## Bispectrum Analysis Essay

Editor–We read the article by Miller and colleagues1 with great interest. They examined the changes of the SynchFastSlow, bispectrum derived component in the BIS calculation, and concluded that bispectral analysis did not give any more information than power spectral‐based analysis. They could not find any changes in EEG bicoherence, which directly indicates the degree of phase coupling.

In contrast, we found significant changes in EEG bicoherence during isoflurane anaesthesia combined with epidural anaesthesia.2 What was the difference between the study by Miller and colleagues and our own? We used the EEG recorded from a unilateral lead (Fp_{1}‐A_{1}), while they used the EEG recorded from a bilateral lead (F7–F8). This is an important point in bispectral analysis of the EEG. We demonstrated the EEG bicoherence patterns obtained from a unilateral lead (Fig. 1a, Fp_{1}‐A_{1}) and from a bilateral lead (Fig. 1b, Fp_{1}‐Fp_{2}). Both EEG signals were recorded simultaneously during isoflurane 0.9% anaesthesia. EEG bicoherence obtained from a unilateral lead showed two significant peaks in the low frequency region, but EEG bicoherence obtained from the bilateral lead was quite low in all bifrequency planes. When we used the bilateral lead, we could not find any increase in EEG bicoherence during either isoflurane, sevoflurane or propofol anaesthesia. Thus, if we want to obtain more information than that provided by power spectral‐based analysis from bispectral analysis, we must use the unilateral lead. As our previous report3 suggested, the increase in EEG bicoherence during anaesthesia seems to be related to the spindle wave and delta wave, whose rhythms are generated in the thalamic nuclei. During anaesthesia, those waves become predominant and EEG waveforms in both hemispherical leads, for example Fp_{1}‐A_{1} and Fp_{2}‐A_{2}, show a similar pattern. However, those waves were cancelled out and disappeared when a bilateral lead was used. The BIS monitor uses a unilateral lead (Fpz‐Ta1), so the conclusions of Miller and colleagues1 would not be applicable for the BIS calculation.

These important issues in applying bispectral analysis are relevant in other situations. Even when using a unilateral lead, the major changes in the bispectrum are caused by changes in the power spectrum. As we pointed out, as far as using bispectral values, there is little advantage in bispectral analysis. We would use bicoherence values. We have also shown that we can obtain other facts from phase information.4 Bispectral analysis would have given more information than power spectral‐based analysis, if the authors had used the EEG recorded from unilateral leads.

S. Hagihira

M. Takashina

T. Mori

T. Mashimo

*Osaka, Japan*

Editor—We thank Professor Hagihira and colleagues for their interest in our work, and their elegant demonstration of how the use of higher order EEG analysis can be extremely sensitive to the subtleties of electrode montage. We used a frontal bipolar electrode placement because of its convenience and its relative resistance to electromyographic interference. An EEG measure that is critically dependent on accurate lead placement is not likely to be of much use in the hustle and bustle of daily clinical practice. The peaks in the bicoherence detected by Professor Hagihara during sleep probably reflect the well‐described coupling between slow waves (∼4 Hz) and spindles (∼10 Hz).5 EEG spindles are largely driven by the thalamus and are distributed synchronously and bilaterally to the cortex. Typically, spindles occur after a slow wave of cortical depolarization, thus phase‐coupling will be seen between these two frequencies of cortical activity. It would seem that the bicoherence is a complicated way to detect sleep spindles and that a unipolar frontal electrode system would be superior to a bipolar frontal electrode system when collecting EEG data for this style of analysis. We agree that spindle detection and phase information analysis in general are potentially very important,6 and at present are missing from most EEG analysis of depth of anaesthesia. More original work is needed in this area.

In contrast to the bicoherence, the bispectrum is sensitive to amplitude changes of the components of the standard EEG frequency spectrum. Clearly, there are gross changes in amplitude of the various frequency components of the EEG in the awake *vs* the asleep or the anaesthetized state. Potentially, these changes of amplitude in the standard spectrum may mask any real changes in the amplitude of the true bispectral components of the bispectrum. The true bicoherence contribution to the bispectral power is minimal. The observations from Professor Hagihira do not change the other conclusions from our paper:

(i) The bispectrum is not significantly better than normal frequency spectrum in quantifying state of consciousness.

(ii) The calculation of higher order statistics from non‐stationary short time series may be misleading because long EEG data segments of at least 180–360 s are necessary for accurate statistical averaging.

