Power spectrum derivations of the electroencephalogram (EEG) have been used as surrogate parameters for the quantification of depth of sedation or anesthesia (1). However, the power spectrum is one-half of the spectral decomposition of a signal; the other half is given by the phase spectrum. It has been argued that the omission of phase information in constructing EEG derivations might lead to inadequate descriptors for monitoring anesthesia. Thus, phase spectrum parameters of the EEG have achieved increasing attention for monitoring anesthesia in clinical and experimental settings (2,3).1 Nontrivial, frequency-dependent values for the bispectrum and its normalized version, the bicoherence, indicate the presence of nonlinear features of the signal. In the past, such nonlinear features played an important role in the analysis of so called “chaotic systems.” (4,5)
The bicoherence estimates the proportion of energy in every possible pair of frequency components, say f1 and f2, that satisfy the definition of quadratic phase coupling (phase of component at f3, which is f1 + f2, equals phase of f1 + phase of f2) (6). It allows us to investigate whether a given time series exhibits such a nonlinear feature or can be considered as generated by a linear random process. In the case of a linear random process, the bicoherence is a mere constant independent of the frequency. This constant is zero if the random process is a Gaussian random process. In this case, both the bispectrum and bicoherence are zero. If the EEG is a linear random process, then the phases are also randomly distributed. Consequently, the power spectrum contains all the information of the original signal. The bispectral index (BIS™) has achieved substantial interest because it is said to add bispectral analysis to more traditional ways of EEG analysis.
This study aimed to investigate whether the EEG during anesthesia with propofol and alfentanil can be considered as a linear random process or whether it contains a large percentage of nonlinear features that should be considered in future EEG derivations by implementation of higher-order spectral analysis. The distinction between linear and nonlinear features of the EEG has been performed with the procedure described by Subba Rao and Gabr (7) and improved by Hinich (8). The test procedure proves the hypothesis that the bicoherence is independent of the frequency and is a mere constant. Therefore, we determined the frequency of EEG epochs with nontrivial bicoherence as an indicator of the degree of nonlinearity in the EEG recorded during anesthesia with propofol and alfentanil.
After institutional approval and informed consent from participants, we reanalyzed 90 h of EEG data from 20 patients without apparent neurological deficit (11 men and 9 women, aged 48.9 ± 10.6 yr, and weighing 73.0 ± 15.3 kg [mean ± sd]) who underwent elective abdominal surgery under anesthesia with propofol and alfentanil (9). Premedication with midazolam 7.5 mg was administered orally to all patients 1 h before the induction of anesthesia. Tracheal intubation was facilitated by rocuronium, and anesthesia was induced and maintained by target-controlled infusions for propofol (10) and alfentanil (11). The choice of the target concentrations for propofol and alfentanil was left to the discretion of the attending anesthesiologist. The EEG was recorded with an Aspect A-1000® monitor (Aspect Medical Systems, Inc, Natick, MA) using one of the two recommended frontal leads. The raw EEG, digitized at 128 Hz, was obtained from the serial port of the Aspect A-1000® monitor and stored on CD-ROM.
The digitized EEG was segmented into epochs of 1024 data points (i.e., 8 s). After automatic and manual artifact rejection, there were 34,562 artifact-free epochs (i.e., 76.8 h). Before testing for nontrivial bispectra, one has to prove whether the investigated signal is stationary (i.e., the statistical properties do not change with time). Most EEG analysis methods, such as Fourier transform and autoregressive modeling, are based on implicit assumptions of stationarity. As the bispectrum depends on the power spectrum, analyzing bispectra of nonstationary EEG epochs is likely to provide misleading information. The statistical properties of the EEG usually change with time so that the EEG is not necessarily stationary within the investigated epoch. Thus, each EEG epoch was submitted to a “run test” (12,13) for testing the hypothesis of stationarity. Once the test for stationarity was passed, the Hinich test for gaussianity and linearity was performed (Appendix). The estimation of the bispectrum, bicoherence, and test statistics was performed with the “glstat.m” routine of the Higher-Order Spectral Analysis Toolbox for use with Matlab® (The Mathworks, Inc., Natick, MA) using default parameter settings.
