Geometric theory predicts bifurcations in minimal wiring cost trees in biology are flat

Strahler Ordering Analyses on Branching Coral Canopies: Stylophora pistillata as a Case Study

The three-dimensional structural complexities generated by living sessile organisms, such as trees and branching corals, embrace distinct communities of dwelling organisms, many of which are adapted to specific niches within the structure. Thus, characterizing the build-up rules and the canopy compartments may clarify small-scale biodiversity patterns and rules for canopy constituents. While biodiversity within tree canopies is usually typified by the vertical axis that is delineated by its main compartments (understory, trunk, crown), traditional studies of coral canopy dwelling species are evaluated only by viewing the whole coral head as a single homogeneous geometric structure. Here, we employ the Strahler number of a mathematical tree for the numerical measurements of the coral’s canopy complexity. We use the branching Indo-Pacific coral species Stylophora pistillata as a model case, revealing five compartments in the whole coral canopy volume (Understory, Base, Middle, Up, and Bifurcation nods). Then, the coral’s dwellers’ diel distribution patterns were quantified and analyzed. We observed 114 natal colonies, containing 32 dwelling species (11 sessile), totaling 1019 individuals during day observations, and 1359 at night (1–41 individuals/colony). Biodiversity and abundance associated with Strahler numbers, diel richness, abundance, and patterns for compartmental distributions differed significantly between day/night. These results demonstrate that the coral-canopy Strahler number is an applicable new tool for assessing canopy landscapes and canopy associated species biodiversity, including the canopy-compartmental utilization by mobile organisms during day/night and young/adult behaviors.

Dissecting Sholl Analysis into Its Functional Components

Sholl analysis has been an important technique in dendritic anatomy for more than 60 years. The Sholl intersection profile is obtained by counting the number of dendritic branches at a given distance from the soma and is a key measure of dendritic complexity; it has applications from evaluating the changes in structure induced by pathologies to estimating the expected number of anatomical synaptic contacts. We find that the Sholl intersection profiles of most neurons can be reproduced from three basic, functional measures: the domain spanned by the dendritic arbor, the total length of the dendrite, and the angular distribution of how far dendritic segments deviate from a direct path to the soma (i.e., the root angle distribution). The first two measures are determined by axon location and hence microcircuit structure; the third arises from optimal wiring and represents a branching statistic estimating the need for conduction speed in a neuron.

Neural arbors are Pareto optimal

Neural arbors (dendrites and axons) can be viewed as graphs connecting the cell body of a neuron to various pre- and post-synaptic partners. Several constraints have been proposed on the topology of these graphs, such as minimizing the amount of wire needed to construct the arbor (wiring cost), and minimizing the graph distances between the cell body and synaptic partners (conduction delay). These two objectives compete with each other-optimizing one results in poorer performance on the other. Here, we describe how well neural arbors resolve this network design trade-off using the theory of Pareto optimality. We develop an algorithm to generate arbors that near-optimally balance between these two objectives, and demonstrate that this algorithm improves over previous algorithms. We then use this algorithm to study how close neural arbors are to being Pareto optimal. Analysing 14 145 arbors across numerous brain regions, species and cell types, we find that neural arbors are much closer to being Pareto optimal than would be expected by chance and other reasonable baselines. We also investigate how the location of the arbor on the Pareto front, and the distance from the arbor to the Pareto front, can be used to classify between some arbor types (e.g. axons versus dendrites, or different cell types), highlighting a new potential connection between arbor structure and function. Finally, using this framework, we find that another biological branching structure-plant shoot architectures used to collect and distribute nutrients-are also Pareto optimal, suggesting shared principles of network design between two systems separated by millions of years of evolution.