J. W. Sleigh

J. Barnard

A. Miller

D. A. Steyn‐Ross

*Hamilton, New Zealand*

## References

1

Br J Anaesth

2004

;92

:8

–13Google Scholar

2

Anesth Analg

2001

;93

:966

–70Google Scholar

3

Anesthesiology

2002

;97

:1409

–15Google Scholar

4

Anesthesiology

, in press.Google Scholar

5

J Physiol

1996

;494

:265

–78Google Scholar

6

Human Brain Mapping

2003

;19

:248

–72Google Scholar

## Author notes

1Osaka, Japan 2Hamilton, New Zealand

British Occupational Hygiene Society

## Abstract

**Background.** Analysis of the bispectrum of EEG waveforms is a component of the proprietary BIS index—a commonly used commercial monitor of depth of anaesthesia. Does the use of the bispectrum give more information about depth of anaesthesia than the power spectrum?

**Methods.** We collected and analysed EEG waveforms during induction of general anaesthesia in 39 patients, comparing the changes in bispectral parameter (SynchFastSlow), with an analogous power spectrum‐based parameter (PowerFastSlow). Both compare the logarithmic ratio of high frequency components (40–47 Hz) with the total (1–47 Hz). Because the changes in bispectrum are affected by signal amplitude, we also calculated a third parameter (SFSbicoh) from the bicoherence, which is an amplitude‐independent statistic.

**Results.** The SynchFastSlow and PowerFastSlow were correlated (*r*=0.84) and neither was superior in predicting the awake or anaesthetized state (area under receiver operating characteristic curves = 0.85 *vs* 0.93). There was no change in the SFSbicoh over the induction period, and it did not correlate with SynchFastSlow (*r*=0.07).

**Conclusions.** We could not show that bispectral analysis gave more information than power spectral‐based analysis. Most of the changes in the bispectral values result from decreases in the relative high frequency content of the EEG caused by anaesthesia.

*Br J Anaesth* 2004; **92**: 8–13

Accepted for publication: June 11, 2003

Intraoperative electroencephalogram (EEG) monitoring has recently moved from experimental to clinical practice, as commercial monitors have been developed that process raw EEG data to provide a single value representing the ‘depth of anaesthesia’. A widely used example is the Bispectral Index Score (BIS), introduced by Aspect Medical Systems in 1997. The BIS is a combination of three weighted parameters: (i) the burst suppression ratio (the proportion of isoelectric EEG signal in an epoch); (ii) the beta ratio (a measure of the proportion of signal power in the high *vs* medium frequency range); and (iii) the SynchFastSlow (which quantifies the relative bispectral power in the 40–47 Hz frequency band).1^{–}3

As anaesthesia deepens, the amplitude of the high frequency portion of the EEG decreases, and the amplitude of the low frequencies increases.3 These changes can be described and quantified statistically. First‐order statistics like the mean can be applied to the raw EEG. More useful information is obtained by applying second‐order statistics like the power spectrum and the autocorrelation. However, it is conceivable that the different frequencies within the signal may not be independent of each other. If the oscillations are linked by a common phase relationship, it is necessary to use third‐order statistics to extract this information. The bispectral power is said to indicate the presence of quadratic phase‐coupling between different frequencies within the signal.4 The question arises, how important is this higher‐order information? Does the use of the bispectrum add information that is clinically useful that could not be obtained from the simpler power spectrum?

We compared changes, during induction of anaesthesia, of the SynchFastSlow value (a measure of bispectral con tent) with a value that is numerically analogous to the SynchFastSlow, but derived from the simple, second‐order, power spectrum. We term this parameter the PowerFast Slow. Because we found that the SynchFastSlow and PowerFastSlow were correlated, we then compared the ability of these measures to distinguish awake from anaesthetized states.

The SynchFastSlow value is intended to quantify the decrease in the relative amount of high‐frequency bispectral power. However the values taken by the bispectrum are not independent of the size of amplitudes in the EEG signal.4 Therefore, the SynchFastSlow parameter may only act as a measure of the anaesthesia‐induced loss of high frequencies in the EEG caused by anaesthesia. To quantify this effect we also calculated SynchFastSlow using signal bicoherence—an amplitude independent measure of phase‐coupling.

