DOI: 10.7274/25601175.v1
title: Revealing the Functional Circuits of Anatomical Neural Networks

description: The mammalian brain is one of the most complex network structures found in nature, being comprised of a staggering number of interconnected neuronal cells (about 86 billion, or 10^11, in humans), responsible for all the complex behaviors exhibited by the species, both at the individual and population levels. 
One of the most fundamental challenges in the sciences is understanding how the brain works, how its structure and composition lead to the complex repertoire of functional behaviors that allow individuals and populations to adapt, survive and evolve over long time scales. Currently, at the cellular level, a complete understanding of the picture is unfeasible since we are talking about as many neurons as there are stars in the Milky Way. Moreover, the neurons in the brain form an interconnected network, with an estimated 10^15 number of links (synaptic connections). Interpreting dynamics and function in a network of such complexity at the neuronal level, i.e., using a bottom-up approach, is an intractable problem, currently. A more feasible approach is the top-down description that studies the brain's connectivity at the mesoscale, which is the scale at which brain regions responsible for different functions (vision, hearing, touch, decision-making, language, etc) can be anatomically discerned. This is based on the observation that function is localized to specific regions of the brain, and in particular, within the cortex. Recent experiments in the brain, such as retrograde tract-tracing was able to generate physical connectivity data, i.e., estimates of the density of axonal connections between the functional brain areas. This data forms the brain network at the mesoscale, which is a set of directed, spatially embedded and weighted (provided by the density of axons) connections. One key observation about such interareal networks is that they are very dense (the number of links in a network of <em>N</em> nodes is O(N^2)), in stark contrast with other real-world networks, which are all sparse (having O(N) links) and have no apparent structural regularity. However, structural heterogeneity is always a hallmark of functional systems and thus we expect that the interareal network (also called connectome) has a non-trivial internal organization, that must be extracted from the data and understood. Another challenge to the study of such networks is that the connection weights span five orders of magnitude for the species studied (macaque monkey and mouse).
Prior research performed by our group in collaboration with neuroscientist collaborators from France have shown that there is indeed a structural/topological constraint imposed by the metabolic/energetic costs of axonal lengths, expressed as an exponentially decaying probability of connections with distance, called the EDR rule. A simple, Maximum Entropy Principle based model using the EDR was able to reproduce global topological properties within the macaque monkey and mouse connectomes, in close agreement with experiments. However, this model falls short in describing weighted network properties, and in particular the organization/regularity of the structural heterogeneity within the network, if any is present. In previous work, neuroscientists have discovered hierarchically organized information pathways within the visual system (which is part of the network), and thus one expects that hierarchical organization is an ubiquitous feature of brain networks. Even more so, as we know from information theory, hierarchies are the most efficient structures for representing and processing information. Typically, hierarchies in complex networks appear through clusters (also called network communities) and their organization into super clusters (clusters of clusters) through several such levels. The challenge, however, is to extract and reveal the hierarchical organization within such dense and strongly heterogeneous datasets. Unfortunately, all existing algorithms for community detection have been designed for sparse networks, unusable for these types of datasets. 
This thesis presents a novel community detection algorithm that we devised for dense, directed and spatially embedded networks. We also introduced an algorithm that uncovers the hierarchical community structure within the brain (if any exists). Importantly, our methodology is applicable to datasets with similar features, beyond neuroscience. Our results reveal several key findings: 
1) There is significant structure beyond what the EDR model captures in the brain, in forms of constellations of areas clustered across several levels, forming indeed, a hierarchy. 
2) The hierarchical organization discovered in the macaque brain is based only on connection strength/weights data, which is different from other approaches that use additional information, related to laminar organization of connections (the cortex is made of several layers, 6 in the primate). 
3) We demonstrate that the brain's organization is optimized in a way as to generate gradual changes in functional (dis)similarity with increasing physical distance across the cortical mantle.
These results have the potential to enhance current brain models and offer alternative, network science based insights into the functional organization of the brain.