Using theoretical models to analyse neural development

Approaches to embryonic neurodevelopment: from neural cell to neural tube formation through mathematical models

The development of the human central nervous system initiates in the early embryonic period until long after delivery. It has been shown that several neurological and neuropsychiatric diseases originate from prenatal incidents. Mathematical models offer a direct way to understand neurodevelopmental processes better. Mathematical modelling of neurodevelopment during the embryonic period is challenging in terms of how to 'Approach', how to initiate modelling and how to propose the appropriate equations that fit the underlying dynamics of neurodevelopment during the embryonic period while including the variety of elements that are built-in naturally during the process of neurodevelopment. It is imperative to answer where and how to start modelling; in other words, what is the appropriate 'Approach'? Therefore, one objective of this study was to tackle the mathematical issue broadly from different aspects and approaches. The approaches were divided into three embryonic categories: cell division, neural tube growth and neural plate growth. We concluded that the neural plate growth approach provides a suitable platform for simulation of brain formation/neurodevelopment compared to cell division and neural tube growth. We devised a novel equation and designed algorithms that include geometrical and topological algorithms that could fit most of the necessary elements of the neurodevelopmental process during the embryonic period. Hence, the proposed equations and defined mathematical structure would be a platform to generate an artificial neural network that autonomously grows and develops.

Neurostimulation targeting the epileptic focus: Current understanding and perspectives for treatment

For the one third of people with epilepsy whose seizures are not controlled with medications, targeting the seizure focus with neurostimulation can be an effective therapeutic strategy. In this focused review, we summarize a discussion of targeted neurostimulation modalities during a workshop held in Frankfurt, Germany in September 2023. Topics covered include: available devices for seizure focus stimulation; alternating current (AC) and direct current (DC) stimulation to reduce focal cortical excitability; modeling approaches to simulate DC stimulation; reconciling the efficacy of focal stimulation with the network theory of epilepsy; and the emerging concept of 'neurostimulation zones,' which are defined as cortical regions where focal stimulation is most effective for reducing seizures and which may or may not directly involve the seizure onset zone. By combining experimental data, modeling results, and clinical outcome analysis, rational selection of target regions and stimulation parameters is increasingly feasible, paving the way for a broader use of neurostimulation for epilepsy in the future.

Distinctive properties of biological neural networks and recent advances in bottom-up approaches toward a better biologically plausible neural network

Although it may appear infeasible and impractical, building artificial intelligence (AI) using a bottom-up approach based on the understanding of neuroscience is straightforward. The lack of a generalized governing principle for biological neural networks (BNNs) forces us to address this problem by converting piecemeal information on the diverse features of neurons, synapses, and neural circuits into AI. In this review, we described recent attempts to build a biologically plausible neural network by following neuroscientifically similar strategies of neural network optimization or by implanting the outcome of the optimization, such as the properties of single computational units and the characteristics of the network architecture. In addition, we proposed a formalism of the relationship between the set of objectives that neural networks attempt to achieve, and neural network classes categorized by how closely their architectural features resemble those of BNN. This formalism is expected to define the potential roles of top-down and bottom-up approaches for building a biologically plausible neural network and offer a map helping the navigation of the gap between neuroscience and AI engineering.

Neuronal filopodia: From stochastic dynamics to robustness of brain morphogenesis

Brain development relies on dynamic morphogenesis and interactions of neurons. Filopodia are thin and highly dynamic membrane protrusions that are critically required for neuronal development and neuronal interactions with the environment. Filopodial interactions are typically characterized by non-deterministic dynamics, yet their involvement in developmental processes leads to stereotypic and robust outcomes. Here, we discuss recent advances in our understanding of how filopodial dynamics contribute to neuronal differentiation, migration, axonal and dendritic growth and synapse formation. Many of these advances are brought about by improved methods of live observation in intact developing brains. Recent findings integrate known and novel roles ranging from exploratory sensors and decision-making agents to pools for selection and mechanical functions. Different types of filopodial dynamics thereby reveal non-deterministic subcellular decision-making processes as part of genetically encoded brain development.

Reinforcement Learning for Central Pattern Generation in Dynamical Recurrent Neural Networks

Lifetime learning, or the change (or acquisition) of behaviors during a lifetime, based on experience, is a hallmark of living organisms. Multiple mechanisms may be involved, but biological neural circuits have repeatedly demonstrated a vital role in the learning process. These neural circuits are recurrent, dynamic, and non-linear and models of neural circuits employed in neuroscience and neuroethology tend to involve, accordingly, continuous-time, non-linear, and recurrently interconnected components. Currently, the main approach for finding configurations of dynamical recurrent neural networks that demonstrate behaviors of interest is using stochastic search techniques, such as evolutionary algorithms. In an evolutionary algorithm, these dynamic recurrent neural networks are evolved to perform the behavior over multiple generations, through selection, inheritance, and mutation, across a population of solutions. Although, these systems can be evolved to exhibit lifetime learning behavior, there are no explicit rules built into these dynamic recurrent neural networks that facilitate learning during their lifetime (e.g., reward signals). In this work, we examine a biologically plausible lifetime learning mechanism for dynamical recurrent neural networks. We focus on a recently proposed reinforcement learning mechanism inspired by neuromodulatory reward signals and ongoing fluctuations in synaptic strengths. Specifically, we extend one of the best-studied and most-commonly used dynamic recurrent neural networks to incorporate the reinforcement learning mechanism. First, we demonstrate that this extended dynamical system (model and learning mechanism) can autonomously learn to perform a central pattern generation task. Second, we compare the robustness and efficiency of the reinforcement learning rules in relation to two baseline models, a random walk and a hill-climbing walk through parameter space. Third, we systematically study the effect of the different meta-parameters of the learning mechanism on the behavioral learning performance. Finally, we report on preliminary results exploring the generality and scalability of this learning mechanism for dynamical neural networks as well as directions for future work.

