**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

**Citation:** Faes, A.; Vantieghem, I.; Van Hulle, M.M. Neural Networks for Directed Connectivity Estimation in Source-Reconstructed EEG Data. *Appl. Sci.* **2022**, *12*, 2889. https:// doi.org/10.3390/app12062889

Academic Editor: Alexander Hramov

Received: 31 January 2022 Accepted: 8 March 2022 Published: 11 March 2022

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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. Artificial Neural Networks (ANNs) were considered as particularly interesting candidates given their flexible way of approximating highly non-linear relationships between variables [9] and the fact that no a priori assumptions need to be made about signal stationarity nor the connectivity pattern (for a clear overview, see [10]). Temporal convolutional networks (TCNs), as well as recurrent neural networks (RNNs), are usually well-suited architectures for time series [11–15]. While RNNs are often seen as the gold standard for sequence modeling, TCNs have also proven their suitability, for instance, in financial forecasting [14], electric power forecasting [16] and language modeling [17]. However, it remains unclear whether ANNs can signal the presence or absence of connections and their strength. Although some authors already used ANNs to derive directed brain connectivity with multilayer perceptrons and recurrent networks [15,18], these approaches did not include source-reconstructed EEG data. As stated before, unlike EEG source reconstruction, analyses based on EEG electrode levels do not allow for trustworthy inferences about interacting regions [19]. Hence, the suitability of ANNs in deriving directed connectivity between reconstructed EEG sources remains unknown.

Our motivation to assess ANNs for directed connectivity estimation between reconstructed EEG sources was two-fold. First, although many connectivity estimators exist, it is not yet known which current ANNs architectures can cope better with source-reconstructed EEG activity and under various circumstances. The authors of [1] were the first to conduct a comprehensive simulation study on the influence of dipole location, noise level, inverse solution and connectivity estimation, as well as the interactions between these factors. It was shown that different circumstances call for different analysis pipelines and that under advanced noise levels and for particular dipole configurations, even well-established methods such as TRGC can return aberrant connectivity estimates. Second, ANNs boast several appealing modeling properties that are potentially relevant to EEG modelers, such as the ability to deal with non-stationarity, non-linearity and, depending on the ANN architecture, to dispense with the prior specification of model order.

In order to assess the ability of ANNs to correctly signal the presence or absence of directed connectivity as well as connectivity strength, we compared several ANN models, including Conv2D, a novel ANN model we propose, with TRGC. We compared their performance for different dipole locations (i.e., Far–Deep/Far–Superficial) as this can inform us whether there is a future for ANN models in brain connectivity estimation. In addition, we evaluated the ANN models relevant for directed connectivity estimation. We investigated these issues by means of a simulation study, thereby making use of a slightly adapted version of the simulation framework developed by [1] in which we manipulated the location of the dipoles and their connectivity while keeping noise level and the choice of the inverse solution constant.

#### **2. Materials and Methods**