For the validation of the Hinich test, we generated artificial synthetic and surrogate data with known bispectrum and bicoherence and determined the sensitivity and specificity of the test. The sensitivity and specificity were calculated as a function of the parameter r = max(Rth/Res, Res/Rth), which serves as a measure of difference between the theoretical and the estimated bicoherence (Appendix). As a model for deterministic nonlinear data, for which the bicoherence is expected to be different from a constant, we used the z-component of the Lorenz attractor (Appendix). As a model for data with known zero bispectrum and zero bicoherence, we used synthetic linear gaussian random data (13). We also generated random surrogate data from the set of artifact-free EEG epochs with the widely applied method of phase randomization (14), which annihilates any information in a signal caused by phase coupling. Because the power spectrum does not depend on the phases, the original and the phase-randomized data have identical power spectra (Fig. 1).
Therefore, the Hinich test was validated with four sets of data before applying it to the original EEG data: the Lorenz data, for which one expects that both hypotheses H0 for gaussianity and H0′ for linearity have to be rejected (Appendix); the linear Gaussian data, for which both hypotheses H0 for gaussianity and H0′ for linearity have to be true; and the phase-randomized Lorenz and phase-randomized EEG data, for which one anticipates that at least the hypothesis H0 for gaussianity is not rejected.
We obtained 23,973 artifact-free and stationary epochs from the original EEG data, which were tested for linearity and gaussianity. The overall characteristics of this material are depicted in Figure 2 as the histogram of the BIS-values and the mean amplitude of the EEG epochs. We obtained 27,553 stationary epochs from the phase-randomized artifact-free EEG. This material was used in addition to 4000 epochs of Lorenz data, 4000 epochs of phase-randomized Lorenz data, and 6869 epochs of synthetic linear gaussian data to determine the sensitivity and specificity of the Hinich test. Figure 3 shows the sensitivity and specificity for the original and phase-randomized Lorenz data as a function of the ratio r. For the value of r0 = 4, the test achieves approximately 0.95 sensitivity and specificity (sensitivity(4) = 0.9476; specificity(4) = 0.9435). Given these results, the final linearity test was performed with r0 = 4. For this value we expect that the percentage of false classified nonlinear epochs and the percentage of false classified phase-randomized epochs is approximately 5%.
The percentages of nonlinear epochs for the original EEG data and for the synthetic data are compared in Table 1. It seems that the number of nonlinear epochs contained in the original EEG does not exceed that identified in the phase-randomized EEG or the Gaussian data. Figure 4 depicts the comparison between frequencies of identified nonlinear epochs contained in the five different data sets as a function of r. Obviously, the original EEG differs from the nonlinear deterministic data.
Our results indicate that the percentage of EEG epochs with nontrivial bispectra is approximately 10%. Thus, during anesthesia with propofol and alfentanil, 90% of EEG epochs can be considered largely as a linear random process. These findings agree with the results and expectations of earlier investigations in the field of nonlinear dynamical analysis (15,16). Other EEG work (17) reported that 98.75% of epochs of human α rhythm cannot be distinguished from a random process. Also, Pritchard and Stam (18) found that linear models accounted for more than 94% variance in human-resting, eyes-closed EEG. Another study supports our findings in that it demonstrates a need to average over 360 or more EEG epochs of two seconds to identify nonzero bispectral contributions to the EEG signal (19). This is similar to the identification of evoked potentials, where one needs considerable averaging to identify small event-related contributions in a signal of much larger variance.
