Optimally wired subnetwork determines neuroanatomy of Caenorhabditis elegans

Evolution of neural circuitry and cognition

Neural circuits govern the interface between the external environment, internal cues and outwardly directed behaviours. To process multiple environmental stimuli and integrate these with internal state requires considerable neural computation. Expansion in neural network size, most readily represented by whole brain size, has historically been linked to behavioural complexity, or the predominance of cognitive behaviours. Yet, it is largely unclear which aspects of circuit variation impact variation in performance. A key question in the field of evolutionary neurobiology is therefore how neural circuits evolve to allow improved behavioural performance or innovation. We discuss this question by first exploring how volumetric changes in brain areas reflect actual neural circuit change. We explore three major axes of neural circuit evolution-replication, restructuring and reconditioning of cells and circuits-and discuss how these could relate to broader phenotypes and behavioural variation. This discussion touches on the relevant uses and limitations of volumetrics, while advocating a more circuit-based view of cognition. We then use this framework to showcase an example from the insect brain, the multi-sensory integration and internal processing that is shared between the mushroom bodies and central complex. We end by identifying future trends in this research area, which promise to advance the field of evolutionary neurobiology.

Why an animal needs a brain

In Principles of Neural Design (2015, MIT Press), inspired by Charles Darwin, Sterling and Laughlin undertook the unfashionable task of distilling principles from facts in the technique-driven, data-saturated domain of neuroscience. Their starting point for deriving the organizing principles of brains are two brainless single-celled organisms, Escherichia coli and Paramecium, and the 302-neuron brain of the nematode Caenorhabditis elegans. The book is an exemplar in how to connect the dots between simpler and (much) more complex organisms in a particular area. Here, they have generously agreed to republish an abridged version of Chapter 2 (Why an Animal Needs a Brain), in which many of their principles are first described.

The synaptic organization in the Caenorhabditis elegans neural network suggests significant local compartmentalized computations

Neurons are characterized by elaborate tree-like dendritic structures that support local computations by integrating multiple inputs from upstream presynaptic neurons. It is less clear whether simple neurons, consisting of a few or even a single neurite, may perform local computations as well. To address this question, we focused on the compact neural network of Caenorhabditis elegans animals for which the full wiring diagram is available, including the coordinates of individual synapses. We find that the positions of the chemical synapses along the neurites are not randomly distributed nor can they be explained by anatomical constraints. Instead, synapses tend to form clusters, an organization that supports local compartmentalized computations. In mutually synapsing neurons, connections of opposite polarity cluster separately, suggesting that positive and negative feedback dynamics may be implemented in discrete compartmentalized regions along neurites. In triple-neuron circuits, the nonrandom synaptic organization may facilitate local functional roles, such as signal integration and coordinated activation of functionally related downstream neurons. These clustered synaptic topologies emerge as a guiding principle in the network, presumably to facilitate distinct parallel functions along a single neurite, which effectively increase the computational capacity of the neural network.

A set of hub neurons and non-local connectivity features support global brain dynamics in C. elegans

The wiring architecture of neuronal networks is assumed to be a strong determinant of their dynamical computations. An ongoing effort in neuroscience is therefore to generate comprehensive synapse-resolution connectomes alongside brain-wide activity maps. However, the structure-function relationship, i.e., how the anatomical connectome and neuronal dynamics relate to each other on a global scale, remains unsolved. Systematically, comparing graph features in the C. elegans connectome with correlations in nervous system-wide neuronal dynamics, we found that few local connectivity motifs and mostly other non-local features such as triplet motifs and input similarities can predict functional relationships between neurons. Surprisingly, quantities such as connection strength and amount of common inputs do not improve these predictions, suggesting that the network's topology is sufficient. We demonstrate that hub neurons in the connectome are key to these relevant graph features. Consistently, inhibition of multiple hub neurons specifically disrupts brain-wide correlations. Thus, we propose that a set of hub neurons and non-local connectivity features provide an anatomical substrate for global brain dynamics.

