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The loop community algorithm
The loop community algorithm
Finding network communities can help us identify hidden patterns relating to structure and function. However, depending on the network, the problem could become challenging. One needs an algorithm with substantial complexity power to uncover meaningful community structures that can sort out the intricacies of real networks. Mentored by Prof. Zoltan Toroczkai, we started studying the promising link community algorithm. Although flexible to uncovering link hierarchical structures encoding vertical (scaling) and horizontal (nodes belonging to multiple communities, i.e., overlapping) community information about the network, we realized that the output was in the language of link communities, which is difficult to understand in relationship with the main characters, the nodes.
I discovered that it is possible to create a dictionary to translate the link hierarchy into a node hierarchy that inherits the scaling and overlapping structural information from the former. This is the story of the loop community algorithm. The algorithm can be implemented in any simple network that is (un)directed, sparse(dense), and (un)weighted.
In this entry, I will show how the algorithm can be used in a real network.
Identifying functional communities from structural information in the braIn
Identifying functional communities from structural information in the braIn
Neuroscience introduction
Neuroscience introduction
We may all be familiar with what a brain is. We think with the brain, but how can we do that? The brain is composed mostly of two types of cells: the glia and neurons (blue and yellow cells in the right picture). The neurons are the most important to thinking (although glial cells are essential to keep good information transmission speeds, interact with the vascular system to provide nutrients and energy to the brain, etc.). Even though neurons work in a "simple" way, transmitting information back and forth through our senses, cortex, limbs, etc., in the form of electric pulses, what happens "inside" the brain is not understood. How the information travels and is processed to create thoughts is still an open question.
Initially, various experiments showed that specific cognitive tasks are located in certain brain areas. The hypothesis was that some areas are related to vision, such as the ones in the back of the brain, some to spatial processing, such as the ones in the parietal region, and others to speech, like the ones in the prefrontal cortex (very in front of the brain), etc. Although some areas indeed have sharply specific functions, using modern tools, such as functional magnetoresonance imaging (fMRI), we have discovered that complex cognitive functions such as memory or visual-motor reaction tasks require the simultaneous activation of distant cortical areas.
The brain was thought to be made of multiple functional blocks: This picture shows how the brain was initially thought to work, where cognitive tasks were modulated in individual brain regions. However, we now know that complex functions such as speech processing and understanding require the simultaneous computation of different brain parts.
Understanding how the electric pulses from our eyes, skin, nose, tongue, and ears travel to create an idea, to bring a memory from the past, or to learn something new is a challenge. There are around 86 billion neurons in the brain, which makes it impossible to track the activity of single neurons from a whole brain. However, the current technology makes measuring the number of neural connections at the area level possible. Cortical areas are little processors with directional wires stacking to other areas. Currently, the best technology to estimate the density of axons (the little neurons' tails) between cortical areas is obtained via retrograde tract-tracing experiments.
In the picture below, we can see what the result of the experiment looks like.
Retrograde tract-tracing injection on areas V1 of the mouse. The figures show pictures of the mouse brain. White dot lines demark the areal parcellation. The injection was made in area V1. In the top center image, the yellow dots outside the V1 are labeled neurons that absorbed the tracer in V1. The tracer can exclusively enter the neuron from its axon, then travel upstream, reaching the neuron's soma and coloring it (taken from Gamanut et al. 2018).
Retrograde tract-tracing injection on areas V1 of the mouse. The figures show pictures of the mouse brain. White dot lines demark the areal parcellation. The injection was made in area V1. In the top center image, the yellow dots outside the V1 are labeled neurons that absorbed the tracer in V1. The tracer can exclusively enter the neuron from its axon, then travel upstream, reaching the neuron's soma and coloring it (taken from Gamanut et al. 2018).
In other words, we can estimate the density of axonal projections projecting towards an area with the injection in another area. It is a density because the number of labeled neurons depends on the surface of the injection site, i.e., injections of different sizes lead to different counts.
