Biomedical Engineering Reference
In-Depth Information
that the examples in Figure 4.1(b, c) should have the highest values, followed by the
example of Figure 4.1(a). Wrong!
A closer look at Figure 4.1(c) shows that the spikes in both channels have a vari-
able time lag. Just picking up the times of the maximum of the spikes in the left and
right channels and calculating the lag between them, we determined that for Figure
4.1(b) the lag was very small and stable, between
5 ms—of the order of the
sampling rate of these signals—and the standard deviation was of 4.7 ms [8]. In con-
trast, for Figure 4.1(c) the lag was much more variable and covered a range between
−
−
5 and
+
20 and 50 ms, with a standard deviation of 14.9. This clearly shows that in exam-
ple B the simultaneous appearance of spikes is due to a generalized synchronization
across hemispheres, whereas in Figure 4.1(c) the bilateral spikes are not synchro-
nized and they reflect local independent generators for each hemisphere.
Interestingly, the signal of Figure 4.1(a) looks very noisy, but a closer look at
both channels shows a strong covariance of these seemingly random fluctuations.
Indeed, in a comprehensive study using several linear and nonlinear measures of
synchronization, it was shown that the synchronization values ranked as follows:
Sync
B
>
Sync
C
. This stresses the need for optimal measures to establish cor-
relation patterns.
Throughout this chapter, we will use these three examples to illustrate the use of
some of the correlation measures to be described. These examples can be down-
loaded from http://www.le.ac.uk/neuroengineering.
Sync
A
>
4.1
Cross-Correlation Function
The cross-correlation function is perhaps the most used measure of interdependence
between signals in neuroscience. It has been, and continues to be, particularly popu-
lar for the analysis of similarities between spike trains of different neurons.
Let us suppose we have two simultaneously measured discrete time series
x
n
and
y
n
,
n
=
1, …,
N
. The cross-correlation function is defined as
⎛
⎞
1
N
−
∑
τ
⎛
⎜
xxy y
−
⎞
⎟
−
()
c
τ
=
i
⎜
i
+
τ
⎟
(4.1)
xy
N
−
τ
σ
σ
⎝
⎠
i
=
1
x
y
σ
x
denote mean and variance and is the time lag. The cross-correla-
tion function is basically the inner product between two normalized signals (that is,
for each signal we subtract the mean and divide by the standard deviation) and it
gives a measure of the linear synchronization between them as a function of the time
lag
where
x
and
1, in the case of complete inverse correlation (that is,
one of the signals is an exact copy of the other with opposite sign), to
. Its value ranges from
−
1 for com-
plete direct correlation. If the signals are not correlated, then the cross-correlation
values will be around zero. Note, however, that noncorrelated signals will not give a
value strictly equal to zero and the significance of nonzero cross-correlation values
should be statistically validated, for example, using surrogate tests [9]. This basi-
cally implies generating signals with the same autocorrelation of the original ones
but independent from each other. A relatively simple way of doing this is to shift one
+
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