Building Models of Functional Interactions Among Brain Domains that Encode Varying Information Complexity: A Schizophrenia Case Study

Abstract

Revealing associations among various structural and functional patterns of the brain can yield highly informative results about the healthy and disordered brain. Studies using neuroimaging data have more recently begun to utilize the information within as well as across various functional and anatomical domains (i.e., groups of brain networks). However, most whole-brain approaches assume similar complexity of interactions throughout the brain. Here we investigate the hypothesis that interactions between brain networks capture varying amounts of complexity, and that we can better capture this information by varying the complexity of the model subspace structure based on available training data. To do this, we employ a Bayesian optimization-based framework known as the Tree Parzen Estimator (TPE) to identify, exploit and analyze patterns of variation in the information encoded by temporal information extracted from functional magnetic resonance imaging (fMRI) subdomains of the brain. Using a repeated cross-validation procedure on a schizophrenia classification task, we demonstrate evidence that interactions between specific functional subdomains are better characterized by more sophisticated model architectures compared to less complicated ones required by the others for optimally contributing towards classification and understanding the brain's functional interactions. We show that functional subdomains known to be involved in schizophrenia require more complex architectures to optimally unravel discriminatory information about the disorder. Our study points to the need for adaptive, hierarchical learning frameworks that cater differently to the features from different subdomains, not only for a better prediction but also for enabling the identification of features predicting the outcome of interest.

Keywords: Bayesian optimization; Functional connectivity; Hyperparameter optimization; Multilayer perceptron; Schizophrenia; Subdomain analysis; fMRI.

Figure 1.

(a) A step-by-step description of the whole analysis. (b) An architecture in the search space that TPE optimizes over, defined by the vector ${\left\{x_i\right\}}_{i=1}^{28}$, with x_i ∈ {0, 1, 2} representing the number of fully connected hidden layers on top of the input node corresponding to data from the i-th subdomain interaction (SDI). Data for each SDI is a sub-matrix of the full static functional connectivity matrix with connections from participating subdomain(s). (c) TPE search space traversal on a toy example with a quadratic cost function (x − 1)² to narrow down the search closer to the optimal value.

Figure 2.

Illustration of how the functional connectivity matrix for the 53 components (53 × 53 in size) is divided into subdomain interactions (SDIs) that form the input to the branched Multilayer Perceptron (MLP) architecture. Each colored sub-matrix represents a certain SDI i.e., the set of functional connectivity values between components of a given pair of subdomains. The 7 subdomains include: default mode network (DMN), visual (VIS), auditory (AU), cognitive control (CC), sensorimotor (SM), cerebellar (CB) and sub-cortical (SC). A total of 28 SDIs, which are sub-matrices of the full functional connectivity matrix, are colored in different colors. Since the number of subdomains is 7, a total of ⁷C₂ = 21 out of the 28 SDIs correspond to inter-network connections (e.g. DMN-VIS) while 7 correspond to intra-network connections (e.g. DMN-DMN).

Figure 3.

A schematic diagram for neural network based methods used for performance comparison with the TPE based approach. In addition to standard machine learning models like SVM, logistic regression (LOG) and random forest classifier (RFC), baseline non-branched neural network architectures used were (a) multilayer perceptron (MLP), (b) feedforward neural network with encoder-decoder architecture (FNN) and (c) BrainNetCNN. Branched neural network architectures included (d) uniform architectures, UNIF0, UNIF1 and UNIF2, representing non-flexible multi-branched architectures with 0, 1 and 2 fully connected layers before the fusion step and above input layer in each SDI branch of the architecture respectively. (e) As the third baseline neural network methods, existing hyper-parameter optimization techniques including random search (RNDS) and grid search (GRDS) were used to optimize the domain of branched architectures with variable branch-depth, representing the same class as TPE.

Figure 4.

