Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations

“…fluctuation analysis has been developed for quantifying correlation properties in nonstationary signals, e.g., in physiological time series, because long-range correlations—revealed by an ACF analysis—can arise also as an artifact of the “patchiness” of nonstationary data (Peng et al., 1994, 1995).”

“Unlike deterministic approaches aimed at finding low-dimensional chaos, the SOC framework allows for a highdimensional character of the dynamics and for the presence of stochastic effects.”

“and Marchesi, 1999), heartbeats (Peng et al., 1995), and human”

“…frequencies are a prominent feature of the spontaneous electroencephalogram (EEG) (Berger, 1929; Connors and Amitai, 1997) and are believed to reflect functional states of the brain (Llinás, 1988; Steriade et al., 1993; Arieli et al., 1996; Herculano-Houzel et al., 1999; Tsodyks et al., 1999).”

“…frequencies are a prominent feature of the spontaneous electroencephalogram (EEG) (Berger, 1929; Connors and Amitai, 1997) and are believed to reflect functional states of the brain (Llinás, 1988; Steriade et al., 1993; Arieli et al., 1996; Herculano-Houzel et al., 1999; Tsodyks et al., 1999).”

“This timescale invariance of wavelet correlations and the brain functional network parameters derived from them is a theoretically predictable corollary of the long-range autocorrelations of neurophysiological time series 82–84 .”

“This 1/f distribution has also been observed in studies of spontaneous electroencephalography (EEG”

“…many people argue for criticality and power law effects in large-scale cortical activity (e.g., Freyer et al., 2009; Kitzbichler et al., 2009; Linkenkaer-Hansen et al., 2001; Stam and de Bruin, 2004), other people do not (Bedard et al., 2006; Miller et al., 2007; Touboul and Des-texhe, 2009).”

“On the other hand, multisecond periodicities or infraslow oscillations have been reported in both local field potential recordings from rat basal ganglia and monkey cortex [31,41,42] and the surface EEG in humans [ 43 ].”

“, 2002), and EEG time series in humans (Linkenkaer-Hansen et al., 2001; Worrell et al., 2002)] our results go beyond the phenomenological description of a power law only.”

“Moreover, long period or very low-frequency coherent oscillations have been described recently in both monkey and human electrophysiological data (Linkenkaer-Hansen et al., 2001; Leopold et al., 2003).”

“These scale free fluctuations of synchronization levels have been demonstrated for local (Linkenkaer-Hansen et al., 2001; Nikulin and Brismar, 2004, 2005) as well as global dynamics (Stam and de Bruin, 2004), and migh be due to self-organized criticality or critical dynamics near a phase transition.”

“Notably, scale-free properties have recently been described in the amplitude and synchronization of oscillatory brain activity (Linkenkaer-Hansen et al., 2001; Stam and de Bruin, 2004), and in the temporal and spatial distributions of negative LFP peaks (Plenz and Thiagarajan, 2007).”

“…noise-like version of the data as a control condition is not to be recommended, because the prominent autocorrelations inherent to neurophysiological signals (see, e.g., (Linkenkaer-Hansen et al., 2001)) turn this kind of surrogates into very unlikely realizations of any neurophysiological process.”

“Several studies have shown that time series of brain activity demonstrate scaling with characteristic exponents, suggesting critical dynamics near a phase transition [12-15].”

“…of a brain is marked by a vast number of ongoing electrical and chemical processes spanning multiple spatial and temporal scales (Pritchard, 1992; Linkenkaer-Hansen et al., 2001; Kozma et al., 2005; Honey et al., 2007; Plenz and Chialvo, 2009; Werner, 2010; Lowen et al., 1997) that both arise…”

“rapid fluctuations (Linkenkaer-Hansen et al., 2001; Stam and de Bruin, 2004; Freyer et al., 2009), possibly indicative of a complex dynamic”

“Unsurprisingly, neuronal oscillations therefore display ongoing variability in power even at baseline (Linkenkaer-Hansen et al., 2001), in addition to showing sensory-driven changes (Bauer et al., 2006; Gross et al., 2007; van Dijk et al., 2008).”

“…for many years in neuroscience in the temporal correlations of single time-series data (e.g. the power spectrum from the electroencephalogram (Linkenkaer-Hansen et al. 2001; Worrell et al. 2002), Fano or Allan factors in spike count statistics (Teich et al. 1997), neurotransmitter secretion…”

“…(Bertschinger and Natschläger, 2004; Haldeman and Beggs, 2005; Boedecker et al., 2012; Lizier, 2013), from non-oscillatory to oscillatory regimes (Linkenkaer-Hansen et al., 2001; Poil et al., 2012), from replay to non-replay of spatiotemporal patterns (Scarpetta and de Candia, 2013), and from a…”

“The discovery of LRTC in the amplitude envelope of ongoing oscillations, was based on 10 subjects recorded with EEG and MEG for 20 min during eyes-closed and eyes-open rest (Linkenkaer-Hansen et al., 2001).”

