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