A Bayesian Model for Brain Network Functional Connectivity using PyMC3

Page Figure 1.1 Venn diagram shows the studies combining functional and anatomical data, with an emphasis on anatomical and functional connectivity data. In this dissertation, we developed a Bayesian hierarchical spatiotemporal model that jointly combined fMRI and DTI data to improve the estimation of resting-state functional connectivity. Structural connectivity from DTI data was used to construct an informative prior for functional connectivity from resting-state fMRI data via the Cholesky decomposition in a mixture model.

Through a case study of MDD, we also applied our model to examine how the estimated functional connectivity was associated with tasks such as episodic memory, executive function, processing speed, and working memory.

Brain Imaging

With all the imaging techniques developed, the studies of brain are focused on the structural and functional connection between two or more brain regions. The relationship between structural and functional connection is illustrated in Figure 1.1 in the form of a Venn diagram [4].

Functional Connectivity

With the deep understanding of fMRI in the past two decades, more and more studies suggest functional connectivity between brain regions such as the level of co-activation of spontaneous fMRI time series recorded at rest [7–9]. In the resting state experiments, their level of spontaneous brain activity was recorded when the subjects were supposed to relax without thinking. Experimental evidence shows that left and right hemispheric regions of the primary motor network at rest show a high correlation between their fMRI time series [9, 10], indicating the ongoing information transport and ongoing functional connectivity between these regions at rest.

Therefore, resting-state fMRI research focuses on mapping functional communication channels between different brain regions by exploiting the correlated dynamics of fMRI time series.

Structural Connectivity

For example, it was found that the degree of brain matter conservation in groups of older adults is strongly associated with their performance on tasks requiring functional integration involving interhemispheric interactions [4]. The axes of these ellipsoids are oriented along the diffusion tensor eigenvectors, and the lengths of these axes are proportional to the amount of diffusivity (corresponding eigenvalues) in the respective dimensions. In the past few years, a new imaging technology, diffusion tensor imaging (DTI), has been developed as a promising method to describe the structural connectivity in vivo.

DTI is important when the nerve axons of the substance in the brain have an internal fibrous structure.

Figure 1.3 Schematic of the basic principles underlying DTI: isotropy and anisotropy of water motion in tissue

Existing Methods

Another statistical method implementing a hierarchical clustering algorithm combined different data sources, including anatomically weighted functional connectivity (awFC), fMRI and DTI data, to determine functional connectivity [ 27 ]. Moreover, DTI data have been used in addition to fMRI information to assess functional connectivity in a multimodal approach. Other studies have also demonstrated the superior advantages of using fMRI and DTI data to investigate functional connectivity in brain networks [29-33].

Spatiotemporal Hierarchical Model

Spatiotemporal Structure
Hierarchical Structure
Double Fusion
Prior Distribution

We combine structural connectivity and naive functional connectivity together because the effect of direct structural connectivity is different from the effect of indirect structural connectivity. And low structural connectivity will indicate very low functional connectivity if structural connectivity does not exist. Then, the prior distribution of the covariance matrix Σ𝑑 is further considered as a function of the structural and naive functional connectivity matrices.

The covariance matrix for functional connectivity and structural connectivity is also constructed via a prior diagonal matrix.

Table 2.1 Various common covariance functions from Ref [35].

Introduction to PyMC3 and NUTS

However, by adopting NUTS in the model, we only need to set the sampling size to 1000 and the number of fitting (or fitting) to 1000. NUTs also have some self-tuning strategies to flexibly set the HMC parameters and allow many complex models to be fitted without special knowledge of fitting algorithms [ 36]. It is also important to note that PyMC3 takes advantage of Theano [39, 40] as a backend to transparently transcode models into C and translate them into machine code.

Thus, it can increase the performance of the sampling procedure by taking advantage of graphics processing units (GPU) architectures. Theano is a numerical computation library for Python, which allows expressions to be like NumPy syntax. When the sampling is completed, the posterior analysis can be inspected by means of trace graph of each parameter and various diagnostics such as Geweke statistics [41] and Gelman Rubin statistics [42, 43].

This is the potential scale reduction factor, which converges to unity when each of the traces is a sample of the rear target. In practice, we look for values of 𝑅̂ close to unity (e.g. less than 1.1) to ensure that a given estimate has converged. Where 𝑚 is the number of chains, 𝑛 is the sampling rate, 𝜌̂𝑡 is the estimated autocorrelation at lag 𝑡, and 𝑇 is the first odd positive integer for which the sum 𝜌̂𝑇+1+ 𝜌̂𝑇+2 is negative [44].

