Commonly used scalp electrodes consist of Ag-AgCl disks, 1 to 3 mm in diameter, with a very flexible long lead that can be plugged into an amplifier. Although it is desirable to obtain a low-impedance contact at the lectrode ski interface(less than 10 kQ2), this objective is confounded by hair and the difficulty of mechanically stabilizing the electrodes Conductive electrode paste helps obtain low impedance and keep the electrodes in place A type of cement( collodion) is used to fix small patches of gauze over electrodes for mechanical stability, and leads are usually taped to the subject to provide some strain relief. Slight abrasion of the skin is sometimes sed to obtain better electrode impedances, but this can cause irritation and sometimes infection(as well as pain in sensitive subjects). For long-term recordings, as in seizure monitoring, electrodes present major problems. Needle electrodes, which must be inserted into the tissue between the surface of the scalp and skull, are sometimes useful. However, ne danger of infection increases significantly. Electrodes with self-contained miniature amplifiers are somewhat more tolerant because they provide a low-impedance source to interconnecting leads, but they are expensive Despite numerous attempts to simplify the electrode application process and to guarantee long-term stability none has been widely accepted Instruments are available for measuring impedance between electrode pairs. The procedure is recommended trongly as good practice, since high impedance leads to distortions that may be difficult to separate from actual EEG signals. In fact, electrode impedance monitors are built into some commercial devices for recording EEGs Standard dc ohmmeters should not be used, since they apply a polarizing current that causes build-up of noisy electrode potential at the skin-electrode interface. Commercial devices apply a known-amplitude sinusoidal voltage(typically 1 kHz) to an electrode pair circuit and measure root mean square(rms)current, which directly related to the magnitude of the impedance From carefully applied electrodes, signal amplitudes of I to 10 uV can be obtained Considerable amplification (gain = 106)is required to bring these levels up to an acceptable level for input to recording devices. Because of long electrode leads and the common electrically noisy environment where recordings take place, differential amplifiers with inherently high input impedance and high common mode rejection ratios are essential for high In some facilities, special electrically shielded rooms minimize environmental electrical noise, particularly 60-Hz alternating current(ac) line noise. Since much of the information of interest in the EEG lies in the frequency bands less than 40 Hz, low-pass filters in the amplifier can be switched into attenuate 60-Hz noise sharply. For attenuating ac noise when the low-pass cutoff is greater than 60 Hz, many EEG amplifiers have notch filters that attenuate only frequencies in a narrow band centered around 60 Hz. Since important signal infor mation may also be attenuated, notch filtering should be used as a last resort; one should try to identify and eliminate the source of interference instead In trying to identify 60-Hz sources to eliminate or minimize their effect, it is sometimes useful to use a lummy source, such as a fixed 100-kQ2 resistor attached to the electrodes. An amplifier output represents only contributions from interfering sources. If noise can be reduced to an acceptable level (at least by a factor of 10 less than EEG signals) under this condition, one is likely to obtain uncontaminated EEG records Different types of recording instruments obtain a temporary or permanent record of the eeg. The most ommon recording device is a pen or chart recorder(usually multichannel) that is an integral part of most commercially available EEG instruments. The bandwidth of clinical EEGs is relatively low (less than 40 Hz) and therefore within the frequency response capabilities of these devices. Recordings are on a long sheet of continuous paper(from a folded stack), fed past the moving pen at one of several selectable constant speed The paper speed translates into distance per unit time or cycles per unit time, to allow EEG interpreters to identify different frequency components or patterns within the EEG. Paper speed is selected according to the monitoring situation at hand: slow speeds(10 mm/s)for observing the spiking characteristically associat with seizures and faster speeds(up to 120 mm/s )for the presence of individual frequency bands in the EEG In addition to (or instead of)a pen recorder, the EEg may be recorded on a multichannel frequency modulated(FM) analog tape recorder. During such recordings, a visual output device such as an oscilloscope or video display is necessary to allow visual monitoring of signals, so that corrective action(reapplying the electrodes and so on) can take place immediately if necessary. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Commonly used scalp electrodes consist of Ag-AgCl disks, 1 to 3 mm in diameter, with a very flexible long lead that can be plugged into an amplifier. Although it is desirable to obtain a low-impedance contact at the electrode ski interface (less than 10 kW), this objective is confounded by hair and the difficulty of mechanically stabilizing the electrodes. Conductive electrode paste helps obtain low impedance and keep the electrodes in place. A type of cement (collodion) is used to fix small patches of gauze over electrodes for mechanical stability, and leads are usually taped to the subject to provide some strain relief. Slight abrasion of the skin is sometimes used to obtain better electrode impedances, but this can cause irritation and sometimes infection (as well as pain in sensitive subjects). For long-term recordings, as in seizure monitoring, electrodes present major problems. Needle electrodes, which must be inserted into the tissue between the surface of the scalp and skull, are sometimes useful. However, the danger of infection increases significantly. Electrodes with self-contained miniature amplifiers are somewhat more tolerant because they provide a low-impedance source to interconnecting leads, but they are expensive. Despite numerous attempts to simplify the electrode application process and to guarantee long-term stability, none has been widely accepted. Instruments are available for measuring impedance between electrode pairs. The procedure is recommended strongly as good practice, since high impedance leads to distortions that may be difficult to separate from actual EEG signals. In fact, electrode impedance monitors are built into some commercial devices for recording EEGs. Standard dc ohmmeters should not be used, since they apply a polarizing current that causes build-up of noisy electrode potential at the skin-electrode interface. Commercial devices apply a known-amplitude sinusoidal voltage (typically 1 kHz) to an electrode pair circuit and measure root mean square (rms) current, which is directly related to the magnitude of the impedance. From carefully applied electrodes, signal amplitudes of 1 to 10 mV can be obtained. Considerable amplification (gain = 106 ) is required to bring these levels up to an acceptable level for input to recording devices. Because of long electrode leads and the common electrically noisy environment where recordings take place, differential amplifiers with inherently high input impedance and high common mode rejection ratios are essential for highquality EEG recordings. In some facilities, special electrically shielded rooms minimize environmental electrical noise, particularly 60-Hz alternating current (ac) line noise. Since much of the information of interest in the EEG lies in the frequency bands less than 40 Hz, low-pass filters in the amplifier can be switched into attenuate 60-Hz noise sharply. For attenuating ac noise when the low-pass cutoff is greater than 60 Hz, many EEG amplifiers have notch filters that attenuate only frequencies in a narrow band centered around 60 Hz. Since important signal information may also be attenuated, notch filtering should be used as a last resort; one should try to identify and eliminate the source of interference instead. In trying to identify 60-Hz sources to eliminate or minimize their effect, it is sometimes useful to use a dummy source, such as a fixed 100-kW resistor attached to the electrodes. An amplifier output represents only contributions from interfering sources. If noise can be reduced to an acceptable level (at least by a factor of 10 less than EEG signals) under this condition, one is likely to obtain uncontaminated EEG records. Different types of recording instruments obtain a temporary or permanent record of the EEG. The most common recording device is a pen or chart recorder (usually multichannel) that is an integral part of most commercially available EEG instruments. The bandwidth of clinical EEGs is relatively low (less than 40 Hz) and therefore within the frequency response capabilities of these devices. Recordings are on a long sheet of continuous paper (from a folded stack), fed past the moving pen at one of several selectable constant speeds. The paper speed translates into distance per unit time or cycles per unit time, to allow EEG interpreters to identify different frequency components or patterns within the EEG. Paper speed is selected according to the monitoring situation at hand: slow speeds (10 mm/s) for observing the spiking characteristically associated with seizures and faster speeds (up to 120 mm/s) for the presence of individual frequency bands in the EEG. In addition to (or instead of) a pen recorder, the EEG may be recorded on a multichannel frequency modulated (FM) analog tape recorder. During such recordings, a visual output device such as an oscilloscope or video display is necessary to allow visual monitoring of signals, so that corrective action (reapplying the electrodes and so on) can take place immediately if necessary
