Dynamics of electroencephalogram entropy and pitfalls of scaling detection

M. Ignaccolo, M. Latka, W. Jernajczyk, P. Grigolini, and B. J. West

Phys. Rev. E 81, 031909 – Published 10 March 2010

Abstract

In recent studies a number of research groups have determined that human electroencephalograms (EEG) have scaling properties. In particular, a crossover between two regions with different scaling exponents has been reported. Herein we study the time evolution of diffusion entropy to elucidate the scaling of EEG time series. For a cohort of 20 awake healthy volunteers with closed eyes, we find that the diffusion entropy of EEG increments (obtained from EEG waveforms by differencing) exhibits three features: short-time growth, an alpha wave related oscillation whose amplitude gradually decays in time, and asymptotic saturation which is achieved after approximately 1 s. This analysis suggests a linear, stochastic Ornstein-Uhlenbeck Langevin equation with a quasiperiodic forcing (whose frequency and/or amplitude may vary in time) as the model for the underlying dynamics. This model captures the salient properties of EEG dynamics. In particular, both the experimental and simulated EEG time series exhibit short-time scaling which is broken by a strong periodic component, such as alpha waves. The saturation of EEG diffusion entropy precludes the existence of asymptotic scaling. We find that the crossover between two scaling regions seen in detrended fluctuation analysis (DFA) of EEG increments does not originate from the underlying dynamics but is merely an artifact of the algorithm. This artifact is rooted in the failure of the “trend plus signal” paradigm of DFA.

Received 12 June 2008

DOI:https://doi.org/10.1103/PhysRevE.81.031909

Authors & Affiliations

M. Ignaccolo¹, M. Latka², W. Jernajczyk³, P. Grigolini⁴, and B. J. West^1,5

¹Physics Department, Duke University, Durham, North Carolina, 27709 USA
²Institute of Biomedical Engineering, Wroclaw University of Technology, Wroclaw, Poland
³Department of Clinical Neurophysiology, Institute of Psychiatry and Neurology, Warsaw, Poland
⁴Center for Nonlinear Science, University of North Texas, Denton, Texas, 76203 USA
⁵Information Science Directorate, Army Research Office, Research Triangle Park, North Carolina, 27709 USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 81, Iss. 3 — March 2010

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Time evolution of diffusion entropy for: channel O1 (solid line) and the increments of the variable $X$ which is the solution of the EEG OU Langevin equation (16) (squares). Inset: the comparison between the asymptotic probability density functions at $t = 8 s$ for: channel O1 (solid line), variable $X$ of Eq. (16) (squares), and the best Gaussian fit to the pdf of the experimental data (thick line).Reuse & Permissions
Figure 2
Time evolution of diffusion entropy $S (t)$ of all 19 EEG channels of one subject.Reuse & Permissions
Figure 3
DEA and DFA of the dynamics of the stochastic variable $X (t)$ which is the solution of the OU Langevin equation Eq. (1) with $λ = 0.055$ and $σ_{η} = 40$ . The triangles denote the values of the diffusion entropy $S (t)$ while open circles represent the values of base-2 logarithm of $F (t)$ . The filled circles correspond to the DFA calculations with nonoverlapping windows. The solid lines indicate the analytical values for $S (t)$ and standard deviation of $X (t)$ , cf. Eqs. (12, 4), respectively. Note that these analytical values were calculated for the effective dissipation rate $λ_{e f f} = λ / 2$ , see the discussion in the text. The dashed lines were drawn to guide the eye. Their slopes are equal to 0.5 which is the value of short-time scaling exponent for the OU Langevin equation (1).Reuse & Permissions
Figure 4
DEA and DFA of the EEG segment extracted from channel O1 of one of the subjects (solid lines). Both analyses were also performed for the stochastic variable $X (t)$ —the solution of the EEG OU Langevin equation (16) with $λ = 0.055$ and $σ_{η} = 40$ . Triangles and circles in this plot correspond to the diffusion entropy $S (t)$ and standard deviation $F (t)$ of the DFA residuals of the simulated EEG, respectively. The thick solid lines indicate three pseudoscaling regions.Reuse & Permissions
Figure 5
DFA of the dynamics of the stochastic variable $X (t)$ which is the solution of the OU Langevin equation (1) with $λ = 0.055$ and $σ_{η} = 40$ . The circles correspond to linear detrending. The squares denote the second moment analysis (“zeroth order detrending”). The solid thin line is the analytical expression for the standard deviation Eq. (4). Note that the standard deviation was calculated for the effective dissipation rate $λ_{e f f} = λ / 2$ , see the discussion in the text. The dashed lines were drawn to guide the eyes. Their slopes are equal to 0.5 which is the value of short-time scaling exponent for the OU Langevin equation (1).Reuse & Permissions
Figure 6
Diffusion entropy of the EEG increments of channel O1 of Fig. 1) (dashed lines) and the increments of the variable $X (t)$ (solid lines). $X (t)$ is the solution of Eq. (1) with $λ = 0.055$ and $σ_{η} = 40$ , as in Fig. 1, for different values of the stability time $t_{s}$ . We consider $t_{s} = 1 / 8$ , 1/4, 1/2, and 1 s. The entropy curves are shifted for clarity.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics