Introduction
The human brain is a massive neural network whose emerging computations result in adaptive cognition and behavior. Neighboring neurons in the cortex have similar functions and form functionally distinct regions leading to cortical functional parcels. Finding functional parcellations of the brain is currently a topic of intense research [1]. There are increasing evidences to suggest that the functional architecture of the brain is organized in a hierarchical modular manner [2], [3]. The hierarchical modular architecture has evolved due to the need of efficient, low cost information transfer among different brain regions.
Functional magnetic resonance imaging (fMRI) provides an opportunity for in vivo investigation of the functional architecture of the brain. Functional connectivity of the human brain is measured by the correlations of time-series of neuronal responses between brain regions and is derived from fMRI images taken at rest or while performing a task. Recent advances in network science provide models and tools to derive brain's functional architecture from functional connectivity obtained from fMRI [4]. Network models such as scale-free networks [6], [7] and modular networks [8], , [10] have been used to explore the functional architecture of the brain. Highly connected nodes or hubs that are associated with brain disorders and network modules that correspond to known functional modules of the brain have been identified [4].
In this paper, as models of functional brain networks, we investigate two types of networks: (i) networks that are defined by their degree distributions such as power-law networks, exponential networks, and stretched exponential networks; and (ii) networks that are defined by the distribution of connectivity weights such as standard stochastic block models (SBM) and degree-corrected SBM (dc-SBM). Nervous system has evolved to conserve two themes of wiring [5]: (a) the tendency to organize network topology into modules that serve specialized functionality; and (b) the general drive to high topological integration by means of short communication paths, hubs and rich hubs. We hypothesize that theme (a) of wiring is represented in network models (ii), namely stochastic block models, and theme (b) of wiring is represented in network models (i) defined with node distributions.
By using resting-state fMRI (rs-fMRI) brain scans collected in the Human Connectome Project 1[12], we will evaluate the a goodness-of-fit of several network models touted for modeling functional brain connectivity. We will first investigate the power-law and exponential network models which are characterized by nodal connectivity and then use stochastic block models to unfold the functional modules of the brain. We will show that the power-law and exponential models fit well on functional brain networks and demonstrate the potential of stochastic block models in detecting functional modules of the brain [11], [13], [14]. In order to reduce the complexity, we used functional connectivity among 264 regions of interests (ROI) identified by Power et al. as functionally relevant cortical regions [15].
Methods
2.1. Models of Brain Connectivity
We consider the functional network of the brain as a graph
The functional architecture of the brain is studied by fitting network models to functional connectivity of the brain. We consider five network models to fit into the functional connectivity of brain: power-law network model, exponential model, stretched exponential model, standard random block model and degree-corrected random block model.
2.1.1. Power-Law and Exponential Networks
A power-law network is defined by the nodal degree ![Right-click on figure for MathML and additional features.](data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="24" height="20"><rect fill-opacity="0"/></svg> "Right-click on figure or equation for MathML and additional features."){kind=small}
\begin{equation*}
p_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}(k)=C_{1}k^{-\alpha}
\end{equation*}
The networks with exponential degree distribution are defined by [17] \begin{equation*}
p_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}1}(k)=C_{2}\exp(-\lambda k)
\end{equation*}
The stretched exponential networks combine power-law and exponential degree distributions and are defined by [18], [17]:\begin{equation*}
Pstretched (k)=C_{3}k^{\beta-1}\exp(-\lambda k^{\beta})
\end{equation*}
2.1.2. Stochastic Block Models
Stochastic block models (SBM) partition the set of nodes into modules (or clusters) with many intra-modular edges and few inter-modular edges. Suppose that the block model partitions the brain network
The stochastic block model is defined by a Poisson distribution of edges. The likelihood of \begin{equation*}
p_{\mathrm{S}\mathrm{B}\mathrm{M}}(G\vert E,\ M)=\frac{1}{\prod_{i < j}w_{ij}!}\prod_{r,s}e^{\frac{m_{rs}}{rs^{2}}}\exp(-n_{r}n_{s}e_{rs})
\end{equation*}
A degree-corrected stochastic block models (dc-SBM) was introduced to account for the degree distribution of edges, which is given by [11]:\begin{equation*}
p_{\mathrm{d}\mathrm{c}-\mathrm{S}\mathrm{B}\mathrm{M}}(G\vert E, \ M)=\frac{1}{\Pi_{i < j}w_{ij}!}\prod_{i}\theta_{i}^{k_{i}}\prod_{r,s}e_{r,s}^{\frac{m_{rs}}{2}}\text{exp}\left(-\frac{e_{rs}}{2}\right)
\end{equation*}
2.2. Model Fitting
In order to find the plausibility of different network models, we first fit different models on functional connectivity matrices evaluated on subjects. For each subject, average in of brain voxels within a sphere of 5mm radius at the 264 functionally relevant ROI defined in [15] were obtained. Functional connectivity of the brain is obtained by calculating partial correlations between resting-state fMRI time-series of the ROIs. We discarded negative edges and thresholded the functional connectivity for a range of thresholds in order to investigate the functional architecture of the brain. We consider unweighted networks with connectivity matrix
For power-law and exponential networks, the parameters
Results
3.1. Data
The dataset included 627 healthy adults from ages 22–36 from the S900 release of the HCP. All HCP rs-fMRI data were acquired on a Siemens Skyra 3T scanner at the Washington University. The details of MR imaging protocols are described in the S900 release manual available at HCP website. We used preprocessed data that had undergone standard preprocessing steps [20] and subsequent ICA-denoising [21] in order to remove artifacts and noises. In addition, 24 head motion parameters were regressed out of the time series.
