Mutual information identifies spurious Hurst phenomena in resting state EEG and fMRI data

Long-range memory in time series is often quantified by the Hurst exponent $H$, a measure of the signal's variance across several time scales. We analyze neurophysiological time series from electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) resting state experiments with two standard Hurst exponent estimators and with the time-lagged mutual information function applied to discretized versions of the signals. A confidence interval for the mutual information function is obtained from surrogate Markov processes with equilibrium distribution and transition matrix identical to the underlying signal. For EEG signals, we construct an additional mutual information confidence interval from a short-range correlated, tenth-order autoregressive model. We reproduce the previously described Hurst phenomenon ($H>0.5$) in the analytical amplitude of alpha frequency band oscillations, in EEG microstate sequences, and in fMRI signals, but we show that the Hurst phenomenon occurs without long-range memory in the information-theoretical sense. We find that the mutual information function of neurophysiological data behaves differently from fractional Gaussian noise (fGn), for which the Hurst phenomenon is a sufficient condition to prove long-range memory. Two other well-characterized, short-range correlated stochastic processes (Ornstein-Uhlenbeck, Cox-Ingersoll-Ross) also yield $H>0.5$, whereas their mutual information functions lie within the Markovian confidence intervals, similar to neural signals. In these processes, which do not have long-range memory by construction, a spurious Hurst phenomenon occurs due to slow relaxation times and heteroscedasticity (time-varying conditional variance). In summary, we find that mutual information correctly distinguishes long-range from short-range dependence in the theoretical and experimental cases discussed. Our results also suggest that the stationary fGn process is not sufficient to describe neural data, which seem to belong to a more general class of stochastic processes, in which multiscale variance effects produce Hurst phenomena without long-range dependence. In our experimental data, the Hurst phenomenon and long-range memory appear as different system properties that should be estimated and interpreted independently.

Figure 1

Sample paths of the fractional Gaussian noise process. (a) Sample path for the uncorrelated, white noise-like case, $H=0.5$. (b) Sample path showing long-range correlations, $H=0.9$. Both samples are of length $50000$ and were synthesized using the covariance matrix method.

Figure 2

Sample paths of stationary, exponentially correlated Gaussian noise, the Ornstein-Uhlenbeck process. (a) Sample path for the shortest relaxation time $\tau =0.1$ resembling white noise. (b) A sample for the largest relaxation time $\tau =10.0$ shows significant intermediate and low-frequency content, leading to variance contributions across many time scales in Hurst analyses.

Figure 3

Sample paths of the heteroscedastic, short-range correlated Cox-Ingersoll-Ross process. (a) Sample path with short relaxation time $\tau =0.1$ showing a sequence of short lasting bursts. (b) For the largest relaxation time $\tau =10.0$, the strictly positive bursts of activity show a longer duration and thus contribute to the global variance across many time scales in Hurst analysis.

Figure 4

Amplitude dynamics for two different resting state EEG recordings. (a) Subject 1, (b) Subject 6, as indexed in Table 1. The respective left occipital EEG channel signals (O1, light gray) and their analytical amplitudes (black lines) are shown. The analytical amplitude shows irregular bursts of large amplitude activity, interlaced with periods of low-amplitude oscillations. In both subjects, we observe similar bursty dynamics, anticipating variance contributions across many time scales in subsequent Hurst analyses.

Figure 5

Sample paths of experimental EEG alpha band (8–12 Hz) oscillations (a, b) and autoregressive AR(10) surrogates (c, d) on two different time scales, 100 s and 8 s. The AR(10) model data reproduce the alpha band dynamics on both time scales. Both, empirical EEG (a, b) and surrogate data (c, d) show irregular bursts of 10 Hz activity. The analytical amplitude (black) of the signal measures the instantaneous power of the alpha band oscillation (gray).

Figure 6

Hurst exponent estimation using the discrete wavelet transform. The figure shows the scaling functions of all ten subjects, analyzing the analytical amplitude of the alpha envelope recorded at the left occipital electrode O1. The blue line above the scaling functions (black) shows the theoretical scaling function using the empirical average Hurst exponent across the ten subjects, $\overline{H}=0.74$, restricted to the fitting interval $j_{\min }=6$ to $j_{\max }$. For two subjects, the sample size determined $j_{\max }=11$, for all other subjects $j_{\max }=12$. The blue line is shifted vertically for visualization purposes. The smallest wavelet scales containing transient short-range correlations are not shown.

Figure 7

Effects of preprocessing on mutual information estimation. Gaussian white noise yields low mutual information at all time lags as shown by the flat confidence interval ($\alpha =0.01$) labeled “unfiltered white noise.” Applying the EEG pre-processing algorithm to white noise, the confidence interval labeled “white noise alpha band envelope” shows short-term memory components up to approximately $200\mathrm{ms}$. Memory effects in EEG time series are therefore considered for time lags $>500\mathrm{ms}$ (gray shaded area).

Figure 8

Hurst exponents for fractional Gaussian noise, the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross process. In (a–c), $n=100$ sample paths of length 50 000 are used per data point. (a) For each nominal Hurst exponent H (abscissa), $\widehat{H}$ is estimated by the wavelet method (ordinate). Average $\widehat{H}$-values are shown as black squares, SEM error bars are so small that they are not visible. The blue line indicates the identity $\widehat{H}=H$. It is observed that long-range correlations in fGn are correctly estimated. (b) For the OU process, the same estimation procedure shows increasing Hurst exponent estimates with increasing relaxation time $\tau$, although the process is short-range correlated. (c) For the CIR process, the same behavior as in (b) is observed. It is concluded that for OU and CIR, the Hurst phenomenon occurs without long-range correlations.

Figure 9

Time-lagged mutual information functions for Freedman-Diaconis discretized stochastic processes, LRD and relaxation time parameters ($H, \tau$) increase from bottom to top. (a) fractional Gaussian noise (fGn), (b) Ornstein-Uhlenbeck (OU), (c) Cox-Ingersoll-Ross (CIR).

Figure 10

Time-lagged mutual information functions (bold black lines) for fGn (a), OU (b), and CIR (c), and their first-order Markov confidence intervals (gray). fGn clearly deviates from the Markovian hypothesis, showing a significantly higher memory content and a much slower decay. For OU and CIR, the mutual information function lies within the Markov confidence interval, demonstrating their short-range correlated nature. For EEG data (d) and the AR(10) model (e), two types of confidence intervals are shown, the first-order Markov null hypothesis as in (a–c) (gray area with black borders) and the AR(10) confidence interval (gray area with blue borders). EEG and AR(10) data do not show a fGn-like mutual information function and are statistically indistinguishable from short-range correlated processes.

Figure 11

Continuous Kozachenko-Leonenko estimates of mutual information for fGn and EEG. (a) Average mutual information function for fGn with Hurst exponent $H=0.75$ ($n=10$). (b) Individual mutual information functions for the ten EEG recordings. While fGn shows a power-law decay of memory effects, the information content of the alpha envelope decays like a short-range correlated process.

Figure 12

Mutual information functions for neurophysiological data. (a) Analytical alpha amplitude of the left occipital EEG channel. Time scales not affected by pre-processing effects are indicated by the gray-shaded area. (b) Memory effects for the EEG microstate random walk. (c) fMRI signal recorded in the left occipital area, the same region where the signal in (a) was recorded using EEG. For (a–c), the empirical mutual information functions (black lines) lie within the Markovian confidence intervals (gray areas, $\alpha =0.01$) supporting the short-range correlated null hypothesis.