Introduction#

Connectomes are maps of neural wiring, and have have become increasingly important in neuroscience [ABC+20, VBP+19]. A rich literature exists on studying connectomes at varying scales, from brain-wide maps of white-matter bundles based on diffusion MRI, to nanoscale connectomes based on the connectivity of individual neurons as measured by electron microscopy [SL16]. For the purposes of this work, we restrict our analysis to so-called nanoscale connectomes, where nodes are neurons and edges represent some number of synapses between them. Henceforth, the term “connectome” can be taken to mean “nanoscale connectome.”

Many of the goals of connectomics rely on being able to compare networks. For instance, to understand how memory relates to connectivity, one would need to measure a connectome which has learned something and one which has not. To understand how a allele affects connectivity, one would need to measure a connectome from an organism with the allele and one from an organism without. Several recent works begun this quest towards comparative connectomics. [WMM+21] studied connectomes in the context of development, collecting connectomes from Caenorhabditis elegans at varying life stages and examining which aspects of the connectome are preserved or varying across development. [CJB+19] extracted connectomes for both a male and hermaphrodite C. elegans worm to understand which aspect of this organism’s wiring diagram differ between the sexes. [VAFS+21] made genetic perturbations to different individual Drosophila melanogaster fly larva, and examined how these perturbations affected the connectivity of a local circuit in the organism’s nerve cord (the insect equivalent of a spinal cord). Viewed through the lens of the wiring diagrams alone (e.g. ignoring morphology, subcellular structures, etc.), these questions all amount to comparing two or more networks.

In addition to those described above, one connectome comparison that has been prevalent in the connectomics literature is that of bilateral symmetry. The term bilaterian refers to an animal which as an anterior-posterior axis of symmetry. This clade is thought to have emerged around 550 million years ago [FW97], making it one of the oldest groups of animals. Most organisms studied in neuroscience (C. elegans, D. melanogaster, mice, rats, monkeys, and humans, to name a few) are all bilaterians. This axis of symmetry extends to the brain - most of these organisms have a distinct left and right side to the brain. This left-right symmetry in the brain demonstrably impacts an organism’s behavior. [LAD+20] found that the number of M-DCN neurons on each side of the brain in D. melanogaster is stochastically determined during development. Moreover, this simple difference in the number of one type of neuron across the two brain hemispheres was shown to robustly predict a movement behavior of the organism, where individuals with more asymmetry in M-DCN neuron numbers were more likely to traverse a circular arena in a coherent manner.

Connectomics studies have also investigated bilateral symmetry in various ways. For instance, [LTWL09] reconstructed the connectome of the axons projecting to the interscutularis muscle on the left and right side of two individual mice. They found that the branching patterns of axons between the left and right sides were no more similar than a comparison between the two animals, and also no more similar than a two random branching patterns generated by a null model. In contrast, [SBSturner+21] found a striking similarity between the morphologies of neurons (as measured by NBLAST) in the left and right hemispheres of the D. melanogaster antennal lobe, and a similar level of stereotypy between the antennal lobes of two different individuals. TODO review Hildebrand et al 2017 paper. [CJB+19] use the observed level of left-right variability in a C. elegans hermaphrodite connectome as a proxy for the amount of variability in connectivity - seemingly assuming that we should expect the connectomes of the left and right to be the same up to developmental and experiential variability. Conversely, they also point out the fact that the ASEL neuron (on the left side) projects more strongly to neuron class AWC than the analogous version on the right, verifying this difference via fluorescent labeling in another animal. They point out that this difference may be related to a lateralized chemosensory behavior TODO look at citations 34/35 in Cook paper, summarize briefly. These studies highlight the complicated relationships between neuroscientists and bilateral symmetry: at times, we may assume that the left and right sides of a nervous system are in some sense the same in expectation, but at other times we find marked, reproducible differences between them. To date, no study (to our knowledge) has framed this question of bilateral symmetry of connectivity as a statistical hypothesis comparing two networks.

The problem of statistically comparing two networks (known as a two-sample test) is a relatively new problem in statistics, though several approaches have been proposed.

TODO Review statistical approaches from the literature

  • nonpar

  • semipar

  • von luxburg 2017 one

  • [BCSL19] present a method for comparing paired or matched networks based on bootstrap samples from various network models. Looks very similar to semipar paper but test statistic based on \(P\) not \(X\)

  • [Aue19] presents what they call a KS test for network data. Based on communities. Mentions a test for reciprocity. Randomization procedure is based on swapping \(ij\)s between the two networks which are being compared. Test statistics based on operator norms. TODO Not sure whether it is worth trying

  • KS test on degree sequences

  • Some way of comparing rooted subgraph counts

  • Some way of comparing global subgraph counts

Each of the approaches described above makes different assumptions about the underlying distribution that each of the two networks were generated from. Thus, these tests may differ in the settings in which they are valid (the probability of type-I error is controlled). Similarly, as these tests all rely on differing models, they may be more or less powerful (able to detect a difference between the two networks when one is present) under different alternatives to the null hypothesis. For a practicing neuroscientist, these differences are more than a mere statistical nuisance, as the choice of a test to compare two networks can change the conclusion one makes about the data.

Motivated by the discussion above, the goals of this paper are threefold: 1) to formally state various notions of bilateral symmetry for connectomes as statistical hypotheses, 2) to present test procedures for each of these null hypotheses of bilateral symmetry, and 3) to show the advantages and drawbacks to each definition/test by studying these tests under various synthetic perturbations to the connectomes. We will present the first rigorous evaluation of bilateral symmetry of connectivity, in this case by studying the brain of the D. melanogaster larval brain. In doing so, we will also provide practical considerations to any neuroscientist wishing to compare two networks.