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 metab￾olism. 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 vari￾ables derivable from FFT offer a measure of the relationship between activity recorded at distinct electrode sites. Coherence (which is a complex number), cal￾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, 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 dia￾gram of measures determined from spectral analysis