Sophisticated FM cassette recording and playback systems allow clinicians to review long EEG recordings over a greatly reduced time, compared to that required to flip through stacks of paper or observe recordings as they occur in real time. Such systems take advantage of time compensation schemes, whereby a signal recorded at one speed(speed of the tape moving past the recording head of the cassette drive)is played back at a different faster speed. The ratio of playback to recording speed is known, so the appropriate correction factor can be applied to played-back data to generate a properly scaled video display. A standard ratio of 60 1 is often used. Thus, a trained clinician can review each minute of real-time EEG in 1 s. The display appears to be scrolled at a high rate horizontally across the display screen. Features of these instruments allow the clinician to freeze segment of EEG on the display and to slow down or accelerate tape speed from the standard playback as needed. A time mark channel is usually displayed as one of the traces as a convenient reference(vertical tick "mark displayed at periodic intervals across the screen) Computers can also be recording devices, digitizing( converting to digital form) one or several amplified EEG channels at a fixed rate. In such sampled data systems, each channel is repeatedly sampled at a fixed time interval(sample interval)and this sample is converted into a binary number representation by an A/D converter The A/D converter is interfaced to a computer system so that each sample can be saved in the computer nemory. A set of such samples, acquired at a sufficient sampling rate(at least two times the highest frequency component in the sampled signal), is sufficient to represent all the information in the waveform. To ensure that the signal is band-limited, a low-pass filter with a cutoff frequency equal to the highest frequency of interest is used. Since physically realizable filters do not have the ideal characteristics, the sampling rate is usually greater than two times the filters cutoff frequency. Furthermore, once converted to a digital format, digital filtering es can On-line computer recordings are only practical for short-term recordings or for situations in which the EEG immediately processed. This limitation is primarily due rage requirements. For example, a typical sampling rate of 128 Hz yields 128 new samples per second that require storage. For an 8-channel recording, ,024 samples are acquired per second. A 10-minute recording period yields 614, 400 data points. Assuming 8- bit resolution per sample, over 0.5 megabyte(MB)of storage is required to save the 10-minute recording Processing can consist of compression for more efficient storage(with associated loss of total information content), as in data record or epoch averaging associated with evoked responses, or feature extraction and subsequent pattern recognition, as in automated spike detection in seizure monitoring. Frequency analysis of the EEG In general, the EEG contains information regarding changes in the electrical pot of the brain obtained om a given set of recording electrodes. These data include the characteristic waveform with its variation in amplitude, frequency, phase, etc and the occurrence of brief electrical patterns, such as spindles. Any analysis procedure cannot simultaneously provide information regarding all of these variables. Consequently, the selec tion of any analytic technique will emphasize changes in one particular variable at the expense of the others This observation is extremely important if one is to properly interpret the results obtained by any analytic chnique. In this chapter, special attention is given to frequency analysis of the EEG In early attempts to correlate the EEG with behavior, analog frequency analyzers were used to examine single channels of EEG data. Although disappointing, these initial efforts did introduce the utilization of frequency analysis to study gross brain wave activity. Although, power spectral analysis, i.e., the