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The Dynamics of EEG Entropy
M. Ignaccolo1, M. Latka2, W. Jernajczyk3, P. Grigolini4 and B.J. West,1'5
1) Physics Department, Duke University, Durham, NC
2) Institute of Biomedical Engineering, Wroclaw University of Technology, Wroclaw, Poland
3) Department of Clinical Neurophysiology, Institute of Psychiatry and Neurology, Warsaw, Poland
4) Center for Nonlinear Science, University of North Texas, Denton, TX
5) Mathematics and Information Science Directorate, Army Research Office
(Dated: March 5, 2009)
EEG time series are analyzed using the diffusion entropy method. The resulting EEG entropy
manifests short-time scaling, asymptotic saturation and an attenuated alpha-rhythm modulation.
These properties are faithfully modeled by a phenomenological Langevin equation interpreted within
a neural network context.
In the nearly one hundred years since the electroencephalogram (EEG) was introduced into neuroscience there
have been a variety of methods used in attempts to establish a taxonomy of EEG patterns in order to delineate
the correspondence between brain wave patterns and brain activity. Over that time the single channel EEG time
series has been interpreted as consisting of relatively slow regular variations called signal and the relatively rapid
erratic fluctuations called noise. This separation implies that the signal contains information about the EEG channels
in the brain, whereas the erratic fluctuations are a property of a channel's environment and does not contain any
useful information. Recent studies have refined this engineering model and extracted information from the random
fluctuations by concentrating on what is believed to be the scaling behavior of the time series. Analysis of the second
moment of the single channel time series shows an algebraic scaling in time, i.e., (X(t)2 c t2H. The brackets denote
a suitably defined averaging procedure and detrended fluctuation analysis (DFA) , which measures the standard
deviation of the detrended fluctuations, has been the method of choice for the recent analysis of EEG data . Buiatti
et al.  use DFA to show that specific task-demands can modify the temporal scale-free dynamics of the ongoing
brain activity as measured by the scaling index.
The 'signal' parts of the EEG time series are called waves or rhythms. The nature and scope of these waves have
been widely investigated, see Basar  for a review. The alpha rhythm (7-12 Hz) has been shown to be typical of
awake individuals under no stimulation. Basar, along with colleagues, have developed an integrative theory of alpha
oscillations in brain functioning . They hypothesize that there is not one, but several alpha generators distributed
within the brain and note that the alpha rhythm may act as a nonlinear clock in the manner suggested by Wiener 
to serve as a gating function to facilitate the association mechanisms in the brain.
As a measure of order/disorder, entropy has been used to characterize EEG signals. Schlogl et al.  measured the
entropy of 16 bits EEG polysomnograhic records and found it in the range of 8-11 bits. Inouye et al.  employed
spectral entropy, as defined by the Fourier power spectrum, but the fact that EEG time series are not stationary, in
the sense that the autocorrelation function is not simply a function of the two-time difference, obviates the use of
Fourier transforms. Subsequently, wavelet entropy was used by Rosso et al.  to study the order/disorder dynamics
in short duration EEG signals including evoked response potentials. Patel et al. , using a combination of MRI and
entropy maximization, demonstrated that the generators of alpha rhythm are mainly concentrated over the posterior
regions of the cortex.
The diffusion entropy (DE) method  has been successfully used to discriminate between the contributions of
the low frequency waves (signal) and the high frequency fluctuations (noise), e.g., the influence of the seasons on the
daily number of teen births in Texas  and the effect of solar cycles on the statistics of solar flares . In this
Letter we use the DE method to provide insight on the low/high frequency dynamics of EEG time series.
II. EEG ANALYSIS
Each single channel recording of the EEG time series consists of a sequence of N + 1 data points, and the difference
between successive data points is denoted by ( for j = 1, 2, .., N. For the DE analysis a set of stochastic variables
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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/m1/1/?rotate=90: accessed June 26, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.