Optimality of syntactic dependency distances

It is often stated that human languages, as other biological systems, are shaped by cost-cutting pressures but, to what extent? Attempts to quantify the degree of optimality of languages by means of an optimality score have been scarce and focused mostly on English. Here we recast the problem of the optimality of the word order of a sentence as an optimization problem on a spatial network where the vertices are words, arcs indicate syntactic dependencies, and the space is defined by the linear order of the words in the sentence. We introduce a score to quantify the cognitive pressure to reduce the distance between linked words in a sentence. The analysis of sentences from 93 languages representing 19 linguistic families reveals that half of languages are optimized to a 70% or more. The score indicates that distances are not significantly reduced in a few languages and confirms two theoretical predictions: that longer sentences are more optimized and that distances are more likely to be longer than expected by chance in short sentences. We present a hierarchical ranking of languages by their degree of optimization. The score has implications for various fields of language research (dependency linguistics, typology, historical linguistics, clinical linguistics, and cognitive science). Finally, the principles behind the design of the score have implications for network science.

Structural optimality and neurogenetic expression mediate functional dynamics in the human brain

The human brain exhibits a rich functional repertoire in terms of complex functional connectivity patterns during rest and tasks. However, how this is developed upon a fixed structural anatomy remains poorly understood. Here we investigated the hypothesis that resting state functional connectivity and the manner in which it changes during tasks related to a set of underlying structural connections that promote optimal communication in the brain. We used a game‐theoretic model to identify such optimal connections in the structural connectome of 50 healthy individuals and subsequently used the optimal structural connections to predict resting‐state functional connectivity with high accuracy. In contrast, we found that nonoptimal connections accurately predicted functional connectivity during a working memory task. We further found that this balance between optimal and nonoptimal connections between brain regions was associated with a specific gene expression linked to neurotransmission. This multimodal evidence shows for the first time that structure–function relationships in the human brain are related to how brain networks navigate information along different white matter connections as well as the brain's underlying genetic profile.

Developmental trajectory of Caenorhabditis elegans nervous system governs its structural organization

A central problem of neuroscience involves uncovering the principles governing the organization of nervous systems which ensure robustness in brain development. The nematode Caenorhabditis elegans provides us with a model organism for studying this question. In this paper, we focus on the invariant connection structure and spatial arrangement of the neurons comprising the somatic neuronal network of this organism to understand the key developmental constraints underlying its design. We observe that neurons with certain shared characteristics-such as, neural process lengths, birth time cohort, lineage and bilateral symmetry-exhibit a preference for connecting to each other. Recognizing the existence of such homophily and their relative degree of importance in determining connection probability within neurons (for example, in synapses, symmetric pairing is the most dominant factor followed by birth time cohort, process length and lineage) helps in connecting specific neuronal attributes to the topological organization of the network. Further, the functional identities of neurons appear to dictate the temporal hierarchy of their appearance during the course of development. Providing crucial insights into principles that may be common across many organisms, our study shows how the trajectory in the developmental landscape constrains the structural organization of a nervous system.

The human connectome from an evolutionary perspective

The connectome describes the comprehensive set of neuronal connections of a species' central nervous system. Identifying the network characteristics of the human macroscale connectome and comparing these features with connectomes of other species provides insight into the evolution of human brain connectivity and its role in brain function. Several network properties of the human connectome are conserved across species, with emerging evidence also indicating potential human-specific adaptations of connectome topology. This review describes the human macroscale structural and functional connectome, focusing on common themes of brain wiring in the animal kingdom and network adaptations that may underlie human brain function. Evidence is drawn from comparative studies across a wide range of animal species, and from research comparing human brain wiring with that of non-human primates. Approaching the human connectome from a comparative perspective paves the way for network-level insights into the evolution of human brain structure and function.

Evidence of Rentian Scaling of Functional Modules in Diverse Biological Networks

Biological networks have long been known to be modular, containing sets of nodes that are highly connected internally. Less emphasis, however, has been placed on understanding how intermodule connections are distributed within a network. Here, we borrow ideas from engineered circuit design and study Rentian scaling, which states that the number of external connections between nodes in different modules is related to the number of nodes inside the modules by a power-law relationship. We tested this property in a broad class of molecular networks, including protein interaction networks for six species and gene regulatory networks for 41 human and 25 mouse cell types. Using evolutionarily defined modules corresponding to known biological processes in the cell, we found that all networks displayed Rentian scaling with a broad range of exponents. We also found evidence for Rentian scaling in functional modules in the Caenorhabditis elegans neural network, but, interestingly, not in three different social networks, suggesting that this property does not inevitably emerge. To understand how such scaling may have arisen evolutionarily, we derived a new graph model that can generate Rentian networks given a target Rent exponent and a module decomposition as inputs. Overall, our work uncovers a new principle shared by engineered circuits and biological networks.

