A comparison of bivariate frequency domain measures of electrophysiological connectivity

1The problem of interest here concerns electrophysiological signals from two cortical sites, acquired as invasive intracranial recordings, or from non-invasive estimates of cortical electric neuronal activity computed from EEG or MEG recordings (see e.g. https://doi.org/10.1101/269753). In the absence of other sources, these measured signals consist of an instantaneous linear mixture of the true, actual, unobserved local signals, due to low spatial resolution and volume conduction. A connectivity measure is unreliable as a true indicator of electrophysiological connectivity if it is not invariant to mixing, or if it reports a significant connection for a mixture of independent signals. In (Vinck et al 2011 Neuroimage 55:1548) it was shown that coherence, imaginary coherence, and phase locking value are not invariant to mixing, while the phase lag index (PLI) and the weighted version (wPLI) are invariant to mixing. Here we show that the lagged coherence (LagCoh) measure (2007, https://arxiv.org/abs/0711.1455), not studied in Vinck et al, is invariant to mixing. Additionally, we present here a new mixture-invariant connectivity statistic: the \"standardized imaginary covariance\" (sImCov). We also include in the comparisons the directed PLI (dPLI) by Stam et al (2012 Neuroimage 62:1415). Fourier coefficients for \"N\" trials are generated from a linear unidirectional causal time domain model with electrophysiological delay \"k\" and regression coefficient \"b\". 1000 random data sets of \"N\" trials are simulated, and for each one, and for each connectivity measure, non-parametric randomization tests are performed. The \"true positive detection rate\" is calculated as the fraction of 1000 cases that have significant connectivity at p<0.05, 0.1, and 0.2. The connectivity methods were compared in terms of detection rates, under non-mixed and mixed conditions, for small and large sample sizes \"N\", with and without jitter for \"k\", and for different values of signal to noise. Under mixing, the results show that LagCoh outperforms wPLI, PLI, dPLI, and sImCov. Without mixing, LagCoh and sImCov outperform wPLI, PLI, and dPLI. Finally, it is shown that dPLI is an invalid estimator of flow direction, i.e. it reverses and \"goes against the flow\" by merely changing the sign of one of the time series, a fact that violates the basic definition of Granger causality. For the sake of reproducible research, the supplementary material includes Delphi Pascal source code and all detailed result files in human readable format.