magnitude square of Fourier transform, provides a quantitative measure of the frequency distribution of the EEG, it does so as mentioned above, at the expense of other details in the EEg such as the amplitude distribution, as well as the presence of specific patterns in the EEG The first systematic application of power spectral analysis by general-purpose computers was reported in 1963 by Walter; however, it was not until the introduction of the fast Fourier transform(FFT) by Cooley and Tukey in the early 1970s that machine computation of the EEG became commonplace. Although an individual FFT is ordinarily calculated for a short section of EEG data(e.g, from 1 to 8 s epoch), such segmentation of a signal with subsequent averaging over individual modified periodograms has been shown to provide a consistent estimator of the power spectrum, and an extension of this technique, the compressed spectral array, has been particularly useful for computing EEG spectra over long periods of time. A detailed review of the c2000 by CRC Press LLC
© 2000 by CRC Press LLC Sophisticated FM cassette recording and playback systems allow clinicians to review long EEG recordings over a greatly reduced time, compared to that required to flip through stacks of paper or observe recordings as they occur in real time. Such systems take advantage of time compensation schemes, whereby a signal recorded at one speed (speed of the tape moving past the recording head of the cassette drive) is played back at a different, faster speed. The ratio of playback to recording speed is known, so the appropriate correction factor can be applied to played-back data to generate a properly scaled video display. A standard ratio of 60:1 is often used. Thus, a trained clinician can review each minute of real-time EEG in 1 s. The display appears to be scrolled at a high rate horizontally across the display screen. Features of these instruments allow the clinician to freeze a segment of EEG on the display and to slow down or accelerate tape speed from the standard playback as needed. A time mark channel is usually displayed as one of the traces as a convenient reference (vertical “tick” mark displayed at periodic intervals across the screen). Computers can also be recording devices, digitizing (converting to digital form) one or several amplified EEG channels at a fixed rate. In such sampled data systems, each channel is repeatedly sampled at a fixed time interval (sample interval) and this sample is converted into a binary number representation by an A/D converter. The A/D converter is interfaced to a computer system so that each sample can be saved in the computer’s memory. A set of such samples, acquired at a sufficient sampling rate (at least two times the highest frequency component in the sampled signal), is sufficient to represent all the information in the waveform. To ensure that the signal is band-limited, a low-pass filter with a cutoff frequency equal to the highest frequency of interest is used. Since physically realizable filters do not have the ideal characteristics, the sampling rate is usually greater than two times the filter’s cutoff frequency. Furthermore, once converted to a digital format, digital filtering techniques can be used. On-line computer recordings are only practical for short-term recordings or for situations in which the EEG is immediately processed. This limitation is primarily due to storage requirements. For example, a typical sampling rate of 128 Hz yields 128 new samples per second that require storage. For an 8-channel recording, 1,024 samples are acquired per second. A 10-minute recording period yields 614,400 data points. Assuming 8- bit resolution per sample, over 0.5 megabyte (MB) of storage is required to save the 10-minute recording. Processing can consist of compression for more efficient storage (with associated loss of total information content), as in data record or epoch averaging associated with evoked responses, or feature extraction and subsequent pattern recognition, as in automated spike detection in seizure monitoring. Frequency Analysis of the EEG In general, the EEG contains information regarding changes in the electrical potential of the brain obtained from a given set of recording electrodes. These data include the characteristic waveform with its variation in amplitude, frequency, phase, etc. and the occurrence of brief electrical patterns, such as spindles. Any analysis procedure cannot simultaneously provide information regarding all of these variables. Consequently, the selection of any analytic technique will emphasize changes in one particular variable at the expense of the others. This observation is extremely