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

Mathematical and statistical standardization of EEG coherence are
best understood using a hand held calculator and then by simulation of the
EEG.
Coherence arises from Joseph Fourier’s 1805 fundamental inequality
where by the ratio of the cross-spectrum/product of auto-spectrum < 1.
Coherence is inherently a statistical estimate of coupling or association
between two time series and is in essence the correlation over trials or
repeated measures. As mentioned previously, the critical concept is “phase
consistency”, i.e., when the phase relationship between two time series is
constant over trials than coherence = 1.
7- How Does One Compute Coherence?
The first step in the calculation of the coherence spectrum is to
describe the activity of each raw time-series in the frequency domain by the
“auto-spectrum” which is a measure of the amount of energy or “activity” at
different frequencies. The second step is to compute the “cross-spectrum”
which is the energy in a frequency band that is in common to the two
different raw data time-series. The third step is to compute coherence
which is a normalization of the cross-spectrum as the ratio of the autospectra
and cross-spectra. To summarize:
1- Compute the auto-spectra of channels X and Y based on the “atoms” of
the spectrum
2- Compute the cross-spectra of X and Y from the “atoms” of the spectrum
3- Compute Coherence as the ratio of the auto-spectra and cross-spectra
8- First Compute the auto-spectra of channels X and Y based on the
“atoms” of the spectrum
Joseph Fourier in his thesis of 1805, benefiting from almost a century
of failed attempts, finally correctly showed that any complex time-series can
be decomposed into elemental “atoms” of individual frequencies (sine and
cosine and linear operations). Fourier defined the autospectrum as the
amount of energy present at a specific frequency band. He showed that the
autospectrum can be computed by multiplying each point of the raw data by
a series of cosines, and independently again by a series of sines, for the
frequency of interest. The average product of the raw-data and cosine is
known as the cosine coefficient of the finite discrete Fourier transform, and
that for the sine and the raw data as a sine coefficient. The relative
contributions of each frequency are expressed by these cosine and sine
(finite discrete Fourier) coefficients. The cosine and sine coefficients