## Methods

### Patient data and EEG signal acquisition

After obtaining approval by the regional ethics committee and informed written consent, we studied 39 ASA I or II adult patients (24 females, mean (range) age = 47 (26–65) yr) undergoing elective surgery, a subset of those previously reported.5 No patient had a history of neurological disorder. The patients received midazolam 1–3 mg and fentanyl 50–100 µg i.v., followed by induction of anaesthesia with propofol 80–200 mg i.v. Four patients received alfentanil 0.5 mg i.v., and five received atracurium 20–35 mg or mivacurium 10–12 mg for neuromuscular block. Anaesthesia was maintained with isoflurane 0.8–2.5%. The clinical anaesthetist was blinded to the EEG monitoring. Electrode skin impedance was reduced to <5000 Ω by cleaning the skin with an abrasive cleaning fluid and using a low impedance electrode paste. We used disposable adhesive silver/silver chloride electrodes. The EEG was recorded using a bipolar bifrontal montage (F7–F8) with the ground electrode placed in the mid forehead position (Fz). An Aspect A‐1000 EEG monitor (software version 3.12, Aspect Medical Systems, Natick, MA, USA) was used to collect the raw EEG Data (sampling frequency 256 Hz), downloaded to a computer for later analysis.

### Signal processing and analysis

EEG signals were processed using algorithms written in Matlab (Matlab 6.0, The Mathworks Inc., Natick, MA, USA). The accuracy of these algorithms was tested using computer‐generated test data with known bispectral properties (using the Matlab function ‘qpcgen.m’). We were able to demonstrate that: (i) the bispectrum and bicoherence could distinguish between quadratically phase‐coupled, and uncoupled oscillations (Fig. 1); (ii) the log_{10} bispectral power (1–47 Hz) of an uncorrelated white noise signal decreased with increasing length of data epoch: from values of 1.63 (using 256 data points or 1 s) to 0.57 (using 8192 data points or 16 s). Signal processing algorithms were designed to emulate as far as possible the published calculation of the SynchFastSlow parameter.3 The raw signal was divided into 2 s epochs, each containing 512 discrete values. The first 160 s of patient EEG signal was analysed. This covered the period of induction of general anaesthesia. Epochs containing signals whose amplitude was >100 µV were excluded as artefact. The signal was filtered using a linear phase‐preserving notch filter to remove 50 Hz, and a Whittaker adaptive smoothing, high‐pass filter to remove fluctuations below 1 Hz.6 The bispectrum was calculated using the direct FFT method with 1 Hz resolution, as implemented with the Matlab Higher Order Signal Analysis toolbox function ‘bispecd.m’. We used a 128 point Blackman window, with 75% overlap. The results were robust to changes in window and overlap.

The SynchFastSlow measure for each epoch was calculated from the bispectrum as follows:

SynchFastSlow=log_{10}[bispectral power (40–47 Hz)/ bispectral power(1–47 Hz)]

We then calculated a measure analogous to the SynchFastSlow, but instead based on the (second‐order) spectral power alone. We have termed this the PowerFastSlow. It is a measure of the relative loss of high (gamma) frequencies that occurs as anaesthesia deepens:

PowerFastSlow=log_{10}[power (40–47 Hz)/power (1–47 Hz)]

Figure 2 illustrates the calculations of the SynchFastSlow and PowerFastSlow ratios using typical awake and anaesthetized epochs. The PowerFastSlow is the logarithm of the ratio of the area ‘B’ divided by area ‘A+B’; the SynchFastSlow is calculated similarly with the volumes A and B for SynchFastSlow. To help visualize the calculation of SynchFastSlow, we have separated the bispectral ‘volume’ ‘B’ from 40 to 47 Hz. This volume is equivalent to area ‘B’ in the upper plots. SynchFastSlow is the logarithm of a ratio of volumes whereas PowerFastSlow is the logarithm of a ratio of areas.