However, the results above do not state that EEG during anesthesia is generated by a random process. The EEG is thought of as a spatially averaged number of postsynaptic potentials (20). The changes in each postsynaptic potential are likely to occur because of some laws of nature and not by chance. As the skewness of the amplitude distribution of a signal is directly related to the integral over the bispectrum (13), one can expect a nonzero bispectrum whenever the amplitude histogram of the EEG is skewed. Such patterns of amplitude distribution occur during epileptic seizures (21). In an early investigation of the EEG by means of bispectrum in newborns, Dummermuth et al. (22) described nontrivial bispectra during the occurrence of monophasic sleep spindles and polispikes, both patterns showing a nonzero skewness of the amplitude histogram. Previous work also described an increase in nonlinear EEG dynamics by increasing depth of anesthesia with propofol in healthy volunteers.1 However, the incidence of identified epochs with nonlinear features remained less than 10%. Also, phase-coupling differences have been reported by Ning and Bronzino (23) in intracerebral, hippocampal EEG of the rat during various vigilance states and by Bullock et al. (6) in human intracerebral EEG during sleeping, waking, and seizure states.
Therefore, it might be possible that the generation of the EEG shows nonlinear features. As these features were found to have a variable and transient character (6), the transmission from the skull to the scalp and its recording may destroy some of this information and introduce some kind of randomization. This may explain the minimal difference between the original EEG curve and the phase-randomized EEG curve of Figure 4 regarding the proportion of epochs with nontrivial bispectra. Nevertheless, the physiological significance and meaning of nonlinear EEG dynamics during anesthesia deserves further research. It might substantially contribute to the understanding of the mechanisms of anesthesia.
Our findings show that EEG epochs with nonlinear characteristics are rarely detected during anesthesia with propofol and alfentanil. Consequently, the amount of information derived from bispectrum and bicoherence could explain only a small percentage of variance in the investigated data. We conclude that for EEG monitoring during anesthesia the spectral information in the frequency domain is largely contained in the power spectrum of the signal and is sufficient for the derivation of EEG descriptors. This calls into question the benefit of EEG bispectral analysis for monitoring anesthesia effect.
The meteorologist Edward Lorenz modeled the location of a particle moving subject to atmospheric forces and obtained a certain system of three coupled nonlinear partial differential equations. Whereas the numerical solution of the system yielded an apparently random motion of the particle, the observation of the motion over time delivered a butterfly-shaped pattern of behavior in a bounded region now known as Lorenz attractor:
where ς = 10, ρ = 28, and b = 8/3 (24). Investigations on complex-dynamic systems use the z-component of the Lorenz attractor for seemingly stochastic time series that obey a deterministic order.
Hinich has shown that under the assumption bispectrum = 0, the squared bicoherence follows a specific statistical distribution (central χ2 distribution with two degrees of freedom). In essence, the Hinich test examines whether the measured squared bicoherence differs from this statistical distribution or not. In a first step, the procedure tests for gaussianity the null hypothesis H0: bispectrum = 0. If this null hypothesis has to be rejected, one enters the test for linearity. The null hypothesis H0′ for this step is: squared bicoherence = constant and independent of the frequency. Under the assumption H0′ is true, the sum of squared bicoherence is a noncentral χ2 distribution with some noncentrality parameter δ. The Hinich test uses for this step the comparison between the interquartile range of the estimated bicoherence (Res) and the interquartile range of the noncentral χ2 distribution (Rth). The hypothesis of linearity has to be rejected if Res is much larger or smaller than Rth. For a ratio r = max (Res/Rth; Rth/Res) and a value r0 > 1 the hypothesis of linearity is rejected if r ∉ [1/r0, r0]. After computing the sensitivity and specificity of the linearity test as a function of r, the final linearity test was performed with the specific value r0 for which sensitivity equals specificity.
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1Jeleazcov C, Bremer F, Schwilden H. The non-linear EEG dynamics increase with depth of anesthesia. Anesthesiology 2000;93:A759.© 2005 International Anesthesia Research Society