The Dynamics of EEG Entropy Page: 3
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12
11
8-I
10
8 "
0
7[
7 - 0
6 O
0.01 0.1 1 10
t (seconds)
FIG. 1: The diffusion entropy S(t) calculated using the increments of the channels 01, 02, C3 and C4 for one of the
20 subjects considered in this study. The inset depicts the pdfs psat(x)=p(x, t = 2000) for each channel: squares
(01), circles (02), upward triangles (C3), and downward triangles (C4).
The values of the frequencies fj and amplitudes Aj are calculated as follow. First, we calculate the spectral density
in the time-frequency domain of time series of EEG increments with a time resolution t and a frequency resolution
Af by means of a Windowed Fourier Transform. The spectral density, called the spectrogram (e.g. [17]), is a three-
dimensional plot of the spectrum of the EEG increments (4 as it changes over time. Then, for each time interval
of duration t we consider the range of frequencies of the alpha waves, 7-12 Hz, and find which frequency has the
maximum amplitude in the spectrogram. This procedure defines the frequency and the amplitude of the time interval
considered.
Panel (a) of Fig. 2 shows the spectrogram relative to the increments (j of the channel 01 for the same subject as
in Fig. 1. Panels (b) and (c) of Fig. 2 show respectively the sequence of amplitudes Aj (normalized to a maximum
amplitude of 1) and of frequencies fj calculated using the procedure described above. Without an a priori knowledge
of the typical duration of an alpha wave packet, we set the stability time t. of Eq. (4) equal to 0.5s. A time resolution
of 0.5s and a frequency resolution of ~ 0.5Hz in the spectrogram represent a reasonable time-frequency localization
for our purposes.
Consider the model case where Aj=O, for all j, no modulation is present, and Eq. (4) is the Ornstein-Uhlenbeck
Langevin equation. In this case the standard deviation of the variable X is o(t) D (1 - e-2At) /. Consequently,
for t <1/A the entropy increases as S(t)= C'+ log2 t, with CC+log2 and a linear-log plot yields a straight line
of slope 6=0.5. For t >/A the entropy reaches the saturation level S(t)=C+log2 ~, yielding an entropy structure
similar to that of the EEG data depicted in Fig. 1 without the modulation being present.
When the modulation is present Aj O, Eq. (4) is numerically integrated, and the increments of the dynamic variable
X are processed using the DE algorithm. In Fig. 3, we compare the DE obtained via Eq. (4) with that of the channels
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Ignaccolo, Massimiliano; Latka, Miroslaw; Jernajczyk, Wojciech; Grigolini, Paolo & West, Bruce J. The Dynamics of EEG Entropy, article, March 5, 2009; [Berlin, Germany]. (https://digital.library.unt.edu/ark:/67531/metadc132967/m1/3/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.