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites—axons and dendrites—to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

Time Course of Homeostatic Structural Plasticity in Response to Optogenetic Stimulation in Mouse Anterior Cingulate Cortex

Plasticity is the mechanistic basis of development, aging, learning, and memory, both in healthy and pathological brains. Structural plasticity is rarely accounted for in computational network models due to a lack of insight into the underlying neuronal mechanisms and processes. Little is known about how the rewiring of networks is dynamically regulated. To inform such models, we characterized the time course of neural activity, the expression of synaptic proteins, and neural morphology employing an in vivo optogenetic mouse model. We stimulated pyramidal neurons in the anterior cingulate cortex of mice and harvested their brains at 1.5 h, 24 h, and $48\,\mathrm{h}$ after stimulation. Stimulus-induced cortical hyperactivity persisted up to 1.5 h and decayed to baseline after $24\,\mathrm{h}$ indicated by c-Fos expression. The synaptic proteins VGLUT1 and PSD-95, in contrast, were upregulated at $24\,\mathrm{h}$ and downregulated at $48\,\mathrm{h}$, respectively. Spine density and spine head volume were also increased at $24\,\mathrm{h}$ and decreased at $48\,\mathrm{h}$. This specific sequence of events reflects a continuous joint evolution of activity and connectivity that is characteristic of the model of homeostatic structural plasticity. Our computer simulations thus corroborate the observed empirical evidence from our animal experiments.

On the role of vesicle transport in neurite growth: Modeling and experiments

The processes that determine the establishment of the complex morphology of neurons during development are still poorly understood. Here, we focus on the question how a difference in the length of neurites affects vesicle transport. We performed live imaging experiments and present a lattice-based model to gain a deeper theoretical understanding of intracellular transport in neurons. After a motivation and appropriate scaling of the model we present numerical simulations showing that initial differences in neurite length result in phenomena of biological relevance, i.e. a positive feedback that enhances transport into the longer neurite and oscillation of vesicles concentrations that can be interpreted as cycles of extension and retraction observed in experiments. Thus, our model is a first step towards a better understanding of the interplay between the transport of vesicles and the spatial organization of cells.

The Neurobiology of Zika Virus: New Models, New Challenges

The Zika virus (ZIKV) attracted attention due to one striking characteristic: the ability to cross the placental barrier and infect the fetus, possibly causing severe neurodevelopmental disruptions included in the Congenital Zika Syndrome (CZS). Few years after the epidemic, the CZS incidence has begun to decline. However, how ZIKV causes a diversity of outcomes is far from being understood. This is probably driven by a chain of complex events that relies on the interaction between ZIKV and environmental and physiological variables. In this review, we address open questions that might lead to an ill-defined diagnosis of CZS. This inaccuracy underestimates a large spectrum of apparent normocephalic cases that remain underdiagnosed, comprising several subtle brain abnormalities frequently masked by a normal head circumference. Therefore, new models using neuroimaging and artificial intelligence are needed to improve our understanding of the neurobiology of ZIKV and its true impact in neurodevelopment.

Spatially resolved dendritic integration: towards a functional classification of neurons

The vast tree-like dendritic structure of neurons allows them to receive and integrate input from many neurons. A wide variety of neuronal morphologies exist, however, their role in dendritic integration, and how it shapes the response of the neuron, is not yet fully understood. Here, we study the evolution and interactions of dendritic spikes in excitable neurons with complex real branch structures. We focus on dozens of digitally reconstructed illustrative neurons from the online repository NeuroMorpho.org, which contains over 130,000 neurons. Yet, our methods can be promptly extended to any other neuron. This approach allows us to estimate and map specific and heterogeneous patterns of activity observed across extensive dendritic trees with thousands of compartments. We propose a classification of neurons based on the location of the soma (centrality) and the number of branches connected to the soma. These are key topological factors in determining the neuron's energy consumption, firing rate, and the dynamic range, which quantifies the range in synaptic input rate that can be reliably encoded by the neuron's firing rate. Moreover, we find that bifurcations, the structural building blocks of complex dendrites, play a major role in increasing the dynamic range of neurons. Our results provide a better understanding of the effects of neuronal morphology in the diversity of neuronal dynamics and function.