The Influence of Wiring Economy on Nervous System Evolution

The activity and maintenance of neurons requires substantial metabolic energy, resulting in selective pressure to decrease resource consumption by the nervous system. The wiring economy principle proposes that animals have evolved mechanisms that wire circuits efficiently by minimizing neurite length. Computational modeling of neuronal morphology, microcircuit organization, and neural networks reveals that wiring economy is a significant determinant of nervous system layout. The strategies for reducing wiring costs are shared across phyla and point to the possibility of generalizable rules that specify the development of efficient nervous systems. As the developmental mechanisms underpinning wiring economy are only now being elucidated, whether the molecular basis of this phenomenon is the result of conserved genetic programs or convergent evolution remains to be determined.

Network Design and the Brain

Neural circuits have evolved to accommodate similar information processing challenges as those faced by engineered systems. Here, we compare neural versus engineering strategies for constructing networks. During circuit development, synapses are overproduced and then pruned back over time, whereas in engineered networks, connections are initially sparse and are then added over time. We provide a computational perspective on these two different approaches, including discussion of how and why they are used, insights that one can provide the other, and areas for future joint investigation. By thinking algorithmically about the goals, constraints, and optimization principles used by neural circuits, we can develop brain-derived strategies for enhancing network design, while also stimulating experimental hypotheses about circuit development and function.

Micro-connectomics: probing the organization of neuronal networks at the cellular scale

Defining the organizational principles of neuronal networks at the cellular scale, or micro-connectomics, is a key challenge of modern neuroscience. In this Review, we focus on graph theoretical parameters of micro-connectome topology, often informed by economical principles that conceptually originated with Ramón y Cajal's conservation laws. First, we summarize results from studies in intact small organisms and in samples from larger nervous systems. We then evaluate the evidence for an economical trade-off between biological cost and functional value in the organization of neuronal networks. Various results suggest that many aspects of neuronal network organization are indeed the outcome of competition between these two fundamental selection pressures.

The C. elegans Connectome Consists of Homogenous Circuits with Defined Functional Roles

A major goal of systems neuroscience is to decipher the structure-function relationship in neural networks. Here we study network functionality in light of the common-neighbor-rule (CNR) in which a pair of neurons is more likely to be connected the more common neighbors it shares. Focusing on the fully-mapped neural network of C. elegans worms, we establish that the CNR is an emerging property in this connectome. Moreover, sets of common neighbors form homogenous structures that appear in defined layers of the network. Simulations of signal propagation reveal their potential functional roles: signal amplification and short-term memory at the sensory/inter-neuron layer, and synchronized activity at the motoneuron layer supporting coordinated movement. A coarse-grained view of the neural network based on homogenous connected sets alone reveals a simple modular network architecture that is intuitive to understand. These findings provide a novel framework for analyzing larger, more complex, connectomes once these become available.

Dendritic and Axonal Wiring Optimization of Cortical GABAergic Interneurons

The way in which a neuronal tree expands plays an important role in its functional and computational characteristics. We aimed to study the existence of an optimal neuronal design for different types of cortical GABAergic neurons. To do this, we hypothesized that both the axonal and dendritic trees of individual neurons optimize brain connectivity in terms of wiring length. We took the branching points of real three-dimensional neuronal reconstructions of the axonal and dendritic trees of different types of cortical interneurons and searched for the minimal wiring arborization structure that respects the branching points. We compared the minimal wiring arborization with real axonal and dendritic trees. We tested this optimization problem using a new approach based on graph theory and evolutionary computation techniques. We concluded that neuronal wiring is near-optimal in most of the tested neurons, although the wiring length of dendritic trees is generally nearer to the optimum. Therefore, wiring economy is related to the way in which neuronal arborizations grow irrespective of the marked differences in the morphology of the examined interneurons.

Lamination Speeds the Functional Development of Visual Circuits

A common feature of the brain is the arrangement of synapses in layers. To examine the significance of this organizational feature, we studied the functional development of direction-selective (DS) circuits in the tectum of astray mutant zebrafish in which lamination of retinal ganglion cell (RGC) axons is lost. We show that although never laminar, the tuning of DS-RGC axons targeting the mutant tectum is normal. Analysis of mutant tectal neurons at late developmental stages reveals that directional tuning is indistinguishable from wild-type larvae. Furthermore, we show that structural plasticity of tectal dendrites and RGC axons compensates for the loss of lamination, establishing connectivity between DS-RGCs and their normal tectal targets. However, tectal direction selectivity is severely perturbed at earlier developmental stages. Thus, the formation of synaptic laminae is ultimately dispensable for the correct wiring of direction-selective tectal circuits, but it is crucial for the rapid assembly of these networks.

