From Correlation to Causation: Estimation of Effective Connectivity from Continuous Brain Signals based on Zero-Lag Covariance

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 $L^1$-minimization. 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 a fMRI dataset. The resulting network exhibits a tendency for close-by areas being connected as well as inter-hemispheric connections between corresponding areas. Also, we found that a large fraction of identified connections were inhibitory.Comment: 18 pages, 10 figure

Organic matter fluxes and biogeochemical processes in the OMZ off Peru, Cruise No. M138, 01 June - 03 July 2017, Callao (Peru) - Bahia Las Minas (Panama)

The oxygen minimum zone (OMZ) in the eastern tropical South Pacific Ocean is tightly connected to the coastal upwelling system off Peru. The high biological productivity off Peru is therefore, driven by the complex interplay between the amount of nutrients recycled by remineralisation processes in the OMZ and the upwelling which brings these nutrients to the surface layer. However, surprisingly little is known about organic matter cycling and its effects on biogeochemical processes in the OMZ off Peru. To this end we conducted a first comprehensive study on the role of organic matter for the biogeochemical processes and the maintenance of the OMZ off Peru. M138 combined measurements of marine biogeochemistry, microbiology, physical oceanography and air chemistry with foci on (i) the efficiency of the biological pump, (ii) the nitrogen cycle processes in the OMZ, (iii) the ventilation of the OMZ as well as (iv) the air/sea gas exchange across the ocean/atmosphere interface and (v) aerosol deposition. The METEOR cruise M138 was performed as part of the third phase of the SFB754 'Climate-Biogeochemistry Interactions in the Tropical Ocean' (www.sfb754.de)

Sparse Estimation of Resting-State Effective Connectivity From fMRI Cross-Spectra

In functional magnetic resonance imaging (fMRI), functional connectivity is conventionally characterized by correlations between fMRI time series, which are intrinsically undirected measures of connectivity. Yet, some information about the directionality of network connections can nevertheless be extracted from the matrix of pairwise temporal correlations between all considered time series, when expressed in the frequency-domain as a cross-spectral density matrix. Using a sparsity prior, it then becomes possible to determine a unique directed network topology that best explains the observed undirected correlations, without having to rely on temporal precedence relationships that may not be valid in fMRI. Applying this method on simulated data with 100 nodes yielded excellent retrieval of the underlying directed networks under a wide variety of conditions. Importantly, the method did not depend on temporal precedence to establish directionality, thus reducing susceptibility to hemodynamic variability. The computational efficiency of the algorithm was sufficient to enable whole-brain estimations, thus circumventing the problem of missing nodes that otherwise occurs in partial-brain analyses. Applying the method to real resting-state fMRI data acquired with a high temporal resolution, the inferred networks showed good consistency with structural connectivity obtained from diffusion tractography in the same subjects. Interestingly, this agreement could also be seen when considering high-frequency rather than low-frequency connectivity (average correlation: r = 0.26 for f < 0.3 Hz, r = 0.43 for 0.3 < f < 5 Hz). Moreover, this concordance was significantly better (p < 0.05) than for networks obtained with conventional functional connectivity based on correlations (average correlation r = 0.18). The presented methodology thus appears to be well-suited for fMRI, particularly given its lack of explicit dependence on temporal lag structure, and is readily applicable to whole-brain effective connectivity estimation

Parameter used for estimation.

<p>Parameter used for estimation.</p

Performance of network inference based on simulated Ornstein-Uhlenbeck processes.

<p>Same colors as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006056#pcbi.1006056.g003" target="_blank">Fig 3</a>. <b>A</b> Performance of the estimation when measurement noise is added. <b>B</b> Performance of the estimation if only parts of the network are observable. The fraction of observed nodes in a network are indicated on the <i>x</i>-axis. The total number of nodes in the network was <i>N</i> = 180.</p

Networks inferred from a simulated Ornstein-Uhlenbeck process.

<p><b>A</b> shows the original network. <b>B</b> shows the network inferred with our new method from the zero-lag covariances. White and black entries indicate true negative (TN) and true positive (TP) connections, blue and red entries indicate false negative (FN) and false positive (FP) connections, respectively. In this example, the performance measures are AUC = 0.98, PRS = 0.97 and PCC = 0.95. <b>C</b> depicts the sample covariance (functional connectivity) matrix directly estimated from the data. In <b>C</b>, as a consequence of symmetry, the number of wrongly estimated connections is quite high, the performance measures are AUC = 0.93, PRS = 0.54, and PCC = 0.29. <b>D</b> shows the Receiver Operating Characteristic Curve and the Precision Recall Curve for the networks estimated from zero-lag covariance <i>G</i>_est in blue/orange and of the functional connectivity <i>C</i> in red/cyan. The areas under these curves are the AUC and PRS, respectively.</p

Performance (color coded) of the estimation depending on data length (<i>y</i>-axis) and time scale of the activity (<i>x</i>-axis).

<p>Both scales are logarithmic. For interpolation a bilinear method is used.</p

Comparison of performance with the Regularized Inverse Covariance (RIC) method based on numerical simulations of Ornstein-Uhlenbeck processes.

<p>Shown are the results from the reconstruction of 20 different networks with Erdős-Rényi connectivity profiles as described before (cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006056#pcbi.1006056.g002" target="_blank">Fig 2</a>). AUC, PRS and PCC of our new method and of the RIC method, respectively, are shown side-by-side.</p

Estimated networks for one representative MREG dataset.

<p>The left panel shows the symmetrized network reconstructed with our estimation method, the middle panel shows the network found with the RIC method. The right panel shows the connections which are identified by both methods (EB, black), by none of the methods (EN, white), the connections found only by the RIC (ERIC, blue) and the connections found only by our method, but not by the RIC method (NERIC, red).</p