Modelling the Relationship Between Structural and Functional Connectomes

As part of my biomedical imaging module, I participated in a group project focused on studying the connectome using graph theory and parametric models. In this post, I’ll briefly overview connectomics, how connectomes are constructed, our team’s experiments, and my big takeaways from the project.

What is connectomics?

If you’ve read a lot about biological research, I’m sure you’ve come across the word “omics.” Omics is the study of some aspect of the biology of the body. For example, genomics is the study of the genome, which is all our DNA. Proteomics studies the proteome, all the proteins in our body. Microbiomics studies the bacteria or microbes living in our bodies. Similarly, connectomics is the study of the connectome – the entire set of neural connections in the brain, essentially the brain’s wiring diagram.

Connectomics is becoming a popular field of research, especially combined with AI and machine learning. It is important because so many aspects of our brain, from general intelligence to psychological disorders and neuropathologies, are all related to how our brains are hard-wired. If we can understand these connections, we can better diagnose neurological conditions, study disease progression patterns, and treat the disease effectively.

Types of connectomes

Concretizing our understanding of connectomics, there are two types of connectomes. The first is the structural connectome: the brain’s physical connections, which we can reconstruct using diffusion imaging. The second type is the functional connectome: this is the connectivity we determine by studying correlations in activity between different brain regions. This is done using functional imaging.

You may be wondering why one should study both structural and functional connectomes – why not just one of them? The reason is that the information from both modalities can supplement each other – the structural connectome tells you what brain regions are physically connected. But maybe the connection isn’t relevant, or perhaps it’s an imaging artifact; the functional connectome could help validate that. Similarly, the functional connectome tells you how much one region correlates with another – using a physical connection could validate that hypothesis better.

How to create a connectome?

Let’s now delve even deeper into how the connectomes are constructed. First, we are provided with a cortical parcellation, a lookup atlas labeling the different regions of the cerebral cortex. An example is shown below. This parcellation is used as a guide for creating the connectomes.

Cortical parcellation. Image taken from here.

As I mentioned previously, the structural connectome is created using diffusion imaging. I discussed this in a previous article (see here). As a quick summary, diffusion imaging creates images where the voxel intensities are sensitive to the diffusion of water molecules in the nerves. By studying diffusion patterns in the nerves, one can create a tractography, a visual representation of nerve tracts based on the diffusion gradient. An example tractography is shown below.

Finally, the connectivity information from the tractography can be compiled into an undirected network where the vertices are the brain regions, and the edges are connections.

For creating the functional connectome, functional MRI (fMRI) is used. fMRI measures the activity of a brain region based on the amount of oxygenated blood perfusing that region. This is also called the blood-oxygen-level-dependent (BOLD) signal. If you’re interested, check out this video on fMRI I made during my first year of undergrad.

Pipeline of constructing functional connectome. Image taken from here.

The pipeline for constructing the functional connectome is shown above. From fMRI, a time series of brain activity is obtained for the different regions of the cortex (steps A-D). This is processed into an $N \times N$ correlation matrix $A$ where $N$ is the number of brain regions. Each element of the matrix $A_{ij}$ represents the correlation between region $i$ and region $j$ of the brain (step E). This is the functional connectome. By binarizing the correlation matrix using a threshold, we can convert it into an undirected network for graph theory analysis (steps F-H).

Experiments

For the project, our team studied the properties of the structural and functional connectomes. First, we analyzed the graphical properties of the connectomes. Second, we implemented parametric models to model the relationship between functional and structural connectivity.

Analyzing graphical properties of connectomes

The structural connectomes were reconstructed using a software called “TractoR” – this converts a tractography into an adjacency matrix $B$ where each element $B_{ij}$ indicates whether the $i$ -th and $j$ -th brain regions are connected (encoded as 1) or not connected (encoded as 0). This matrix can be represented as an undirected network, for example see below.

Example of how a network (right) can be represented as an adjacency matrix (left) and vice versa. Image taken from here.

From these undirected graphs, we computed metrics such as the edge density, mean shortest path, efficiency, and clustering coefficient, which basically describe the graph connectivity. Additionally, we analyzed the effect of varying thresholds on connectivity.