Video Abstract

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Robo2 directs lamination of direction-selective retinal axons and tectal dendrites

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Tectal lamination is required for rapid assembly of direction-selective circuits

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Functional direction-selective circuits eventually form when lamination is lost

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Structural plasticity compensates for the loss of tectal lamination

Cortical Composition Hierarchy Driven by Spine Proportion Economical Maximization or Wire Volume Minimization

The structure and quantitative composition of the cerebral cortex are interrelated with its computational capacity. Empirical data analyzed here indicate a certain hierarchy in local cortical composition. Specifically, neural wire, i.e., axons and dendrites take each about 1/3 of cortical space, spines and glia/astrocytes occupy each about (1/3)², and capillaries around (1/3)⁴. Moreover, data analysis across species reveals that these fractions are roughly brain size independent, which suggests that they could be in some sense optimal and thus important for brain function. Is there any principle that sets them in this invariant way? This study first builds a model of local circuit in which neural wire, spines, astrocytes, and capillaries are mutually coupled elements and are treated within a single mathematical framework. Next, various forms of wire minimization rule (wire length, surface area, volume, or conduction delays) are analyzed, of which, only minimization of wire volume provides realistic results that are very close to the empirical cortical fractions. As an alternative, a new principle called “spine economy maximization” is proposed and investigated, which is associated with maximization of spine proportion in the cortex per spine size that yields equally good but more robust results. Additionally, a combination of wire cost and spine economy notions is considered as a meta-principle, and it is found that this proposition gives only marginally better results than either pure wire volume minimization or pure spine economy maximization, but only if spine economy component dominates. However, such a combined meta-principle yields much better results than the constraints related solely to minimization of wire length, wire surface area, and conduction delays. Interestingly, the type of spine size distribution also plays a role, and better agreement with the data is achieved for distributions with long tails. In sum, these results suggest that for the efficiency of local circuits wire volume may be more primary variable than wire length or temporal delays, and moreover, the new spine economy principle may be important for brain evolutionary design in a broader context.

Cerebral cortex is an outer layer of the brain in mammals, and it plays a critical part in various cognitive processes such as learning, memory, attention, language, and consciousness. The cerebral cortex contains a number of neuroanatomical parameters whose values are essentially conserved across species and brain sizes, which suggests that these particular parameters are somehow important for brain efficient functioning. This study shows that the fractional volumes of five major cortical components both neuronal and non-neuronal (axons, dendrites, spines, glia/astrocytes, capillaries) are also approximately conserved across mammals, and neural wire (axons and dendrites) occupies the most of cortical space. Moreover, the fractional volumes form a special hierarchy of dependencies, being approximately equal to integer powers of 1/3. Is there any evolutionary principle of cortical organization that would explain these properties? This study finds that there are two different theoretical principles that can provide answers: one standard related to minimization of neural wire fractional volume, and a new proposition associated with economical maximization of spine content. However, the latter principle produces more robust results, which suggests that spine economical maximization is potentially an alternative to the more common “wire minimization” in explaining the cortical layout. Therefore, the current study can become an important contribution to our understanding (or debating) of the main factors influencing the evolution of local cortical circuits in the brain.

Phase transitions in Pareto optimal complex networks

The organization of interactions in complex systems can be described by networks connecting different units. These graphs are useful representations of the local and global complexity of the underlying systems. The origin of their topological structure can be diverse, resulting from different mechanisms including multiplicative processes and optimization. In spatial networks or in graphs where cost constraints are at work, as it occurs in a plethora of situations from power grids to the wiring of neurons in the brain, optimization plays an important part in shaping their organization. In this paper we study network designs resulting from a Pareto optimization process, where different simultaneous constraints are the targets of selection. We analyze three variations on a problem, finding phase transitions of different kinds. Distinct phases are associated with different arrangements of the connections, but the need of drastic topological changes does not determine the presence or the nature of the phase transitions encountered. Instead, the functions under optimization do play a determinant role. This reinforces the view that phase transitions do not arise from intrinsic properties of a system alone, but from the interplay of that system with its external constraints.

