Fig 1.
Collider structures are encoded in the inverse covariance matrix.
Upper row: Three simple network architectures. Lower row: The corresponding inverse covariance matrices, red color represents positive entries, blue color stands for negative ones. In the left and middle column, the entries (2, 3) and (3, 2) are 0. The only difference between the right column and the middle column is that the connection between node 1 and 2 is flipped, such that nodes 1, 2 and 3 form a collider structure. Although there is still only an indirect connection between node 2 and 3, the entry in the corresponding inverse covariance matrix is now non-zero.
Table 1.
Variables used for the estimation method and simulation.
Table 2.
Parameter used for estimation.
Table 3.
Parameter used for simulations.
Fig 2.
Networks inferred from a simulated Ornstein-Uhlenbeck process.
A shows the original network. 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. C depicts the sample covariance (functional connectivity) matrix directly estimated from the data. In C, 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. D shows the Receiver Operating Characteristic Curve and the Precision Recall Curve for the networks estimated from zero-lag covariance Gest in blue/orange and of the functional connectivity C in red/cyan. The areas under these curves are the AUC and PRS, respectively.
Fig 3.
Effects of spectral radius ρ, connection probability p and network size N.
Here we consider the case of noise-free covariance matrices, which were created based on the theory of the underlying model. The quantities considered are the area under the ROC curve (AUC; green), the precision recall score (PRS; orange) and the Pearson correlation coefficient (PCC; purple). The shaded areas indicate the mean ± standard deviation computed over 20 realizations. A If the network interaction is larger than ρmin, it has relatively little effect on the performance of the estimation. Even in the extreme case, where ρ is close to 1, the estimation works well. B Performance of the estimation for different sparsity levels, encoded by the respective connection probabilities p. As expected, for non-sparse networks the performance of the algorithm degrades dramatically. C Performance of the estimation for increasing network size. Our results indicate clearly that bigger networks can be better reconstructed. Applicability may be limited by the numerical effort associated with the optimization. D Performance of the estitmation in presence of weak background connections. It is nevertheless possible to infer the skeleton of strong connections with high fidelity.
Fig 4.
Performance of network inference based on simulated Ornstein-Uhlenbeck processes.
Same colors as in Fig 3. A Performance of the estimation when measurement noise is added. 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 x-axis. The total number of nodes in the network was N = 180.
Fig 5.
Performance (color coded) of the estimation depending on data length (y-axis) and time scale of the activity (x-axis).
Both scales are logarithmic. For interpolation a bilinear method is used.
Fig 6.
Left panel: Directed connectivity estimated with our new method from one sample MREG data set.
Voxels were parceled using the AAL90-atlas. In the top-left block of the connectivity matrix connections within the left hemisphere are shown, in the lower-right block connections within the right hemisphere. The off-diagonal blocks represent the inter-hemispheric connections from the left to the right hemisphere (lower left) and from the right to the left hemisphere (top right). The strength of all connections is color coded, with red representing positive (excitatory) connections and blue representing negative (inhibitory) connections. Only the strongest 10% of connections are shown. Right panel: Functional connectivity matrix derived from the same data.
Fig 7.
Comparison of performance with the Regularized Inverse Covariance (RIC) method based on numerical simulations of Ornstein-Uhlenbeck processes.
Shown are the results from the reconstruction of 20 different networks with Erdős-Rényi connectivity profiles as described before (cf. Fig 2). AUC, PRS and PCC of our new method and of the RIC method, respectively, are shown side-by-side.
Fig 8.
Estimated networks for one representative MREG dataset.
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).