## 1 Introduction  
A challenging problem in neuroimaging is to estimate directed connectivity between brain regions reconstructed from scalp EEG recordings but important to unveil their joint dynamics. Due to volume conduction, a given EEG electrode can pick up signals from several sources simultaneously, distorted along the way due to the presence of tissues with different electrical properties. Resolving these sources is called the \"inverse problem\", and it consists of estimating the source parameters given the scalp EEG recordings. The number of sources is higher than the number of electrodes, rendering an ill-posed problem. Valid brain connectivity estimation critically depends on the correct localization and time series reconstruction in this stage. Several localization methods have been proposed, often yielding differing outcomes. In a comprehensive set of simulations, [1] studied the influence of several inverse solutions, the depth of the sources, their reciprocal distance, and the Signal-to-Noise Ratio (SNR) of the recordings. They found that all these factors had a significant impact on the resulting connectivity pattern and that the number of spurious connectivity estimations depends heavily on the combinations of these factors. In addition to the said factors, the choice of the connectivity estimator also has a significant impact. Our interest lies in directed connectivity estimation, of which partial directed coherence [2], dynamic causal modeling [3], structural equation modeling [4] and (conditional) Granger causality (GC) [5] are well-known methods. However, they rely on statistical assumptions that usually do not hold for EEG data, such as linearity [6], stationarity and prior assumptions on connectivity being expressible as a relation between time series. However, even though some of these assumptions are violated, these methods still are best practice cases of directed connectivity estimation. In what follows, we focused on variations in traditional Granger Causality, given that it does not rely on an a priori assumed connectivity pattern. Granger Causality is a statistical hypothesis used to determine temporal causal effects between two time series. If the past of a second time series (Z) together with the past of a first time series (Y) (i.e., the \"full\" model) results in an improved prediction of the future value of the first time series, then the past of the first time series alone (the \"reduced\" model), it is said that time series Z \"Granger-causes\" Y. Two main problems with this bivariate model can be discerned. Firstly, bivariate GC does not account for other time series that may be causing both Y and Z, resulting in spurious connectivity patterns. Secondly, even when bivariate GC is extended towards multiple time series by conditioning on these other variables, it is still possible that the found influence is actually caused by a linear mixture of non-interacting sources. This is because the signal measured from one electrode usually contains contributions of several sources [7]. Important to note is the proposal of Time-Reversed Granger Causality (TRGC) by [8], further validated by [7], to reduce the impact of additive correlational noise due to source mixing. The idea is that when connectivity is based on temporal delay, directed connectivity should be reversed when the temporal order is reversed. Concretely, it is checked whether the obtained GC scores for non-reversed and reversed data have opposing directions and are both significant [1]. This is clearly different from a classical way to determine significance (i.e., a likelihood ratio test). Hence, the main difference between TRGC and traditional GC is the proposed significance procedure. Still, even with TRGC, errors in connectivity estimation are here to stay. The question remains whether a totally different approach could cope with the above-mentioned problems and could perform better, or at least equally well, in comparison with the standard approaches.