Wiring cost and topological participation of the mouse brain connectome

Brain connectomes are topologically complex systems, anatomically embedded in 3D space. Anatomical conservation of "wiring cost" explains many but not all aspects of these networks. Here, we examined the relationship between topology and wiring cost in the mouse connectome by using data from 461 systematically acquired anterograde-tracer injections into the right cortical and subcortical regions of the mouse brain. We estimated brain-wide weights, distances, and wiring costs of axonal projections and performed a multiscale topological and spatial analysis of the resulting weighted and directed mouse brain connectome. Our analysis showed that the mouse connectome has small-world properties, a hierarchical modular structure, and greater-than-minimal wiring costs. High-participation hubs of this connectome mediated communication between functionally specialized and anatomically localized modules, had especially high wiring costs, and closely corresponded to regions of the default mode network. Analyses of independently acquired histological and gene-expression data showed that nodal participation colocalized with low neuronal density and high expression of genes enriched for cognition, learning and memory, and behavior. The mouse connectome contains high-participation hubs, which are not explained by wiring-cost minimization but instead reflect competitive selection pressures for integrated network topology as a basis for higher cognitive and behavioral functions.

The strength of weak connections in the macaque cortico-cortical network

Examination of the cortico-cortical network of mammals has unraveled key topological features and their role in the function of the healthy and diseased brain. Recent findings from social and biological networks pinpoint the significant role of weak connections in network coherence and mediation of information from segregated parts of the network. In the current study, inspired by such findings and proposed architectures pertaining to social networks, we examine the structure of weak connections in the macaque cortico-cortical network by employing a tract-tracing dataset. We demonstrate that the cortico-cortical connections as a whole, as well as connections between segregated communities of brain areas, comply with the architecture suggested by the so-called strength-of-weak-ties hypothesis. However, we find that the wiring of these connections is not optimal with respect to the aforementioned architecture. This configuration is not attributable to a trade-off with factors known to constrain brain wiring, i.e., wiring cost and efficiency. Lastly, weak connections, but not strong ones, appear important for network cohesion. Our findings relate a topological property to the strength of cortico-cortical connections, highlight the prominent role of weak connections in the cortico-cortical structural network and pinpoint their potential functional significance. These findings suggest that certain neuroimaging studies, despite methodological challenges, should explicitly take them into account and not treat them as negligible.

Neural circuit assembly: economically wired by a single cadherin

Neurons are thought to acquire shapes and configurations consistent with the wiring optimization principle. A new study sheds light on the underlying molecular mechanisms by demonstrating that N-cadherin-mediated differential adhesion determines relative neurite positioning in developing columnar synaptic modules.

Synaptic laminae in the visual system: molecular mechanisms forming layers of perception

Synaptic connections between neurons form the basis for perception and behavior. Synapses are often clustered in space, forming stereotyped layers. In the retina and optic tectum, multiple such synaptic laminae are stacked on top of each other, giving rise to stratified neuropil regions in which each layer combines synapses responsive to a particular sensory feature. Recently, several cellular and molecular mechanisms that underlie the development of multilaminar arrays of synapses have been discovered. These mechanisms include neurite guidance and cell-cell recognition. Molecules of the Slit, Semaphorin, Netrin, and Hedgehog families, binding to their matching receptors, bring axons and dendrites into spatial register. These guidance cues may diffuse over short distances or bind to sheets of extracellular matrix, thus conditioning the local extracellular milieu, or are presented on the surface of cells bordering the future neuropil. In addition, mutual recognition of axons and dendrites through adhesion molecules with immunoglobulin domains ensures cell type-specific connections within a given layer. Thus, an elaborate genetic program assembles the parallel processing channels that underlie visual perception.

Phase transition in the economically modeled growth of a cellular nervous system

Spatially embedded complex networks, such as nervous systems, the Internet, and transportation networks, generally have nontrivial topological patterns of connections combined with nearly minimal wiring costs. However, the growth rules shaping these economical tradeoffs between cost and topology are not well understood. Here, we study the cellular nervous system of the nematode worm Caenorhabditis elegans, together with information on the birth times of neurons and on their spatial locations. We find that the growth of this network undergoes a transition from an accelerated to a constant increase in the number of links (synaptic connections) as a function of the number of nodes (neurons). The time of this phase transition coincides closely with the observed moment of hatching, when development switches metamorphically from oval to larval stages. We use graph analysis and generative modeling to show that the transition between different growth regimes, as well as its coincidence with the moment of hatching, may be explained by a dynamic economical model that incorporates a tradeoff between topology and cost that is continuously negotiated and renegotiated over developmental time. As the body of the animal progressively elongates, the cost of longer-distance connections is increasingly penalized. This growth process regenerates many aspects of the adult nervous system's organization, including the neuronal membership of anatomically predefined ganglia. We expect that similar economical principles may be found in the development of other biological or manmade spatially embedded complex systems.

