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 in equation 31 of the same 2007 publication (which trivially includes the bivariate case).

翻译：基于活动信号测量皮层区域之间的关联性，为研究脑功能连接提供了重要信息。然而，在处理电神经活动信号时存在困难——由于容积传导和低空间分辨率，观测信号实际上是所有区域真实（不可观测）信号的瞬时加权混合。正因如此，滞后关联的测度备受关注，因为从理论上讲，"滞后关联"具有生理学起源。相比之下，实际生理学意义上的瞬时零滞后关联会被混合伪迹掩盖和混淆。滞后关联测度的最低要求是：当真实瞬时生理关联增强时，其取值不应趋向于零。此类存在偏倚的测度无法区分其数值变化究竟源于滞后关联还是瞬时关联的改变。本文针对两个多元时间序列提出了频域滞后连通性的显式可检验定义。该定义具有两个重要特性：分别对每个向量时间序列的非奇异线性变换具有不变性，且对瞬时关联具有不变性。作为初级验证：当处理两个单变量时间序列时，该新定义可还原为2007年提出的双变量滞后相干性（doi:10.48550/arXiv.0706.1776中的式25和26）。作为更强的验证：当处理单变量与多元时间序列时，本文提出的新测度可还原为同一2007年论文中第31式的原始多元滞后相干性（该式自然包含双变量情形）。