“Power law scaling of human neurophysiological processes has been previously described in both functional MRI and MEG or electrophysiological datasets [ 14 ,15,43].”

“Employing an analysis called detrended fluctuation analysis (DFA), Linkenkaer-Hauser and colleagues reported that the fluctuating amplitudes of the two dominant oscillations of the human brain – the alpha and beta rhythms – exhibited scale-free temporal statistics (Linkenkaer-Hansen et al., 2001).”

“POWER LAWS AND NEURAL DATA Critio: Well, there were several early reports that the nervous system could produce power law distributions (Chen et al., 1997; Teich et al., 1997; Linkenkaer-Hansen et al., 2001; Worrell et al., 2002).”

“As shown by human and animal studies, the dynamics of envelope ICMs exhibits these characteristic features (Linkenkaer-Hansen et al., 2001; He et al., 2010; Palva and Palva, 2011; Tagliazucchi et al., 2012b).”

“We believe that these perspectives provide a strong incentive for more experimental and theoretical work in the area of self-organized criticality.”

“Second, the observations of LRTCs and neuronal avalanches with power-law size and lifetime distributions add to the growing body of data suggesting that the human brain operates near a critical regime (12, 30, 32, 33), which may fundamentally determine the dynamics of human perceptual, cognitive, and behavioral proces…”

“To more robustly investigate the scalefree behavior of the fMRI signal, we used the well-established detrended fluctuation analysis (DFA) method (Linkenkaer-Hansen et al., 2001; Eke et al., 2006), which was initially introduced to study correlations in DNA sequences (Peng et al., 1994).”

“…of 5–20 Hz oscillations are autocorrelated over tens to hundreds of seconds (LinkenkaerHansen et al., 2001; Linkenkaer-Hansen et al., 2005), and their amplitude dynamics show 1/f-type power distributions both in human (Linkenkaer-Hansen et al., 2001) and monkey (Leopold et al., 2003) cortex.”

“If the oscillations wax and wane at random (e.g., as bandpass filtered white noise), the DFA exponent will be 0.5, whereas a DFA exponent between 0.5 and 1.0 imply that oscillations have a scale-free amplitude modulation (Linkenkaer-Hansen et al., 2001).”

“1C), suggesting that the underlying networks were in a highly dynamic state (Linkenkaer-Hansen et al., 2001).”

“The scaling behavior of temporal patterns has been well studied (Ingber, 1995), particularly in the alpha and beta ranges (Nunez et al., 1997; Linkenkaer-Hansen et al., 2001).”

“, 2010), and corresponding long-range temporal correlations (Linkenkaer-Hansen et al., 2001).”

“Yet, ongoing activity fluctuations contribute to variability of evoked responses (Arieli et al., 1996) and impact on perceptual performance (Linkenkaer-Hansen et al., 2001; Boly et al., 2007; Hesselmann et al., 2008a,b).”

“Amplitude fluctuations of interhemispherically coherent high-frequency activity have been demonstrated at much slower time scales ranging from seconds to minutes (Linkenkaer-Hansen et al., 2001; Nikouline et al., 2001) and such variations correlate well with hemodynamic fluctuations (Logothetis et al.”

“The main steps from the broadband MEG signal to the quantification of LRTC using DFA have been explained in detail elsewhere (23, 49, 50).”

“At low frequencies (less than 1.5 Hz) the slope is shallower and it is likely that the amplitude or power enve-lopes of faster frequencies are scale-free (Kitzbichler et al., 2009; Linkenkaer-Hansen et al., 2001) or heavy-tailed (Freyer et al., 2009).”

“For instance, the distribution of endogenous EEG and MEG oscillations are inversely powerlaw distributed (Linkenkaer-Hansen et al., 2001).”

“Neuronal activity was described in the SOC framework in human electroencephalogram [34,35] (but see also [36]) and in local field potentials from cultured and acute cortical rat slices [27], but this is the first time to our knowledge that similar phenomena are found: i - at the spike level, ii - in a random neuronal…”

“Fractal dimensions and 1/f spectral properties have been measured in electroencephalographic (EEG) signals (Bullmore et al., 1994a; Linkenkaer-Hansen et al., 2001; Senhadji et al., 1995).”

“Long-range temporal memory is a landmark of power fluctuations in faster EEG and magnetoencephalography frequencies, such as α (8–12 Hz) or β (13–30 Hz) (21); in contrast, a decreased H exponent relative to the exponent of wakefulness is associated with the slow rhythms that herald the onset of deep sleep (36).”

“The importance of long-distance correlations has been emphasized by numerous brain theorists (e.g., Ingber, 1995; Hoppenstaedt and Izhkevich, 1998; Haken, 1999; Friston, 2000; Linkenkaer-Hansen et al., 2001; Vitiello, 2001; Kaneko and Tsuda, 2001; Kozma, Freeman and Erdí, 2003; Stam et al., 2003).”

