Theoretical models of neural circuit development

Nonlinear Growth Dynamics of Neuronal Cells Cultured on Directional Surfaces

During the development of the nervous system, neuronal cells extend axons and dendrites that form complex neuronal networks, which are essential for transmitting and processing information. Understanding the physical processes that underlie the formation of neuronal networks is essential for gaining a deeper insight into higher-order brain functions such as sensory processing, learning, and memory. In the process of creating networks, axons travel towards other recipient neurons, directed by a combination of internal and external cues that include genetic instructions, biochemical signals, as well as external mechanical and geometrical stimuli. Although there have been significant recent advances, the basic principles governing axonal growth, collective dynamics, and the development of neuronal networks remain poorly understood. In this paper, we present a detailed analysis of nonlinear dynamics for axonal growth on surfaces with periodic geometrical patterns. We show that axonal growth on these surfaces is described by nonlinear Langevin equations with speed-dependent deterministic terms and gaussian stochastic noise. This theoretical model yields a comprehensive description of axonal growth at both intermediate and long time scales (tens of hours after cell plating), and predicts key dynamical parameters, such as speed and angular correlation functions, axonal mean squared lengths, and diffusion (cell motility) coefficients. We use this model to perform simulations of axonal trajectories on the growth surfaces, in turn demonstrating very good agreement between simulated growth and the experimental results. These results provide important insights into the current understanding of the dynamical behavior of neurons, the self-wiring of the nervous system, as well as for designing innovative biomimetic neural network models.

Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns

Neuronal networks are complex systems of interconnected neurons responsible for transmitting and processing information throughout the nervous system. The building blocks of neuronal networks consist of individual neurons, specialized cells that receive, process, and transmit electrical and chemical signals throughout the body. The formation of neuronal networks in the developing nervous system is a process of fundamental importance for understanding brain activity, including perception, memory, and cognition. To form networks, neuronal cells extend long processes called axons, which navigate toward other target neurons guided by both intrinsic and extrinsic factors, including genetic programming, chemical signaling, intercellular interactions, and mechanical and geometrical cues. Despite important recent advances, the basic mechanisms underlying collective neuron behavior and the formation of functional neuronal networks are not entirely understood. In this paper, we present a combined experimental and theoretical analysis of neuronal growth on surfaces with micropatterned periodic geometrical features. We demonstrate that the extension of axons on these surfaces is described by a biased random walk model, in which the surface geometry imparts a constant drift term to the axon, and the stochastic cues produce a random walk around the average growth direction. We show that the model predicts key parameters that describe axonal dynamics: diffusion (cell motility) coefficient, average growth velocity, and axonal mean squared length, and we compare these parameters with the results of experimental measurements. Our findings indicate that neuronal growth is governed by a contact-guidance mechanism, in which the axons respond to external geometrical cues by aligning their motion along the surface micropatterns. These results have a significant impact on developing novel neural network models, as well as biomimetic substrates, to stimulate nerve regeneration and repair after injury.

A Guide for the Multiplexed: The Development of Visual Feature Maps in the Brain

Neural maps are found ubiquitously in the brain, where they encode a wide range of behaviourally relevant features into neural space. Developmental studies have shown that animals devote a great deal of resources to establish consistently patterned organization in neural circuits throughout the nervous system, but what purposes maps serve beneath their often intricate appearance and composition is a topic of active debate and exploration. In this article, we review the general mechanisms of map formation, with a focus on the visual system, and then survey notable organizational properties of neural maps: the multiplexing of feature representations through a nested architecture, the interspersing of fine-scale heterogeneity within a globally smooth organization, and the complex integration at the microcircuit level that enables a high dimensionality of information encoding. Finally, we discuss the roles of maps in cortical functions, including input segregation, feature extraction and routing of circuit outputs for higher order processing, as well as the evolutionary basis for the properties we observe in neural maps.

