Mathematical models of neuronal growth

Correction to: Mathematical models of neuronal growth

Correction to: Biomechanics and Modeling in Mechanobiology (2022) 21:89-118 https://doi.org/10.1007/s10237-021-01539-0.

A memristive circuit for self-organized network topology formation based on guided axon growth

Circuit implementations of neuronal networks so far have been focusing on synaptic weight changes as network growth principles. Besides these weight changes, however, it is also useful to incorporate additional network growth principles such as guided axon growth and pruning. These allow for dynamical signal delays and a higher degree of self-organization, and can thus lead to novel circuit design principles. In this work we develop an ideal, bio-inspired electrical circuit mimicking growth and pruning controlled by guidance cues. The circuit is based on memristively coupled neuronal oscillators. As coupling element, we use memsensors consisting of a general sensor, two gradient sensors, and two memristors. The oscillators and memsensors are arranged in a grid structure, where oscillators and memsensors realize nodes and edges, respectively. This allows for arbitrary 2D growth scenarios with axon growth controlled by guidance cues. Simulation results show that the circuit successfully mimics a biological example in which two neurons initially grow towards two target neurons, where undesired connections are pruned later on.

A simple active fluid model unites cytokinesis, cell crawling, and axonal outgrowth

Axonal outgrowth, cell crawling, and cytokinesis utilize actomyosin, microtubule-based motors, cytoskeletal dynamics, and substrate adhesions to produce traction forces and bulk cellular motion. While it has long been appreciated that growth cones resemble crawling cells and that the mechanisms that drive cytokinesis help power cell crawling, they are typically viewed as unique processes. To better understand the relationship between these modes of motility, here, we developed a unified active fluid model of cytokinesis, amoeboid migration, mesenchymal migration, neuronal migration, and axonal outgrowth in terms of cytoskeletal flow, adhesions, viscosity, and force generation. Using numerical modeling, we fit subcellular velocity profiles of the motions of cytoskeletal structures and docked organelles from previously published studies to infer underlying patterns of force generation and adhesion. Our results indicate that, during cytokinesis, there is a primary converge zone at the cleavage furrow that drives flow towards it; adhesions are symmetric across the cell, and as a result, cells are stationary. In mesenchymal, amoeboid, and neuronal migration, the site of the converge zone shifts, and differences in adhesion between the front and back of the cell drive crawling. During neuronal migration and axonal outgrowth, the primary convergence zone lies within the growth cone, which drives actin retrograde flow in the P-domain and bulk anterograde flow of the axonal shaft. They differ in that during neuronal migration, the cell body is weakly attached to the substrate and thus moves forward at the same velocity as the axon. In contrast, during axonal outgrowth, the cell body strongly adheres to the substrate and remains stationary, resulting in a decrease in flow velocity away from the growth cone. The simplicity with which cytokinesis, cell crawling, and axonal outgrowth can be modeled by varying coefficients in a simple model suggests a deep connection between them.

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.

Axonal tension contributes to consistent fold placement

Cortical folding is a critical process during brain development, resulting in morphologies that are both consistent and distinct between individuals and species. While earlier studies have highlighted important aspects of cortical folding, most existing computational models, based on the differential growth theory, fall short of explaining why folds tend to appear in particular locations. The axon tension hypothesis may provide insight into this conundrum; however, there has been significant controversy about a potential role of axonal tension during the gyrification. The common opinion in the field is that axonal tension is inadequate to drive gyrification, but we currently run the risk of discarding this hypothesis without comprehensively studying the role of axonal tension. Here we propose a novel bi-layered finite element model incorporating the two theories, including characteristic axonal tension in the subcortex and differential cortical growth. We show that axon tension can serve as a perturbation sufficient to trigger buckling in simulations; similarly to other types of perturbations, the natural stability behavior of the system tends to determine some characteristics of the folding morphology (e.g. the wavelength) while the perturbation determines the location of folds. Certain geometries, however, can interact or compete with the natural stability of the system to change the wavelength. When multiple perturbations are present, they similarly compete with each other. We found that an axon bundle of reasonable size will overpower up to a 5% thickness perturbation (typical in the literature) and determine fold placement. Finally, when multiple axon tracts are present, even a slight difference in axon stiffness, representing the heterogeneity of axonal connections, is enough to significantly change the folding pattern. While the simulations presented here are a very simple representation of white matter connectivity, our findings point to urgent future research on the role of axon connectivity in cortical folding.

