interrelations between EEG channels

pucefakeAI and Robotics

Nov 30, 2013 (3 years and 7 months ago)

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Amir Omidvarnia

22 Oct. 2010

Multivariate approaches to extract neural
interrelations between EEG channels


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 [1]:

Multivariate AR models


x
1
(n), . . .,
x
N
(n)
are the current values of each time series.


a
11
(
i
) . . .
a
NN
(
i
)
are predictor coefficients at delay
i
.


M

is the model order, indicating the number of previous data
points used for modelling.


e
1
(n) . . .
e
N
(n)
are one
-
step prediction errors [1]



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 [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].

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 [6]:

∑: 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 [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.

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 (
σ
jj
=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 [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

Multivariate connectivity based on
time
-
invariant methods (cont.)


Example of DTF and PDC functions [6]:

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
).

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 [18].

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)
[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].

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.


Multivariate connectivity based on
time
-
varying methods (cont.)


Instantaneous EEG coherence [16]


K’th

epoch of the M
-
channel system

W
n

= (Y
n1
,…,
Y
np
)

All MVAR parameters in time
n

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.


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.



Multivariate connectivity based on
time
-
varying methods (cont.)


Time
-
varying PDC based on Extended Kalman Filter
[21]



1

2

3

Multivariate connectivity based on
time
-
varying methods (cont.)


Time
-
varying PDC based on Extended Kalman Filter
[21]



General form of the Kalman filter


Multivariate connectivity based on
time
-
varying methods (cont.)


Time
-
varying PDC based on Extended Kalman Filter
[21]


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.

References


1.

Hytti
, H., R.
Takalo
, and H.
Ihalainen
,
Tutorial on Multivariate
Autoregressive Modelling.

Journal of Clinical Monitoring and Computing,
2006. 20(2): p. 101
-
108.



2.

Schelter
, B.,
Analyzing

multivariate dynamical processes


From linear to
nonlinear approaches.

PhD thesis, University of Freiburg, 2006.



3.

Deshpande
, G., et al.,
Multivariate Granger causality analysis of
fMRI

data.

Human Brain Mapping, 2009. 30(4): p. 1361
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1373.



4.

Franaszczuk
, P.J., K.J.
Blinowska
, and M.
Kowalczyk
,
The application
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, L.A. and K.
Sameshima
,
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Blinowska
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, A., et al.,
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, R., et al.,
Methods for Determination of Functional Connectivity in
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References


11.

Baccala
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in
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12.

Granger, C.W.J.,
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13.

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, M., et al.,
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14.

Hesse, W., et al.,
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-
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, J.,
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, E., et al.,
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-
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