariant methods Multivariate connectivity based on time-invariant methods The input signal is considered as stationary and statistically time-invariant. These methods can be divided into two main groups; Non-parametric measures Extract multivariate Cross-Power Spectral Density matrix using Fourier transforms of the signals directly. Parametric measures Extract multivariate Cross-Power Spectral Density matrix using the fitted MVAR model on the multichannel data. Multivariate connectivity based on time-invariant methods (cont.) Non-parametric measures Ordinary Coherence: Reflects the correlation (linear relationship) between channels k and j in the frequency domain . Partial Coherence: Removes linear influences from all other channels in order to detect directly interaction between channels i and j [2,3]. Multiple Coherence: Describes the proportion of the power of the i’th channel at a certain frequency which is explained by the influences of all other channels (the rest) [4,5]. Multivariate connectivity based on time-invariant methods (cont.) Corresponding multichannel matrices of the previously indicated criteria are symmetric. There is no difference between the measures of channel i- channel j and channel j-channel i pairs. In other words, none of the ordinary, partial and multiple coherence measures show the direction of the information flow between channels. Multivariate connectivity based on time-invariant methods (cont.) Parametric approach MVAR coefficient matrices need to be transferred into the frequency domain: Multivariate connectivity based on time-invariant methods (cont.) Parametric approach Cross-Power Spectral Density and Transfer Function matrices can be estimated based on a fitted MVAR model on the multichannel data : ∑: Noise covariance matrix of the fitted MVAR model Multivariate connectivity based on time-invariant methods (cont.) Granger causality: The main idea originates from this fact that a cause must precede its effect [12,13]. A dynamical process X is said to Granger-cause a dynamical process Y, if information of the past of process X enhances the prediction of the process Y compared to the knowledge of the past of process Y alone. Granger causality can be investigated by using MVAR models. Multivariate connectivity based on time-invariant methods (cont.) Parametric measures Granger Causality Index (GCI): A time-domain criterion which investigates directed influences from channel i to channel j in a multichannel dynamical system . In an AR(2) model including two channels, if channel X causes channel Y, the variance of the prediction error decreases for two-dimensional modelling, because the past of channel X improves the prediction of channel Y [14,15]. If X Granger-causes Y, F will be positive, otherwise F is negative. Multivariate connectivity based on time-invariant methods (cont.) Parametric measures All parametric measures are defined in the frequency domain based on S, A and H matrices. Directed Coherence: A unique decomposition of the ordinary coherence function which represents the feedback aspects of the interaction between channels [6,7]. Directed Transfer Function (DTF): The same as Directed Coherence when the effect of the noise is ignored (σjj=1) [6,8]. Multivariate connectivity based on time-invariant methods (cont.) Parametric measures direct Directed Transfer Function (dDTF): DTF shows all direct and cascade flows together. For example, both propagation 1→2→3 and propagation 1→3 are reflected in the DTF results. dDTF can separate direct flows from indirect flows [9,10]. dDTF is the product of the non-normalized DTF and partial coherence over frequency : Partial Directed Coherence (PDC): Provides a frequency description of Granger causality. This criterion is defined using the MVAR –derived form of the partial coherence function . Partial Coherence Partial Directed Coherence Multivariate connectivity based on time-invariant methods (cont.) Example of DTF and PDC functions : Multivariate connectivity based on time-invariant methods (cont.) Difference of the DTF and PDC : Directed Transfer Function is normalized by the sum of the influencing processes (i’th row of the Transfer Function matrix H). Partial Directed Coherence is normalized by the sum of the influenced processes (j’th column of the MVAR matrix A). Multivariate connectivity based on time-invariant methods (cont.) Generalized Partial Directed Coherence (GPDC) This criterion combines the idea of DTF (to show the influencing effects) and PDC (to reflect influenced effects) between channel i and channel j [10,11]. Time-frequency representations of the coherence measures Time-Frequency Coherence Estimate (TFCE) Ordinary coherence measure can be extended to the time- frequency domain for the class of positive TFDs . Time-frequency representations of the coherence measures (cont.) Short-time DFT and PDC The whole data is divided into short overlapping time windows. Then either the DFT function or the PDC function is extracted in each window. Finally, a time-frequency representation of the information flow can be obtained for each pair combination of channels. Bootstrap or surrogate data approaches can be used to obtain statistical significance of the results [19,20]. Multivariate connectivity based on time-varying methods Multivariate connectivity based on time-varying methods Time-varying MVAR model estimation Least-Squared based algorithms have been suggested to estimate time-varying MVAR coefficient matrices for several realizations of the multichannel signal (e.g., ERP and VEP signal analysis) . If there is only one realization of the signal in each step (e.g., spontaneous EEG), both Least-square approaches and Kalman filtering based algorithms have been proposed . Multivariate connectivity based on time-varying methods (cont.) Instantaneous EEG coherence  Similar to the previous study , time-varying MVAR matrix is updated in each step for a batch of ERP signals using a RLS-based approach. In each step, ordinary coherence and multiple coherence measures are extracted from the MVAR model parameters. Finally, time-frequency representations of the coherence values can be plotted. Multivariate connectivity based on time-varying methods (cont.) Instantaneous EEG coherence  K’th epoch of the M-channel system Wn = (Yn1,…,Ynp) All MVAR parameters in time n Multivariate connectivity based on time-varying methods (cont.) Time-varying Granger Causality  In a recursive method based on RLS algorithm and for a batch of multichannel signals (ERP data), noise covariance matrix of the MVAR model is updated and Granger causality index is computed using the time-varying covariance matrix ∑. This algorithm is not applicable for spontaneous EEG, as there is only one realization of the signal in each step. Multivariate connectivity based on time-varying methods (cont.) Time-varying PDC based on Extended Kalman Filter  MVAR(M,p) is re-written as M*p AR(1) models. State space equations are extracted using the equivalent AR(1) models. Another state space is considered for AR coefficients (the coefficients are considered as time-varying processes). Two Kalman filters are applied on two state spaces to estimate time- varying AR(1) coefficients and states. Multivariate connectivity based on time-varying methods (cont.) Time-varying PDC based on Extended Kalman Filter  1 2 3 Multivariate connectivity based on time-varying methods (cont.) Time-varying PDC based on Extended Kalman Filter  General form of the Kalman filter Multivariate connectivity based on time-varying methods (cont.) Time-varying PDC based on Extended Kalman Filter  Conclusion Time-invariant coherence measures based on the time- invariant MVAR models are not sufficient to investigate the interrelations of the brain. Least-Square based algorithms as well as Kalman filtering tools have been suggested for adaptive estimation of timevarying MVAR coefficients in spontaneous EEG signals. 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