In computing the bispectrum, the EEG signal was first divided into a series of epochs. There is no rule how to divide the signal to a series of epochs. Overlapping of epochs is often used for increase the number of epochs in restricted EEG signal. For each epoch, Fourier transform was computed after applying appropriate window function.

Bispectrum: \(B\left(_, _\right)\) is calculated using the following equation.

$$_\left(_, _\right)= _\left(_\right)\bullet _\left(_\right)\bullet _^\left(_+_\right)$$

$$B\left(__\right)= \left|\sum__\left(_, _\right)\right|$$

Here, \(X\left(f\right)\) is a frequency component calculated using Fourier transform and is a complex-value. \(^\left(f\right)\), the conjugate of \(X\left(f\right)\). \(_\left(_, _\right)\), is a triple product. And j indicated the j-th of TP or X.

This summation is the heart of bispectral calculation. The reason is as follows:

When a complex-value is expressed as a polar coordinate, \(X\left(f\right)\) is expressed as follows:

$$X\left(f\right)=r\bullet \left\\left(ft+\theta \right)+i\bullet \text\left(ft+\theta \right)\right\}$$

Here, ‘t’ is the time parameter. Subsequently, each frequency component of triple products is expressed as follows:

$$X\left(_\right)=_\bullet \left\\left(_t+_\right)+i\bullet \text\left(_t+_\right)\right\}$$

$$X\left(_\right)=_\bullet \left\\left(_t+_\right)+i\bullet \text\left(_t+_\right)\right\}$$

$$^\left(_+_\right)=_\bullet \left[\text\left\_+_\right)t+_\right\}-i\bullet \text\left\_+_\right)t+_\right\}\right]$$

Here, \(_, _,_\) represent the magnitudes of \(X\left(_\right)\), \(X\left(_\right)\), \(X\left(_+_\right)\), and \(_,_,_\) represent the phase angle of \(X\left(_\right)\), \(X\left(_\right)\), \(X\left(_+_\right)\).

Using the characteristics of polar coordinates, the magnitude of \(_\left(_, _\right)\) becomes \(_\bullet _\bullet _\), and the phase angle of \(_\left(_, _\right)\) are calculated as follows:

$$\left(_t+_\right)+ \left(_t+_\right)-\left\_+_\right)t+_\right\}= _+_-_$$

And

$$TP\left(_+_\right)= _\bullet _\bullet _\left\\left(_+_-_\right)+i\bullet \text\left(_+_-_\right)\right\}$$

The fact that time parameter ‘t’ is canceled is important because it means that the phase angle of the triple product is independent of the EEG epoch that was used to calculate the frequency components.

It also follows that the form \(\left(_+ _-_\right)\) is suitable for investigating the relationship between (\(_+ _)\) and \(_\). Furthermore, the equation above also shows that the size of the triple product is independent of the \(_,_,_\) alignment.

Consequently, it is impossible to tell from a single epoch result whether or not a signal is phase-coupled.

Incidentally, bispectrum is the magnitude of the sum of triple products. The bispectrum value is determined by the distribution pattern of each \(\left(_+_-_\right)\). If there was no relationship among \(_,_,_\), \(\left(_+_-_\right)\) would be randomly distributed between \(\left[\text\pi \right]\). As a result, the sum of the triple products would tend toward zero. On the other hand, bispectrum would show non-zero values if there were some relationships among \(_,_,_\). Thus, bispectral analysis can detect phase coupling among the frequency components of a signal.

Because bispectrum is also influenced by the amplitudes of the frequency components, to evaluate the degree of phase coupling, it is necessary to normalize bispectrum values. Normalized bispectrum values are called bicoherence. Several methods for calculating bicoherence have been described in the literature. As described in a previous report [3], the authors hold that the following equation is most mathematically reasonable:

$$BIC\left(_,_\right)= \frac_,_\right)}\left|_\left(_,_\right)\right|}\bullet 100 (\%)$$

$$= \frac_+_\right)}\sqrt_\left(_\right)\bullet _\left(_\right)\bullet _\left(_+_\right)}}\bullet 100(\%)$$

Since bispectrum is defined as the magnitude of the sum of complex-numbers, using the equation below, it should be scaled using the sum of the magnitudes of each complex-number:

$$\left|_+_\right| \le \left|_\right|+ \left|_\right|$$

Another important parameter is average BIC (aBIC). As defined in the previous study [4], aBIC(f) was calculated as follows:

$$aBIC\left(f\right)= \frac\left[BIC\left(f,f\right)+2\left\\right]$$

And maximum value among \(aBIC\left(f\right) 2.0\le f\le 6.0\) Hz was defined as pBIC-low, and the maximum value among \(aBIC\left(f\right) 7.0\le f\le 13.0\) Hz was defined as pBIC-high.

In BSA, the calculating method for bispectrum followed that of the BIS monitor. Namely, EEG signal was first divided into two-second length of epoch, each epoch was overlapped by 75% each other, Fourier transform was computed after applying Blackman window.