Activity-dependent Organization of Topographic Neural Circuits

Sensory information in the brain is organized into spatial representations, including retinotopic, somatotopic, and tonotopic maps, as well as ocular dominance columns. The spatial representation of sensory inputs is thought to be a fundamental organizational principle that is important for information processing. Topographic maps are plastic throughout an animal's life, reflecting changes in development and aging of brain circuitry, changes in the periphery and sensory input, and changes in circuitry, for instance in response to experience and learning. Here, we review mechanisms underlying the role of activity in the development, stability and plasticity of topographic maps, focusing on recent work suggesting that the spatial information in the visual field, and the resulting spatiotemporal patterns of activity, provide instructive cues that organize visual projections.

Feedback-controlled dynamics of neuronal cells on directional surfaces

The formation of neuronal networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body and axonal navigation toward target neurons. Axonal growth is guided by the interactions between the tip of the axon (growth cone) and its extracellular environmental cues, which include intercellular interactions, the biochemical landscape around the neuron, and the mechanical and geometrical features of the growth substrate. Here, we present a comprehensive experimental and theoretical analysis of axonal growth for neurons cultured on micropatterned polydimethylsiloxane (PDMS) surfaces. We demonstrate that closed-loop feedback is an essential component of axonal dynamics on these surfaces: the growth cone continuously measures environmental cues and adjusts its motion in response to external geometrical features. We show that this model captures all the characteristics of axonal dynamics on PDMS surfaces for both untreated and chemically modified neurons. We combine experimental data with theoretical analysis to measure key parameters that describe axonal dynamics: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. The experiments performed on neurons treated with Taxol (inhibitor of microtubule dynamics) and Y-27632 (disruptor of actin filaments) indicate that the internal dynamics of microtubules and actin filaments plays a critical role for the proper function of the feedback mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrate to guide neuronal growth and promote nerve repair.

Crossed Entorhino-Dentate Projections Form and Terminate With Correct Layer-Specificity in Organotypic Slice Cultures of the Mouse Hippocampus

The entorhino-dentate projection, i.e., the perforant pathway, terminates in a highly ordered and laminated fashion in the rodent dentate gyrus (DG): fibers arising from the medial entorhinal cortex (MEC) terminate in the middle molecular layer, whereas fibers arising from the lateral entorhinal cortex (LEC) terminate in the outer molecular layer of the DG. In rats and rabbits, a crossed entorhino-dentate projection exists, which originates from the entorhinal cortex (EC) and terminates in the contralateral DG. In contrast, in mice, such a crossed projection is reportedly absent. Using single and double mouse organotypic entorhino-hippocampal slice cultures, we studied the ipsi- and crossed entorhino-dentate projections. Viral tracing revealed that entorhino-dentate projections terminate with a high degree of lamina-specificity in single as well as in double cultures. Furthermore, in double cultures, entorhinal axons arising from one slice freely intermingled with entorhinal axons originating from the other slice. In single as well as in double cultures, entorhinal axons exhibited a correct topographical projection to the DG: medial entorhinal axons terminated in the middle and lateral entorhinal axons terminated in the outer molecular layer. Finally, entorhinal neurons were virally transduced with Channelrhodopsin2-YFP and stimulated with light, revealing functional connections between the EC and dentate granule cells. We conclude from our findings that entorhino-dentate projections form bilaterally in the mouse hippocampus in vitro and that the mouse DG provides a permissive environment for crossed entorhinal fibers.

A stochastic framework to model axon interactions within growing neuronal populations

The confined and crowded environment of developing brains imposes spatial constraints on neuronal cells that have evolved individual and collective strategies to optimize their growth. These include organizing neurons into populations extending their axons to common target territories. How individual axons interact with each other within such populations to optimize innervation is currently unclear and difficult to analyze experimentally in vivo. Here, we developed a stochastic model of 3D axon growth that takes into account spatial environmental constraints, physical interactions between neighboring axons, and branch formation. This general, predictive and robust model, when fed with parameters estimated on real neurons from the Drosophila brain, enabled the study of the mechanistic principles underlying the growth of axonal populations. First, it provided a novel explanation for the diversity of growth and branching patterns observed in vivo within populations of genetically identical neurons. Second, it uncovered that axon branching could be a strategy optimizing the overall growth of axons competing with others in contexts of high axonal density. The flexibility of this framework will make it possible to investigate the rules underlying axon growth and regeneration in the context of various neuronal populations.