3.2. Goodness-of-Fit
Connectivity matrices of individual subjects were the data points for network analysis. Each subject's data was fitted with power-law, exponential, and stretched exponential (stretched) distributions and standard and degree-corrected SBM. The power-law and exponential models were implemented using the power-law python package [19]. The standard and degree-corrected random block models were implemented by using the graph-tool package [24].
The goodness-of-fit values of each subject scan to different models were evaluated by computing the log likelihood values given the models. The averages of log likelihoods on all subjects for each model at different thresholds of connectivity are given in figure 1. As seen, the stretched exponential had the highest likelihood, so the best fitting. The power-law and exponential networks attempt to fit the corresponding degree distribution to the data while stochastic block models attempt to fit distributions of connections to brain networks.
Average log-likelihoods of all the subjects by different models at different threshold values of connectivity.
Figure 2 shows connectivity network for a representative subject at different thresholds. The number of connections reduce as the connectivity threshold is increased. 1 gives us the standard deviation in the log likelihood values across all subjects for different models and threshold values.
3.3. Comparison of Models
In order to compare different models, we compute log likelihood ratios between two models and corresponding p-values, which are shown in Table 2. We assumed each subject to be a data sample and calculated the p-values by computing the average and standard deviation of the log likelihood ratios as described in [23]. As seen, the model fits were significantly different from one another
3.4. Functional Modules
We calculated the modules by using both standard and degree-corrected stochastic block models. Since the quality of fit for random block models depends on initial parameters, we chose the best fit over 25 random initializations. The modules detected for a representative subject by stochastic block models are shown in figure 3. The variation of the number of modules of all subjects recovered at a threshold of 1.3 is shown in the histogram for standard and degree-corrected block models figure 4.
Functional brain networks of a representative subject at different thresholds
As seen, standard and degree-corrected block models give different numbers of modules for different subjects. At connectivity threshold of 1.3, the SBM favored 3 modules while dc-SBM favored 6 modules. By visual inspection, the functional modules detected were similar to those obtained by modularity maximization algorithm [22].
Discussion and Conclusion
We fit network models based on degree distributions and weight distributions to rs-fMRI scans gathered in the HCP. The functional connectivity of the brain was calculated as partial correlations of fMRI time-series on 264 cortical ROI earlier identified as functionally relevant. The functional connectivity matrices had to be thresholded in order to evaluate goodness-of-fit of network models on brain scans. We investigated two types of network models: networks that are defined by distribution of (i) node degree and (ii) connection weights.
By applying rs-fMRI scans on a database of 627 subjects, we showed the validity of several network models on functional brain networks. Our results confirm that the two types of network models, those defined on degree distribution and those defined on connectivity distribution, capture different aspects of network topology. This supports our hypothesis that brain networks have the tendency to organize network topology into functional modules by preserving distribution of connection strength and the drive of brain networks to high topological integration by means of short communication paths, hubs and rich hubs by networks preserving the networks of nodal distribution.
Modules in the functional brain network for a representative subject for degree-corrected (on left) and non-degree corrected (on right) stochastic block model at connectivity-theshold of 1.3: Top row are sagittal left views; middle are sagittal right views; and bottom row are axial top view.
Our study should help to understand how the brain organization compromise to have an architecture between small world networks and modular networks. As a future of this work, individual variability of different models across subjects can be studied. Further, thresholding step of connectivity need to be avoided by using analysis based on weighted networks.
ACKNOWLEDGMENT
This work was partially supported by AcRF Tier 1 grant RG 19/15 of Ministry of Education, Singapore. Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.