Optimization and Decomposition

For a positive definite covariance matrix Σ, the Cholesky decomposition expresses Σ as 𝑈𝑇𝑈, where 𝑈 is a unique upper triangular matrix with positive diagonal entries. For example, to generate correlated random variables that follow the 𝑛 dimensional multivariate normal distribution 𝑋 ~ N(𝜇, Σ) with mean vector 𝜇 (𝑛 × 1) and covariance matrix Σ (𝑛 × 𝑛 positive definite matrix), we can decompose the covariance matrix Σ to 𝑈𝑇𝑈 and produces a vector as 𝑍 with 𝑛 independent 𝑁(0, 1) random variables. Moreover, we can generate a (𝑚 × 𝑛) matrix from the univariate random variable 𝑍 ~ N(0, 1) to speed up the sampling process.

Simulation Study

Data Generation

Estimation

This result based on the NUTS algorithm in PyMC3 is quite different from that of the previous report using the Metropolis-Hastings algorithm in PyMC2 [34], implying the superior advantage of the NUTS algorithm even though the Bayesian independence assumption may be incorrect. To explore more about the estimated functional connectivity, we plot the histograms of the 10 parameters from their posterior distributions in Figure 2.2. We can clearly observe smaller variances in the exact Bayesian structural association than those in the Bayesian independence assumption.

Furthermore, to ensure convergence with the appropriate number of samples using NUTS algorithms, we conduct an experiment with 3000 posterior samples under the appropriate correlation matrix in PyMC3. In practice, the typical sampling using NUTS is 500 or 1000, which is more economical than the Metropolis-Hastings algorithm (usually more than 100,000).

Figure 2.2 Histogram plots of 10 parameters from their posterior distribution.

Case Study

Background

Exploratory Analysis

In addition, we conducted correlation tests between assessed functional connectivity and several cognitive domains, including episodic memory, executive function, processing speed, and working memory. Here, the p values from the tests were adjusted to control for the false discovery rate (FDR), which is the expected proportion of false discoveries among. Across all ninety-one functional connectivity, we found no adjusted p value below 0.1, meaning no significant evidence that any functional connectivity is associated with any of the cognitive domains at FDR = 0.1.

In the MDD group, there is no adjusted p-value below 0.1, indicating that we see no association between the functional connectivity and cognitive domains at FDR = 0.1. However, in the control group, we find that “FC80” is correlated with executive function domain after we adjusted the correlation tests using the FDR method to 0.1.

Figure 2.4 Correlation plot of covariates and estimated functional connectivity.

Regression Analysis

Moreover, we examined which of the assessed functional associations might influence each cognitive function using multiple variable selection methods. Due to the large number of explanatory variables, including four covariates (age, gender, education, and depression), we decided to always force the four covariates in the model setting and then select the top six variables from “FC1” to. Another comparable variable selection method is to use the LASSO (least absolute shrinkage and selection operator) method.

We calculated the percentage of each functional compound selected in each bootstrapping sample. After we obtained the variable selection results from all the methods, we listed six of the selected variables in Table 2.4. For processing speed domain, “FC10” and “FC29” were chosen frequently in all the five methods.

For working memory domain, "FC26" was selected five times, which warrants more research on the network connectivity between the corresponding brain regions. For executive function domain, "FC26" is included in five methods, the same frequency for working memory domain. Through five different methods of variable selection, certain estimated functional connectivity can be expanded to investigate the correlated links between ROIs and depression in the future.

For example, we listed a set of covariance functions to capture local spatial correlation, which can be extended to compare the estimated results. Another explorable work is to combine all the variables, including the covariates, estimated functional connectivity and multiple cognitive domains, and build a classifier to identify whether this subject is under MDD or not.

Table 2.4 Variable selection results from each method.

Setting of PyMC3 and Theano

Spatiotemporal Hierarchical Model in PyMC3

Cov_temp = tt.set_subtensor(Cov_temp[np.triu_indices(n)], rhonn) Cov_mat_v = tt.dot(Cov_temp.T, Cov_temp).

Histogram Plots of Parameters with 1000, 2000 and 3000 Sample Draws

Regression Coefficients

Variable Selection with Lasso Method

Plots of Subset Selection

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