A multi-component model of the developing retinocollicular pathway incorporating axonal and synaptic growth

Neural Field Continuum Limits and the Structure–Function Partitioning of Cognitive–Emotional Brain Networks

In The cognitive-emotional brain, Pessoa overlooks continuum effects on nonlinear brain network connectivity by eschewing neural field theories and physiologically derived constructs representative of neuronal plasticity. The absence of this content, which is so very important for understanding the dynamic structure-function embedding and partitioning of brains, diminishes the rich competitive and cooperative nature of neural networks and trivializes Pessoa’s arguments, and similar arguments by other authors, on the phylogenetic and operational significance of an optimally integrated brain filled with variable-strength neural connections. Riemannian neuromanifolds, containing limit-imposing metaplastic Hebbian- and antiHebbian-type control variables, simulate scalable network behavior that is difficult to capture from the simpler graph-theoretic analysis preferred by Pessoa and other neuroscientists. Field theories suggest the partitioning and performance benefits of embedded cognitive-emotional networks that optimally evolve between exotic classical and quantum computational phases, where matrix singularities and condensations produce degenerate structure-function homogeneities unrealistic of healthy brains. Some network partitioning, as opposed to unconstrained embeddedness, is thus required for effective execution of cognitive-emotional network functions and, in our new era of neuroscience, should be considered a critical aspect of proper brain organization and operation.

On the Role of LGN/V1 Spontaneous Activity as an Innate Learning Pattern for Visual Development

Correlated, spontaneous neural activity is known to play a necessary role in visual development, but the higher-order statistical structure of these coherent, amorphous patterns has only begun to emerge in the past decade. Several computational studies have demonstrated how this endogenous activity can be used to train a developing visual system. Models that generate spontaneous activity analogous to retinal waves have shown that these waves can serve as stimuli for efficient coding models of V1. This general strategy in development has one clear advantage: The same learning algorithm can be used both before and after eye-opening. This same insight can be applied to understanding LGN/V1 spontaneous activity. Although lateral geniculate nucleus (LGN) activity has been less discussed in the literature than retinal waves, here we argue that the waves found in the LGN have a number of properties that fill the role of a training pattern. We make the case that the role of “innate learning” with spontaneous activity is not only possible, but likely in later stages of visual development, and worth pursuing further using an efficient coding paradigm.

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.

Developmental time windows for axon growth influence neuronal network topology

Early brain connectivity development consists of multiple stages: birth of neurons, their migration and the subsequent growth of axons and dendrites. Each stage occurs within a certain period of time depending on types of neurons and cortical layers. Forming synapses between neurons either by growing axons starting at similar times for all neurons (much-overlapped time windows) or at different time points (less-overlapped) may affect the topological and spatial properties of neuronal networks. Here, we explore the extreme cases of axon formation during early development, either starting at the same time for all neurons (parallel, i.e., maximally overlapped time windows) or occurring for each neuron separately one neuron after another (serial, i.e., no overlaps in time windows). For both cases, the number of potential and established synapses remained comparable. Topological and spatial properties, however, differed: Neurons that started axon growth early on in serial growth achieved higher out-degrees, higher local efficiency and longer axon lengths while neurons demonstrated more homogeneous connectivity patterns for parallel growth. Second, connection probability decreased more rapidly with distance between neurons for parallel growth than for serial growth. Third, bidirectional connections were more numerous for parallel growth. Finally, we tested our predictions with C. elegans data. Together, this indicates that time windows for axon growth influence the topological and spatial properties of neuronal networks opening up the possibility to a posteriori estimate developmental mechanisms based on network properties of a developed network.

A reaction-diffusion model of cholinergic retinal waves

Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability.

Visual circuit development requires patterned activity mediated by retinal acetylcholine receptors

The elaboration of nascent synaptic connections into highly ordered neural circuits is an integral feature of the developing vertebrate nervous system. In sensory systems, patterned spontaneous activity before the onset of sensation is thought to influence this process, but this conclusion remains controversial, largely due to the inherent difficulty recording neural activity in early development. Here, we describe genetic and pharmacological manipulations of spontaneous retinal activity, assayed in vivo, that demonstrate a causal link between retinal waves and visual circuit refinement. We also report a decoupling of downstream activity in retinorecipient regions of the developing brain after retinal wave disruption. Significantly, we show that the spatiotemporal characteristics of retinal waves affect the development of specific visual circuits. These results conclusively establish retinal waves as necessary and instructive for circuit refinement in the developing nervous system and reveal how neural circuits adjust to altered patterns of activity prior to experience.

Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting

Morphogenesis and axonal targeting are key processes during development that depend on complex interactions at molecular, cellular and tissue level. Mathematical modeling is essential to bridge this multi-scale gap in order to understand how the emergence of large structures is controlled at molecular level by interactions between various signaling pathways. We summarize mathematical modeling and computational methods for time evolution and precision of morphogenetic gradient formation. We discuss tissue patterning and the formation of borders between regions labeled by different morphogens. Finally, we review models and algorithms that reveal the interplay between morphogenetic gradients and patterned activity for axonal pathfinding and the generation of the retinotopic map in the visual system.