Representation of an injection on area V1 in the macaque brain. This is an old flat map of the macaque cortex, which will suit to understand how the retrograde tracers work. In a big yellow ellipse, we see an illustration of the injection site of area V1, the primary sensory area of the visual system, i.e., it is the first cortical structure to receive visual information from the eyes. The yellow circles outside V1 represent places where labeled neurons were identified. Note that only some areas project to V1. The arrows represent axon terminals. This is a vital observation since retrograde tract-tracing experiments reveal all the areas (and their strength) projecting the injected area but not vice versa. The injection also reveals axonal densities, which measure the information bandwidth (proportional to the number of pulses propagated from an area). Picture edited from Kandel E. R. (2013). Principles of neural science (5th ed.). McGraw-Hill.
To quantify the connection strength between area pairs measured by the injection, the fraction of extrinsic labeled neurons (FLNe) is used. It is the number of labeled neurons from one area to another (in the case of the equation below, from area A to B) divided by the total number of labeled neurons outside area B.
FLNe from neural counts. This example shows what the FLNe between areas LIP and V1. The dots represent other areas not shown in the sketch.
FLNe consistency. In the picture above, we want to illustrate that even though two different injections (orange and green) in V1 can measure two different neural counts, the FLNe between the areas is the same.
Notice that the FLNe is associated with the density of axons, which does not depend on the injected area's site and extension. Injections in different places and with varying amounts of ink in the target area change the neural counts, but it has been observed that it does not change the density.
By repeating injections in different areas, we can record the information on axonal densities between the whole brain in the set of injected areas. Currently, there is no complete retrograde tracing dataset with the information of all the cortical areas, but it will be measured in the future. Even though we can not study the whole network yet, we can explore the subnetwork formed by the subset of injected areas in which all the edges' existence and weights are known.
On the left, we see an example of an edge-complete subgraph. We do not know the connections between the nodes outside the dashed circle (uninjected); however, we know everything about the connections of the nodes inside (injected), shown by the light orange bidirected links.
Network definition
Network definition
The information from the edge-complete subgraph of a dataset of the macaque brain with 29 injected areas from an atlas of 91 shows that the network is dense (0.6), meaning that most of the links between area pairs exist. This makes this network unique because biological and artificial networks are typically sparse. This property is very intriguing since the brain splits its functions into different areas; however, if the network is so dense, the functional specificity must not be encoded in the presence or absence of connections but in their weights. By observing the distribution of the weights in the dataset, we can observe that they cover several orders of magnitude. This shows that strong and weak coactivation events influence the interareal communication orchestration since the FLNe distribution is close to the lognormal distribution, which is heavy-tailed. (Ercsey-Ravasz et al., 2013).
Edge-complete subgraph of the macaque network. There are 29 areas with their connections. Distances are inversely proportional to the -log(FLNe) between the areas.
Distribution of log(FLNe). Weights span over several orders of magnitude and have a wide distribution.
In such a scenario, it is difficult to distinguish how areas are related or differ from each other. The relationship between FLNes and physical distances shows that the connection strength decays as a function of interareal distances. Still, the trend is heteroscedastic, meaning that the variance of strengths is not constant, making it hard to infer the structural properties of the weights since they constitute several systems.
All of the above shows that the network of cortical areas is heterogeneous, with properties that make it unique. However, we know that brains are not only determined by "static" anatomical connections but their function emerges from the spatio-temporal interactions between neurons. Nowadays, many techniques measure the activity of areas in the awake or sleep brain (fMRI, EEG, MEG, etc.). These measurements show that areas in the brain activate simultaneously in different ways depending on the cognitive process. Nevertheless, these are only macroscopic views of what is happening inside. Still, we believe brain functionality must be related and constrained by the anatomy and geometry of the brain.
Community analysis
Community analysis
If we want to better understand how structure and function in the brain are related, we need to find the communities of anatomical neural networks. In network science, identifying the communities of a network means fragmenting the network into homogeneous components, at least more homogenous than between different groups. This is done with the hope that those communities represent different functional parts of a system.
Bridging the anatomy (hardware) with the physiology (software) of the brain. We understand how metallic computers work because "we" made them. A metallic computer has two different components: hardware and software. We understand the relationship, but what is their relationship in the brain? Experiments such as fMRI, EEG, or MEG have discovered several insightful properties of the dynamic brain. How those observables are related to the underlying brain circuitry is still an open question.
