[FieldTrip] Coherence in FieldTrip and mscohere in Matlab

[Clara A. Scholl]

Dear FieldTrip Community,

I'd like to better understand the requirement that the calculation of
coherence be averaged over > 1 trials, a topic previously discussed on
the FieldTrip discussion site (e.g.
http://mailman.science.ru.nl/pipermail/fieldtrip/2012-November/005923.html).

I worked through the calculation of coherence for n=1 trials at two
distinct channels manually (dividing the cross-spectrum by the mean of
the power spectra), and indeed the value resolves to 1 as others have
previously said it would (where the power spectra and cross spectrum
were estimated with the multitaper method).  However, I'm still
confused because the Matlab function mscohere returns coherence values
between 0 and 1 (less than 1) at each calculated frequency for an
input of two single time series X and Y, e.g. two single time courses
(from the same trial) at two distinct channels.

I suspect this may be related to the method for estimating the
power/cross spectral density in mscohere (Welch's averaged modified
periodogram method) because if I estimate the cross-spectrum and power
spectra in FieldTrip with cfg.method = 'mtmwelch_old' (a deprecated
option), then the coherence estimates are between 0 and 1 (not 1).
Can someone please help me to understand this -- why would coherence
be 1 using the multitaper method but not using Welch's method?

I'd like to understand if coherence is fundamentally required to be
estimated over multiple trials (at two distinct channels), or if the
requirement for multiple trials is related to the method of
calculating the spectra, and if so why that is the case.

Thanks immensely,
Clara