Neuropil is a fundamental form of tissue organization within brains 1 . In neuropils, densely packed neurons synaptically interconnect into precise circuit architecture 2 , 3 , yet the structural and developmental principles governing this nanoscale precision remain largely unknown 4 , 5 . Here, we use diffusion condensation, an iterative data coarse-graining algorithm 6 , to identify nested circuit structures within the C. elegans neuropil (called the nerve ring). We show that the nerve ring neuropil is largely organized into four strata composed of related behavioral circuits. The stratified architecture of the neuropil is a geometrical representation of the functional segregation of sensory information and motor outputs, with specific sensory organs and muscle quadrants mapping onto particular neuropil strata. We identify groups of neurons with unique morphologies that integrate information across strata and that create neural structures that cage the strata within the nerve ring. We use high resolution light-sheet microscopy 7 , 8 , coupled with lineage-tracing and cell-tracking algorithms 9 , 10 , to resolve the developmental sequence and reveal principles of cell position, migration and outgrowth that guide stratified neuropil organization. Our results uncover conserved structural design principles underlying nerve ring neuropil architecture and function, and a pioneer-neuron-based, temporal progression of outgrowth that guides the hierarchical development of the layered neuropil. Our findings provide a systematic blueprint for using structural and developmental approaches to understand neuropil organization within brains.

Big data often has emergent structure that exists at multiple levels of abstraction, which are useful for characterizing complex interactions and dynamics of the observations. Here, we consider multiple levels of abstraction via a multiresolution geometry of data points at different granularities. To construct this geometry we define a time-inhomogeneous diffusion process that effectively condenses data points together to uncover nested groupings at larger and larger granularities. This inhomogeneous process creates a deep cascade of intrinsic low pass filters on the data affinity graph that are applied in sequence to gradually eliminate local variability while adjusting the learned data geometry to increasingly coarser resolutions. We provide visualizations to exhibit our method as a "continuously-hierarchical" clustering with directions of eliminated variation highlighted at each step. The utility of our algorithm is demonstrated via neuronal data condensation, where the constructed multiresolution data geometry uncovers the organization, grouping, and connectivity between neurons.

To visualize DC cluster relationships we developed C-PHATE, an extension of PHATE 30 capable of generating a 3D representation of the multigranular structures in the DC hierarchical tree ( Fig. 1a,b; see Methods). Quantitative comparisons of DC/C-PHATE results revealed similar clustering behaviors between the two separately analyzed nervering reconstructions (Adjusted Rand Index (ARI) of 0.7; Extended Data Fig. 1a,b), consistent with the long-accepted qualitative descriptions of the stereotyped structure for the C. elegans nerve ring 3,31 , and recent analyses of neurite adjacencies 22 . Examination of the cluster patterns through the iterations of DC revealed known cell-cell interactions and behavioral circuits ( Fig. 1b,c; Extended Data Fig. 2a,b; [32][33][34][35][36][37][38][39] ), consistent with behavioral and connectomic studies 10,19,20 . The multigranular outputs of the DC/C-PHATE results enabled understanding of the cell-cell interactions within the context of functional circuits, and the functional circuits within the context of the larger bundles of the neuropil.Comparisons of the modularity score, a measure of cluster robustness 40,41 , was highest when the majority of neurites were grouped into four super-clusters ( Fig. 1b; Extended Data Fig. 1a,b; Supplemental Video 1,2; see Methods). Overlaying our analysis of these four super-clusters (see Methods) onto the anatomy of the nerve ring revealed that they correspond to four distinct and tightly grouped bundles within the greater neuropil, circling the pharynx isthmus and stacked along the anterior-posterior axis of the animal. These clusters are herein called S1, S2, S3 and S4 for Stratum 1 etc. (Fig. 1c; Extended Data Fig. 1c-h; Supplemental Video 3 show details of these structures and single-cell identities). The stratified organization of the nerve ring neuropil, resolved here at single neurite level, is reminiscent of laminar organizations in the Drosophila nervous system 42,43 , and in the vertebrate retina and the brain cortex 44,45 .We observed that not all computationally clustered neurites followed simple bundled paths through the neuropil. In Strata 1 (S1), 32 anterior sensory neurons project perpendicular to the nerve ring bundle before making a 180º curl and returning to the anterior limits of the neuropil, where they terminate as synaptic endplates (Fig. 1d; Extended Data Fig. 3a-d; 3,46 ). These uniquely shaped S1 neurons looped at the borders of the S2/S3/S4 bundles. The anterior loops encase ~90% of S2, and the posterior loops encase ~84% of S3 and 100% of S4 ( Fig. 1d-g; Extended Data Fig. 3e-k ; SupplementalVideo 5; Supplemental Table 1). We found that the 32 axons form a six-fold symmetrical honeycomb structure along the neuropil's arc, creating a scaffold that encapsulates the different strata ( Fig. 1g; Extended Data Fig. 3e-h; Supplemental Video 4).The architecture of the neuropil reflects functional segregation of distinct sensory information streams and motor outputs.C. elegans neurons can be divided into three categories ...