A NOVEL SPATIO-TEMPORAL HUB IDENTIFICATION METHOD FOR DYNAMIC FUNCTIONAL NETWORKS

Abstract

Functional connectivity (FC) has been widely investigated to understand the cognition and behavior that emerge from human brain. Recently, there is overwhelming evidence showing that quantifying temporal changes in FC may provide greater insight into fundamental properties of brain network. However, scant attentions has been given to characterize the functional dynamics of network organization. To address this challenge, we propose a novel spatio-temporal hub identification method for functional brain networks by simultaneously identifying hub nodes in each static sliding window and maintaining the reasonable dynamics across the sliding windows, which allows us to further characterize the full-spectrum evolution of hub nodes along with the subject-specific functional dynamics. We have evaluated our spatio-temporal hub identification method on resting-state functional resonance imaging (fMRI) data from an obsessive-compulsive disease (OCD) study, where our new functional hub detection method outperforms current methods (without considering functional dynamics) in terms of accuracy and consistency.

1. INTRODUCTION

Resting state fMRI provides a non-invasive way to measure changes of cerebral blood oxygenation when a subject is not performing any explicit task. In the resting state, fluctuations in spontaneous neural activity are thought to underlie the spontaneous BOLD (Blood Oxygenation level Dependent) signal fluctuation. Synchrony, or correlation, between the fluctuations among regions are used to assess inter-region functional connectivity (FC) that forms the backbone of functional brain network [1, 2].

Recently, there is a growing consensus in the neuroimaging area that there exist changes in the functional brain networks over the entire scanning time even in a task-free environment [3, 4]. Since multiple lines of evidence show the dynamic patterns are more relevant to certain neuropsychiatric or neurodegeneration diseases, dynamic FC has been widely investigated by mainly using the sliding window technique [5].

Network organization is an interesting topic in network analysis. For example, various graph theory tools have been proposed to detect community and modularity in the brain network since the densely connected sub-networks often correspond to the specialized functional components in human brain. Likewise, recent studies demonstrate the existence of a number densely connected hub nodes [6] that play a key role in global information integration between different parts of the network [7, 8].

In our previous work [9], we have developed a multivariate hub detection method to simultaneously identify a set of hub nodes such that the removal of these marked hub nodes results in the largest damage to the original network, i.e., breaking the originally connected network into a set of disconnected components (sub-networks). Similar to the current hub detection methods, however, our method is designed only for static functional brain network. Although we can obtain a spatio-temporal setting of hub nodes by applying current hub detection method to each sliding window separately, it is difficult to (1) maintain the temporal consistency of hub nodes and (2) assure that the evolution of hub nodes is aligned with the cognition change presented in the functional dynamics.

To address these challenges, we propose a spatio-temporal hub identification method that is tailored to functional dynamics. Specifically, our spatio-temporal hub detection method is built upon our previous work in [9] since the alternative optimization framework allows us to regularize the consistency of hub nodes across sliding windows. The overview of our spatio-temporal hub identification method is shown in Fig. 1. First, we partition the BOLD signal into several sliding windows. At each sliding window, we estimate the likelihood of each node in the brain network being a hub node. Then, we form a trajectory of hub likelihood, at each node, along the sliding windows, which allows us to regulate the biologically unusual transitions (e.g., the pulse-like peak shown in Fig. 1) due to noisy BOLD signal or sub-optimal estimation in certain sliding windows. After applying the temporal regularization, the refined estimation of hub likelihood provides the chance for each node to refine the hub identification result in each sliding window. By alternating these two steps, we can (1) identify more reasonable hub nodes within each sliding window and (2) improve the temporal consistency of hub identification results.

Fig. 1.

The overall framework of our approach.

We have evaluated our spatio-temporal hub identification method on the resting-state fMRI data in the group comparison between normal control (NC) and OCD. Compared to conventional independent hub detection method, our proposed method achieves more accurate results in the context of dynamic functional network, which suggests the great applicability of our method in neuroscience network area.