important if one is to properly interpret the results obtained by any analytic technique. In this chapter, special attention is given to frequency analysis of the EEG. In early attempts to correlate the EEG with behavior, analog frequency analyzers were used to examine single channels of EEG data. Although disappointing, these initial efforts did introduce the utilization of frequency analysis to study gross brain wave activity. Although, power spectral analysis, i.e., the magnitude square of Fourier transform, provides a quantitative measure of the frequency distribution of the EEG, it does so as mentioned above, at the expense of other details in the EEG such as the amplitude distribution, as well as the presence of specific patterns in the EEG. The first systematic application of power spectral analysis by general-purpose computers was reported in 1963 by Walter; however, it was not until the introduction of the fast Fourier transform (FFT) by Cooley and Tukey in the early 1970s that machine computation of the EEG became commonplace. Although an individual FFT is ordinarily calculated for a short section of EEG data (e.g., from 1 to 8 s epoch), such segmentation of a signal with subsequent averaging over individual modified periodograms has been shown to provide a consistent estimator of the power spectrum, and an extension of this technique, the compressed spectral array, has been particularly useful for computing EEG spectra over long periods of time. A detailed review of the
development and use of various methods to analyze the EEG is provided by Givens and Redmond [1987 Figure 115.3 provides an overview of the computational processes involved in 28 performing spectral analysis of the EEG, i.e, including computation of auto and cross spectra [Bronzino, 1984. It is to be noted that the power spectrum is the utocorrellogram, i.e. the correlation of the signal with itself. As a result, the power CALCULATE spectrum provides only magnitude information in the frequency domain; it does RAW SPECTRA not provide any data regarding phase. The power spectrum is computed by: P(=Re2X(]+Im2[X(0) (1151) CALCULATE POWER SPECTRAL DENSITY where X() is the Fourier transform of the EEG. Power spectral analysis not only provides a summary of the EEG in a convenient graphic form, but also facilitates statistical analysis of EEG changes which may not ALCULATE be evident on simple inspection of the records. In addition to absolute power CROSS SPECTRA derived directly from the power spectrum, other measures calculated from absolute power have been demonstrated to be of value in quantifying various aspects of the <EG. Relative power expresses the percent contribution of each frequency band to SMOOTH DATA total power and is calculated by dividing the power within a band by the total power across all bands Relative power has the benefit of reducing the intersubject variance associated with absolute power that arises from intersubject differences in skull and scalp conductance. The disadvantage of relative power is that an increase in one frequency band will be reflected in the calculation by a decrease in CALCULATE COHERENCE other bands; for example, it has been reported that directional shifts between high and low frequencies are associated with changes in cerebral blood flow and metab olism. Power ratios between low(0-7 Hz) and high(10-20 Hz) frequency bands CALCULATE PHASE SHIFT have been demonstrated to be an accurate estimator of changes in cerebral activity during these metabolic changes. Although the power spectrum quantifies activity at each electrode, other vari- FIGURE 115.3 Block dia. ables derivable from FFT offer a measure of the relationship between activity gram of measures determined recorded at distinct electrode sites. Coherence( which is a complex number), cal- from spectral analysis. culated from the cross-spectrum analysis of two signals, is similar to cross-corre- lation in the time domain. The magnitude squared coherence(MSC) values range from 1 to 0, indicating maximum or no synchrony, respectively, and are independent of power. The temporal relationship between two signals is expressed by phase, which is a measure of the lag between two signals for common frequency components or bands. Phase is expressed in units of degrees, 0o indicating no time lag between signals or 180 if the signals are of opposite polarity. Phase can also be transformed into the time domain, giving a measure of the time difference between two frequencies Cross spectrum is computed by Cross spectrum=X(r where X(, Y) are Fourier transforms and* indicates complex conjugates and coherence is calculated by Coherence Cross spectrum (115.3) PX()-PY) Since coherence is a complex number, the phase is simply the angle associated with the polar expression of that number. MSC and phase represent measures that can be employed to investigate the cortical interactions of cerebral activity. For example, short(intracortical)and long( cortico-cortical) pathways have been proposed e 2000 by CRC Press LLC
