From correlation to causation: Estimating effective connectivity from zero-lag covariances of brain signals

Knowing brain connectivity is of great importance both in basic research and for clinical applications. We are proposing a method to infer directed connectivity from zero-lag covariances of neuronal activity recorded at multiple sites. This allows us to identify causal relations that are reflected in neuronal population activity. To derive our strategy, we assume a generic linear model of interacting continuous variables, the components of which represent the activity of local neuronal populations. The suggested method for inferring connectivity from recorded signals exploits the fact that the covariance matrix derived from the observed activity contains information about the existence, the direction and the sign of connections. Assuming a sparsely coupled network, we disambiguate the underlying causal structure via L1-minimization, which is known to prefer sparse solutions. In general, this method is suited to infer effective connectivity from resting state data of various types. We show that our method is applicable over a broad range of structural parameters regarding network size and connection probability of the network. We also explored parameters affecting its activity dynamics, like the eigenvalue spectrum. Also, based on the simulation of suitable Ornstein-Uhlenbeck processes to model BOLD dynamics, we show that with our method it is possible to estimate directed connectivity from zero-lag covariances derived from such signals. In this study, we consider measurement noise and unobserved nodes as additional confounding factors. Furthermore, we investigate the amount of data required for a reliable estimate. Additionally, we apply the proposed method on full-brain resting-state fast fMRI datasets. The resulting network exhibits a tendency for close-by areas being connected as well as inter-hemispheric connections between corresponding areas. In addition, we found that a surprisingly large fraction of more than one third of all identified connections were of inhibitory nature.Author summary: Changes in brain connectivity are considered an important biomarker for certain brain diseases. This directly raises the question of accessibility of connectivity from measured brain signals. Here we show how directed effective connectivity can be inferred from continuous brain signals, like fMRI. The main idea is to extract the connectivity from the inverse zero-lag covariance matrix of the measured signals. This is done using L1-minimization via gradient descent algorithm on the manifold of unitary matrices. This ensures that the resulting network always fits the same covariance structure as the measured data, assuming a canonical linear model. Applying the estimation method on noise-free covariance matrices shows that the method works nicely on sparsely coupled networks with more than 40 nodes, provided network interaction is strong enough. Applying the estimation on simulated Ornstein-Uhlenbeck processes supposed to model BOLD signals demonstrates robustness against observation noise and unobserved nodes. In general, the proposed method can be applied to time-resolved covariance matrices in the frequency domain (cross-spectral densities), leading to frequency-resolved networks. We are able to demonstrate that our method leads to reliable results, if the sampled signals are long enough.