To construct the functional connectome, we performed the steps in the pipeline described earlier. The fMRI time series were already collected, and we processed them to create the correlation matrices using a “shrinkage approach” as described in this paper. From these correlation matrices, we computed the same graph metrics.

Modeling the relationship between structural and functional connectome

In the second experiment, we modeled the relationship between the structural and functional connectome. Here, we used functional and structural connectivity weights – these are matrices containing elements which represent the strength of a connection between the $i$ -th and $j$ -th brain regions.

We implemented six parametric models listed below:

$f_{ij} = \alpha_{ij} + \beta_{ij}s_{ij}$
$f_{ij} = \alpha_{ij} + \beta_{ij}s_{ij} + \gamma_{ij}s_{ij}^2$
$f_{ij} = \alpha_{ij} + \beta_{ij}t_{ij}$
$f_{ij} = \alpha_{ij} + \beta_{ij}t_{ij} + \gamma_{ij}t_{ij}^2$
$f_{ij} = \alpha_{ij} + \beta_{ij}s_{ij} + \gamma_{ij}t_{ij}$
$f_{ij} = \alpha_{ij} + \beta_{ij}s_{ij} + \gamma_{ij}t_{ij} + \delta_{ij}u_{ij} + \eta_{ij}s_{ij}t_{ij}u_{ij}$

$f_{ij}$ is the functional connectivity weight for the $i$ -th and $j$ -th brain regions while $s_{ij}$ is the structural connectivity weight for the same. $t_{ij}$ and $u_{ij}$ are new terms – they represent the strength of “two-hop” connections and “three-hop” connections respectively – these represent indirect connections between two regions via exactly one or two other regions (like 1st degree, 2nd degree and 3rd degree connections on your LinkedIn network!). Finally, the terms $\alpha$ , $\beta$ , $\gamma$ , $\delta$ and $\eta$ are just the coefficients for these models.

Results

We performed many sub-experiments for these two studies, but I’ll summarize the major findings for convenience.

First, for the graph metrics analysis, the results are shown below. The first row of plots is the structural connectome’s graph metrics, and the second row is for the functional connectome. These are plotted over varying thresholds (for structural connectome) and shrinkage parameter $\lambda$ values (for functional connectome).

The key finding from these results is that when the threshold or $\lambda$ value increases, the connectivity in the graph decreases. For example, observe the plots for edge density, efficiency, and clustering coefficient metrics which summarize how densely connected the network is – when we increase the threshold on a network, we strip away some of the connections, essentially making a sparser graph, reducing the connectivity. On the other hand, the mean shortest path increases because increasing the threshold eliminates shortcut paths and retain longer paths. The functional connectome also demonstrates a similar behavior.

For the modeling task, we evaluated the models using many different metrics like the sum of squared error (SSE), leave-one-out cross-validation (LOOCV) error, AIC, and BIC. The results for one of our sub-experiments are shown below.

Model	SSE	LOOCV	AIC	BIC
1	244.7	15.86	52.6	54.44
2	236.3	155.44	53.89	56.72
3	239.9	16.30	52.18	54.07
4	225.5	94.98	53.00	55.84
5	230.2	17.31	53.40	56.23
6	208.1	59.05	60.10	60.2

The results are quite mixed and hard to interpret, although the third model seems to be the best as it has the lowest AIC and BIC scores.

Big takeaways

In this project, I learned a lot about connectomics, its importance in neuroscience, and the computational techniques used to analyze connectomes. Out of all the experiments, I found the network analysis study to be the most interesting as I explored the data processing pipelines – how the MRI images are converted into these graph-based representations – which I found fascinating. Furthermore, the research overlapped with some projects I did during my undergraduate studies, making it even more exciting. Through team meetings and Github code walkthroughs, I also engaged in interesting discussions with my teammates. At the end of the project, we had the opportunity to present our findings to our peers and hear about other projects, which was an enriching experience.

In retrospect, however, there were challenges too – the biggest one was performing the modeling task as there were many sub-experiments for which the implementations were quite involved. Also, while I have worked on projects involving deep learning and machine learning, graph theory was still quite new to me, particularly writing programs for interpreting network models. The time constraint of two weeks also made it all the more hectic because we all worked remotely and had to set convenient times to organize meetings and discuss findings. Overall, with support from our mentors and apps like Gathertowns and Teams, we could stay in constant touch and collaborate effectively.