© 2000 by CRC Press LLC development and use of various methods to analyze the EEG is provided by Givens and Redmond [1987]. Figure 115.3 provides an overview of the computational processes involved in performing spectral analysis of the EEG, i.e., including computation of auto and cross spectra [Bronzino, 1984]. It is to be noted that the power spectrum is the autocorrellogram, i.e., the correlation of the signal with itself.As a result, the power spectrum provides only magnitude information in the frequency domain; it does not provide any data regarding phase. The power spectrum is computed by: P(f) = Re2[X(f)] + Im2[X(f)] (115.1) where X(f) is the Fourier transform of the EEG. Power spectral analysis not only provides a summary of the EEG in a convenient graphic form, but also facilitates statistical analysis of EEG changes which may not be evident on simple inspection of the records. In addition to absolute power derived directly from the power spectrum, other measures calculated from absolute power have been demonstrated to be of value in quantifying various aspects of the EEG. Relative power expresses the percent contribution of each frequency band to the total power and is calculated by dividing the power within a band by the total power across all bands. Relative power has the benefit of reducing the intersubject variance associated with absolute power that arises from intersubject differences in skull and scalp conductance. The disadvantage of relative power is that an increase in one frequency band will be reflected in the calculation by a decrease in other bands; for example, it has been reported that directional shifts between high and low frequencies are associated with changes in cerebral blood flow and metabolism. Power ratios between low (0–7 Hz) and high (10–20 Hz) frequency bands have been demonstrated to be an accurate estimator of changes in cerebral activity during these metabolic changes. Although the power spectrum quantifies activity at each electrode, other variables derivable from FFT offer a measure of the relationship between activity recorded at distinct electrode sites. Coherence (which is a complex number), calculated from the cross-spectrum analysis of two signals, is similar to cross-correlation in the time domain. The magnitude squared coherence (MSC) values range from 1 to 0, indicating maximum or no synchrony, respectively, and are independent of power. The temporal relationship between two signals is expressed by phase, which is a measure of the lag between two signals for common frequency components or bands. Phase is expressed in units of degrees, 0° indicating no time lag between signals or 180° if the signals are of opposite polarity. Phase can also be transformed into the time domain, giving a measure of the time difference between two frequencies. Cross spectrum is computed by: Cross spectrum = X(f) Y*(f) (115.2) where X(f), Y(f) are Fourier transforms and * indicates complex conjugates and coherence is calculated by (115.3) Since coherence is a complex number, the phase is simply the angle associated with the polar expression of that number. MSC and phase represent measures that can be employed to investigate the cortical interactions of cerebral activity. For example, short (intracortical) and long (cortico-cortical) pathways have been proposed Coherence = Cross spectrum PX( )f - PY(f ) FIGURE 115.3 Block diagram of measures determined from spectral analysis
as the anatomic substrate underlying the spatial frequency and patterns of coherence. Therefore, discrete cortical regions linked by such fiber systems should demonstrate a relatively high degree of synchrony, whereas the time lag between signals, as represented by phase, quantifies the extent to which one signal leads another Nonlinear Analysis of the EEG As mentioned earlier, the EEG has been studied extensively using signal-processing schemes, most of which are based on the assumption that the Eeg is a linear, gaussian process. Although linear analysis schemes are computationally efficient and useful, they only utilize information retained in the autocorrelation function(ie the second-order cumulant). Additional information stored in higher-order cumulants is therefore ignored by linear analysis of the EEG. Thus, while the power spectrum provides the energy distribution of a stationary process in the frequency