Biomedical Engineering Reference
In-Depth Information
where
c
is a constant of the order of unity. There
is a trade-off between the frequency and time
resolutions, small values of
k
0
provide better time
resolution, while using big
k
0
improves frequency
resolution. The commonly adopted value is
k
0
=
1, and the limit
k
→ ∞
corresponds to the Fou-
rier transform. As we shall show further for our
purpose
Data Analysis
Summed peristimulus time histograms (PSTHs)
were calculated off-line with Spike 2 software
running on a PC computer, using 2 ms bins. We
considered a cellular response when the PSTH
area in first 50 ms after the stimulus onset was at
least 3 times bigger than the area corresponding
to 50 ms preceding the stimulus. The latency of
the sensory responses was measured as the time
elapsing between the sensory stimulus onset and
the largest peak in the PSTH. All data are shown
as mean ± standard error.
k
=
is more suitable.
Since we deal with finite-length time series
(spike trains) the evaluation of the wavelet spec-
trum (1) will have edge artifacts at the begin-
ning and the end of the time interval. The cone
of influence (COI) is the region in (
p, z
) plane
where edge effects cannot be ignored. We define
the size of the COI when the wavelet power is
dropped by
e
2
(Torrence & Compo, 1998), which
gives
2
0
Wavelet Transform of a Spike Train
z k
=
.
The spiking output (point process) of a neuron
can be represented as a series of δ-functions at
the times when action potentials occur:
The continuous wavelet transform (WT) of a signal
( )
0
x t
(e.g. spike train) involves its projection onto
a set of soliton-like basis functions obtained by
rescaling and translating along the time axis the
so called “mother wavelet”
Ψ
:
∑
x t
( )
=
(
t
−
t
)
.
(4)
i
i
1
t
−
z
∞
∫
*
W
(
p z
,
)
=
x t
( )
Ψ
d
t
Representation (4) allows us to estimate ana-
lytically the wavelet-coefficients:
,
(1)
p
p
−∞
where parameters
p
and
z
define the wavelet time
scale and localization, respectively. The choice
of the function ψ depends on the research aim.
To study rhythmic components of a signal the
Morlet-wavelet is well suited:
2
1
t
−
z
(
t
−
z
)
∑
W
(
p z
,
)
=
exp
−
j
2
i
exp
−
i
2
2
a
2
k p
p
i
0
(5)
Using the wavelet-transform (5) we can per-
form the time-frequency analysis of rhythmic
components hidden in the spike train. Wavelet-
coefficients can be considered as a parameterized
function
)
(
)
(
2
2
0
Ψ
(
y
)
=
exp
j
2
y
exp
−
y
/ 2
k
, (2)
z
, where
z
plays the role of time.
where
k
0
is a parameter, which can be tuned ac-
cording to physical phenomena under study. In
the wavelet transform (1) the time scale
p
plays
the role of the period of the rhythmic component.
Given a characteristic time scale (i.e., period)
p
the resolution of the wavelet in the time and
frequency domains is given by
W
( )
p
Wavelet Power Spectrum and
Coherence
The wavelet power spectrum of a spike train can
be defined by
c
t
=
ck p
,
=
1
,
(3)
0
2
k p
E p z
(
,
)
=
W
(
p z
,
)
(6)
0
rk
0
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