Measuring functional connectivity of the brain

The rich temporal content of measurements of electromagnetic activity, including electroencephalography (EEG) and magnetoencephalography (MEG), allow researchers to study dynamic functional networks in the human brain. However, it is difficult to turn this data into meaningful conclusions about those brain networks. In this dissertation, we describe the theoretical relationships between different interaction measures, followed by development of novel measures to address classical nuisance of cross-talk in brain electrophysiological recordings. ❧ Coherence and phase locking value (PLV) are widely used measures that can reveal interactions between electrophysiological signals within a frequency range of interest. We investigate the statistical properties of the PLV by describing two distributions that are widely used to a priori model phase interactions. The first of these is the von Mises distribution, for which the standard sample PLV is a maximum likelihood estimator. The second is the relative phase distribution derived from bivariate circularly symmetric complex Gaussian data. We derive an explicit expression for the PLV for this distribution and show that it is a function of the coherence between the two signals. We then compare results via local field potential data from a visually-cued motor study in macaque for the two different PLV estimators and conclude that, for this data, the sample PLV provides equivalent information to the coherence of the two complex time series. This result reduces the analysis of time-locked activity between signals to the computation of coherence rather than coherence and PLV. ❧ Since the PLV is a bivariate measure (that is, it is computed pairwise between signals) it cannot differentiate between direct and indirect connections in a multidimensional network. A non-parametric partial phase synchronization index attempted to resolve this problem by extending sample PLV to the multivariate case using the same mechanism relating correlation to partial correlation. Here we derive an analytical expression for partial PLV for a multivariate circular complex Gaussian model and show that partial PLV can be computed from partial coherence. We demonstrate our method in simulations with Roessler oscillators and experimental data of multichannel local field potentials from a macaque monkey. We show that the multivariate circular complex Gaussian model suggests similar synchronization networks. In addition, the circular complex Gaussian model has a lower variance in the estimation of partial PLV. ❧ Interpretation of functional connectivity from EEG/MEG data is challenging due to cross-talk problem between signals of interest. For example, coherence may yield spuriously large values leading false positive connections. Approaches such as imaginary coherence, phase lag index, lagged coherence and orthogonal coherence have been proposed to overcome this problem. The common assumption of these measures is that time-lagged interactions are more robust to cross-talk than instantaneous interactions. However, none of these measures account for the remaining nodes in a multivariate network. While partial coherence quantifies the direct relationship between signals after excluding the linear effect from the remaining signals, it is still significantly affected by cross-talk. Here we combine the cross-talk robustness of lagged coherence with the multivariate framework of partial coherence to form a new measure called partial lagged coherence. Briefly, partial lagged coherence regresses our signals of interest onto the remaining signals so that only the instantaneous contributions from other signals are removed before computing lagged coherence. Our findings on realistic simulations of MEG data indicate a better performance of partial lagged coherence than other approaches in distinguishing direct from indirect connections in the presence of cross-talk.

脑电图（electroencephalography, EEG）与脑磁图（magnetoencephalography, MEG）所记录的电磁活动数据蕴含丰富的时序信息，可供研究者探索人类大脑的动态功能网络。然而，将这类数据转化为关于大脑网络的可靠结论却颇具挑战。本论文首先阐述不同交互度量间的理论关联，随后提出全新的度量方法，以解决脑电生理记录中普遍存在的串扰干扰难题。

相干性（coherence）与锁相值（phase locking value, PLV）是两类常用的度量指标，可用于揭示特定频率范围内电生理信号间的交互关系。本研究通过两种常用于相位交互先验建模的分布，探究锁相值的统计特性：第一种为冯·米塞斯分布（von Mises distribution），标准样本锁相值正是该分布下的极大似然估计量；第二种则是由双变量圆对称复高斯数据推导得到的相对相位分布。我们推导了该分布下锁相值的显式表达式，并证明其与两信号间的相干性相关。随后，我们借助猕猴视觉提示运动实验中的局部场电位（local field potential, LFP）数据，对比两种锁相值估计方法的结果，得出结论：在该数据集下，样本锁相值所能提供的信息与两复时间序列的相干性等价。这一结果将信号间时锁活动的分析简化为仅需计算相干性，而非同时计算相干性与锁相值。

由于锁相值属于双变量度量（即仅在成对信号间计算），因此无法在多维网络中区分直接与间接连接。此前有非参数偏相位同步指数尝试通过类似相关性到偏相关的扩展机制，将样本锁相值推广至多变量场景以解决该问题。本研究推导了多变量圆复高斯模型下偏锁相值的解析表达式，并证明偏锁相值可通过偏相干性计算得到。我们通过勒斯特勒振荡器（Roessler oscillators）仿真数据与猕猴多通道局部场电位实验数据验证了所提方法，结果表明多变量圆复高斯模型可得到一致的同步网络结构，且该模型在偏锁相值估计中具有更低的方差。

由于目标信号间存在串扰问题，基于脑电图与脑磁图数据的功能连接解读颇具挑战。例如，相干性可能产生虚假的高值，导致连接假阳性。为此，学界已提出虚部相干性、相位滞后指数、滞后相干性与正交相干性等方法以缓解该问题。这类方法的共同假设为：与时域瞬时交互相比，时滞交互对串扰具有更强的鲁棒性。但这类方法均未考虑多变量网络中的其余节点。尽管偏相干性可在排除其余信号的线性影响后量化信号间的直接关联，但仍易受串扰的显著干扰。本研究将滞后相干性的串扰鲁棒性与偏相干性的多变量框架相结合，提出一种名为偏滞后相干性的全新度量指标。简言之，偏滞后相干性首先将目标信号对其余信号进行回归，从而在计算滞后相干性前仅保留去除其他信号瞬时贡献后的分量。我们在逼真的脑磁图仿真数据中得到的结果表明：在存在串扰的场景下，偏滞后相干性在区分直接与间接连接方面的性能优于其他现有方法。