There are several technical issues in the calculation of the bispectrum. (i) Because the bispectrum is contaminated by the effects of signal amplitude, we also calculated the SynchFastSlow using the bicoherence rather than the bispectrum (called SFSbicoh). The bicoherence is similar to the bispectrum except it is constructed to be independent of the signal amplitude. If the bicoherence is non‐zero at a certain pair of frequencies, then it implies that the frequency pair have a common phase—they are said to be quadratically phase‐coupled. Therefore, if there was more information to be obtained from the use of third‐order statistics than from the simple power spectrum (which measures of the amplitude at each frequency), there should be observable changes in the pattern of the bicoherence during induction of general anaesthesia. (ii) Because calculation of the bispectrum requires averaging of the product of three quantities, rather than two (for the power spectrum), it requires many more data for accurate estimation of parameters. Our algorithm testing reported in the previous paragraph, indicates that short data lengths bias the bispectrum and bicoherence away from zero—thus falsely indicating the presence of interfrequency quadratic phase‐coupling. In the Aspect algorithm, data are analysed in 2 s (or 4 s in the original paper by Sigl and Chamoun) epochs. However because the variance of the bispectral estimation decreases with the length of data, the SynchFastSlow is probably a moving average, incorporating information from the last ∼60 s of EEG, the exact nature of which remains proprietary.34 To exclude the possibility that our results contained errors caused by the use of data epochs that were too short, we reanalysed the data using longer windowed epochs (following Akgul and Ning, 8 s = 2048 data points).78 We found no substantial changes in the results (see Results section).

### Statistical analysis

We compared the PowerFastSlow and SynchFastSlow using Bland–Altman plots separately for each patient and also for the pooled data. The within‐patient variation was calculated as the mean square of the SynchFastSlow–PowerFastSlow difference for each data point, minus the mean square difference for each patient. The between‐patient contribution was the mean square of the SynchFastSlow–PowerFastSlow difference for each patient subtracted from the grand mean square difference. We used the Pearson correlation coefficient (*r*) as a measure of association between variables, and the area under receiver operating curve analysis (ROC) to compare the ability of SynchFastSlow and PowerFastSlow to predict awake and anaesthetized states for each of the 39 patients. The distribution of values of SynchFastSlow and PowerFastSlow for awake and anaesthetized groups had a normal distribution (K‐S test), and so a parametric method of comparison of ROC areas was used (*t*‐test). The area under the ROC curve measures the discrimination (percentage of correctly classified patients) for each method. Awake epochs for each patient were defined as the mean of three non‐overlapping epochs from the start of the recorded EEG; and anaesthetized epochs as three non‐overlapping epochs before the end of the signal respectively; which corresponded to clinical states of (i) consciousness and (ii) surgical anaesthesia.

## Results

### Analysis of patient data

Figure 2 shows a typical power spectrum and bispectrum for an awake and an anaesthetized epoch. The changes in both the power spectrum and bispectrum are similar, with a loss of power (bispectral and spectral) in the high frequency bands when anaesthetized (i.e. area B, or volume B is diminished). Figure 3 illustrates, in a typical patient, the decrease in SynchFastSlow, and PowerFastSlow over the course of the induction of anaesthesia. The reduction in SynchFastSlow and PowerFastSlow closely track each other over time. However, when we remove the confounding effects of anaesthetic‐induced changes in signal amplitude from the bispectrum, by using bicoherence, the SFSbicoh did not change at all with induction of anaesthesia. Figure 4a is a scatterplot of SynchFastSlow *vs* PowerFastSlow for all patients and epochs (*r*=0.84 for 2 s epochs, *r*=0.88 for 8 s epochs, and *r*=0.93 for unfiltered data). In contrast there was no correlation between SynchFastSlow and SFSbicoh (*r*=0.07), or between PowerFastSlow and SFSbicoh (*r*=0.06). The Bland–Altman plot for the combined data (Fig. 4b) demonstrates that mean (sd) values of SynchFastSlow–PowerFastSlow=0.91 (0.71). Using the 2 s epochs the mean (sd) within‐patient mean‐square difference was 0.49 (0.13), and between patients was 0.06 (0.08). Using 8 s epochs the equivalent values were 0.40 (0.16) and 0.07 (0.08), respectively.

The mean (sd) values of the PowerFastSlow for the awake epochs were –1.19 (0.41), and for the anaesthetized epochs were –2.17 (0.46). The corresponding changes in SynchFastSlow were –0.81 (0.31) *vs* –1.32 (0.27). The area (se) under the awake‐anaesthetized ROC curves (Fig. 5) did not differ significantly between the two measurements, with a trend in favour of the PowerFastSlow (PowerFast Slow area=0.93 (0.16) *vs* SynchFastSlow area=0.85 (0.15); *P*=0.09, *t*‐test); suggesting that the SynchFastSlow in these patients was no better at predicting the anaesthetized state than the PowerFastSlow. Our additional ROC analysis was done on an additional sample of 88 patients from other EEG studies—anaesthetized using a variety of different anaesthetic techniques. With these additional data, the respective areas under the curves were 0.91 (0.23) and 0.92 (0.17).