Robust emergence of sharply tuned place-cell responses in hippocampal neurons with structural and biophysical heterogeneities

Hippocampal pyramidal neurons sustain propagation of fast electrical signals and are electrotonically non-compact structures exhibiting cell-to-cell variability in their complex dendritic arborization. In this study, we demonstrate that sharp place-field tuning and several somatodendritic functional maps concomitantly emerge despite the presence of geometrical heterogeneities in these neurons. We establish this employing an unbiased stochastic search strategy involving thousands of models that spanned several morphologies and distinct profiles of dispersed synaptic localization and channel expression. Mechanistically, employing virtual knockout models (VKMs), we explored the impact of bidirectional modulation in dendritic spike prevalence on place-field tuning sharpness. Consistent with the prior literature, we found that across all morphologies, virtual knockout of either dendritic fast sodium channels or N-methyl-D-aspartate receptors led to a reduction in dendritic spike prevalence, whereas A-type potassium channel knockouts resulted in a non-specific increase in dendritic spike prevalence. However, place-field tuning sharpness was critically impaired in all three sets of VKMs, demonstrating that sharpness in feature tuning is maintained by an intricate balance between mechanisms that promote and those that prevent dendritic spike initiation. From the functional standpoint of the emergence of sharp feature tuning and intrinsic functional maps, within this framework, geometric variability was compensated by a combination of synaptic democracy, the ability of randomly dispersed synapses to yield sharp tuning through dendritic spike initiation, and ion-channel degeneracy. Our results suggest electrotonically non-compact neurons to be endowed with several degrees of freedom, encompassing channel expression, synaptic localization and morphological microstructure, in achieving sharp feature encoding and excitability homeostasis.

Morphological and functional properties of the conducting human airways investigated by in vivo computed tomography and in vitro MRI

The accurate representation of the human airway anatomy is crucial for understanding and modeling the structure-function relationship in both healthy and diseased lungs. The present knowledge in this area is based on morphometric studies of excised lung casts, partially complemented by in vivo studies in which computed tomography (CT) was used on a small number of subjects. In the present study, we analyzed CT scans of a cohort of healthy subjects and obtained comprehensive morphometric information down to the seventh generation of bronchial branching, including airway diameter, length, branching angle, and rotation angle. Although some of the geometric parameters (such as the child-to-parent branch diameter ratio) are found to be in line with accepted values, for others (such as the branch length-to-diameter ratio) our findings challenge the common assumptions. We also evaluated several metrics of self-similarity, including the fractal dimension of the airway tree. Additionally, we used phase-contrast magnetic resonance imaging (MRI) to obtain the volumetric flow field in the three-dimensional-printed airway model of one of the subjects during steady inhalation. This is used to relate structural and functional parameters and, in particular, to close the power-law relationship between branch flow rate and diameter. The diameter exponent is found to be significantly lower than in the usually assumed Poiseuille regime, which we attribute to the strong secondary (i.e., transverse) velocity component. The strength of the secondary velocity with respect to the axial component exceeds the levels found in idealized airway models and persists within the first seven generations. NEW & NOTEWORTHY We performed a comprehensive computed tomography-based study of the conductive airway morphology in normal human subjects, including branch diameter, length, and mutual angles. We found significant departure from classic homothetic relationships. We also carried out MRI measurements of the three-dimensional inspiratory flow in an anatomy-based model and directly assessed structure-function relationships that have so far been assumed. We found that strong secondary flows (i.e., transverse velocity components) persist through the first seven generations of bronchial branching.

High-Resolution Laser Scanning Reveals Plant Architectures that Reflect Universal Network Design Principles

Transport networks serve critical functions in biological and engineered systems, and yet their design requires trade-offs between competing objectives. Due to their sessile lifestyle, plants need to optimize their architecture to efficiently acquire and distribute resources while also minimizing costs in building infrastructure. To understand how plants resolve this design trade-off, we used high-precision three-dimensional laser scanning to map the architectures of tomato, tobacco, or sorghum plants grown in several environmental conditions and through multiple developmental time points, scanning in total 505 architectures from 37 plants. Using a graph-theoretic algorithm that we developed to evaluate design strategies, we find that plant architectures lie along the Pareto front between two simple length-based objectives-minimizing total branch length and minimizing nutrient transport distance-thereby conferring a selective fitness advantage for plant transport processes. The location along the Pareto front can distinguish among species and conditions, suggesting that during evolution, natural selection may employ common network design principles despite different optimization trade-offs.