As explained before, due to the complexity of interareal cortical networks, it is hard even for current state-of-the-art algorithms to discover their community structure. To find the communities of these networks, I made the loop community algorithm. To sort out the puzzle, I developed new criteria about what a "community" is. The current definition of a community as a set of nodes with more shared connections between them than with the rest of the network is no longer significant when graphs are so dense as in the brain.
The graph below shows the five node communities of the edge-complete subgraph of 29 areas in the macaque brain computed by the loop community algorithm. A remarkable thing about this partition is that nodes within a community project/receive connections to/from different areas similarly. Noticeably, three areas (7a, 7b, and 7m) belong to more than one community, so they may be part of different functional circuits. Interestingly, those areas belong to the associative cortex, which is linked to abstract thinking.
The cover partition of the macaque edge-complete subgraph with 29 injected areas. The network layout was computed using -log(FLNe) weights and the Kamada-Kawai algorithm. Areas closer to each other have stronger weights. Using the loop community algorithm, we have identified five communities in this network. Three areas (7a, 7b, and 7m) are nodes with overlapping community membership (NOCs), meaning they belong to more than one community. Arrows represent connections and are shown in gray and red. Red arrows are links between very different areas in their function or connectivity. Notice that red arrows are mostly between the periphery (not purple and orange) and core areas (purple and orange), denoting a sharp structural change.
Cover assignment on the M132_LH atlas (12 MB) [interactive]: This 3D surface reconstruction belongs to the left hemisphere of the subject M132. Hover an area to display its name. Black lines divide the surface into 91 areas + medium wall. Non-white colors represent cover memberships, the same ones used in the network above. In contrast, white-colored areas were not injected, so we could not cluster them. Notice that clusters are mostly local, except for some NOCs (see, for example, the purple community and area 7a). This suggests that long connections are relevant in shaping the functional (community) brain structure. Nevertheless, this result might change by adding new injections.
A remarkable property of the loop community algorithm is that it computes a node community hierarch, where areas are ensembled from the most similar (bottom), making bigger communities until all the areas and functional components merge at the top. The node community hierarchy from this network is shown below. Notice the branch inside the turquoise rectangle. That branch has a physiological interpretation related to the visual ventral stream, which works in figure recognition. Surprisingly, that branch was observed through physiological and lesion experiments, but how it emerged from brain connectivity was unclear. Analysis before showed that the stream could be explained by axon distributions between areas in different cortical layers (Barone et al., 2000). We prove that physiological structure can be recovered using simpler physical connectivity information. Moreover, this suggests a deep connection between the brain's information flow and community organization. Furthermore, it also suggests that we can discover functional components in the brain by measuring the connectivity strength, at least at the interareal level.
The node community hierarchy of the macaque brain. The most similar areas merge first and are at the bottom. As the hierarchy grows, communities become less similar but increase their size. The length of the branches is proportional to the dissimilarity/distinguishability change between communities. For example, branch lengths between areas 8l and 8m in the prefrontal cortex are very short, so distinguishing those areas by their axonal densities is difficult. Rectangles coat branches with potential functional interpretation. For example, the light blue rectangle with areas from V1 to TEPd resembles the visual ventral stream in charge of object recognition. Notice that areas are ordered from low- to high-level areas, showing that the branches also encode information flow details. The orange (purple) rectangle close (in) the auditory (prefrontal) cortex could also be a parallel pathway of information flow from another region besides the visual system.
Our analysis gets deeper and explores several extra new properties of the macaque dataset derived from community analysis (distinguishability, optimal partition, etc.). For the moment, we will stop here. In the end, brain areas form clusters that could be linked to cognitive functions. More details will be shown once the results are published.
Results
Results
The edge-complete subgraph of 29 injected areas exhibits a complex community and hierarchical structure.
In particular, the branch within the turquoise rectangle arranges areas from low- to high-level visual areas and shows the sequence from V1 to TEPd, which is associated with the visual ventral stream.
This shows a connection between community/similarity structure and information flow in the brain.
We find evidence of other parallel pathways as branches in the orange (auditory) and purple (prefrontal) rectangles.
A hierarchical structure in the prefrontal region is a new finding in neuroscience.
More results and details will be shown when the analysis on the macaque brain gets published!
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