Understanding how neuronal cells establish complex circuits with specific functions within a developing brain is a major current challenge. Over the last past years, enormous progress has been done to precisely resolve brain anatomy and to dissect the mechanisms controlling the establishment of precise neuronal networks. However, due to the extreme complexity of the brain, it is still experimentally difficult to investigate in vivo how neurons interact with each other and with their physical environments to innervate target territories during development. Here, we have developed a framework that integrates a dynamic 3D mathematical model of single axonal growth with parameters estimated from neurons grown in vivo and simulations of entire populations of growing axons. The emergent properties of our model enable the study of the mechanistic principles underlying the growth of axonal population in developing brains. Specifically, our results highlight the impact of mechanical interactions on both individual and collective axon growth, and uncover how branching regulate this process.

Theoretical Models of Neural Development

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

Ephrin-A/EphA specific co-adaptation as a novel mechanism in topographic axon guidance

Genetic hardwiring during brain development provides computational architectures for innate neuronal processing. Thus, the paradigmatic chick retinotectal projection, due to its neighborhood preserving, topographic organization, establishes millions of parallel channels for incremental visual field analysis. Retinal axons receive targeting information from quantitative guidance cue gradients. Surprisingly, novel adaptation assays demonstrate that retinal growth cones robustly adapt towards ephrin-A/EphA forward and reverse signals, which provide the major mapping cues. Computational modeling suggests that topographic accuracy and adaptability, though seemingly incompatible, could be reconciled by a novel mechanism of coupled adaptation of signaling channels. Experimentally, we find such 'co-adaptation' in retinal growth cones specifically for ephrin-A/EphA signaling. Co-adaptation involves trafficking of unliganded sensors between the surface membrane and recycling endosomes, and is presumably triggered by changes in the lipid composition of membrane microdomains. We propose that co-adaptative desensitization eventually relies on guidance sensor translocation into cis-signaling endosomes to outbalance repulsive trans-signaling.

What, if anything, are topological maps for?

What, if anything, is the functional significance of spatial patterning in cortical feature maps? We ask this question of four major theories of cortical map formation: self-organizing maps, wiring optimization, place coding, and reaction-diffusion. We argue that (i) self-organizing maps yield spatial patterning only as a by-product of efficient mechanisms for developing environmentally appropriate distributions of feature preferences, (ii) wiring optimization assumes rather than explains a map-like organization, (iii) place-coding mechanisms can at best explain only a subset of maps in functional terms, and (iv) reaction-diffusion models suggest two factors in the evolution of maps, the first based on efficient development of feature distributions, and the second based on generating feature-specific long-range recurrent cortical circuitry. None of these explanations for the existence of topological maps requires spatial patterning in maps to be useful. Thus despite these useful frameworks for understanding how maps form and how they are wired, the possibility that patterns are merely epiphenomena in the evolution of mammalian neocortex cannot be rejected. The article is intended as a nontechnical introduction to the assumptions and predictions of these four important classes of models, along with other possible functional explanations for maps.

Neuronal growth as diffusion in an effective potential

Current understanding of neuronal growth is mostly qualitative, as the staggering number of physical and chemical guidance cues involved prohibit a fully quantitative description of axonal dynamics. We report on a general approach that describes axonal growth in vitro, on poly-D-lysine-coated glass substrates, as diffusion in an effective external potential, representing the collective contribution of all causal influences on the growth cone. We use this approach to obtain effective growth rules that reveal an emergent regulatory mechanism for axonal pathfinding on these substrates.

