  Identifying dynamic transactions between brain regions has become
increasingly important. Measurements within and across brain structures,
demonstrating the occurrence of bursts of beta/gamma oscillations only during
one specific phase of each theta/alpha cycle, have motivated the need to
advance beyond linear and stationary time series models. Here we offer a novel
measure, namely, the "dual frequency RV-coupling coefficient", for assessing
different types of frequency-frequency interactions that subserve information
flow in the brain. This is a measure of coherence between two complex-valued
vectors, consisting of the set of Fourier coefficients for two different
frequency bands, within or across two brain regions. RV-coupling is expressed
in terms of instantaneous and lagged components. Furthermore, by using
normalized Fourier coefficients (unit modulus), phase-type couplings can also
be measured. The dual frequency RV-coupling coefficient is based on previous
work: the second order bispectrum, i.e. the dual-frequency coherence (Thomson
1982; Haykin & Thomson 1998); the RV-coefficient (Escoufier 1973); Gorrostieta
et al (2012); and Pascual-Marqui et al (2011). This paper presents the new
measure, and outlines relevant statistical tests. The novel aspects of the
"dual frequency RV-coupling coefficient" are: (1) it can be applied to two
multivariate time series; (2) the method is not limited to single discrete
frequencies, and in addition, the frequency bands are treated by means of
appropriate multivariate statistical methodology; (3) the method makes use of a
novel generalization of the RV-coefficient for complex-valued multivariate
data; (4) real and imaginary covariance contributions to the RV-coherence are
obtained, allowing the definition of a "lagged-coupling" measure that is
minimally affected by the low spatial resolution of estimated cortical electric
neuronal activity.