2. METHOD

2.1. Multivariate Hub Identification for Single Network Preliminary.

Suppose that we use Pearson’s correlation to construct the functional network in each sliding window, encoded in a graph data structure $\mathbb{G}=(V,W)$, where V denotes for the set of N nodes and $\mathit{W}={\left[w_{ij}\right]}_{i,j=1}^N$ is a N × N adjacency matrix (symmetric). Next, we can calculate the Laplacian matrix L = D − W, where D is a diagonal matrix with each diagonal element $d_{ii}={\sum }_{j=1}^N{w}_{ij}$. In graph spectrum theory, the spectrum of the underlying graph $\mathbb{G}$ is formed by a set of orthogonal bases Φ which can be obtained by applying eigen-decomposition to the Laplacian matrix L, i.e., L = Φ^TΛΦ, where the eigenvalues are sorted in a ascending order in the diagonal matrix Λ = diag[λ₁, λ₂, …, λ_N]. If all the nodes are connected in the network, the smallest eigenvalue λ₁ is zero. Furthermore, the multiplicity of zero eigenvalues equals to the number of connected components in the graph, which sets the stage for the following multivariate hub identification method.

Multivariate hub identification.

Each node v_i in the network is associated with a binary index s_i, where s_i = 0 indicate v_i is hub node and s_i = 1 otherwise. The criterion of selecting hub node is based on the damage of removing the underlying node v_i out of the network, where the damage can be quantitatively measured by whether the remaining network undergoes increased number of zero eigenvalues after the removal. Although it is easy to check each node independently using such criterion, joint estimation of the selection index vector s = [s₁, s₂, … , s_n] is a NP-hard combinatorial optimization problem. To alleviate this issue, we resort to minimizing the summation of the top K smallest eigenvalues, i.e., $\text{min}{\sum }_{i=1}^K{\lambda }_i$, assuming there are in total K hub nodes in the network. According to Ky Fan’s [10] theorem, we further derive a continuous objective function as $\underset{{}_{F,S}}{\text{argmin}}\mathit{\text{trace}}\left(F^T{L}_{s}F\right)$, where we assemble the selection vector s into the diagonal line of the diagonal matrix S. L_s = D − S^TWS denotes the Laplacian matrix of the remaining network. $F\in {ℝ}^{N\times K}$ forms the spectrum bases of the remaining network S^TWS and subjects to the orthogonal constraint F^TF = I . We solve F and S in an alternative manner.

Optimizing F.

By fixing S, we have the closed-form solution for F which is the K eigenvectors of L_s associated with top K smallest eigenvalues.

Optimizing S.

By fixing F, the objective function for S becomes $\underset{s}{\text{min}}\mathit{\text{trace}}\left(S^{T}AS\right)$, where $A={\left[a_{ij}\right]}_{i,j=1}^N$ is a N × N matrix with a_ij = w_ij ‖f_i – f_j‖², and f_i denotes the i^th row of F. Since this objective function is not strictly convex, we introduce a likelihood vector $p={\left[p_i\right]}_{i=1}^N$ as the continuous variable to the underlying binary selection vector s and relax the above objective function to:

$$\underset{S,P}{\text{min}}\mathit{\text{trace}}\left(S^{T}AP\right),s.t.,P=S,$$ (1)

where P is the diagonal matrix derived from variable p. Intuitively, the selection vector s is the binarized result of likelihood vector p. Similarly, we alternatively solve P and S in Eq. (1). The entire workflow of our multivariate hub identification is summarized in the blue box of Fig. 2.

Fig. 2.

The workflow of our spatio-temporal hub identification method where we alternatively optimize the hub likelihood vector in each sliding window (blue boxes) and regulate the temporal consistency of hub likelihood vectors across sliding windows (red cube).

2.2. Spatio-Temporal Hub Identification for Dynamic Functional Network

The adaptation of our hub identification to dynamic functional network is shown in Fig. 2. Specifically, we first segment the entire BOLD signal into a set of T overlapped sliding windows. At each time t , we can estimate the likelihood p_i for each node v_i. Thus, we form a likelihood trajectory ${\left\{p_i^t\right\}}_{t=1,\dots ,T}$, where $p_i^t$ is the discrete observations of a continuous function p_i(τ) sampled at time points {τ_t}. In this regard, p(τ) at any arbitrary time τ can be extrapolated using radial basis functions (RBFs) by: $p_i(\tau )={\sum }_{t=1}^T{w}_i^t{e}^{-\frac{{\left(\tau -{\tau }_t\right)}^2}{2{\sigma }^2}}$, where σ controls the smoothing strength. Given ${\left\{p_i^t\right\}}_{t=1,\dots ,T}$, the RBF parameters $w_i={\left[w_i^t\right]}_{t=1,\dots ,T}$ can be estimated by:

$$\text{arg}\underset{{\mathit{w}}_i}{\text{min}}{{\sum }_{t=1}^T\Vert p_i^t-p_i\left({\tau }_t\right)\Vert }_2^2+\lambda {\Vert w_i\Vert }_2^2,$$ (2)

where λ controls the strength of temporal consistency in the likelihood trajectory p_i(τ).