New model of retinocollicular mapping predicts the mechanisms of axonal competition and explains the role of reverse molecular signaling during development

Precise connections in the brain result from elaborate processes during development. In the visual system, axonal projections from retinal ganglion cells (RGCs) onto the superior colliculus form a precise retinotopic map. Studies have revealed that the development of retinocollicular maps involves three main factors: graded expression of molecular guidance cues such as EphAs and ephrin-As, activity-dependent processes driven by spontaneous activity in RGCs, and different forms of axonal competition. In this study, we developed a new, versatile model including these factors. We first modeled the selective arborization of RGC axons, mediated by EphA/ephrin-A signaling, without assuming that this initial process instructed the map's final topology. We also derived an integro-differential equation modeling a second, dynamic phase in which activity-dependent plasticity of axonal arbors combined with their competition for collicular resources can deeply remodel the topology of immature maps. Our model hence challenges the view that retinotopic maps are instructed by matching molecular gradients and then merely refined by activity-dependent processes. We reproduce fine features of retinotopic map development in wild-type and various transgenic mice, allowing a new understanding of the underlying mechanisms. Our model predicts that competition is not based on comparisons of axonal EphA receptor levels but rather relies on the optimization of collicular resources mediated by neurotrophic receptors such as p75(NTR). Our model finally clarifies the elusive role of reverse signaling between retinal ephrin-As and collicular EphAs by reproducing for the first time the phenotypes of two mouse genotypes in which this function is altered.

Tumor suppressor menin mediates peripheral nerve injury-induced neuropathic pain through potentiating synaptic plasticity

Synaptic plasticity is a crucial step in the development of central sensitization in the pathogenesis of neuropathic hyperalgesia. Menin, the product of the multiple endocrine neoplasia type 1 (MEN1) gene, possesses the property of synaptogenesis which plays an essential role in neuronal activity. We tested the contributing role of spinal menin in peripheral nerve injury-induced neuropathic hypersensitivity through modulating neuronal synaptic plasticity. After approval by the Institutional Animal Care and Use Committee, nociceptive responses were detected with von Frey filaments and thermal plate after spared nerve injury in C57BL/6 mice who were treated with either intrathecal antisense oligonucleotide of MEN1 (ASO) or vehicle. Extracellular spontaneous discharge frequency, field excitatory postsynaptic potential (fEPSP), and monosynaptic excitatory postsynaptic currents (EPSCs) were measured electrophysiologically. Intrathecal ASO alleviated nerve injury-induced mechanical and thermal hypersensitivity. Upregulated spinal menin after nerve injury colocalized with NeuN in the superficial laminae; genetic knockdown of spinal menin reduced nerve injury induced in vivo spontaneous activity and instantaneous frequency and in vitro field potentials; ASO decreased the frequency and amplitude of monosynaptic EPSCs, and reduced synaptic strength and total charge. Collectively, these findings highlight the role of upregulated neuronal menin in the spinal cord in potentiating spinal synaptic plasticity in peripheral nerve injury-induced neuropathic hypersensitivity.

Calcium and cAMP levels interact to determine attraction versus repulsion in axon guidance

Correct guidance of axons to their targets depends on an intricate network of signaling molecules in the growth cone. Calcium and cAMP are two key regulators of whether axons are attracted or repelled by molecular gradients, but how these molecules interact to determine guidance responses remains unclear. Here, we constructed a mathematical model for the relevant signaling network, which explained a large range of previous biological data and made predictions for when axons will be attracted or repelled. We then confirmed these predictions experimentally, in particular showing that while small increases in cAMP levels promote attraction large increases do not, and that under some circumstances reducing cAMP levels promotes attraction. Together, these results show that a relatively simple mathematical model can quantitatively predict guidance decisions across a wide range of conditions, and that calcium and cAMP levels play a more complex role in these decisions than previously determined.

EphA3 expressed in the chicken tectum stimulates nasal retinal ganglion cell axon growth and is required for retinotectal topographic map formation

Background

Retinotopic projection onto the tectum/colliculus constitutes the most studied model of topographic mapping and Eph receptors and their ligands, the ephrins, are the best characterized molecular system involved in this process. Ephrin-As, expressed in an increasing rostro-caudal gradient in the tectum/colliculus, repel temporal retinal ganglion cell (RGC) axons from the caudal tectum and inhibit their branching posterior to their termination zones. However, there are conflicting data regarding the nature of the second force that guides nasal axons to invade and branch only in the caudal tectum/colliculus. The predominant model postulates that this second force is produced by a decreasing rostro-caudal gradient of EphA7 which repels nasal optic fibers and prevents their branching in the rostral tectum/colliculus. However, as optic fibers invade the tectum/colliculus growing throughout this gradient, this model cannot explain how the axons grow throughout this repellent molecule.

