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.
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