Multivariate approaches to extract neural interrelations between EEG channels

1. Multivariate approaches to extract neural interrelations between EEG channels Amir Omidvarnia 22 Oct. 2010

2. 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

3. 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.

4. Multivariate AR models • The MVAR model with N variables is defined by the equations [1]:

5. 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 [1]

6. Multivariate connectivity based on time-invariant methods

7. 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.

8. 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 [2]. • 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].

9. 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 ipairs. • In other words, none of the ordinary, partial and multiple coherence measures show the direction of the information flow between channels.

10. Multivariate connectivity based on time-invariant methods (cont.) • Parametric approach • MVAR coefficient matrices need to be transferred into the frequency domain:

11. 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 [6]: ∑: Noise covariance matrix of the fitted MVAR model

12. 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.

13. 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 [13]. • 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.

14. 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].

15. 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 [3]: • 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 [6]. Partial Coherence Partial Directed Coherence

16. Multivariate connectivity based on time-invariant methods (cont.) • Example of DTF and PDC functions [6]:

17. Multivariate connectivity based on time-invariant methods (cont.) • Difference of the DTF and PDC [2]: • 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).

18. 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].

19. 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 [18].

20. 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].

21. Multivariate connectivity based on time-varying methods

22. 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) [16]. • 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 [17].

23. Multivariate connectivity based on time-varying methods (cont.) • Instantaneous EEG coherence [16] • Similar to the previous study [14], 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.

24. Multivariate connectivity based on time-varying methods (cont.) • Instantaneous EEG coherence [16] K’th epoch of the M-channel system Wn = (Yn1,…,Ynp) All MVAR parameters in time n

25. Multivariate connectivity based on time-varying methods (cont.) • Time-varying Granger Causality [14] • 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.

26. Multivariate connectivity based on time-varying methods (cont.) • Time-varying PDC based on Extended Kalman Filter [21] • 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.

27. Multivariate connectivity based on time-varying methods (cont.) • Time-varying PDC based on Extended Kalman Filter [21] 1 3 2

28. Multivariate connectivity based on time-varying methods (cont.) • Time-varying PDC based on Extended Kalman Filter [21] • General form of the Kalman filter

29. Multivariate connectivity based on time-varying methods (cont.) • Time-varying PDC based on Extended Kalman Filter [21]

30. 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 time-varying MVAR coefficients in spontaneous EEG signals. • Extended Kalman filtering seems to be a good candidate for the problem, as it will consider both non-stationarity and non-linearity.

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