Methodology/Principal Findings

By using chicken retinal cultures we showed that EphA3 ectodomain stimulates nasal RGC axon growth in a concentration dependent way. Moreover, we showed that nasal axons choose growing on EphA3-expressing cells and that EphA3 diminishes the density of interstitial filopodia in nasal RGC axons. Accordingly, in vivo EphA3 ectodomain misexpression directs nasal optic fibers toward the caudal tectum preventing their branching in the rostral tectum.

Conclusions

We demonstrated in vitro and in vivo that EphA3 ectodomain (which is expressed in a decreasing rostro-caudal gradient in the tectum) is necessary for topographic mapping by stimulating the nasal axon growth toward the caudal tectum and inhibiting their branching in the rostral tectum. Furthermore, the ability of EphA3 of stimulating axon growth allows understanding how optic fibers invade the tectum growing throughout this molecular gradient. Therefore, opposing tectal gradients of repellent ephrin-As and of axon growth stimulating EphA3 complement each other to map optic fibers along the rostro-caudal tectal axis.

An uncertainty principle underlying the functional architecture of V1

We present a model of the morphology of orientation maps in V1 based on the uncertainty principle of the SE(2) group. Starting from the symmetries of the cortex, suitable harmonic analysis instruments are used to obtain coherent states in the Fourier domain as minimizers of the uncertainty. Cortical activities related to orientation maps are then obtained by projection on a suitable cortical Fourier basis.

Coverage, continuity, and visual cortical architecture

BACKGROUND

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far.

RESULTS

We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all the previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise.

CONCLUSIONS

Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.

Computational modeling of neuronal map development: insights into disease

The study of the formation of neuronal maps in the brain has greatly increased our understanding of how the brain develops and, in some cases, regenerates. Computational modeling of neuronal map development has been invaluable in integrating complex biological phenomena and synthesizing them into quantitative and predictive frameworks. These models allow us to investigate how neuronal map development is perturbed under conditions of altered development, disease and regeneration. In this article, we use examples of activity-dependent and activity-independent models of retinotopic map formation to illustrate how they can aid our understanding of developmental and acquired disease processes. We note that fully extending these models to specific clinically relevant problems is a largely unexplored domain and suggest future work in this direction. We argue that this type of modeling will be necessary in furthering our understanding of the pathophysiology of neurological diseases and in developing treatments for them. Furthermore, we discuss how the nature of computational and theoretical approaches uniquely places them to bridge the gap between the bench and the clinic.

A simple model can unify a broad range of phenomena in retinotectal map development

A paradigm model system for studying the development of patterned connections in the nervous system is the topographic map formed by retinal axons in the optic tectum/superior colliculus. Starting in the 1970s, a series of computational models have been proposed to explain map development in both normal conditions, and perturbed conditions where the retina and/or tectum/superior colliculus are altered. This stands in contrast to more recent models that have often been simpler than older ones, and tend to address more limited data sets, but include more recent genetic manipulations. The original exploration of many of the early models was one-dimensional and limited by the computational resources of the time. This leaves open the ability of these early models to explain both map development in two dimensions, and the genetic manipulation data that have only appeared more recently. In this article, we show that a two-dimensional and updated version of the XBAM model (eXtended Branch Arrow Model), first proposed in 1982, reproduces a range of surgical map manipulations not yet demonstrated by more modern models. A systematic exploration of the parameter space of this model in two dimensions also reveals richer behavior than that apparent from the original one-dimensional versions. Furthermore, we show that including a specific type of axon-axon interaction can account for the map collapse recently observed when particular receptor levels are genetically manipulated in a subset of retinal ganglion cells. Together these results demonstrate that balancing multiple influences on map development seems to be necessary to explain many biological phenomena in retinotectal map formation, and suggest important constraints on the underlying biological variables.