A generative growth model for thalamocortical axonal branching in primary visual cortex

Axonal morphology displays large variability and complexity, yet the canonical regularities of the cortex suggest that such wiring is based on the repeated initiation of a small set of genetically encoded rules. Extracting underlying developmental principles can hence shed light on what genetically encoded instructions must be available during cortical development. Within a generative model, we investigate growth rules for axonal branching patterns in cat area 17, originating from the lateral geniculate nucleus of the thalamus. This target area of synaptic connections is characterized by extensive ramifications and a high bouton density, characteristics thought to preserve the spatial resolution of receptive fields and to enable connections for the ocular dominance columns. We compare individual and global statistics, such as a newly introduced length-weighted asymmetry index and the global segment-length distribution, of generated and biological branching patterns as the benchmark for growth rules. We show that the proposed model surpasses the statistical accuracy of the Galton-Watson model, which is the most commonly employed model for biological growth processes. In contrast to the Galton-Watson model, our model can recreate the log-normal segment-length distribution of the experimental dataset and is considerably more accurate in recreating individual axonal morphologies. To provide a biophysical interpretation for statistical quantifications of the axonal branching patterns, the generative model is ported into the physically accurate simulation framework of Cx3D. In this 3D simulation environment we demonstrate how the proposed growth process can be formulated as an interactive process between genetic growth rules and chemical cues in the local environment.

Constructing multilayered neural networks with sparse, data-driven connectivity using biologically-inspired, complementary, homeostatic mechanisms

The immense complexity of the brain requires that it be built and controlled by intrinsic, self-regulating mechanisms. One such mechanism, the formation of new connections via synaptogenesis, plays a central role in neuronal connectivity and, ultimately, performance. Adaptive synaptogenesis networks combine synaptogenesis, associative synaptic modification, and synaptic shedding to construct sparse networks. Here, inspired by neuroscientific observations, novel aspects of brain development are incorporated into adaptive synaptogenesis. The extensions include: (i) multiple layers, (ii) neuron survival and death based on information transmission, and (iii) bigrade growth factor signaling to control the onset of synaptogenesis in succeeding layers and to control neuron survival and death in preceding layers. Also guiding this research is the assumption that brains must achieve a compromise between good performance and low energy expenditures. Simulations of the network model demonstrate the parametric and functional control of both performance and energy expenditures, where performance is measured in terms of information loss and classification errors, and energy expenditures are assumed to be a monotonically increasing function of the number of neurons. Major insights from this study include (a) the key role a neural layer between two other layers has in controlling synaptogenesis and neuron elimination, (b) the performance and energy-savings benefits of delaying the onset of synaptogenesis in a succeeding layer, and (c) how the elimination of neurons in a preceding layer provides energy savings, code compression, and can be accomplished without significantly degrading information transfer or classification performance.

Network remodeling induced by transcranial brain stimulation: A computational model of tDCS-triggered cell assembly formation

Transcranial direct current stimulation (tDCS) is a variant of noninvasive neuromodulation, which promises treatment for brain diseases like major depressive disorder. In experiments, long-lasting aftereffects were observed, suggesting that persistent plastic changes are induced. The mechanism underlying the emergence of lasting aftereffects, however, remains elusive. Here we propose a model, which assumes that tDCS triggers a homeostatic response of the network involving growth and decay of synapses. The cortical tissue exposed to tDCS is conceived as a recurrent network of excitatory and inhibitory neurons, with synapses subject to homeostatically regulated structural plasticity. We systematically tested various aspects of stimulation, including electrode size and montage, as well as stimulation intensity and duration. Our results suggest that transcranial stimulation perturbs the homeostatic equilibrium and leads to a pronounced growth response of the network. The stimulated population eventually eliminates excitatory synapses with the unstimulated population, and new synapses among stimulated neurons are grown to form a cell assembly. Strong focal stimulation tends to enhance the connectivity within new cell assemblies, and repetitive stimulation with well-chosen duty cycles can increase the impact of stimulation even further. One long-term goal of our work is to help in optimizing the use of tDCS in clinical applications.

Noninvasive brain stimulation techniques like tDCS have the potential to directly interfere with neural activity, but may also trigger activity-dependent plasticity. We propose a model to study the mechanism of tDCS and persistent aftereffects that may be induced as a consequence of homeostatic structural plasticity. Based on the idea that tDCS perturbs the ongoing activity of neurons, our model predicts that the stimulation also triggers a rearrangement of synapses among stimulated and unstimulated neurons, eventually leading to network remodeling and cell assembly formation. Focal and strong stimulation leads to stronger cell assemblies, and so does repetitive stimulation with optimized stimulation protocols. This is the first original work studying possible long-lasting aftereffects of transcranial stimulation at the mesoscopic neuronal network level using a computational model.

Single-cell branching morphogenesis in the Drosophila trachea

The terminal cells of the tracheal epithelium in Drosophila melanogaster are one of the few known cell types that undergo subcellular morphogenesis to achieve a stable, branched shape. During the animal's larval stages, the cells repeatedly sprout new cytoplasmic processes. These grow very long, wrapping around target tissues to which the terminal cells adhere, and are hollowed by a gas-filled subcellular tube for oxygen delivery. Our understanding of this ramification process remains rudimentary. This review aims to provide a comprehensive summary of studies on terminal cells to date, and attempts to extrapolate how terminal branches might be formed based on the known genetic and molecular components. Next to this cell-intrinsic branching mechanism, we examine the extrinsic regulation of terminal branching by the target tissue and the animal's environment. Finally, we assess the degree of similarity between the patterns established by the branching programs of terminal cells and other branched cells and tissues from a mathematical and conceptual point of view.