The components obtained from the scICA procedure Neuromark are shown in different colors. Each map shows the components belonging to a particular subdomains i.e., networks of brain. The 7 subdomains include: default mode network (DMN), visual (VIS), auditory (AU), cognitive control (CC), sensorimotor (SM), cerebellar (CB) and sub-cortical (SC). subdomain interaction (SDI) features built using these subdomains and components were used as input to the multi-branch MLP architecture optimized for variable branch-depth by the TPE algorithm. The SDI features are comprised of functional connectivity (static time-series correlation) between pairs of components belonging the same subdomain (intra-network connections) or different subdomains (inter-network conenctions). The 7 subdomains shown above lead to the creation of 28 SDIs, with ⁷C₂ = 21 corresponding to the inter-network connections (Eg. DMN-VIS) and 7 corresponding to the intra-network connections (Eg. DMN-DMN).

Figure 5.

Mean validation accuracy vs. time point (iterations) for 50 repetitions of the TPE algorithm over the architecture search space depicted in Figure 1b. The mean test accuracy using the final architecture on held-out data for each repetition is also shown. The points traversed in the search space are selected based on expected improvement (Equation 2).

Figure 6.

Mean validation accuracy with error-bar for 50 repetitions of the TPE-optimized final architecture in comparison to baseline methods for (a) fBIRN and (b) COBRE datasets along with p-values for two-sample t-test on the mean test accuracy across 50 repetitions, shown in (c),(d). Methods used for performance comparison with TPE approach include simple machine learning models (SVM, LOG, RFC), Non-Branched Neural Network Architectures (MLP, FNN, BCNN), Branched Neural Network Architectures without Flexibility (UNIF0, UNIF1, UNIF2) and also Branched Neural Network Architectures with Non-Uniform Branch-Depth (GRDS, RNDS). See Figure 3 and subsection 2.6 for visualization and detailed explanation of these methods. The architecture created from the repeated optimizations using the TPE procedure is termed as TPE in the plots. It can be noted that for both the datasets, the accuracy obtained by using the TPE-optimized architecture is significantly higher than the accuracies from uniformly branched architectures (UNIF0, UNIF1), indicating the need for flexible architectures. Moreover, the accuracy with TPE is slightly higher than the accuracy for the other baseline methods, showing the scope of interpretability in the optimized model in terms of certain subdomain interactions (SDIs) with higher complexity requiring deeper while others requiring shallow architectures.

Figure 7.

To check whether the relative importance of SDI parameters learned in the fusion layer of MLP architecture is similar to the functional connectivity features of the corresponding SDI, Random Forest classifier was used to compute purity-based feature importance on the parameters in the fusion layer as well as the input layer (connectivity features). The prediction power vector obtained for both these cases was divided into 28 bins corresponding to each SDI and summed to get the cumulative prediction power of each of the SDIs. The above plots show the cumulative prediction power of SDIs, averaged over 50 repetitions, in the learned parameters inside the fusion layer (marked as TPE on the y-axis) and in the functional connectivity features in the input layer (marked as RFC on the x-axis). There was a high correlation of 0.9 and 0.81 between these prediction power values for (a) fBIRN and (b) COBRE datasets respectively. This means that in addition to being consistent in terms of the prediction accuracy, the TPE algorithm is also consistent in terms of the importance that the SDIs have for the prediction task.

Figure 8.

(a),(b) Visualization of the depth (in terms of the number of fully connected layers) in the final optimized architecture required by various subdomain interactions (SDIs) (intra-network or inter-network functional connectivity). SDIs consistently requiring higher depth across repetitions of the algorithm can be said to carry more complex information towards the classification objective.

Figure 9.

Connectogram showing SDIs requiring the same depth i.e., number of fully connected layers, in the TPE-optimized architectures for both COBRE and fBIRN datasets. The depth required by SDIs was the same for 19 out of 28 SDIs, indicating a common pattern across datasets in terms of certain SDIs requiring deeper models while others requiring shallower ones for a better prediction.