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 of 2007 (eq 31 in https://doi.org/10.48550/arXiv.0711.1455), which again trivially includes the bivariate case.

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，由电神经活动产生的信号存在固有问题：由于容积传导和低空间分辨率，观测信号是真实未观测信号（来自所有脑区）的瞬时加权平均混合体。正因如此，滞后关联度量备受关注——至少从理论层面而言，“滞后关联”具有生理学起源。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：其值不得随真实瞬时生理关联强度的增强而趋近于零。有偏的度量将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。本文提出一种针对频域中两个多元时间序列之间滞后连通性的可检验显式定义。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，且对瞬时关联保持不变。作为第一项基本验证：在双变量时间序列情形下，新定义可退化为2007年提出的双变量滞后相干性（参见https://doi.org/10.48550/arXiv.0706.1776中的公式25-26）。作为第二项更强验证：在单变量与多元向量时间序列情形下，本文提出的新度量可退化为2007年提出的原始多元滞后相干性（参见https://doi.org/10.48550/arXiv.0711.1455中的公式31），而该多元度量自然包含双变量情形。