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.

Numerical characterization of regenerative axons growing along a spherical multifunctional scaffold after spinal cord injury

Spinal cord injury (SCI) followed by extensive cell loss, inflammation, and scarring, often permanently damages neurological function. Biomaterial scaffolds are promising but currently have limited applicability in SCI because after entering the scaffold, regenerating axons tend to become trapped and rarelyre-enter the host tissue, the reasons for which remain to be completely explored. Here, we propose a mathematical model and computer simulation for characterizing regenerative axons growing along a scaffold following SCI, and how their growth may be guided. The model assumed a solid, spherical, multifunctional, biomaterial scaffold, that would bridge the rostral and caudal stumps of a completely transected spinal cord in a rat model and would guide the rostral regenerative axons toward the caudal tissue. Other assumptions include the whole scaffold being coated with extracellular matrix components, and the caudal area being additionally seeded with chemoattractants. The chemical factors on and around the scaffold were formulated to several coupled variables, and the parameter values were derived fromexisting experimental data. Special attention was given to the effects of coating strength, seeding location, and seeding density, as well as the ramp slope of the scaffold, on axonal regeneration. In numerical simulations, a slimmer scaffold provided a small slope at the entry "on-ramp" area that improved the success rate of axonal regeneration. If success rates are high, an increased number of regenerative axons traverse through the narrow channels, causing congestion and lowering the growth rate. An increase in the number of severed axons (300-12000) did not significantly affect the growth rate, but it reduced the success rate of axonal regeneration. However, an increase in the seeding densities of the complexes on the whole scaffold, and that in the seeding densities of the chemoattractants on the caudal area, improved both the success and growth rates. However, an increase in the density of thecomplexes on the whole scaffold risks an over-eutrophic surface that harms axonal regeneration.Although theoretical predictions are yet to be validated directly by experiments, this theoretical tool can advance the treatment of SCI, and is also applicable to scaffolds with other architectures.

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.

Growing Critical: Self-Organized Criticality in a Developing Neural System

Experiments in various neural systems found avalanches: bursts of activity with characteristics typical for critical dynamics. A possible explanation for their occurrence is an underlying network that self-organizes into a critical state. We propose a simple spiking model for developing neural networks, showing how these may "grow into" criticality. Avalanches generated by our model correspond to clusters of widely applied Hawkes processes. We analytically derive the cluster size and duration distributions and find that they agree with those of experimentally observed neuronal avalanches.

Challenges in Reproducibility, Replicability, and Comparability of Computational Models and Tools for Neuronal and Glial Networks, Cells, and Subcellular Structures

The possibility to replicate and reproduce published research results is one of the biggest challenges in all areas of science. In computational neuroscience, there are thousands of models available. However, it is rarely possible to reimplement the models based on the information in the original publication, let alone rerun the models just because the model implementations have not been made publicly available. We evaluate and discuss the comparability of a versatile choice of simulation tools: tools for biochemical reactions and spiking neuronal networks, and relatively new tools for growth in cell cultures. The replicability and reproducibility issues are considered for computational models that are equally diverse, including the models for intracellular signal transduction of neurons and glial cells, in addition to single glial cells, neuron-glia interactions, and selected examples of spiking neuronal networks. We also address the comparability of the simulation results with one another to comprehend if the studied models can be used to answer similar research questions. In addition to presenting the challenges in reproducibility and replicability of published results in computational neuroscience, we highlight the need for developing recommendations and good practices for publishing simulation tools and computational models. Model validation and flexible model description must be an integral part of the tool used to simulate and develop computational models. Constant improvement on experimental techniques and recording protocols leads to increasing knowledge about the biophysical mechanisms in neural systems. This poses new challenges for computational neuroscience: extended or completely new computational methods and models may be required. Careful evaluation and categorization of the existing models and tools provide a foundation for these future needs, for constructing multiscale models or extending the models to incorporate additional or more detailed biophysical mechanisms. Improving the quality of publications in computational neuroscience, enabling progressive building of advanced computational models and tools, can be achieved only through adopting publishing standards which underline replicability and reproducibility of research results.

Activity-Dependent Downscaling of Subthreshold Synaptic Inputs during Slow-Wave-Sleep-like Activity In Vivo

Activity-dependent synaptic plasticity is critical for cortical circuit refinement. The synaptic homeostasis hypothesis suggests that synaptic connections are strengthened during wake and downscaled during sleep; however, it is not obvious how the same plasticity rules could explain both outcomes. Using whole-cell recordings and optogenetic stimulation of presynaptic input in urethane-anesthetized mice, which exhibit slow-wave-sleep (SWS)-like activity, we show that synaptic plasticity rules are gated by cortical dynamics in vivo. While Down states support conventional spike timing-dependent plasticity, Up states are biased toward depression such that presynaptic stimulation alone leads to synaptic depression, while connections contributing to postsynaptic spiking are protected against this synaptic weakening. We find that this novel activity-dependent and input-specific downscaling mechanism has two important computational advantages: (1) improved signal-to-noise ratio, and (2) preservation of previously stored information. Thus, these synaptic plasticity rules provide an attractive mechanism for SWS-related synaptic downscaling and circuit refinement.

