Detrended fluctuation analysis: a scale-free view on neuronal oscillations

Abstract

Recent years of research have shown that the complex temporal structure of ongoing oscillations is scale-free and characterized by long-range temporal correlations. Detrended fluctuation analysis (DFA) has proven particularly useful, revealing that genetic variation, normal development, or disease can lead to differences in the scale-free amplitude modulation of oscillations. Furthermore, amplitude dynamics is remarkably independent of the time-averaged oscillation power, indicating that the DFA provides unique insights into the functional organization of neuronal systems. To facilitate understanding and encourage wider use of scaling analysis of neuronal oscillations, we provide a pedagogical explanation of the DFA algorithm and its underlying theory. Practical advice on applying DFA to oscillations is supported by MATLAB scripts from the Neurophysiological Biomarker Toolbox (NBT) and links to the NBT tutorial website http://www.nbtwiki.net/. Finally, we provide a brief overview of insights derived from the application of DFA to ongoing oscillations in health and disease, and discuss the putative relevance of criticality for understanding the mechanism underlying scale-free modulation of oscillations.

Keywords: criticality; detrended fluctuation analysis; long-range temporal correlations; ongoing oscillations; scale-free dynamics.

Figure 1

The Roman cauliflower is a striking example of self-similarity in nature. (A) The cauliflower is composed of flowers that are similar to the entire cauliflower. These smaller flowers, in turn, are composed of flowers that are similar to the smaller flowers. The self-similarity is apparent on at least four levels of magnification, thereby illustrating the scale-free property that is a consequence of self-similarity (bottom left). A hypothetical distribution of the likelihood of flowers on a cauliflower having a certain size. This property is captured by the power-law function. The mean or median of a power-law, however, provide a poor representation of the scale-free distribution (and in a mathematical sense is not defined) (bottom right). The power-law function makes a straight line in double-logarithmic coordinates. The slope of this line is the exponent of the power-law, which captures an important property of scale-free systems, namely the relationship between the size of objects or fluctuations on different scales. (B) As the size of apples shows smaller variation they are well described by taking an average value such as the mean or median. (bottom left) Hypothetical distribution showing the likelihood of apples having a certain size. Both the mean and median are good statistics to convey the size of the apples. (bottom right) Plotting the normal distribution on double-logarithmic coordinates has little effect on the appearance of the distribution, which still shows a characteristic scale. (C) Time-signals can also be viewed as self-affine as they can be transformed into a set of sine-waves of different frequencies. In a 1/f signal the lower frequency objects have larger amplitude than the higher frequency objects which we can compare with there being fewer large cauliflowers than there are small cauliflowers. (D) A white-noise signal is also self-affine, but now the lower frequency objects have the same amplitude as the higher frequency objects meaning that only the high-frequency fluctuations are visible in the signal.

Figure 2

The “random walk”: the signal profile of a stationary time series may reveal self-affinity. (A) At each time step a walker moves randomly to the left (−1) or right (+1) with equal probability. At any time step the probability of being at a certain displacement from the origin depends on the number of different paths that could take the walker there. (B) The walker’s steps form a time series that is stationary as its value does not depend on time. (C) The signal profile can take arbitrarily large values as the time increases. (D) Looking at the walker time series on a longer time-scale the standard deviation does not change as the signal cannot take larger values. (E) The cumulative sum, or random walk process, on a longer time-scale shows larger variance than on the shorter time-scale (C) therefore the walker may exhibit self-affinity or scale-free behavior.

Figure 3

Processes with a memory produce qualitatively, and quantitatively, different fluctuations compared to a random walk process. (A) Correlations occur when the “walker’s” decision to follow a certain direction is influenced by its past actions. (Left) Path of an anti-correlated walker shown over time. At each time step the walker makes a decision based on a weighted choice between left and right. The weighted choice can be seen by the sum of the areas of the arrows pointing left and right. Each action the walker takes continues to influence future actions, with the walker being more likely to take the opposite action. This is illustrated as a gradual accumulation of arrows that refer to past actions, but also decrease in size over time, because the bias contributions of those actions decay over time. The green arrows show how the first action the walker takes (going Right) persists over time, with the influence getting smaller as time goes on seen by the green arrow size decreasing. (Center) Path of a random walker shown over time. The random walker is not influenced by previous actions and so always has equal probability of going left or right. (Right) Path of a correlated walker shown over time. Here each action the walker takes influences future actions by making the walker more likely to take that action. The green arrows show that by taking the action of going right at time 0, the walker is more likely to go right in future time steps with the influence getting smaller as time goes on. (B) Cumulative signal for a positively correlated process (red, circle) shows larger fluctuations over time than a random walker (blue, triangle). An anti-correlated signal (green, square) shows smaller fluctuations over time. (C) By looking at the average fluctuations for these different processes at different time-scales, we can quantify this difference. A random walker shows a scaling exponent of 0.5, with the positively correlated process having a larger exponent, and the anti-correlated process having a smaller exponent.