## Discussion

The simple PowerFastSlow compared well with the SynchFastSlow over the induction period. Change in one measure was closely tracked by changes in the other, and we found no clear superiority with higher‐order methods to separate the awake and anaesthetized states.

Only limited work directly examines how higher‐order EEG spectral measures alter in response to changes in consciousness. Published work has not shown higher‐order statistics to be of much use. In the original paper, Barnett and colleagues found that during sleep, no clear patterns in bicoherence emerged, except some interactions at frequencies <7 Hz.9 Using the peak value of the bicoherence in the low frequency range (6–12 Hz), Muthuswamy could not distinguish between awake and anaesthetized dogs unless he also incorporated information about the end‐tidal halothane (see Fig. 6 in his paper).10 Similarly Hagihara and colleagues found modest changes in low‐frequency bicoherence as the concentration of isoflurane was altered, but suggested that at least 360 separate 2 s epochs were required for accurate estimation of the bicoherence.11 Bullock and colleagues measured the EEG from subdural electrode arrays in patients during periods of sleep, wakefulness, and seizures.12 They found that the magnitude of the bicoherence was extremely variable and transient.

We suggest that the traditional explanation of the bispectrum as a measure of interfrequency phase‐coupling is misleading when applied to most real EEG signals. There are two reasons for this statement. First, the EEG signal does not have multiple discrete oscillatory peaks, it is a stochastic, broad‐band power spectrum. With this signal, the bispectrum is a statistic that merely indicates a skewed probability distribution, and does not necessarily imply interfrequency quadratic phase‐coupling.13 Second, the magnitude of the bispectrum depends strongly on the amplitude of each frequency component.4 The bicoherence, a statistic that is independent of signal amplitude, did not change at all in the transition from consciousness to unconsciousness. We conclude that almost all of the change in the bispectrum with induction of anaesthesia is most simply explained by the decrease in relative high‐frequency spectral power caused by anaesthesia.

The original choice to use the SynchFastSlow measure as an important component of the BIS was empirically based, at least in part, on statistical analysis of a large database of thousands of EEG records. It was not published as to whether some parameter, similar to the PowerFastSlow, was ever considered as a possible measure of depth of anaesthesia. Our study does not have sufficient power to distinguish small differences between the two candidate measures. However we can be certain that there is no large difference. The actual contribution of the SynchFastSlow value to the total bispectral index, at the point of loss of consciousness is not published. It is likely that the SynchFastSlow becomes relatively more important with deeper levels of anaesthesia.

We were worried that the effects that we observed were artefacts caused by the specifics of data processing. As described in the methods, we therefore reanalysed the data using different smoothing windows, lengths of epochs, window overlap, and pre‐processing/filtering. The PowerFastSlow:SynchFastSlow correlation did not change substantially. Technically the most important component of the BIS monitor is probably the sophisticated artefact rejection and data repair algorithms, rather than the use of a measure based on the bispectrum.

Fourier analysis assumes that the input signal consists of a sum of sine waves of fixed amplitude, phase, and frequency; and that each sine wave persists for an infinite time. Because in practice we are limited to a signal of finite length, the bispectral power is biased away from zero, even for signals that have no phase‐coupling.14 Another assumption in the calculation of any Fourier‐based parameter is that the signal is said to be ‘stationary’. This term has various technical definitions, but intuitively it implies that the underlying processes that generate the signal are the same from the beginning to the end of the section under analysis. This assumption is difficult to justify for even short 2 s segments of EEG. It would be very unlikely to be consistently true over the course of the 60–720 s necessary for accurate calculation of the EEG bispectrum. These implicit assumptions are a problem for the calculation of the simple power spectrum; and the problems are magnified when attempting to estimate accurately the higher order spectra in the EEG. Short segments of data are more likely to be stationary, but less accurately calculated, whereas longer data lengths allow more accurate estimation of the bispectrum, but the data are less likely to be stationary. The inclusion of the SynchFastSlow parameter, which is based on the analysis of a large number of epochs, means that the BIS will lag behind the clinical state of the patient because of an irreducible calculation lag. The same information could be provided by the simpler, more quickly calculated, power spectrum. We were unable to demonstrate any clear advantages in using a higher‐order spectral measure over that using a simple power spectral measure, in the patients we studied.