Fast wide-volume functional imaging of engineered in vitro brain tissues

The need for in vitro models that mimic the human brain to replace animal testing and allow high-throughput screening has driven scientists to develop new tools that reproduce tissue-like features on a chip. Three-dimensional (3D) in vitro cultures are emerging as an unmatched platform that preserves the complexity of cell-to-cell connections within a tissue, improves cell survival, and boosts neuronal differentiation. In this context, new and flexible imaging approaches are required to monitor the functional states of 3D networks. Herein, we propose an experimental model based on 3D neuronal networks in an alginate hydrogel, a tunable wide-volume imaging approach, and an efficient denoising algorithm to resolve, down to single cell resolution, the 3D activity of hundreds of neurons expressing the calcium sensor GCaMP6s. Furthermore, we implemented a 3D co-culture system mimicking the contiguous interfaces of distinct brain tissues such as the cortical-hippocampal interface. The analysis of the network activity of single and layered neuronal co-cultures revealed cell-type-specific activities and an organization of neuronal subpopulations that changed in the two culture configurations. Overall, our experimental platform represents a simple, powerful and cost-effective platform for developing and monitoring living 3D layered brain tissue on chip structures with high resolution and high throughput.

Recurrently connected and localized neuronal communities initiate coordinated spontaneous activity in neuronal networks

Developing neuronal systems intrinsically generate coordinated spontaneous activity that propagates by involving a large number of synchronously firing neurons. In vivo, waves of spikes transiently characterize the activity of developing brain circuits and are fundamental for activity-dependent circuit formation. In vitro, coordinated spontaneous spiking activity, or network bursts (NBs), interleaved within periods of asynchronous spikes emerge during the development of 2D and 3D neuronal cultures. Several studies have investigated this type of activity and its dynamics, but how a neuronal system generates these coordinated events remains unclear. Here, we investigate at a cellular level the generation of network bursts in spontaneously active neuronal cultures by exploiting high-resolution multielectrode array recordings and computational network modelling. Our analysis reveals that NBs are generated in specialized regions of the network (functional neuronal communities) that feature neuronal links with high cross-correlation peak values, sub-millisecond lags and that share very similar structural connectivity motifs providing recurrent interactions. We show that the particular properties of these local structures enable locally amplifying spontaneous asynchronous spikes and that this mechanism can lead to the initiation of NBs. Through the analysis of simulated and experimental data, we also show that AMPA currents drive the coordinated activity, while NMDA and GABA currents are only involved in shaping the dynamics of NBs. Overall, our results suggest that the presence of functional neuronal communities with recurrent local connections allows a neuronal system to generate spontaneous coordinated spiking activity events. As suggested by the rules used for implementing our computational model, such functional communities might naturally emerge during network development by following simple constraints on distance-based connectivity.

Coordinated spontaneous spiking activity is fundamental for the normal formation of brain circuits during development. However, how ensembles of neurons generate these events remains unclear. To address this question, in the present study, we investigated the network properties that might be required to a neuronal system for the generation of these spontaneous waves of spikes. We performed our study on spontaneously active neuronal cell cultures using high-resolution electrical recordings and a computational network model developed to reproduce our experimental data both quantitatively and qualitatively. Through the analysis of both experimental and simulated data, we found that network bursts are initiated in regions of the network, or “functional communities”, characterized by particular local connectivity properties. We also found that these regions can amplify the background asynchronous spiking activity preceding a network burst and, in this way, can give rise to coordinated spiking events.

As a whole, our results suggest the presence of functional communities of neurons in a developing neuronal system that might naturally emerge by following simple constraints on distance-based connectivity. These regions are most likely required for the generation of the spontaneous coordinated activity that can drive activity-dependent circuit formation.

Mechanisms of Connectome Development

At the centenary of D'Arcy Thompson's seminal work 'On Growth and Form', pioneering the description of principles of morphological changes during development and evolution, recent experimental advances allow us to study change in anatomical brain networks. Here, we outline potential principles for connectome development. We will describe recent results on how spatial and temporal factors shape connectome development in health and disease. Understanding the developmental origins of brain diseases in individuals will be crucial for deciding on personalized treatment options. We argue that longitudinal studies, experimentally derived parameters for connection formation, and biologically realistic computational models are needed to better understand the link between brain network development, network structure, and network 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.

Biomaterials and computation: a strategic alliance to investigate emergent responses of neural cells

Synergistic use of biomaterials and computation allows to identify and unravel neural cell responses.

Multi-phasic bi-directional chemotactic responses of the growth cone

The nerve growth cone is bi-directionally attracted and repelled by the same cue molecules depending on the situations, while other non-neural chemotactic cells usually show uni-directional attraction or repulsion toward their specific cue molecules. However, how the growth cone differs from other non-neural cells remains unclear. Toward this question, we developed a theory for describing chemotactic response based on a mathematical model of intracellular signaling of activator and inhibitor. Our theory was first able to clarify the conditions of attraction and repulsion, which are determined by balance between activator and inhibitor, and the conditions of uni- and bi-directional responses, which are determined by dose-response profiles of activator and inhibitor to the guidance cue. With biologically realistic sigmoidal dose-responses, our model predicted tri-phasic turning response depending on intracellular Ca²⁺ level, which was then experimentally confirmed by growth cone turning assays and Ca²⁺ imaging. Furthermore, we took a reverse-engineering analysis to identify balanced regulation between CaMKII (activator) and PP1 (inhibitor) and then the model performance was validated by reproducing turning assays with inhibitions of CaMKII and PP1. Thus, our study implies that the balance between activator and inhibitor underlies the multi-phasic bi-directional turning response of the growth cone.

