GP CaKe: Effective brain connectivity with causal kernels
A fundamental goal in network neuroscience is to understand how activity in one brain region drives activity elsewhere, a process referred to as effective connectivity.
Here we propose to model this causal interaction using integro-differential equations and causal kernels that allow for a rich analysis of effective connectivity. The approach combines the tractability and flexibility of autoregressive modeling with the biophysical interpretability of dynamic causal modeling. The causal kernels are learned nonparametrically using Gaussian process regression, yielding an efficient framework for causal inference. We construct a novel class of causal covariance functions that enforce the desired properties of the causal kernels, an approach which we call GP CaKe. By construction, the model and its hyperparameters have biophysical meaning and are therefore easily interpretable. We demonstrate the efficacy of GP CaKe on a number of simulations and give an example of a realistic application on magnetoencephalography (MEG) data.
This paper addresses the problem of understanding brain connectivity (i.e. how activity in one region from the brain drives activity in other regions) from brain activity observations. More generally, perhaps, the paper attempts to uncover causal structure and uses neuroscience insights to specifically apply the model to brain connectivity. The model can be seen as an extension of (linear) dynamic causal models (DCMs) and assumes that the observations are a linear combination of latent activities, which have a GP prior, plus Gaussian noise (Eq 11).
Overall the paper is readable but more clarity in the details of how the posterior over the influence from i->j is actually computed (paragraph just below Eq 12).
I write this review as a machine learning researcher (i.e. non-expert in neuroscience) so I am not entirely sure about the significance of the contribution in terms of neuroscience. For the machine learning perspective, the paper does not really provide a significant contribution as it applies standard GP regression. Indeed, it seems that the claimed novelty arises from the specific kernel used in order to incorporate localization, causality, and smoothness. However, none of these components seem actually new as they use, respectively, (i) the squared exponential (SE) covariance function, (iii) a previously published kernel, and (iii) some additional smoothing also previously known. With this regards, it is unclear why this additionally smoothing is necessary as the SE kernel is infinitely differentiable. 
The method is evaluated on a toy problem and on a real dataset. There are a couple of major deficiencies in the approach from the machine learning (ML) perspective. Firstly, the mixing functions are assumed to be known. This is very surprising as these functions will obviously have an important effect on the predictions. Second, for the only real dataset used, most (if not all) the hyperparameters are fixed, which also goes against standard ML practice. These hyperparameters are very important, as shown in the paper for the toy dataset, and ultimately affect the connectivities discovered.
There also seems to be a significant amount of pre-processing to actually fit the problem to the method, for example, as explained in section 5, running ICA. It is unclear how generalizable such an approach would be in practice.