## References

1

1995

Google Scholar

2

1999

;53

–74Google Scholar

3

Anesthesiology

1998

;89

:981

–1001Google Scholar

4

J Clin Monit

1994

;10

:392

–401Google Scholar

5

Br J Anaesth

2001

;86

:50

–8Google Scholar

6

Proc Edinborough Math Soc

1923

;41

:63

–75Google Scholar

7

IEEE Trans Biomed Eng

2000

;47

:997

–1009Google Scholar

8

IEEE Trans Biomed Eng

1989

;36

:497

–9Google Scholar

9

Science

1971

;172

:401

–2Google Scholar

10

J Clin Monit

1996

;12

:353

–64Google Scholar

11

Anesth Analg

2001

;93

:966

–70Google Scholar

12

Electroencephalogr Clin Neurophysiol

1997

;103

:661

–78Google Scholar

13

IEEE Trans Biomed Eng

1999

;46

:92

–9Google Scholar

14

J Geophys Res

1989

;94

:10993

–8Google Scholar

The Board of Management and Trustees of the British Journal of Anaesthesia

**Fig 1** Spectra and bispectra of artificially generated test signals. Data were generated using the ‘qpcgen.m’ program (1024 data points). (a) Uncoupled oscillations, and (b) quadratically‐coupled oscillations. The figure shows that although the power spectra for the uncoupled and coupled signals have identical magnitude, the bispectrum of the time‐series of uncoupled oscillations is many orders of magnitude less than that of the coupled oscillations. It clearly shows a peak at the phase coupled harmonic frequency (38 Hz).

**Fig 1** Spectra and bispectra of artificially generated test signals. Data were generated using the ‘qpcgen.m’ program (1024 data points). (a) Uncoupled oscillations, and (b) quadratically‐coupled oscillations. The figure shows that although the power spectra for the uncoupled and coupled signals have identical magnitude, the bispectrum of the time‐series of uncoupled oscillations is many orders of magnitude less than that of the coupled oscillations. It clearly shows a peak at the phase coupled harmonic frequency (38 Hz).

**Fig 2** Calculation of PowerFastSlow and SynchFastSlow for a typical awake and anaesthetized epoch. PowerFastSlow=log_{10}[Area (B)/Area (A+B)] (upper graphs); SynchFastSlow=log_{10}[Volume(A)/Volume (A+B)] (lower graphs).

**Fig 2** Calculation of PowerFastSlow and SynchFastSlow for a typical awake and anaesthetized epoch. PowerFastSlow=log_{10}[Area (B)/Area (A+B)] (upper graphs); SynchFastSlow=log_{10}[Volume(A)/Volume (A+B)] (lower graphs).

**Fig 3** Typical changes in PowerFastSlow, SynchFastSlow, and SFSbicoh during induction of anaesthesia. Bispectral SFS=SynchFastSlow calculated using the bispectrum; Bicoherence SFS=SynchFastSlow calculated using the bicoherence; PowerFastSlow=PowerFastSlow calculated using the power spectrum.

**Fig 3** Typical changes in PowerFastSlow, SynchFastSlow, and SFSbicoh during induction of anaesthesia. Bispectral SFS=SynchFastSlow calculated using the bispectrum; Bicoherence SFS=SynchFastSlow calculated using the bicoherence; PowerFastSlow=PowerFastSlow calculated using the power spectrum.

**Fig 4** (a and b) Correlation of PowerFastSlow (PFS) and SynchFastSlow (SFS). Combined data from all patients and epochs. (b) Bland–Altman plot. The solid line is the mean difference, and the dashed lines are the limits of agreement.

**Fig 4** (a and b) Correlation of PowerFastSlow (PFS) and SynchFastSlow (SFS). Combined data from all patients and epochs. (b) Bland–Altman plot. The solid line is the mean difference, and the dashed lines are the limits of agreement.

**Fig 5** Receiver operating characteristic curves for the SynchFastSlow and PowerFastSlow, testing awake *vs* anaesthetized EEG samples.

**Fig 5** Receiver operating characteristic curves for the SynchFastSlow and PowerFastSlow, testing awake *vs* anaesthetized EEG samples.

## One thought on “Bispectrum Analysis Essay”