Biomedical Engineering Reference
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where
r
is the neuron mean firing rate. The nor-
malization factor in (6) ensures unit energy of
the white-noise or “random” spike train (with
equiprobable randomly distributed inter-spike
intervals). Thus for a random spike train the energy
is homogeneously distributed over all frequencies
(periods)
Two linearly independent spike trains have
vanishing coherence, whereas
C
(
p, z
) = 1 indicates
a perfect linear relationship between the spike
trains at the scale p and localization
z
. Since we
are interested in studying the coherence level (or
functional coupling) between the stimulus events
and neural response we focus on frequency band
corresponding to the stimulus frequency, i.e., on
f
=
1 Hz, which corresponds to the scale
p =
1s. To
resolve well this frequency with minimal loosing
in time resolution we set
k
0
= 2. Then from (3)
δw
∼ ½ and
δt ∼ 2
.
Although large amplitude of the coherence
usually indicates the presence of a consistent phase
relationship (coupling) between two spike trains
in a given time interval, it is also possible that this
may be a random casual variation in spike trains.
Thus one should cross-check statistical signifi-
cance of the observed coherence. The statistical
significance of the wavelet coherence can be as-
sessed relative to the null hypotheses that the two
spike trains generated by independent stationary
processes with given distribution of inter-spike
intervals (ISIs) are not coherent. To evaluate the
level of significance one can use surrogate data
test (Theiler et al., 1992; Schreiber and Schmitz,
2000) with Monte-Carlo simulation to establish
95% confidence interval. The surrogate spike
trains can be obtained from the original one by
randomizing phase relations keeping intact other
first order characteristics by means of, for instance,
shuffling ISIs. To conclude positively on the con-
nectivity between the stimulus train and neuronal
response their coherence should be higher than
the obtained significance level.
E p
=
. Consequently, we quantify
the power distribution in the train under study in
units of the power of the random spike train with
the same mean firing rate.
The global wavelet spectrum can be obtained
from (6) by time averaging of the local (time
dependent) spectrum:
(
)
1
z
T
=
∫
E
(
p
)
E p z
(
,
) d
z
(7)
0
It provides an unbiased and consistent esti-
mation of the true power spectrum (Percival,
1995).
Dealing with two spike trains N and M
by analogy with the Fourier cross-spectrum
we can introduce the wavelet cross-spectrum
*
W p z W W k r
=
. Then a normalized
measure of association between two spike trains
is the wavelet coherence (Grinsted et al., 2004):
(
,
)
/
NM
N
M
0
N
M
2
(
)
S W
(
p z
,
) /
p
NM
C
(
p z
,
)
=
NM
(
) (
)
S
E
(
p z
,
) /
p S
E
(
p z
,
) /
p
N
M
(8)
where S(•)is a smoothing operator (see for details:
Torrence and Webster, 1998; Grinsted et al., 2004).
The coherence definition (8) can eventually give
artificially high values of coherence in the case of
infinitesimally small values of the power spectrum
of either or both signals (i.e., when
E(
p,z
) ∼ 0
). To
avoid this problem in numerical calculations we
employ thresholding procedure setting to zero
the coherence when either of the power values is
below a threshold.
Stimulus Period Band and Power
Spectrum of Ultraslow Oscillation of the
Wavelet Coherence
The wavelet coherence allows studying tem-
poral structure and variation of the functional
coupling among stimuli and neural response.
To quantify this variation we average the neu-
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