Trade-off between multiple constraints enables simultaneous formation of modules and hubs in neural systems

The formation of the complex network architecture of neural systems is subject to multiple structural and functional constraints. Two obvious but apparently contradictory constraints are low wiring cost and high processing efficiency, characterized by short overall wiring length and a small average number of processing steps, respectively. Growing evidence shows that neural networks are results from a trade-off between physical cost and functional value of the topology. However, the relationship between these competing constraints and complex topology is not well understood quantitatively. We explored this relationship systematically by reconstructing two known neural networks, Macaque cortical connectivity and C. elegans neuronal connections, from combinatory optimization of wiring cost and processing efficiency constraints, using a control parameter α, and comparing the reconstructed networks to the real networks. We found that in both neural systems, the reconstructed networks derived from the two constraints can reveal some important relations between the spatial layout of nodes and the topological connectivity, and match several properties of the real networks. The reconstructed and real networks had a similar modular organization in a broad range of α, resulting from spatial clustering of network nodes. Hubs emerged due to the competition of the two constraints, and their positions were close to, and partly coincided, with the real hubs in a range of α values. The degree of nodes was correlated with the density of nodes in their spatial neighborhood in both reconstructed and real networks. Generally, the rebuilt network matched a significant portion of real links, especially short-distant ones. These findings provide clear evidence to support the hypothesis of trade-off between multiple constraints on brain networks. The two constraints of wiring cost and processing efficiency, however, cannot explain all salient features in the real networks. The discrepancy suggests that there are further relevant factors that are not yet captured here.

The evolutionary origins of modularity

A central biological question is how natural organisms are so evolvable (capable of quickly adapting to new environments). A key driver of evolvability is the widespread modularity of biological networks--their organization as functional, sparsely connected subunits--but there is no consensus regarding why modularity itself evolved. Although most hypotheses assume indirect selection for evolvability, here we demonstrate that the ubiquitous, direct selection pressure to reduce the cost of connections between network nodes causes the emergence of modular networks. Computational evolution experiments with selection pressures to maximize network performance and minimize connection costs yield networks that are significantly more modular and more evolvable than control experiments that only select for performance. These results will catalyse research in numerous disciplines, such as neuroscience and genetics, and enhance our ability to harness evolution for engineering purposes.

Synchronization waves in geometric networks

We report synchronization of networked excitable nodes embedded in a metric space, where the connectivity properties are mostly determined by the distance between units. Such a high clustered structure, combined with the lack of long-range connections, prevents full synchronization and yields instead the emergence of synchronization waves. We show that this regime is optimal for information transmission through the system, as it enhances the options of reconstructing the topology from the dynamics. Measurements of topological and functional centralities reveal that the wave-synchronization state allows detection of the most structurally relevant nodes from a single observation of the dynamics, without any a priori information on the model equations ruling the evolution of the ensemble.

Wiring economy and volume exclusion determine neuronal placement in the Drosophila brain

Wiring economy has successfully explained the individual placement of neurons in simple nervous systems like that of Caenorhabditis elegans [1-3] and the locations of coarser structures like cortical areas in complex vertebrate brains [4]. However, it remains unclear whether wiring economy can explain the placement of individual neurons in brains larger than that of C. elegans. Indeed, given the greater number of neuronal interconnections in larger brains, simply minimizing the length of connections results in unrealistic configurations, with multiple neurons occupying the same position in space. Avoiding such configurations, or volume exclusion, repels neurons from each other, thus counteracting wiring economy. Here we test whether wiring economy together with volume exclusion can explain the placement of neurons in a module of the Drosophila melanogaster brain known as lamina cartridge [5-13]. We used newly developed techniques for semiautomated reconstruction from serial electron microscopy (EM) [14] to obtain the shapes of neurons, the location of synapses, and the resultant synaptic connectivity. We show that wiring length minimization and volume exclusion together can explain the structure of the lamina microcircuit. Therefore, even in brains larger than that of C. elegans, at least for some circuits, optimization can play an important role in individual neuron placement.

