Characterizing Network Search Algorithms Developed for Dynamic Causal Modeling

Abstract

Dynamic causal modeling (DCM) is a widely used tool to estimate the effective connectivity of specified models of a brain network. Finding the model explaining measured data is one of the most important outstanding problems in Bayesian modeling. Using heuristic model search algorithms enables us to find an optimal model without having to define a model set a priori. However, the development of such methods is cumbersome in the case of large model-spaces. We aimed to utilize commonly used graph theoretical search algorithms for DCM to create a framework for characterizing them, and to investigate relevance of such methods for single-subject and group-level studies. Because of the enormous computational demand of DCM calculations, we separated the model estimation procedure from the search algorithm by providing a database containing the parameters of all models in a full model-space. For test data a publicly available fMRI dataset of 60 subjects was used. First, we reimplemented the deterministic bilinear DCM algorithm in the ReDCM R package, increasing computational speed during model estimation. Then, three network search algorithms have been adapted for DCM, and we demonstrated how modifications to these methods, based on DCM posterior parameter estimates, can enhance search performance. Comparison of the results are based on model evidence, structural similarities and the number of model estimations needed during search. An analytical approach using Bayesian model reduction (BMR) for efficient network discovery is already available for DCM. Comparing model search methods we found that topological algorithms often outperform analytical methods for single-subject analysis and achieve similar results for recovering common network properties of the winning model family, or set of models, obtained by multi-subject family-wise analysis. However, network search methods show their limitations in higher level statistical analysis of parametric empirical Bayes. Optimizing such linear modeling schemes the BMR methods are still considered the recommended approach. We envision the freely available database of estimated model-spaces to help further studies of the DCM model-space, and the ReDCM package to be a useful contribution for Bayesian inference within and beyond the field of neuroscience.

Keywords: dynamic causal modeling; fMRI; model-space; network topology; search algorithm.

Figure 1

Network scheme for semantic decision task. A DCM model example to explain brain functions during semantic processing of words and pictures, originally investigated by /Seghier 2011/. The experiment consists of four frontal brain regions: left ventral (lvF), left dorsal (ldF), right ventral (rvF) and right dorsal (rdF). This network is examined in a task designed to involve three conditions: “Pictures” and “Words” includes onsets for the corresponding semantic decision trials and “Task” includes all trials. We generated a model-space of 65,536 models with every possible combination of endogenous connectivity matrix A, along with their experimental modulation, denoted by B1, B2, and B3. We fixed the direct effects, described by matrix C, to “Task” driving each region as experimental input. On the figure the network nodes are overlain on a coronal section of the brain captured from the average brain template of the Montreal Neurological Institute (Grabner et al., 2006).

Figure 2

Flowchart of model search algorithms for DCM. The flowchart on the left side panel depicts the most simple schematic of model search methods in a DCM model-space. On the right side we show the changes to the regular procedure that may optimize the algorithm in terms of number of estimated models or search results (green parts), and helps rapid characterization of model search methods using model-space lookup (red parts). In any case the algorithm starts with an initially selected model (or set of models) M₀ that is used to select models $M_0^1,M_0^2,\dots ,M_0^n$ with an arbitrary method for DCM computation. Then we can select the best fitting model (or models) M₁ of the selected population with Bayesian model selection, which is used to generate the next set of models to compare in the next iteration. This procedure continues until we cannot find an improved variation of the previously estimated models. As shown on the right side image, posterior parameter estimates of previously reached models $P_0^1,P_0^2,\dots ,P_0^n$ can also be used to reduce or manipulate the selected population and improve search efficiency.

Figure 3

Runtime comparison of DCM12 and ReDCM. We measured the average one-threaded computation runtime of the variational update cycles for both implementations. The comparison was based on varying scan length (200, 400, 600, 800, 1,000, and 1,200 frames) of a DCM model with five regions of interest (ROIs), and different model sizes (3, 5, and 7 ROIs) fitting fMRI data with scan length of 400 frames. There is a linear dependence between scan length and runtime, however, computations are exponentially longer with higher model sizes.

Figure 4

Summary of model search results for topological search algorithms. The accumulated number of estimated models, the dFe and the Hamming-distance of each of the three model search methods are displayed for all 20 runs on each of the 10 subject data. Box-and-whisker diagrams show the median, first and third quartiles of search results with the 95% confidence interval of the median. The improved version of the algorithms needed significantly lower number of models to calculate during search, with p < 0.001 for each algorithm.

Figure 5

The joint distribution density of dFe and Hd of an example subject. Joint density images reveal any topological structure over the model-space by characterizing the distance between graph structure, as scored with the Hamming-distance, in terms of differences in model evidence or free-energy. The joint density for the model-space estimated by DCM is shown on (A), and the distribution in the BMR space on (B). In cases where there are no relationship between topological structure and model evidence, the joint distribution would look more evenly distributed over Hd in any range of dFe. In this shown example model structure appears to be correlated to model evidence by r = 0.33 in the DCM model-space and by r = 0.68 in the BMR space. This reflects the assumption of Pyka et al. (2011), being models with higher Fe are also close to the topological structure of the best model. On the (0,0) coordinates the best model of their corresponding space can be found. Model search results are labeled according to their description. As the Bayesian post-hoc model selection is not available in the fully estimated DCM model-space, we indicated the model with the same ID on both panels.

Figure 6

Family-wise inference and family matched by model search methods. The top row shows random effects (RFX) family-wise model selection results over a group of subjects according to laterality, dorsoventral differences and task based differences. The bottom row shows percentage of model search results matching each of the families. The genetic algorithm (GA), the greedy Hamming-distance search (GHD), and post-hoc (PH) methods have the highest chance to match properties of the winning model family.

Figure 7

Model search in group-level PEB. The BMR method removed only the modulatory effect of “Words” on the self-inhibition of rdF region. The GES and GHD algorithms also removed the modulatory effect of “Pictures” from lvF self-connection. The GA method found the fully connected model each time regardless of the initial model.