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：基于活动信号的皮层区域间关联度量，为研究脑功能连接提供了有用信息。然而，在处理电神经活动信号时会出现困难——由于容积传导和低空间分辨率，观测到的信号是各区域真实未观测信号的混合，即瞬时加权平均。这解释了为何滞后关联度量备受关注：至少理论上，“滞后关联”源于生理机制。相比之下，真实的生理性瞬时零滞后关联会被混合伪迹掩盖和混淆。滞后关联度量的最低要求是：其值不能随真实瞬时生理关联强度的增加而趋近于零。具有此类偏差的度量无法区分其数值变化是源于滞后关联还是瞬时关联的变化。本文提出了一种针对多变量时间序列频域滞后连接的可检验显式定义。该定义具备两个重要特性：对每个向量时间序列分别进行非奇异线性变换具有不变性，且对瞬时关联具有不变性。作为验证，当处理两个单变量时间序列时，新定义可回归至2007年提出的双变量滞后一致性（见https://doi.org/10.48550/arXiv.0706.1776 第25-26式）。