01.05.2011 | Special Issue - Review | Ausgabe 5/2011 Open Access

# Review of the methods of determination of directed connectivity from multichannel data

- Zeitschrift:
- Medical & Biological Engineering & Computing > Ausgabe 5/2011

## 1 Introduction

## 2 Bivariate measures of connectivity

### 2.1 Correlation and coherence

_{ xy }has maximum, we can assume that signal Y is delayed by τ in respect to signal X.

### 2.2 Non-linear measures of connectivity

## 3 Granger causality and granger causality index

_{1}:

_{1}(after including series Y to the prediction) is lower than the variance of e we say that Y causes X in the sense of Granger causality. Similarly we can say that X causes Y in the sense of Granger causality when the variance of e

_{ 2 }is reduced after including series X in the prediction of series Y:

## 4 Multivariate estimators of directedness

### 4.1 Multivariate autoregressive model

_{ ij }is a minor of spectral matrix (matrix of spectra and cross-spectra) with the i-th row and j-th column removed. Partial coherence is non-zero only when the given relation between channels is direct. If a signal in a given channel can be explained by a linear combination of some other signals of the set, the partial coherence between them will be low.

### 4.2 Directed transfer function

_{ ij }(f)) showing direct propagation from channel j to i is defined as:

_{ ij }(f) is partial coherence. χ

_{ ij }(f) has a nonzero value when both functions F

_{ ij }

^{2}(f) and C

_{ ij }

^{2}(f) are non-zero, in that case there exists a direct causal relation between channels j → i.

### 4.3 PDC

_{ ij }(f) is an element of A(f)—a Fourier transform of MVAR model coefficients A(t), where a

_{ j }(f) is j-th column of A(f) and the asterisk denotes the transpose and complex conjugate operation. Although it is a function operating in the frequency domain, the dependence of A(f) on the frequency has not a direct correspondence to the power spectrum. From normalization condition it follows that PDC takes values from the interval [0,1]. PDC shows only direct flows between channels. Unlike DTF, PDC is normalized to show a ratio between the outflow from channel j to channel i to all the outflows from the source channel j, so it emphasizes rather the sinks, not the sources.

### 4.4 Bivariate versus multivariate estimators of connectivity

## 5 Estimators of dynamical propagation

### 5.1 Short-time DTF

_{ T }(k-number of channels, N

_{ T }-number of points in the data window) has to be bigger (preferably by order of magnitude) than the number of parameters, which in case of MVAR is equal to k

^{ 2 }p (p model order). In order to evaluate dynamics of the process short data window has to be applied, which requires the increase of the number of the data points. This may achieved by means of the repetition of the experiment. In case when multiple trials are available, it is possible to apply ensemble averaging over realizations. We divide a non-stationary recording into shorter time windows, short enough to treat the data within a window as quasi-stationary. The estimation of MVAR coefficients is based on calculation of the correlation matrix R

_{ ij }of k signals X

_{ i }from multivariate set [21]. We calculate the correlation matrix between channels for each trial separately. The resulting model coefficients are based on the correlation matrix averaged over trials. The correlation matrix has a form: