  Measures of linear dependence (coherence) and nonlinear dependence (phase
synchronization) between any number of multivariate time series are defined.
The measures are expressed as the sum of lagged dependence and instantaneous
dependence. The measures are non-negative, and take the value zero only when
there is independence of the pertinent type. These measures are defined in the
frequency domain and are applicable to stationary and non-stationary time
series. These new results extend and refine significantly those presented in a
previous technical report (Pascual-Marqui 2007, arXiv:0706.1776 [stat.ME],
http://arxiv.org/abs/0706.1776), and have been largely motivated by the seminal
paper on linear feedback by Geweke (1982 JASA 77:304-313). One important field
of application is neurophysiology, where the time series consist of electric
neuronal activity at several brain locations. Coherence and phase
synchronization are interpreted as "connectivity" between locations. However,
any measure of dependence is highly contaminated with an instantaneous,
non-physiological contribution due to volume conduction and low spatial
resolution. The new techniques remove this confounding factor considerably.
Moreover, the measures of dependence can be applied to any number of brain
areas jointly, i.e. distributed cortical networks, whose activity can be
estimated with eLORETA (Pascual-Marqui 2007, arXiv:0710.3341 [math-ph]).