Modeling development in retinal afferents: retinotopy, segregation, and ephrinA/EphA mutants

During neural development, neurons extend axons to target areas of the brain. Through processes of growth, branching and retraction these axons establish stereotypic patterns of connectivity. In the visual system, these patterns include retinotopic organization and the segregation of individual axons onto different subsets of target neurons based on the eye of origin (ocular dominance) or receptive field type (ON or OFF). Characteristic disruptions to these patterns occur when neural activity or guidance molecule expression is perturbed. In this paper we present a model that explains how these developmental patterns might emerge as a result of the coordinated growth and retraction of individual axons and synapses responding to position-specific markers, trophic factors and spontaneous neural activity. This model derives from one presented earlier (Godfrey et al., 2009) but which is here extended to account for a wider range of phenomena than previously described. These include ocular dominance and ON-OFF segregation and the results of altered ephrinA and EphA guidance molecule expression. The model takes into account molecular guidance factors, realistic patterns of spontaneous retinal wave activity, trophic molecules, homeostatic mechanisms, axon branching and retraction rules and intra-axonal signaling mechanisms that contribute to the survival of nearby synapses on an axon. We show that, collectively, these mechanisms can account for a wider range of phenomena than previous models of retino-tectal development.

A developmental approach to predicting neuronal connectivity from small biological datasets: a gradient-based neuron growth model

Relating structure and function of neuronal circuits is a challenging problem. It requires demonstrating how dynamical patterns of spiking activity lead to functions like cognitive behaviour and identifying the neurons and connections that lead to appropriate activity of a circuit. We apply a “developmental approach” to define the connectome of a simple nervous system, where connections between neurons are not prescribed but appear as a result of neuron growth. A gradient based mathematical model of two-dimensional axon growth from rows of undifferentiated neurons is derived for the different types of neurons in the brainstem and spinal cord of young tadpoles of the frog Xenopus. Model parameters define a two-dimensional CNS growth environment with three gradient cues and the specific responsiveness of the axons of each neuron type to these cues. The model is described by a nonlinear system of three difference equations; it includes a random variable, and takes specific neuron characteristics into account. Anatomical measurements are first used to position cell bodies in rows and define axon origins. Then a generalization procedure allows information on the axons of individual neurons from small anatomical datasets to be used to generate larger artificial datasets. To specify parameters in the axon growth model we use a stochastic optimization procedure, derive a cost function and find the optimal parameters for each type of neuron. Our biologically realistic model of axon growth starts from axon outgrowth from the cell body and generates multiple axons for each different neuron type with statistical properties matching those of real axons. We illustrate how the axon growth model works for neurons with axons which grow to the same and the opposite side of the CNS. We then show how, by adding a simple specification for dendrite morphology, our model “developmental approach” allows us to generate biologically-realistic connectomes.

A role for correlated spontaneous activity in the assembly of neural circuits

Before the onset of sensory transduction, developing neural circuits spontaneously generate correlated activity in distinct spatial and temporal patterns. During this period of patterned activity, sensory maps develop and initial coarse connections are refined, which are critical steps in the establishment of adult neural circuits. Over the last decade, there has been substantial evidence that altering the pattern of spontaneous activity disrupts refinement, but the mechanistic understanding of this process remains incomplete. In this review, we discuss recent experimental and theoretical progress toward the process of activity-dependent refinement, focusing on circuits in the visual, auditory, and motor systems. Although many outstanding questions remain, the combination of several novel approaches has brought us closer to a comprehensive understanding of how complex neural circuits are established by patterned spontaneous activity during development.

Flowers and weeds: cell-type specific pruning in the developing visual thalamus

In the first weeks of vertebrate postnatal life, neural networks in the visual thalamus undergo activity-dependent refinement thought to be important for the development of functional vision. This process involves pruning of synaptic connections between retinal ganglion cells and excitatory thalamic neurons that relay signals on to visual areas of the cortex. A recent report in Neural Development shows that this does not occur in inhibitory neurons, questioning our current understanding of the development of mature neural circuits.

Retinocollicular mapping explained?

We review and comment on the recent model of Grimbert and Cang of the development of topographically ordered maps from the retina to the superior colliculus. This model posits a phase in which arbors are created in zones permitted by Eph and ephrin signaling, followed by a phase in which activity-dependent synaptic plasticity refines the map. We show that it is not possible to generate the arborization probability functions used in the simulations of Grimbert and Cang using gradients of Ephs and ephrins and the interaction mechanism that Grimbert and Cang propose in their results. Furthermore, the arborization probabilities we do generate are far less sharp than we imagine truly “permissive” ones would be. It remains to be seen if maps can be generated from the nonpermissive arborization probabilities generated from gradients.

