Application to biomedical signals - Practical Biomedical Signal Analysis Using MATLAB - page 158

cially to correlate their characteristics with the spike occurrence.

4.1.7.3.5 Event-related time-varying functional connectivity The methods de-
scribed in previous paragraphs concentrated on the analysis of ERD/ERS that is on
the variations of power related to a certain event. The other phenomenon, which is
equally interesting, is related to the event variation of couplings between the differ-
ent distant populations. In this respect the methods derived from multivariate auto-
regressive model (Sect. 3.2) can be very useful.
Phenomenon involving the phase information, namely a time varying directional
connectivity between neural population is recently the focus of interest. In the past
the bivariate methods usually based on coherences were used, however the demon-
stration that they give misleading results (Sect. 3.5) turned attention to the methods
based on extension of Granger causality defined in the framework of MVAR.
When applying MVAR one has to keep in mind that the number of model pa-
rameters should be preferably smaller by an order of magnitude than the number of
samples in the data window. Number of MVAR parameters is pk 2 (where p —model
order, k —number of channels), number of data points is kN (where N —number of
points in the window). Effectively we get a condition: pk 1.

In order to get a time-varying estimate we need to use a short time window, which
is in contradiction with the above relation. The number of data points may be ef-
fectively increased by means of ensemble averaging over realizations when they
are available from repeated trials of the experiment. To solve the problem of the
time-varying estimate two approaches may be distinguished: sliding window or time-
continuous fit of MVAR model, which may be performed adaptively, e.g., by means
of a Kalman filter. Usually in both approaches ensemble averaging is applied.
The Kalman filter approach was extended for multivariate non-stationary pro-
cesses by Arnold et al. [Arnold et al., 1998]. However, the computation effort of
the technique is high as was shown, e.g., in [Kaminski et al., 2010]. In the case
of the Kalman filter the computation time rises fast with the number of channels
and number of realizations. Taking into account that the estimation of the errors is
usually based on bootstrap methods, which involves repetition of computations hun-
dreds of times, the computation time in the Kalman filter approach (much longer
than in case of SDTF) can be a serious drawback, which hampers its wide appli-
cation. In [Kaminski et al., 2010] it was also reported that the Kalman method has
difficulties in adaptation for a large number of channels.
Another adaptive method of MVAR fitting is the least mean square (LMS) al-
gorithm. The adaptation capability was found to be better for its modification—
recursive least-square (RLS) algorithm with forgetting factor [Moller et al., 2001].
This algorithm takes into consideration a set of EEG epochs and the RLS estima-
tion is controlled (similarly to Kalman filter) by the value of adaptation factor. The
algorithm was initially used for estimation of multivariate coherences [Moller et al.,
2001]. It was also applied for estimation of bivariate Granger causality in the Stoop

/ N < 0 .