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 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).

翻译：基于活动信号的大脑区域间关联度量，为研究脑功能连接提供了有用信息。但电神经活动信号存在困难——由于容积传导和低空间分辨率，观测信号是各区域真实未观测信号的混合（即瞬时加权平均值）。因此滞后关联度量备受关注，因为至少在理论上，"滞后关联"具有生理起源。而实际生理性的瞬时零滞后关联则被混合伪迹掩盖和混淆。滞后关联度量的基本要求是：其数值不得随真实瞬时生理关联强度的增加而趋近于零。若度量存在此类偏差，则无法区分其数值变化源于滞后关联变化还是瞬时关联变化。本文提出了一种针对两个多元时间序列的频域滞后连接性的显式可检验定义。该定义具有两个重要性质：对每个向量时间序列分别进行非奇异线性变换时保持不变，且对瞬时关联具有不变性。作为自洽性检验，在双变量时间序列情形下，该新定义可还原为2007年的双变量滞后相干公式（详见https://doi.org/10.48550/arXiv.0706.1776中的等式25和26）。