The effect of the neural activity on topological properties of growing neural networks

The connectivity structure in cortical networks defines how information is transmitted and processed, and it is a source of the complex spatiotemporal patterns of network's development, and the process of creation and deletion of connections is continuous in the whole life of the organism. In this paper, we study how neural activity influences the growth process in neural networks. By using a two-dimensional activity-dependent growth model we demonstrated the neural network growth process from disconnected neurons to fully connected networks. For making quantitative investigation of the network's activity influence on its topological properties we compared it with the random growth network not depending on network's activity. By using the random graphs theory methods for the analysis of the network's connections structure it is shown that the growth in neural networks results in the formation of a well-known "small-world" network.

Opposing Effects of Neuronal Activity on Structural Plasticity

The connectivity of the brain is continuously adjusted to new environmental influences by several activity-dependent adaptive processes. The most investigated adaptive mechanism is activity-dependent functional or synaptic plasticity regulating the transmission efficacy of existing synapses. Another important but less prominently discussed adaptive process is structural plasticity, which changes the connectivity by the formation and deletion of synapses. In this review, we show, based on experimental evidence, that structural plasticity can be classified similar to synaptic plasticity into two categories: (i) Hebbian structural plasticity, which leads to an increase (decrease) of the number of synapses during phases of high (low) neuronal activity and (ii) homeostatic structural plasticity, which balances these changes by removing and adding synapses. Furthermore, based on experimental and theoretical insights, we argue that each type of structural plasticity fulfills a different function. While Hebbian structural changes enhance memory lifetime, storage capacity, and memory robustness, homeostatic structural plasticity self-organizes the connectivity of the neural network to assure stability. However, the link between functional synaptic and structural plasticity as well as the detailed interactions between Hebbian and homeostatic structural plasticity are more complex. This implies even richer dynamics requiring further experimental and theoretical investigations.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.

Sharing Neuron Data: Carrots, Sticks, and Digital Records

Routine data sharing is greatly benefiting several scientific disciplines, such as molecular biology, particle physics, and astronomy. Neuroscience data, in contrast, are still rarely shared, greatly limiting the potential for secondary discovery and the acceleration of research progress. Although the attitude toward data sharing is non-uniform across neuroscience subdomains, widespread adoption of data sharing practice will require a cultural shift in the community. Digital reconstructions of axonal and dendritic morphology constitute a particularly "sharable" kind of data. The popularity of the public repository NeuroMorpho.Org demonstrates that data sharing can benefit both users and contributors. Increased data availability is also catalyzing the grassroots development and spontaneous integration of complementary resources, research tools, and community initiatives. Even in this rare successful subfield, however, more data are still unshared than shared. Our experience as developers and curators of NeuroMorpho.Org suggests that greater transparency regarding the expectations and consequences of sharing (or not sharing) data, combined with public disclosure of which datasets are shared and which are not, may expedite the transition to community-wide data sharing.

Effects of homeostatic constraints on associative memory storage and synaptic connectivity of cortical circuits

The impact of learning and long-term memory storage on synaptic connectivity is not completely understood. In this study, we examine the effects of associative learning on synaptic connectivity in adult cortical circuits by hypothesizing that these circuits function in a steady-state, in which the memory capacity of a circuit is maximal and learning must be accompanied by forgetting. Steady-state circuits should be characterized by unique connectivity features. To uncover such features we developed a biologically constrained, exactly solvable model of associative memory storage. The model is applicable to networks of multiple excitatory and inhibitory neuron classes and can account for homeostatic constraints on the number and the overall weight of functional connections received by each neuron. The results show that in spite of a large number of neuron classes, functional connections between potentially connected cells are realized with less than 50% probability if the presynaptic cell is excitatory and generally a much greater probability if it is inhibitory. We also find that constraining the overall weight of presynaptic connections leads to Gaussian connection weight distributions that are truncated at zero. In contrast, constraining the total number of functional presynaptic connections leads to non-Gaussian distributions, in which weak connections are absent. These theoretical predictions are compared with a large dataset of published experimental studies reporting amplitudes of unitary postsynaptic potentials and probabilities of connections between various classes of excitatory and inhibitory neurons in the cerebellum, neocortex, and hippocampus.

A hybrid computational model to predict chemotactic guidance of growth cones

The overall strategy used by growing axons to find their correct paths during the nervous system development is not yet completely understood. Indeed, some emergent and counterintuitive phenomena were recently described during axon pathfinding in presence of chemical gradients. Here, a novel computational model is presented together with its ability to reproduce both regular and counterintuitive axonal behaviours. In this model, the key role of intracellular calcium was phenomenologically modelled through a non standard Gierer-Meinhardt system, as a crucial factor influencing the growth cone behaviour both in regular and complex conditions. This model was able to explicitly reproduce neuritic paths accounting for the complex interplay between extracellular and intracellular environments, through the sensing capability of the growth cone. The reliability of this approach was proven by using quantitative metrics, numerically supporting the similarity between in silico and biological results in regular conditions (control and attraction). Finally, the model was able to qualitatively predict emergent and counterintuitive phenomena resulting from complex boundary conditions.