Simulating cortical development as a self constructing process: a novel multi-scale approach combining molecular and physical aspects

Current models of embryological development focus on intracellular processes such as gene expression and protein networks, rather than on the complex relationship between subcellular processes and the collective cellular organization these processes support. We have explored this collective behavior in the context of neocortical development, by modeling the expansion of a small number of progenitor cells into a laminated cortex with layer and cell type specific projections. The developmental process is steered by a formal language analogous to genomic instructions, and takes place in a physically realistic three-dimensional environment. A common genome inserted into individual cells control their individual behaviors, and thereby gives rise to collective developmental sequences in a biologically plausible manner. The simulation begins with a single progenitor cell containing the artificial genome. This progenitor then gives rise through a lineage of offspring to distinct populations of neuronal precursors that migrate to form the cortical laminae. The precursors differentiate by extending dendrites and axons, which reproduce the experimentally determined branching patterns of a number of different neuronal cell types observed in the cat visual cortex. This result is the first comprehensive demonstration of the principles of self-construction whereby the cortical architecture develops. In addition, our model makes several testable predictions concerning cell migration and branching mechanisms.

The proper operation of the brain depends on the correct developmental wiring of billions of neurons. Understanding this process of living self-construction is crucial not only for biological explanation and medical therapy, but could also provide an entirely new approach to industrial fabrication. We are approaching this problem through detailed simulation of cortical development. We have previously presented a software package that allows for simulation of cellular growth in a 3D space that respects physical forces and diffusion of substances, as well as an instruction language for specifying biologically plausible ‘genetic codes’. Here we apply this novel formalism to understanding the principles of cortical development in the context of multiple, spatially distributed agents that communicate only by local metabolic messages.

On the Importance of Countergradients for the Development of Retinotopy: Insights from a Generalised Gierer Model

During the development of the topographic map from vertebrate retina to superior colliculus (SC), EphA receptors are expressed in a gradient along the nasotemporal retinal axis. Their ligands, ephrin-As, are expressed in a gradient along the rostrocaudal axis of the SC. Countergradients of ephrin-As in the retina and EphAs in the SC are also expressed. Disruption of any of these gradients leads to mapping errors. Gierer's (1981) model, which uses well-matched pairs of gradients and countergradients to establish the mapping, can account for the formation of wild type maps, but not the double maps found in EphA knock-in experiments. I show that these maps can be explained by models, such as Gierer's (1983), which have gradients and no countergradients, together with a powerful compensatory mechanism that helps to distribute connections evenly over the target region. However, this type of model cannot explain mapping errors found when the countergradients are knocked out partially. I examine the relative importance of countergradients as against compensatory mechanisms by generalising Gierer's (1983) model so that the strength of compensation is adjustable. Either matching gradients and countergradients alone or poorly matching gradients and countergradients together with a strong compensatory mechanism are sufficient to establish an ordered mapping. With a weaker compensatory mechanism, gradients without countergradients lead to a poorer map, but the addition of countergradients improves the mapping. This model produces the double maps in simulated EphA knock-in experiments and a map consistent with the Math5 knock-out phenotype. Simulations of a set of phenotypes from the literature substantiate the finding that countergradients and compensation can be traded off against each other to give similar maps. I conclude that a successful model of retinotopy should contain countergradients and some form of compensation mechanism, but not in the strong form put forward by Gierer.

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.

The interplay between branching and pruning on neuronal target search during developmental growth: functional role and implications

Regenerative strategies that facilitate the regrowth and reconnection of neurons are some of the most promising methods in spinal cord injury research. An essential part of these strategies is an increased understanding of the mechanisms by which growing neurites seek out and synapse with viable targets. In this paper, we use computational and theoretical tools to examine the targeting efficiency of growing neurites subject to limited resources, such as maximum total neural tree length. We find that in order to efficiently reach a particular target, growing neurites must achieve balance between pruning and branching: rapidly growing neurites that do not prune will exhaust their resources, and frequently pruning neurites will fail to explore space effectively. We also find that the optimal branching/pruning balance must shift as the target distance changes: different strategies are called for to reach nearby vs. distant targets. This suggests the existence of a currently unidentified higher-level regulatory factor to control arborization dynamics. We propose that these findings may be useful in future therapies seeking to improve targeting rates through manipulation of arborization behaviors.

Modeling developmental patterns of spontaneous activity

Spontaneous activity is found in many regions of the developing nervous system; such activity is thought to be instructive for guiding developmental processes. In particular, the developing retina generates correlated patterns of activity known as retinal waves. We review the main theoretical models that have been developed to study the mechanisms for generation and propagation of retinal waves. Much of the progress in this field has been due to the close interaction between experimentalists and theorists in analyzing and modeling spontaneous activity. We conclude by describing spontaneous activity models in other systems and suggestions for future modeling work.

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.