Topological cluster analysis reveals the systemic organization of the Caenorhabditis elegans connectome

The modular organization of networks of individual neurons interwoven through synapses has not been fully explored due to the incredible complexity of the connectivity architecture. Here we use the modularity-based community detection method for directed, weighted networks to examine hierarchically organized modules in the complete wiring diagram (connectome) of Caenorhabditis elegans (C. elegans) and to investigate their topological properties. Incorporating bilateral symmetry of the network as an important cue for proper cluster assignment, we identified anatomical clusters in the C. elegans connectome, including a body-spanning cluster, which correspond to experimentally identified functional circuits. Moreover, the hierarchical organization of the five clusters explains the systemic cooperation (e.g., mechanosensation, chemosensation, and navigation) that occurs among the structurally segregated biological circuits to produce higher-order complex behaviors.

Caenorhabditis elegans (C. elegans) is a tiny worm whose neuronal network is fully revealed. Since the modular organization in a network of individual neurons interwoven through synapses is not yet fully explored owing to incredibly complex connectivity architecture, this study is designed to investigate hierarchically organized modules in this complete wiring diagram (connectome) of this worm. We used the modularity-based community detection algorithm and found that C. elegans had 5 anatomical clusters in the C. elegans connectome, which corresponded to experimentally-identified functional circuits. We found that the hierarchical organization of the 5 clusters explains the systemic cooperation including mechanosensation, chemosensation, and navigation that occurs among the structurally-segregated biological circuits to produce higher-order complex behaviors.

Relationships between gene expression and brain wiring in the adult rodent brain

We studied the global relationship between gene expression and neuroanatomical connectivity in the adult rodent brain. We utilized a large data set of the rat brain “connectome” from the Brain Architecture Management System (942 brain regions and over 5000 connections) and used statistical approaches to relate the data to the gene expression signatures of 17,530 genes in 142 anatomical regions from the Allen Brain Atlas. Our analysis shows that adult gene expression signatures have a statistically significant relationship to connectivity. In particular, brain regions that have similar expression profiles tend to have similar connectivity profiles, and this effect is not entirely attributable to spatial correlations. In addition, brain regions which are connected have more similar expression patterns. Using a simple optimization approach, we identified a set of genes most correlated with neuroanatomical connectivity, and find that this set is enriched for genes involved in neuronal development and axon guidance. A number of the genes have been implicated in neurodevelopmental disorders such as autistic spectrum disorder. Our results have the potential to shed light on the role of gene expression patterns in influencing neuronal activity and connectivity, with potential applications to our understanding of brain disorders. Supplementary data are available at http://www.chibi.ubc.ca/ABAMS.

We tested the idea that the “wiring diagram” of the adult brain has a relationship with where genes are expressed. We were inspired by similar work carried out by groups examining the nematode worm Caenorhabditis elegans. By using large-scale databases of brain connectivity and gene expression in rodents, we found that many genes involved in the development of the brain show correlations with anatomical connectivity patterns. Some of the genes we found have been implicated in disorders such as autism, which is suspected to affect brain wiring. While the biological causes of the patterns we found are not yet known, we believe they provide new insight into the patterns of gene expression in the brain and will spur further study of this problem.

Mesoscopic organization reveals the constraints governing Caenorhabditis elegans nervous system

One of the biggest challenges in biology is to understand how activity at the cellular level of neurons, as a result of their mutual interactions, leads to the observed behavior of an organism responding to a variety of environmental stimuli. Investigating the intermediate or mesoscopic level of organization in the nervous system is a vital step towards understanding how the integration of micro-level dynamics results in macro-level functioning. The coordination of many different co-occurring processes at this level underlies the command and control of overall network activity. In this paper, we have considered the somatic nervous system of the nematode Caenorhabditis elegans, for which the entire neuronal connectivity diagram is known. We focus on the organization of the system into modules, i.e., neuronal groups having relatively higher connection density compared to that of the overall network. We show that this mesoscopic feature cannot be explained exclusively in terms of considerations such as, optimizing for resource constraints (viz., total wiring cost) and communication efficiency (i.e., network path length). Even including information about the genetic relatedness of the cells cannot account for the observed modular structure. Comparison with other complex networks designed for efficient transport (of signals or resources) implies that neuronal networks form a distinct class. This suggests that the principal function of the network, viz., processing of sensory information resulting in appropriate motor response, may be playing a vital role in determining the connection topology. Using modular spectral analysis we make explicit the intimate relation between function and structure in the nervous system. This is further brought out by identifying functionally critical neurons purely on the basis of patterns of intra- and inter-modular connections. Our study reveals how the design of the nervous system reflects several constraints, including its key functional role as a processor of information.