domain, it cannot distinguish nonlinearly coupled frequencies from spontaneously generated signals with the same resonance condition [Nikias and Raghvveer, 1987] There is evidence showing that the amplitude distribution of the EEg often deviates from gaussian behavior. It has been reported, for example, that the EEG of humans involved in the performance of mental arithmetic ask exhibits significant nongaussian behavior. In addition, the degree of deviation from gaussian behavior of the EEG has been shown to depend to the behavioral state, with the state of slow-wave sleep showing less gaussian behavior than quiet waking, which is less gaussian than rapid eye movement(REM)sleep [Ning and Bronzino, 1989, b]. Nonlinear signal-processing algorithms such as bispectral analysis are therefore necessary to address nongaussian and nonlinear behavior of the EEG in order to better describe it in the frequency domain. But what exactly is the bispectrum? For a zero-mean, stationary process (X(K)l, the bispectrum, by definition, is the Fourier transform of its third-order cumulant(TOC)sequence B(a,02)=∑∑(mn)m 朋三一n三一 The TOC sequence C(m, n)) is defined as the expected value of the triple product C(m, m)=x(k)(k+ m x(k+ n) (115.5) If process X(k)is purely gaussian, then its third-order cumulant C(m, n)is zero for each(m, n), and consequently, Fourier transform, the bispectrum, B(O,, o, )is also zero. This property makes the estimated bispectrum an immediate measure describing the degree of deviation from gaussian behavior. In our studies [Ning and Bronzino, 1989, b], the sum of magnitude of the estimated bispectrum was used as a measure to describe the EEG's deviation from gaussian behavior, that is, D=∑, (o @ 2 Using bispectral analysis, the existence of significant quadratic phase coupling(QPC) in the hippocampa EEG obtained during REM sleep in the adult rat was demonstrated [Ning and Bronzino, 1989a, b, 1990]. The result of this nonlinear coupling is the appearance, in the frequency spectrum, of a small peck centered at approximately 13 to 14 Hz(beta range) that reflects the summation of the two theta frequency(i.e, in the 6- 7-Hz range)waves. Conventional power spectral (linear)approaches are incapable of distinguishing the fact that this peak results from the interaction of these two generators and is not intrinsic to either. To examine the phase relationship between nonlinear signals collected at different sites, the cross-bispectrum is also a useful tool. For example, given three zero-mean, stationary processes"x(n)j=1, 2, 31, there are two nventional methods for determining the cross-bispectral relationship, direct and indirect. Both methods first divide these three processes into M segments of shorter but equal length. The direct method computes the c2000 by CRC Press LLC
© 2000 by CRC Press LLC as the anatomic substrate underlying the spatial frequency and patterns of coherence. Therefore, discrete cortical regions linked by such fiber systems should demonstrate a relatively high degree of synchrony, whereas the time lag between signals, as represented by phase, quantifies the extent to which one signal leads another. Nonlinear Analysis of the EEG As mentioned earlier, the EEG has been studied extensively using signal-processing schemes, most of which are based on the assumption that the EEG is a linear, gaussian process. Although linear analysis schemes are computationally efficient and useful, they only utilize information retained in the autocorrelation function (i.e., the second-order cumulant). Additional information stored in higher-order cumulants is therefore ignored by linear analysis of the EEG. Thus, while the power spectrum provides the energy distribution of a stationary process in the frequency domain, it cannot distinguish nonlinearly coupled frequencies from spontaneously generated signals with the same resonance condition [Nikias and Raghvveer, 1987]. There is evidence showing that the amplitude distribution of the EEG often deviates from gaussian behavior. It has been reported, for example, that the EEG of humans involved in the performance of mental arithmetic task exhibits significant nongaussian behavior. In addition, the degree of deviation from gaussian behavior of the EEG has been shown to depend to the behavioral state, with the state of slow-wave sleep showing less gaussian behavior than quiet waking, which is less gaussian than rapid eye movement (REM) sleep [Ning and Bronzino, 1989a,b]. Nonlinear signal-processing algorithms such as bispectral analysis are therefore necessary to address nongaussian and nonlinear behavior of the EEG in order to better describe it in the frequency domain. But what exactly is the