The Dynamics of EEG Entropy Page: 2
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Xk (t) is constructed from the differenced data points in the following way
k+t
Xk (t) j, k = 1,2,..,N- t +1 (1)
to obtain M = N- t +1 replicas of a stochastic trajectory by means of overlapping windows of length t. This ensemble
of trajectories, generated by the EEG time series, is used to construct the histogram using the number of trajectories
falling in a specified interval to estimate the pdf p(x, t). The pdf is then used to calculated the information entropy, a
quantity introduced in discrete form for coding information by Shannon [14] and in continuous form for studying the
problem of noise and messages in electrical filters by Wiener [15]. We use the latter form here,
S(t) = - p(x, t) log2 p(x, t)dx. (2)
The entropy S(t) of Eq. (2) assumes a simple analytical form if the pdf of the diffusion process satisfies the scaling
relation:
)p(xt F -> S(t) C + log2 o(t) (3)
where C=- f F(y) log2 F(y)dy is a constant and a (t) for a gaussian diffusion process is the time-dependent standard
deviation a(t). More generally an a- stable Levy process also scales in this way, in which case a(t) is more general
than the standard deviation of the underlying process.
If the time series were scaling, as assumed in a number of analyses of EEG data [2], the 'variance' would be a(t)octs
and the entropy graphed versus time on log-linear graph paper would increase linearly with slope b. Consequently,
the way in which the entropy for a time series scales is indicative of the scaling behavior of the time series. Note that
in a simple diffusive process this index is equal to the one obtained from the second moment, that is, b=H. However,
in general, even when there is scaling 6-H and in the case of EEG time series we establish that there is no scaling at
all.
We now consider EEG signals of 20 awake individuals in the absence of external stimulations (quiet, closed eyes).
EEG signals were recorded using the 10-20 international recording scheme. For 8 individuals only the channels
O1,02,C3 and C4 were recorded, for the remaining 12 individuals all the channels are available. To have a coherent
database, we restrict our analysis to the channels O1,02,C3 and C4, which are the channels traditionally used in sleep
studies. The sampling frequency of all EEG records is 250Hz. Durations of EEG records vary from 55s to 400s with
an average duration of 128s.
Fig. 1 shows the DE of the EEG increments for the somnographic channels O1,02,C3 and C4 of a single individual.
We see how for each channel the DE: 1) reaches a saturation level for each channel, 2) has an "alpha" (~7.6 HZ in the
case of this individual) modulation which is attenuated with time, and 3) has a small amplitude residual asymptotic
modulation. The early-time modulation, with variable frequency in the alpha range and variable amplitude, is observed
in the somnographic channels for all subjects. The saturation effect is present in all channels for all subjects and it
should be pointed out that this saturation is neither a consequence of the finite length of the time series, nor of the
finite amplitude of the EEG signal. In fact if the data points were randomly rearranged, thereby destroying any long
time correlation in the time series, the EEG entropy does not saturate. Consequently, this saturation effect is due to
brain dynamics and is not an artifact of the data processing. Robinson [16] observed this saturation in the calculation
of the EEG second moment and interpreted it as being due to dendritic filtering. The inset in Fig. 1 depicts the pdfs
psat (x), after the entropy saturation is attained. These distributions have approximatively exponential tails.
III. EEG MODEL
The simplest dynamic model, which includes fluctuations, modulation and dissipation, in short, all the properties
displayed in Fig. 1, has the form of a Langevin equation. We assume a dissipative linear dynamic process X(t), i.e.,
an Ornstein-Uhlenbeck process, with a periodic driver having a random amplitude and frequency and an additive
random force rj (t) which is a delta correlated Gaussian process of strength D:
dX(t) = -AX(t) + (t) + ZAjx [Iy,s] sin [2 fjt] (4)
j=0
The coefficient A>0 defines a negative feedback, x[Ij,s]= when its argument is in the time interval Ij,s =[jt5, (j + 1)t5]
and is zero otherwise, and t5 is the 'stability' time after which a new frequency fj and a new amplitude Aj are selected.
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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]. (digital.library.unt.edu/ark:/67531/metadc132967/m1/2/: accessed October 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.