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Graph theoretic analysis on large DCM models

Oiwi Parker Jones1 and Mohamed L. Seghier2*
  • 1 Oxford University, Oxford Centre for Functional MRI of the Brain, United Kingdom
  • 2 Emirates College for Advanced Education, Cognitive Neuroimaging Unit, United Arab Emirates

Introduction There is growing interest in the neuroimaging community to implement methods that can derive robust systemic explanations of brain function (Sporns, 2014). Dynamic Causal Modelling (DCM) (Friston et al., 2003) is a generative model that can make mechanistic explanations of brain activation at the neuronal level. Likewise, graph theoretic analysis can characterize network architecture with complex measures such as characteristic path length, modularity, centrality and network resilience (Rubinov and Sporns, 2010). However, DCM typically deals with subgraphs with a small number of nodes whereas graph theory analyses rest upon the statistics on a relatively large number of edges. Recently, it has been shown that it is possible to invert large DCMs (i.e. 20-node models) that can then be submitted to graph theory and related analyses (Seghier and Friston, 2013). Here we investigate in a systematic way how to compute meaningful graph theoretic measures from very large DCM models. Specifically, we aim to examine many methodological issues that can impact on how DCM output is fed to graph theory. Methods The possibility to combine DCM and graph theory raises many practical challenges, including (i) how to invert very large DCMs, (ii) how to transform (reduce) very large DCMs into useful sparse model structures, (iii) how to convert DCM connectivity parameters into quantities that can be read by graph theory, and (iv) how to make inferences at the group level. Here we used fMRI data from 6 subjects during overt reading (for more details, see (Parker Jones et al., 2013)). In each subject, timeseries were extracted at n=40 regions (nodes) that were activated during reading aloud. In each subject, a single densely-connected model was specified without modulatory parameters (B-matrix was empty), yielding models with 1,336 edges (A-matrix), a subset of edges with low anatomical priors having been excluded. All first-level, second-level and DCM analyses were carried out with SPM12. Here we run DCM12 with the following options: deterministic, one-state equation per node, bilinear, and mean-centered inputs. Results 1- Inversion of large DCMs: the computational time required to invert large models grows exponentially with the number of free parameters. The Bayesian inversion of our 40-node models was done using constrained priors that bound the number of effective free parameters as detailed in (Seghier and Friston, 2013). This allowed the inversion of very large models to be done in a reasonable time (i.e. around 41-78 EM iterations per model). 2- Bayesian model reduction: the inverted 40-node models were then reduced using post-hoc model optimization (Friston et al., 2011). This procedure allows the reduction of a dense or even fully-connected model by assessing the impact of absent edges or connections (i.e. discover the sparsity structure) in a graph that best explains the observed time-series. The optimized structure of the reduced model at the group level was computed using Bayesian parameter averaging over our 6 subjects. Although group effects were assessed here during model reduction, it is possible to perform group inferences at later stages (e.g. on the DCM parameters or even after graph theory analyses). 3- DCM matrices: DCM estimates different parameters depending on the inversion scheme (e.g. deterministic, stochastic, or nonlinear). Here we limited our analyses to between-node endogenous connectivity parameters that are stored in the A-matrix. If modulatory parameters are also of interest, users can estimate total effectivity connectivity at each edge by adding B-matrix to A-matrix. Self-connections (diagonal of the A-matrix) were ignored. 4- Generate an adjacency matrix: sparse binary undirected networks are widely used in graph theory analysis. It is possible to generate such networks by thresholding and then symmetrizing the posterior probabilities of the connectivity parameters. It is also possible to generate different types of networks depending on the question of interest. For instance, it is possible to generate binary directed networks (e.g. binarized posterior probabilities after thresholding), weighted directed networks (e.g. posterior expectations in the A-matrix), or weighted undirected networks (e.g. by taking the maximum between the absolute coupling parameters of a given connection in the A-matrix and its reciprocal connection, cf. (Seghier and Friston, 2013)). 5- Measures of network topology: we estimated many complex measures of centrality and modularity using graph theoretic analyses depending on the network type (Bullmore and Sporns, 2009; Rubinov and Sporns, 2010). Conclusion Here we investigate the possibility to combine DCM with graph theory. Our aim is to provide a multi-step procedure that can be used in future studies to analyze task-free or task-induced fMRI activations. Some of the steps detailed above can be achieved using different schemes, for instance spectral DCM to speed up model inversion when dealing with resting-state fMRI data (Razi et al., 2015), and group-based Bayesian reduction schemes for multi-subject datasets (Friston et al., 2016). We hope that the ability to invert large DCMs, in an efficient way, will provide a new opportunity for analyses using graph theory.

References

Bullmore, E., Sporns, O., 2009. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neurosci 10, 186-198.
Friston, K.J., Harrison, L., Penny, W., 2003. Dynamic causal modelling. Neuroimage 19, 1273-1302.
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Friston, K.J., Litvak, V., Oswal, A., Razi, A., Stephan, K.E., van Wijk, B.C., Ziegler, G., Zeidman, P., 2016. Bayesian model reduction and empirical Bayes for group (DCM) studies. Neuroimage (in press).
Parker Jones, O., Seghier, M.L., Kawabata Duncan, K., Leff, A.P., Green, D., Price, C.J., 2013. Auditory-motor interactions for the production of native and non-native speech. J Neurosci 33, 2376-2387.
Razi, A., Kahan, J., Rees, G., Friston, K.J., 2015. Construct validation of a DCM for resting state fMRI. Neuroimage 106, 1-14.
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Keywords: connectivity analysis, networks, graph theory, effective connectivity, fMRI, dynamic causal modeling

Conference: International Conference - Educational Neuroscience, Abu Dhabi, United Arab Emirates, 28 Feb - 29 Feb, 2016.

Presentation Type: Poster Presentation

Topic: Educational Neuroscience

Citation: Parker Jones O and Seghier ML (2016). Graph theoretic analysis on large DCM models. Front. Neurosci. Conference Abstract: International Conference - Educational Neuroscience. doi: 10.3389/conf.fnins.2016.92.00018

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Received: 16 Mar 2016; Published Online: 23 Mar 2016.

* Correspondence: Dr. Mohamed L Seghier, Emirates College for Advanced Education, Cognitive Neuroimaging Unit, Abu Dhabi, United Arab Emirates, mseghier@gmail.com