Figure 4

Trends on longer time-scales can introduce false correlations into the signal. (A) For a signal with a trend, the standard deviation will be larger (σ = 0.41) than the same signal with no trend (σ = 0.29). (B) Average fluctuations for a window size shown for a white-noise signal (blue crosses) and the same signal with a trend added (red crosses) show different scaling. By removing the linear trend of the integrated signal from each window before calculating the standard deviation (circles), we recover the scaling seen without the long-time-scale trend. (C) Importantly, detrending self-similar signals with trends (red crosses) also recovers the scaling of the original signal (blue circles). (D) Self-similar signal (α = 0.75) with trend (red) and without trend (blue) used in (C).

Figure 5

Step-wise explanation of Detrended Fluctuation Analysis. (A) Original time series. Taken from a 1/f signal sampled at 5 Hz with a duration of 100 s. (B) Cumulative sum of original signal shows large fluctuations away from the mean. (C) For each window size looked at, remove the linear trend from the signal, and then calculate the fluctuation. Two example window sizes shown with signal shown as solid line, and detrended signal shown as dotted line. (D) Plot the mean fluctuation per window size against window size on logarithmic axes. The DFA exponent is the slope of the best-fit line (α = 1).

Figure 6

Step-wise explanation of applying DFA to neuronal oscillations. (A) EEG recording from electrode Oz shows clear oscillations during a 15 min eyes-closed rest session. Data were recorded at 250 Hz and band-passed filtered between 0.1 and 200 Hz. (B) Power spectrum (Welch method, zero padded) shown in logarithmic (left) and double-logarithmic axes (right), shows clear peak in the alpha band. (C) Signal in (B) filtered in the alpha band (8–13 Hz) using a fir filter with an order corresponding to the length of two 8 Hz cycles (blue). Amplitude envelope (red) calculated using the Hilbert transform. (D) DFA applied to the amplitude envelope of white-noise signal filtered using the same filter as in (C). At time windows <2 s, filter-induced correlations are visible through a bend away from the 0.5 slope. (E) DFA applied to the amplitude envelope of the alpha band filtered EEG signal shows long-range temporal correlations between 2 and 90 s with exponent α = 0.71.

Figure 7

Results of applying DFA to neuronal oscillations. (A) Robust long-range temporal correlations are observed in the amplitude envelope of human EEG alpha oscillations using the DFA. Circles, eyes-closed rest condition; Dots, surrogate data (Figure modified from Linkenkaer-Hansen et al., 2001). (B) Differences in the scale-free modulation of the amplitude envelope of neuronal oscillations are prominent among individuals and can be quantified using DFA. Here shown for three filtered EEG signals (6–13 Hz) with weak (top), medium (middle), and strong (bottom) LRTC (from channel O2). The gray lines represent the amplitude envelope (low-pass filtered, 1 Hz). DFA fluctuation functions are shown to the right, with signal (circles), and white-noise (crosses). The DFA exponent is the slope of the fluctuation function. (Figure modified from Smit et al., 2011). (C) Individual differences in long-range temporal correlations in alpha oscillations are to a large extent accounted for by genetic variation, as seen by the difference in correlations of DFA exponents between monozygotic and dizygotic twins (Figure modified from Linkenkaer-Hansen et al., 2007). (D) DFA has high test-retest reliability. DFA exponents from the amplitude modulation of alpha oscillations, two sessions with an interval of 6–28 days, symbols indicates different subjects (Figure modified from Nikulin and Brismar, 2004). (E) The DFA exponent is independent of oscillation power. Data were recorded using EEG on 368 subjects during a 3 min eyes-closed rest session (Figure modified from Linkenkaer-Hansen et al., 2007).

Figure 8

DFA is a promising biomarker for pre-clinical studies. (A) DFA exponents of theta oscillations in left sensorimotor region correlate with the severity of depression based on the Hamilton score. Data recorded from 12 depressed patients with MEG, during an eyes-closed rest session of 16 min. (Figure modified from Linkenkaer-Hansen et al., 2005). (B) DFA of alpha oscillations shows a significant decrease in the parietal area in patients with Alzheimer’s disease than in controls. MEG was recorded during 4 min of eyes-closed rest and the DFA exponent estimated in the time range of 1–25 s. (Right) Individual-subject DFA exponents averaged across significant channels are shown for the patients diagnosed with early stage Alzheimer’s disease (n = 19) and the age-matched control subjects (n = 16; Figure modified from Montez et al., 2009). (C) Difference in the DFA exponent of high frequencies (beta band) and low frequencies (alpha band) indicates the location of the epileptic focus (white box). Data recorded from an epileptic subject using subdural EEG during seizure-free activity (modified from Monto et al., by permission of Oxford University Press).