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.

Retinogenesis: stochasticity and the competency model

The vertebrate retina is made up of seven principal cell types. These seven retinal cell types arise from multipotent retinal progenitor cells (RPCs). The competency model was proposed suggesting that RPCs undergo a series of irreversible transitions between competency states, in each of which the RPCs are competent to generate a different subset of cell types, but not retinal cells generated at previous moments. In this work, we generalize the stochastic model of neurogenesis of Barton et al. (2014), assuming that the same factor that regulates the differentiation, regulates the competency. The model reproduces the timing of production of different retinal cell types in rats such as it was experimentally measured. The results show that the evolution of the competency during retinogenesis could be explained by a single factor. Its evolution during the cell cycle and the stochastic inheritance in cell divisions determine the sequence and the overlap of production of different retinal cell types during development.

Kinetics of enzymatic reactions in lipid membranes containing domains

An appreciable part of enzymes operating in vivo is associated with lipid membranes. The function of such enzymes can be influenced by the presence of domains containing proteins and/or composed of different lipids. The corresponding experimental model-system studies can be performed under well controlled conditions, e.g., on a planar supported lipid bilayer or surface-immobilized vesicles. To clarify what may happen in such systems, we propose general kinetic equations describing the enzyme-catalyzed substrate conversion occurring via the Michaelis-Menten (MM) mechanism on a membrane with domains which do not directly participate in reaction. For two generic situations when a relatively slow reaction takes place primarily in or outside domains, we take substrate saturation and lateral substrate-substrate interactions at domains into account and scrutinize the dependence of the reaction rate on the average substrate coverage. With increasing coverage, depending on the details, the reaction rate reaches saturation via an inflection point or monotonously as in the conventional MM case. In addition, we show analytically the types of reaction kinetics occurring primarily at domain boundaries. In the physically interesting situation when the domain growth is fast on the reaction time scale, the latter kinetics are far from conventional. The opposite situation when the reaction is fast and controlled by diffusion has been studied by using the Monte Carlo technique. The corresponding results indicate that the dependence of the reaction kinetics on the domain size may be weak.

Statistical analysis and data mining of digital reconstructions of dendritic morphologies

Neuronal morphology is diverse among animal species, developmental stages, brain regions, and cell types. The geometry of individual neurons also varies substantially even within the same cell class. Moreover, specific histological, imaging, and reconstruction methodologies can differentially affect morphometric measures. The quantitative characterization of neuronal arbors is necessary for in-depth understanding of the structure-function relationship in nervous systems. The large collection of community-contributed digitally reconstructed neurons available at NeuroMorpho.Org constitutes a “big data” research opportunity for neuroscience discovery beyond the approaches typically pursued in single laboratories. To illustrate these potential and related challenges, we present a database-wide statistical analysis of dendritic arbors enabling the quantification of major morphological similarities and differences across broadly adopted metadata categories. Furthermore, we adopt a complementary unsupervised approach based on clustering and dimensionality reduction to identify the main morphological parameters leading to the most statistically informative structural classification. We find that specific combinations of measures related to branching density, overall size, tortuosity, bifurcation angles, arbor flatness, and topological asymmetry can capture anatomically and functionally relevant features of dendritic trees. The reported results only represent a small fraction of the relationships available for data exploration and hypothesis testing enabled by sharing of digital morphological reconstructions.

Quantitative assessment of computational models for retinotopic map formation

Molecular and activity-based cues acting together are thought to guide retinal axons to their terminal sites in vertebrate optic tectum or superior colliculus (SC) to form an ordered map of connections. The details of mechanisms involved, and the degree to which they might interact, are still not well understood. We have developed a framework within which existing computational models can be assessed in an unbiased and quantitative manner against a set of experimental data curated from the mouse retinocollicular system. Our framework facilitates comparison between models, testing new models against known phenotypes and simulating new phenotypes in existing models. We have used this framework to assess four representative models that combine Eph/ephrin gradients and/or activity-based mechanisms and competition. Two of the models were updated from their original form to fit into our framework. The models were tested against five different phenotypes: wild type, Isl2-EphA3(ki/ki), Isl2-EphA3(ki/+), ephrin-A2,A3,A5 triple knock-out (TKO), and Math5(-/-) (Atoh7). Two models successfully reproduced the extent of the Math5(-/-) anteromedial projection, but only one of those could account for the collapse point in Isl2-EphA3(ki/+). The models needed a weak anteroposterior gradient in the SC to reproduce the residual order in the ephrin-A2,A3,A5 TKO phenotype, suggesting either an incomplete knock-out or the presence of another guidance molecule. Our article demonstrates the importance of testing retinotopic models against as full a range of phenotypes as possible, and we have made available MATLAB software, we wrote to facilitate this process.