“5 establishes a missing link between studies of growth processes in economic or social systems (16–18) and studies of long-term correlations, such as in finance and the economy (23), Ethernet traffic (24), and human brain (25) or motor activity (26).”

“Moreover , 1/f fluctuations have been found in electrophysiological measures that are not subject to fMRI artifacts (Bhattacharya, Edwards, Mamelak, & Schuman, 2005; Leopold, Murayama, & Logothetis, 2003; Le Van Quyen, 2003; Linkenkaer-Hansen et al., 2001).”

“Finally, although in this study we apply this technqiue to fMRI data in order to extract the neural signatures of the LSD state, application of the connectome-harmonic decomposition can also be easily extended to other functional neuroimaging datasets such as MEG or high density scalp EEG.”

“The evidence indicates, however, that neuronal activity patterns, including their long-range temporal correlations (Linkenkaer-Hansen et al., 2001; Nikulin and Brismar, 2004), are highly replicable within and across subjects (Gusnard and Raichle, 2001).”

“Extracellular electrophysiological systems and their fluctuations in power have all been shown to follow power laws (He et al., 2010; Linkenkaer-Hansen et al., 2001; Miller et al., 2009; Poil et al., 2008), as does global functional connectivity (Botcharova et al.”

“, 2011), magnetoencephalography, and electroencephalography (EEG) (Linkenkaer-Hansen et al., 2001).”

“Also the complex temporal dynamics of ongoing activity has received a great deal of interest (Linkenkaer-Hansen et al., 2001; He, 2011; Hardstone et al., 2012) with recent studies showing that that intrinsic neuronal dynamics during rest can predict individual variability in behavior (Palva et al.,…”

“For example, measurements of human brain oscillations revealed such critical dynamics of neural networks, implying their ability to effectively propagate information and rapidly reorganize (8).”

“Furthermore, detrended fluctuation analysis characterizing long-range temporal correlations (Linkenkaer-Hansen et al., 2001; Hardstone et al., 2012; Poil et al., 2012), multifractality spectral width (Kantelhardt et al., 2002; Ihlen, 2012) and oscillation bursts 95th percentile durations and sizes…”

“These include results obtained at the whole-brain scale, where functional networks compatible with a critical brain were observed via functional magnetic resonance imaging (fMRI) [14,15], magnetoencephalographic (MEG) [16] and electroencephalographic (EEG) [17] data.”

“An important study of nontrivial temporal correlations in cortical activity is that of Linkenkaer-Hansen et al. (2001) who studied modulations in alpha (8 –13 Hz) and beta (15–25 Hz) power in both EEG and magnetoencephalographic data.”

“Moreover, a detrended fluctuation analysis (75, 76) of the time series reveals the emergence of long-range temporal correlations right at ξc (SI Appendix) as expected at a continuous phase transition.”

“After segmentation, the inter-brain phase synchrony index (phase-locking value, PLV) were quantified based on a wavelet decomposition of the signal 65between 8 and 13 Hz in 1 Hz steps, similar to previous studies (Linkenkaer-Hansen et al., 2001; Mu et al., 2008; Mu and Han, 2010; 2013).”

“, 2013), EEG (Linkenkaer-Hansen et al., 2001; Meisel et al., 2013), and MEG (Linkenkaer-Hansen et al.”

“…incorporated elements of related theories of the EEG that postulated its origin in local distributed feedback [Basar, 1998] and in noise: Gaussian [Bullock, 1969; Elul, 1972], 1/f [Linkenkaer-Hansen et al., 2001; Hwa and Ferree, 2002; Wakeling, 2004], and chaotic [Tsuda, 2001; Stam et al., 2003].”

“Neural dynamics with timescales across many orders of magnitude are also observed across numerous recording methodologies, input stimuli, and organisms (Baria et al. 2011; He et al. 2010; Honey et al. 2012; Lerner et al. 2011; Linkenkaer-Hansen et al. 2001; Schneidman et al. 2006).”

“Several electrophysiology studies suggest that rhythmic neuronal activity, although oscillating at high frequencies (up to 250 Hz), may exhibit a much slower fluctuation in power (or modulation amplitude) that generally coincides with the frequency range of spontaneous BOLD fluctuations (Anderson, 2008; Leopold et al.…”

“Despite the different time scales of fMRI and electrophysiological measurements, one might argue that long memory structure is typically fractal or statistically self-similar and it is, therefore, unsurprising that it should be consistently observable over a range of time scales (Linkenkaer-Hansen et al., 2001; Goldbe…”

“The importance of long-distance correlations has been emphasized by numerous brain theorists (e.g., Ingber, 1995; Hoppenstaedt and Izhkevich, 1998; Haken, 1999; Friston, 2000; Linkenkaer-Hansen et al., 2001; Kaneko and Tsuda, 2001; Kozma et al., 2003; Stam et al., 2003).”