  Measures of association between cortical regions based on activity signals
provide useful information for studying brain functional connectivity.
Difficulties occur with signals of electric neuronal activity, where an
observed signal is a mixture, i.e. an instantaneous weighted average of the
true, unobserved signals from all regions, due to volume conduction and low
spatial resolution. This is why measures of lagged association are of interest,
since at least theoretically, "lagged association" is of physiological origin.
In contrast, the actual physiological instantaneous zero-lag association is
masked and confounded by the mixing artifact. A minimum requirement for a
measure of lagged association is that it must not tend to zero with an increase
of strength of true instantaneous physiological association. Such biased
measures cannot tell apart if a change in its value is due to a change in
lagged or a change in instantaneous association. An explicit testable
definition for frequency domain lagged connectivity between two multivariate
time series is proposed. It is endowed with two important properties: it is
invariant to non-singular linear transformations of each vector time series
separately, and it is invariant to instantaneous association. As a first sanity
check: in the case of two univariate time series, the new definition leads back
to the bivariate lagged coherence of 2007 (eqs 25 and 26 in
https://doi.org/10.48550/arXiv.0706.1776). As a second stronger sanity check:
in the case of a univariate and multivariate vector time series, the new
measure presented here leads back to the original multivariate lagged coherence
of 2007 (eq 31 in https://doi.org/10.48550/arXiv.0711.1455), which again
trivially includes the bivariate case.