Homeostatic structural plasticity can account for topology changes following deafferentation and focal stroke

After brain lesions caused by tumors or stroke, or after lasting loss of input (deafferentation), inter- and intra-regional brain networks respond with complex changes in topology. Not only areas directly affected by the lesion but also regions remote from the lesion may alter their connectivity—a phenomenon known as diaschisis. Changes in network topology after brain lesions can lead to cognitive decline and increasing functional disability. However, the principles governing changes in network topology are poorly understood. Here, we investigated whether homeostatic structural plasticity can account for changes in network topology after deafferentation and brain lesions. Homeostatic structural plasticity postulates that neurons aim to maintain a desired level of electrical activity by deleting synapses when neuronal activity is too high and by providing new synaptic contacts when activity is too low. Using our Model of Structural Plasticity, we explored how local changes in connectivity induced by a focal loss of input affected global network topology. In accordance with experimental and clinical data, we found that after partial deafferentation, the network as a whole became more random, although it maintained its small-world topology, while deafferentated neurons increased their betweenness centrality as they rewired and returned to the homeostatic range of activity. Furthermore, deafferentated neurons increased their global but decreased their local efficiency and got longer tailed degree distributions, indicating the emergence of hub neurons. Together, our results suggest that homeostatic structural plasticity may be an important driving force for lesion-induced network reorganization and that the increase in betweenness centrality of deafferentated areas may hold as a biomarker for brain repair.

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.

Slow synaptic dynamics in a network: from exponential to power-law forgetting

We investigate a mean-field model of interacting synapses on a directed neural network. Our interest lies in the slow adaptive dynamics of synapses, which are driven by the fast dynamics of the neurons they connect. Cooperation is modeled from the usual Hebbian perspective, while competition is modeled by an original polarity-driven rule. The emergence of a critical manifold culminating in a tricritical point is crucially dependent on the presence of synaptic competition. This leads to a universal 1/t power-law relaxation of the mean synaptic strength along the critical manifold and an equally universal 1/√[t] relaxation at the tricritical point, to be contrasted with the exponential relaxation that is otherwise generic. In turn, this leads to the natural emergence of long- and short-term memory from different parts of parameter space in a synaptic network, which is the most original and important result of our present investigations.

Development and matching of binocular orientation preference in mouse V1

Eye-specific thalamic inputs converge in the primary visual cortex (V1) and form the basis of binocular vision. For normal binocular perceptions, such as depth and stereopsis, binocularly matched orientation preference between the two eyes is required. A critical period of binocular matching of orientation preference in mice during normal development is reported in literature. Using a reaction diffusion model we present the development of RF and orientation selectivity in mouse V1 and investigate the binocular orientation preference matching during the critical period. At the onset of the critical period the preferred orientations of the modeled cells are mostly mismatched in the two eyes and the mismatch decreases and reaches levels reported in juvenile mouse by the end of the critical period. At the end of critical period 39% of cells in binocular zone in our model cortex is orientation selective. In literature around 40% cortical cells are reported as orientation selective in mouse V1. The starting and the closing time for critical period determine the orientation preference alignment between the two eyes and orientation tuning in cortical cells. The absence of near neighbor interaction among cortical cells during the development of thalamo-cortical wiring causes a salt and pepper organization in the orientation preference map in mice. It also results in much lower % of orientation selective cells in mice as compared to ferrets and cats having organized orientation maps with pinwheels.

Cell types, network homeostasis, and pathological compensation from a biologically plausible ion channel expression model

How do neurons develop, control, and maintain their electrical signaling properties in spite of ongoing protein turnover and perturbations to activity? From generic assumptions about the molecular biology underlying channel expression, we derive a simple model and show how it encodes an "activity set point" in single neurons. The model generates diverse self-regulating cell types and relates correlations in conductance expression observed in vivo to underlying channel expression rates. Synaptic as well as intrinsic conductances can be regulated to make a self-assembling central pattern generator network; thus, network-level homeostasis can emerge from cell-autonomous regulation rules. Finally, we demonstrate that the outcome of homeostatic regulation depends on the complement of ion channels expressed in cells: in some cases, loss of specific ion channels can be compensated; in others, the homeostatic mechanism itself causes pathological loss of function.

Quantitative investigations of axonal and dendritic arbors: development, structure, function, and pathology

The branching structures of neurons are a long-standing focus of neuroscience. Axonal and dendritic morphology affect synaptic signaling, integration, and connectivity, and their diversity reflects the computational specialization of neural circuits. Altered neuronal morphology accompanies functional changes during development, experience, aging, and disease. Technological improvements continuously accelerate high-throughput tissue processing, image acquisition, and morphological reconstruction. Digital reconstructions of neuronal morphologies allow for complex quantitative analyses that are unattainable from raw images or two-dimensional tracings. Furthermore, digitized morphologies enable computational modeling of biophysically realistic neuronal dynamics. Additionally, reconstructions generated to address specific scientific questions have the potential for continued investigations beyond the original reason for their acquisition. Facilitating multiple reuse are repositories like NeuroMorpho.Org, which ease the sharing of reconstructions. Here, we review selected scientific literature reporting the reconstruction of axonal or dendritic morphology with diverse goals including establishment of neuronal identity, examination of physiological properties, and quantification of developmental or pathological changes. These reconstructions, deposited in NeuroMorpho.Org, have since been used by other investigators in additional research, of which we highlight representative examples. This cycle of data generation, analysis, sharing, and reuse reveals the vast potential of digital reconstructions in quantitative investigations of neuronal morphology.