A Statistical Description of Plant Shoot Architecture

Plant architectures can be characterized statistically by their spatial density function, which specifies the probability of finding a branch at each location in the territory occupied by a plant. Using high-precision 3D scanning, we analyzed 557 plant shoot architectures, representing three species, grown across three to five environmental conditions, and through 20-30 developmental time points. We found two elegant properties in the spatial density functions of these architectures: all functions could be nearly modified in one direction without affecting the density in orthogonal directions (called "separability"), and all functions shared the same underlying shape, aside from stretching and compression (called "self-similarity"). Surprisingly, despite their striking visual diversity, we discovered that all architectures could be described as variations on a single underlying function: a Gaussian density function truncated at roughly two SDs. We also observed systematic variation in the spatial density functions across species, growth conditions, and time, which suggests functional specialization despite following the same general design form.

Sloppy morphological tuning in identified neurons of the crustacean stomatogastric ganglion

Neuronal physiology depends on a neuron’s ion channel composition and unique morphology. Variable ion channel compositions can produce similar neuronal physiologies across animals. Less is known regarding the morphological precision required to produce reliable neuronal physiology. Theoretical studies suggest that moraphology is tightly tuned to minimize wiring and conduction delay of synaptic events. We utilize high-resolution confocal microscopy and custom computational tools to characterize the morphologies of four neuron types in the stomatogastric ganglion (STG) of the crab Cancer borealis. Macroscopic branching patterns and fine cable properties are variable within and across neuron types. We compare these neuronal structures to synthetic minimal spanning neurite trees constrained by a wiring cost equation and find that STG neurons do not adhere to prevailing hypotheses regarding wiring optimization principles. In this highly modulated and oscillating circuit, neuronal structures appear to be governed by a space-filling mechanism that outweighs the cost of inefficient wiring.

DOI:http://dx.doi.org/10.7554/eLife.22352.001

When complex neuronal structures may not matter

Much work has explored animal-to-animal variability and compensation in ion channel expression. Yet, little is known regarding the physiological consequences of morphological variability. We quantify animal-to-animal variability in cable lengths (CV = 0.4) and branching patterns in the Gastric Mill (GM) neuron, an identified neuron type with highly-conserved physiological properties in the crustacean stomatogastric ganglion (STG) of Cancer borealis. We examined passive GM electrotonic structure by measuring the amplitudes and apparent reversal potentials (E_revs) of inhibitory responses evoked with focal glutamate photo-uncaging in the presence of TTX. Apparent E_revs were relatively invariant across sites (mean CV ± SD = 0.04 ± 0.01; 7–20 sites in each of 10 neurons), which ranged between 100–800 µm from the somatic recording site. Thus, GM neurons are remarkably electrotonically compact (estimated λ > 1.5 mm). Electrotonically compact structures, in consort with graded transmission, provide an elegant solution to observed morphological variability in the STG.

DOI:http://dx.doi.org/10.7554/eLife.23508.001

Dendritic branching angles of pyramidal cells across layers of the juvenile rat somatosensory cortex

The characterization of the structural design of cortical microcircuits is essential for understanding how they contribute to function in both health and disease. Since pyramidal neurons represent the most abundant neuronal type and their dendritic spines constitute the major postsynaptic elements of cortical excitatory synapses, our understanding of the synaptic organization of the neocortex largely depends on the available knowledge regarding the structure of pyramidal cells. Previous studies have identified several apparently common rules in dendritic geometry. We study the dendritic branching angles of pyramidal cells across layers to further shed light on the principles that determine the geometric shapes of these cells. We find that the dendritic branching angles of pyramidal cells from layers II-VI of the juvenile rat somatosensory cortex suggest common design principles, despite the particular morphological and functional features that are characteristic of pyramidal cells in each cortical layer. J. Comp. Neurol. 524:2567-2576, 2016. © 2016 Wiley Periodicals, Inc.

Branching angles of pyramidal cell dendrites follow common geometrical design principles in different cortical areas

Unraveling pyramidal cell structure is crucial to understanding cortical circuit computations. Although it is well known that pyramidal cell branching structure differs in the various cortical areas, the principles that determine the geometric shapes of these cells are not fully understood. Here we analyzed and modeled with a von Mises distribution the branching angles in 3D reconstructed basal dendritic arbors of hundreds of intracellularly injected cortical pyramidal cells in seven different cortical regions of the frontal, parietal, and occipital cortex of the mouse. We found that, despite the differences in the structure of the pyramidal cells in these distinct functional and cytoarchitectonic cortical areas, there are common design principles that govern the geometry of dendritic branching angles of pyramidal cells in all cortical areas.