Random distance dependent attachment as a model for neural network generation in the Caenorhabditis elegans

MOTIVATION

The topology of the network induced by the neurons connectivity's in the Caenorhabditis elegans differs from most common random networks. The neurons positions of the C.elegans have been previously explained as being optimal to induce the required network wiring. We here propose a complementary explanation that the network wiring is the direct result of a local stochastic synapse formation process.

RESULTS

We show that a model based on the physical distance between neurons can explain the C.elegans neural network structure, specifically, we demonstrate that a simple model based on a geometrical synapse formation probability and the inhibition of short coherent cycles can explain the properties of the C.elegans' neural network. We suggest this model as an initial framework to discuss neural network generation and as a first step toward the development of models for more advanced creatures. In order to measure the circle frequency in the network, a novel graph-theory circle length measurement algorithm is proposed.

Structure of deviations from optimality in biological systems

Optimization theory has been used to analyze evolutionary adaptation. This theory has explained many features of biological systems, from the genetic code to animal behavior. However, these systems show important deviations from optimality. Typically, these deviations are large in some particular components of the system, whereas others seem to be almost optimal. Deviations from optimality may be due to many factors in evolution, including stochastic effects and finite time, that may not allow the system to reach the ideal optimum. However, we still expect the system to have a higher probability of reaching a state with a higher value of the proposed indirect measure of fitness. In systems of many components, this implies that the largest deviations are expected in those components with less impact on the indirect measure of fitness. Here, we show that this simple probabilistic rule explains deviations from optimality in two very different biological systems. In Caenorhabditis elegans, this rule successfully explains the experimental deviations of the position of neurons from the configuration of minimal wiring cost. In Escherichia coli, the probabilistic rule correctly obtains the structure of the experimental deviations of metabolic fluxes from the configuration that maximizes biomass production. This approach is proposed to explain or predict more data than optimization theory while using no extra parameters. Thus, it can also be used to find and refine hypotheses about which constraints have shaped biological structures in evolution.

The interscutularis muscle connectome

The complete connectional map (connectome) of a neural circuit is essential for understanding its structure and function. Such maps have only been obtained in Caenorhabditis elegans. As an attempt at solving mammalian circuits, we reconstructed the connectomes of six interscutularis muscles from adult transgenic mice expressing fluorescent proteins in all motor axons. The reconstruction revealed several organizational principles of the neuromuscular circuit. First, the connectomes demonstrate the anatomical basis of the graded tensions in the size principle. Second, they reveal a robust quantitative relationship between axonal caliber, length, and synapse number. Third, they permit a direct comparison of the same neuron on the left and right sides of the same vertebrate animal, and reveal significant structural variations among such neurons, which contrast with the stereotypy of identified neurons in invertebrates. Finally, the wiring length of axons is often longer than necessary, contrary to the widely held view that neural wiring length should be minimized. These results show that mammalian muscle function is implemented with a variety of wiring diagrams that share certain global features but differ substantially in anatomical form. This variability may arise from the dominant role of synaptic competition in establishing the final circuit.

Conventionally, the organization of a neural circuit is studied by sparsely labeling its constituent neurons and pooling data from multiple samples. If significant variation exists among circuits, this approach may not answer how each neuron integrates into the circuit's functional organization. An alternative is to solve the complete wiring diagram (connectome) of each instantiation of the circuit, which would enable the identification and characterization of each neuron and its relationship with all others. We obtained six connectomes from the same muscle in adult transgenic mice expressing fluorescent protein in motor axons. Certain quantitative features were found to be common to each connectome, but the branching structure of each axon was unique, including the left and right copies of the same neuron in the same animal. We also found that axonal arbor length is often not minimized, contrary to expectation. Thus mammalian muscle function is implemented with a variety of wiring diagrams that share certain global features but differ substantially in anatomical form, even within a common genetic background.

Reconstruction of the complete set of motor axons innervating a muscle reveals wiring variability, even among corresponding neurons on the left and right sides of the same animal.