bispectrum? For a zero-mean, stationary process {X(k)}, the bispectrum, by definition, is the Fourier transform of its third-order cumulant (TOC) sequence: (115.4) The TOC sequence {C(m, n)} is defined as the expected value of the triple product (115.5) If process X(k) is purely gaussian, then its third-order cumulant C(m, n) is zero for each (m, n), and consequently, its Fourier transform, the bispectrum, B(w1, w2) is also zero. This property makes the estimated bispectrum an immediate measure describing the degree of deviation from gaussian behavior. In our studies [Ning and Bronzino, 1989a,b], the sum of magnitude of the estimated bispectrum was used as a measure to describe the EEG’s deviation from gaussian behavior, that is, (115.6) Using bispectral analysis, the existence of significant quadratic phase coupling (QPC) in the hippocampal EEG obtained during REM sleep in the adult rat was demonstrated [Ning and Bronzino, 1989a,b, 1990]. The result of this nonlinear coupling is the appearance, in the frequency spectrum, of a small peck centered at approximately 13 to 14 Hz (beta range) that reflects the summation of the two theta frequency (i.e., in the 6- to 7-Hz range) waves. Conventional power spectral (linear) approaches are incapable of distinguishing the fact that this peak results from the interaction of these two generators and is not intrinsic to either. To examine the phase relationship between nonlinear signals collected at different sites, the cross-bispectrum is also a useful tool. For example, given three zero-mean, stationary processes ”xj (n)j = 1, 2, 3}, there are two conventional methods for determining the cross-bispectral relationship, direct and indirect. Both methods first divide these three processes into M segments of shorter but equal length. The direct method computes the B Cm ne j wm wn n a m a w w aa 1 2 1 2 ( , , ) = ( ) - + ( ) = -= - ÂÂ Cm n E XkXk mXk n ( , ) = { } ( ) ( + ) ( + ) D B = ( ) ( ) Â w w w w 1 2 1 2
Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is ∑x(o)xo)x"on+2) (1157) x12 3 where X (o) is the Fourier transform of the mth segment of x(n)l, and*indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments Cm(=∑增小)(m+A)r(+ (1158) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate k,1) (1159) xrx The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant (,1) (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment omputation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between x,(n) and Ix2(n)) and its effects on x(n). For example, a peak at Bx, x2*(O1,o2) suggests that the energy component O2 of [,(n) is generated due to the QPC betw of x(n)). In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) P(,P(OP(o, +02 where Pr (o) is the power spectrum of process x(n)). The theoretical value of the bicoherence index ranges In situations where the interest is the presence of QPC and its effects on ix( n)l, the cross-bispectrul equations can be modified by replacing Ix(n)) and x(n)) with (x(n)) and (x,(n)I with n(n)b, that is, Bn(o,0)=∑xo)y"(o) X"(o1+0 (115.12) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is, (115.7) where Xj m(w) is the Fourier transform of the mth segment of {xj (n)}, and * indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments: (115.8) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate: (115.9) The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant sequence: (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment, computation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between {x1(n)} and {x2(n)} and its effects on {x3(n)}. For example, a peak at Bx1x2x3(w1, w2) suggests that the energy component at frequency w1 + w2 of {x3(n)} is generated due to the QPC between frequency w1 of {x1(n)} and frequency w2 of {x2(n)}. In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) where Pxj(w) is the power spectrum of process {xj (n)}. The theoretical value of the bicoherence index ranges between 0 and 1, i.e., from nonsignificant to highly significant. In situations where the interest is the presence of QPC and its effects on {x(n)}, the cross-bispectrum equations can be modified by replacing {x1(n)} and {x3(n)} with {x(n)} and {x2(n)} with {y(n)}, that is, (115.12) B M xxx XXX m m M m m 123 12 1 1 12 23 1 2 1 ww w w w w , * ( ) = ( ) ( ) ( + ) = Â C k l x nx n kx n l xxx m m n m m 123 12 3 ( , ) = Â ( ) ( + ) ( + ) et C kl M C kl xxx xxx m m M 123 123 1 1 ( , , ) = ( ) = Â B C kl xxx xxx jk l lk 123 123 1 2 w w 1 2 w w a a a a ( , , ) = ( ) - + ( ) = -= - ÂÂ bic B PP P xxx xxx xx x 123 123 12 3 1 2 1 2 1 2 12 w w w w w w ww , , ( ) = ( ) ( ) ( ) ( + ) B M xyz XY X m m M m m ww w w w w 12 1 1 2 12 1 , * ( ) = ( ) ( ) ( + ) = Â