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, 68, 181-91

Network Discovery With Large DCMs

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Network Discovery With Large DCMs

Mohamed L Seghier et al. Neuroimage.

Abstract

In this work, we address the problem of using dynamic causal modelling (DCM) to estimate the coupling parameters (effective connectivity) of large models with many regions. This is a potentially important problem because meaningful graph theoretic analyses of effective connectivity rest upon the statistics of the connections (edges). This calls for characterisations of networks with an appreciable number of regions (nodes). The problem here is that the number of coupling parameters grows quadratically with the number of nodes-leading to severe conditional dependencies among their estimates and a computational load that quickly becomes unsustainable. Here, we describe a simple solution, in which we use functional connectivity to provide prior constraints that bound the effective number of free parameters. In brief, we assume that priors over connections between individual nodes can be replaced by priors over connections between modes (patterns over nodes). By using a small number of modes, we can reduce the dimensionality of the problem in an informed way. The modes we use are the principal components or eigenvectors of the functional connectivity matrix. However, this approach begs the question of how many modes to use. This question can be addressed using Bayesian model comparison to optimise the number of modes. We imagine that this form of prior - over the extrinsic (endogenous) connections in large DCMs - may be useful for people interested in applying graph theory to distributed networks in the brain or to characterise connectivity beyond the subgraphs normally examined in DCM.

Figures

Fig. 1
Fig. 1
Illustration of the regions of interest for the contrast “reading > fixation” (in red) and “fixation > all tasks” (in green) at p < 0.05 FWE-corrected in a group level whole brain SPM analysis. LH = left hemisphere, RH = right hemisphere. See Table 1 for a list of the anatomical location and MNI coordinate of these 20 regions.
Fig. 2
Fig. 2
(A) The prior covariances matrix: the 400 endogenous connectivity parameters with constrained priors at m = 10 (left) or without constraints (right) of a typical subject. (B) The use of functional connectivity to constrain the priors induced strong prior dependencies among parameters—that increased with the number of minor modes removed. This is illustrated by transforming the covariances to correlations: (left) illustration of histograms of correlations for m = 2 and m = 10; (right) mean correlations (black dots) and their standard deviations (gray vertical lines) at each number of modes from m = 1 to m = 20.
Fig. 3
Fig. 3
(A) plots the average free energy over our 10 subjects (that constitute a fixed-effect group model comparison) over the number of modes m—relative to the free energy at m = 1 (gray bar graph). The value of the free energy for each subject is shown as black dots. A zoom of the free energy (right bar graph) illustrates how the free energy (log evidence) initially increases with the number of modes and then decreases with larger numbers of modes. The winning model was at m = 15 with a difference in log evidence of 6, in relation to the next best model at m = 12. (B) illustrates the increase of the accuracy with the number of modes (left bar graph) but in the context of increasing model complexity (right bar graph). The complexity is just the difference between the free energy and accuracy.
Fig. 4
Fig. 4
Posterior expectations (left) and variances (right) of endogenous coupling parameters over different number of modes. These were assessed at the group level by estimating the average posterior parameters and the pooled variances over 10 subjects using Bayesian parameter averaging. Each row of the connectivity map represents one connection among the 400 connections of our fully connected models (with the self-connections displayed as dark lines). The strength of some connections (i) increased monotonically with the number of modes (e.g. connections 2 to 40 with the driving regions), (ii) higher at intermediate number of modes (e.g. connections 50 to 60 or connections 240 to 250), and (iii) higher for m = 20 (e.g. connection 160). The posterior variances however showed a consistent pattern across connections, increasing with the number of modes (right graph).
Fig. 5
Fig. 5
(Top-left) plots the average free energy of the 10 simulated datasets over the number of modes m—relative to the free energy at m = 1 (gray bar graph). The value of the free energy for each dataset is shown in black stars. The maximum free energy is clearly identified at intermediate m values. The remaining bar graphs (from top-right to bottom-right) illustrate the posterior expectations (in gray) after inverting the simulated DCMs (the bars in black represent the true coupling values used during the generation of the synthetic data). The error bars represent 95% confidence interval for the average over the 10 synthetic datasets.
Fig. 6
Fig. 6
(A) The strength of the endogenous coupling parameters (displayed as a 20-by-20 connectivity matrix) of the reduced model after post-hoc model optimisation (left) and their posterior probabilities thresholded at p > 0.95 (posterior confidence—right). The coupling parameters of the reduced model ranged from − 0.7 Hz to 0.8 Hz. Both maps can be used to generate weighted or unweighted adjacency matrices respectively for graph theory analyses. (B) provides a description of the structure (or graph) of the reduced model in anatomical space. Here, we defined a weighted adjacency matrix that indicates the maximum between the absolute coupling parameters of a given connection and its reciprocal connection, excluding the self-connections (for a similar rationale see Friston et al., 2011). (C) provides a projection of the nodes into a functional space – using spectral embedding – where the distances reflect the strength of (bidirectional) coupling. The functional space was defined here using the first three principle components of the weighted adjacency matrix (c.f. Pages 1214–1215 of Friston et al., 2011). Regions are labelled from 1 to 20 in the same order as in Table 1.
Fig. 7
Fig. 7
This figure illustrates the difference between the free energy at a given number of modes m and the free energy at m = 20 in each subject. Strong evidence (difference in log evidence of more than 5) is shown in white. The number of subjects that showed strong evidence at a given m compared to m = 20 is provided in the right bar graph, showing for instance that nine out of 10 subjects had DCMs with higher negative free energy at m = 12.

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