After we regulate the likelihood trajectory at each node, we use the refined likelihood estimation to initialize the selection vector s at each sliding window, as shown by arrows in Fig. 2. We repeat this spatio-temporal optimization until converge. The output of our method is an evolution of hub nodes along the dynamic functional network, which are not only accurate at each time point but also consistent along time.

3. EXPERIMENTAL RESULTS

3.1. Subject Information and experiment setup

In total 63 normal control (NC) and 62 OCD subjects [11] are selected in the experiment. Each subject has T1-weighted magnetic resonance (MR) images (TR = 8ms, TE = 1.7ms, flip angle = 20°, resolution = 1.0×1.0×1.0mm³) and resting-state fMRI data (TR=2s, TE=60ms, flip angle=90°, resolution = 3.0×3.0×4.0 mm³), and each subject produced 230 time points at a repetition time. We further processed all these data into the AAL template with 116 ROIs to the subject image domain. The sliding window size is set to 10% of the entire time course for conventional hub identification method. The shift of sliding window is 1TR. Based on the distribution of connectivity degree, we set the number of hub nodes to 12. We empirically set the footprint in RBF as σ = 0.7. We use grid search to find the optimal parameter for λ in optimizing the objective function in Eq. (2). We fixed λ = 0.6 and λ = 0.7 in the following experiments, where we compare our spatio-temporal hub identification method with the conventional independent hub identification approach [9].

3.2. Spatio-temporal hub identification results

For each subject, we apply the conventional hub identification approach in each sliding window. Then, we go ahead to construct a count histogram where each network node is considered as a histogram bin and the value in each bin is the count of the underlying node being selected as hub node across the sliding windows. We use entropy to quantify the temporal consistency where lower entropy degree indicates the less transition of hub nodes in the dynamic functional networks. Similarly, we can construct another count histogram based on the spatio-temporal hub identification method and measure the entropy. Fig. 3 shows the joint distribution of entropy degrees by conventional (vertical axis) and our method (horizontal axis) for NC subjects (left) and OCD subjects (right), respectively, where our method achieves significant lower entropy value than conventional method for both NC and OCD cohorts (p<0.05). Thus, it is clear that the hub identification results by our method is statistically more consistent than the conventional method as the distribution of entropy value is all above the dash diagonal line.

Fig. 3.

The joint distribution of entropy from the count histogram of hub selection for NC (left) and OCD (right) cohorts, where the vertical and horizontal axes stand for conventional method and our method. The statistics (mean and stand deviation) of entropy degree by both methods are displayed in the bottom-right of each joint distribution plot.

Of note, the lowest entropy of such histogram is 2.48 (log_e 12) for the extreme case that no dynamics occurs in the entire time period. The average entropy degrees based on temporal hub identification results by our conventional method and our new method are 4.28 and 4.16, respectively. Thus, the quantitative results shown in Fig. 3 partially demonstrate that our proposed spatio-temporal hub identification method can effectively suppress the unusual unphysical transitions of hub nodes (as shown in Figs. 4–5) along time while maintaining the dynamics across functional networks.

Fig. 4.

The evolution of hub nodes of one typical NC subject by conventional method (bottom) and our method (top).

Fig. 5.

The evolution of hub nodes of one typical OCD subject by conventional method (bottom) and our method (top).

To further evaluate the consistency of hub identification results, we visualize the evolution hub nodes for NC subject and OCD subject by conventional method and our method in Fig. 4 and Fig. 5, respectively. Specifically, we focus on the pulse-like transitions that occur within short TR (2 seconds) window. As shown in the bottom of Fig. 4 and Fig. 5, the setting of hub nodes in the previous TR and next TR time is exactly the same but there is one hub node shifts from the dash box (hub nodes present in previous and next TR time) to the new location (solid box) at the current TR time, which is not only spatially less optimal but also exhibits unusual change that does not follow the normal cognition change in the resting stage. Since we apply the temporal consistency constraint in our energy function, our spatio-temporal hub identification achieves not only more reasonable but also consistent hub nodes across sliding windows, as displayed in the top of Fig. 4–5.

4. CONCLUSION

In this paper, we propose a novel spatio-temporal hub identification method tailored for dynamic functional networks. To the best of our knowledge, our method is the first computational method that can simultaneously identify hub nodes across sliding windows and maintain the temporal consistency. More reasonable hub detection results have been obtained by our method compared to conventional method that identify hub nodes independently in each sliding window. In future, we will apply our method to functional neuroimaging studies that seek for biomarker patterns of functional dynamics.