EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

measure of the area under the curve and is also a measure of the duration of the spindle or
“burst” activity. Example produced using NeuroGuide demo software from
www.appliedneuroscience.com.
24 – The Bi-Spectrum and Bi-Coherence and Bi-Phase Difference
Another method to quantify spindle activity and brain connectivity as
it relates to spindles is the Bi-Spectrum which is divided into auto-channel
cross-frequency (single channels different frequencies) and cross-channel
cross-frequency (different frequencies in different channels). There are
several different definitions of the Bi-Spectrum. One is by Hasselman et al,
(1963) as the 3 rd moment statistical property called “skewness” which was
used to detect nonlinear interacting ocean waves. Brillinger and Rosenblatt
(1967) elaborated and described the computation of the tri-spectrum as the
4 th power statistical moment or “kurtosis”. The application of this
definition of bi-spectra is purely statistical and it is primarily used to detect
non-linearities. The second definition of bi-spectra is by Bendat and
Pearsol (1980) in which bi-spectra are produced by partial-coherence
analyses in order to isolate the covariances between different frequencies
and locations. The bi-spectrum using partial-coherence is a measure of the
association between different frequencies and different inputs, for example,
a measure of the phase consistency and the phase difference between theta
and beta frequencies (Helbig et al, 2006). Witte et al (1997) and Helbig et al
(2006) provide detailed time-series analysis and mathematics of the bispectrum,
bi-amplitude, bi-coherence and phase bi-coherence. In the
present paper we use the Bendat and Piersol (1980) approach to bi-spectra
and bi-coherence to develop measures of coherence and phase differences
between different frequencies within a single channel (auto bi-coherence and
bi-phase) and the correlation between frequencies in different EEG channels
or sensors (cross bi-coherence and cross bi-phase).
To calculate bi-coherence, it is necessary to multiply two complex
domain transforms of the digital time series to obtain a 3 rd order transform
and because of the linearity of the transforms and the need for real-time
computations we transform each instant of time for X t to the complex
domain by multiplying a time series by a sine and cosine sine wave at a
specific center frequency and band width followed by low-pass filtering.
This well established signal processing method is called “complex
demodulation” (Otnes and Enochson, 1972) and is equivalent to a Hilbert
transform that refer to it as a complex demodulation transform or “CDF”
where each time point is represented as a point on the unit circle 0 to 2pi.