Oscillatory potential (OP) visualization is hampered by intrusion from
the a-wave and b-wave. As a consequence, we isolated the OP by
digitally subtracting the a-wave and b-wave from the raw data and
band-pass filtering the resultant waveform
21 (55–250 Hz,
512-tap, finite impulse response filter, Blackman window).
After OP extraction, we modeled the data in the time domain using a
Gabor function
(equation 2c) , which represents the multiplication of a
Gaussian envelope
(equation 2a) with a sine wave carrier
(equation 2b) .
22 \[g(x){=}a\ {\cdot}\ \mathrm{e}^{\mathrm{-1/2\ {\cdot}\ }{[}((x-m)/s)^{2}{]}}\]
\[f(x){=}\mathrm{sin\ 2\ {\cdot}\ }h\ {\cdot}\ x\ {\cdot}\ {\pi}{+}p\]
\[\mathrm{Gabor}(x){=}g(x)\ {\cdot}\ f(x)\]
As a function of time (
x), the Gaussian
envelope
(equation 2a) is described by its maximum amplitude
(
a, OP amplitude in microvolts), peak envelope position
(
m, OP implicit time, in milliseconds), and spread (seconds,
milliseconds). The sine wave carrier
(equation 2b) is described by its
frequency (
h, in hertz) and phase relative to the start of
the waveform (
p, in degrees). Fitting was achieved by
floating all parameters and minimizing the mean-square-error term,
using a customized Levenberg-Marquardt optimization routine. Such
modeling presumes no physiological basis but allows OPs to be easily
compared between control and treatment groups. The model provides an
excellent fit to the extracted waveforms, as shown in
Figure 4A .
Importantly, an excellent correlation is found between the amplitude of
the Gaussian envelope (
a in
equation 2a ) and the normal
parameters used to describe these oscillations, such as the amplitude
of the largest OP (
r 2 = 0.90) and the
root-mean-square amplitude for all OPs
(
r 2 = 0.91).