Bisphenol S impairs mitochondrial function by targeting Myo19/oxidative phosphorylation pathway contributing to axonal and dendritic injury

Exposure to bisphenol S (BPS) is known to adversely affect neuronal development. As pivotal components of neuronal polarization, axons and dendrites are indispensable structures within neurons, crucial for the maintenance of nervous system function. Here, we investigated the impact of BPS exposure on axonal and dendritic development both in vivo and in vitro. Our results revealed that exposure to BPS during pregnancy and lactation led to a reduction in the complexity, density, and length of axons and dendrites in the prefrontal cortex (PFC) of offspring. Employing RNA sequencing technology to elucidate the underlying mechanisms of axonal and dendritic damage induced by BPS, Kyoto Encyclopedia of Genes and Genomes (KEGG) analysis highlighted a significant alteration in the oxidative phosphorylation (OXPHOS) pathway, essential for mitochondrial function. Subsequent experiments demonstrate BPS-induced impairment in mitochondrial function, including damaged morphology, decreased adenosine triphosphate (ATP) and superoxide dismutase (SOD) levels, and increased reactive oxygen species and malondialdehyde (MDA). These alterations coincided with the downregulated expression of OXPHOS pathway-related genes (ATP6V1B1, ATP5K, NDUFC1, NDUFC2, NDUFA3, COX6B1) and Myosin 19 (Myo19). Notably, Myo19 overexpression restored the BPS-induced mitochondrial dysfunction by alleviating the inhibition of OXPHOS pathway. Consequently, this amelioration was associated with a reduction in BPS-induced axonal and dendritic injury observed in cultured neurons of the PFC.

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.

A generative model of the connectome with dynamic axon growth

Connectome generative models, otherwise known as generative network models, provide insight into the wiring principles underpinning brain network organization. While these models can approximate numerous statistical properties of empirical networks, they typically fail to explicitly characterize an important contributor to brain organization - axonal growth. Emulating the chemoaffinity guided axonal growth, we provide a novel generative model in which axons dynamically steer the direction of propagation based on distance-dependent chemoattractive forces acting on their growth cones. This simple dynamic growth mechanism, despite being solely geometry-dependent, is shown to generate axonal fiber bundles with brain-like geometry and features of complex network architecture consistent with the human brain, including lognormally distributed connectivity weights, scale-free nodal degrees, small-worldness, and modularity. We demonstrate that our model parameters can be fitted to individual connectomes, enabling connectome dimensionality reduction and comparison of parameters between groups. Our work offers an opportunity to bridge studies of axon guidance and connectome development, providing new avenues for understanding neural development from a computational perspective.

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.

Chemical and mechanical control of axon fasciculation and defasciculation

Neural networks are constructed through the development of robust axonal projections from individual neurons, which ultimately establish connections with their targets. In most animals, developing axons assemble in bundles to navigate collectively across various areas within the central nervous system or the periphery, before they separate from these bundles in order to find their specific targets. These processes, called fasciculation and defasciculation respectively, were thought for many years to be controlled chemically: while guidance cues may attract or repulse axonal growth cones, adhesion molecules expressed at the surface of axons mediate their fasciculation. Recently, an additional non-chemical parameter, the mechanical longitudinal tension of axons, turned out to play a role in axon fasciculation and defasciculation, through zippering and unzippering of axon shafts. In this review, we present an integrated view of the currently known chemical and mechanical control of axon:axon dynamic interactions. We highlight the facts that the decision to cross or not to cross another axon depends on a combination of chemical, mechanical and geometrical parameters, and that the decision to fasciculate/defasciculate through zippering/unzippering relies on the balance between axon:axon adhesion and their mechanical tension. Finally, we speculate about possible functional implications of zippering-dependent axon shaft fasciculation, in the collective migration of axons, and in the sorting of subpopulations of axons.

The role of mechanics in axonal stability and development

Much of the focus of neuronal cell biology has been devoted to growth cone guidance, synaptogenesis, synaptic activity, plasticity, etc. The axonal shaft too has received much attention, mainly for its astounding ability to transmit action potentials and the transport of material over long distances. For these functions, the axonal cytoskeleton and membrane have been often assumed to play static structural roles. Recent experiments have changed this view by revealing an ultrastructure much richer in features than previously perceived and one that seems to be maintained at a dynamic steady state. The role of mechanics in this is only beginning to be broadly appreciated and appears to involve passive and active modes of coupling different biopolymer filaments, filament turnover dynamics and membrane biophysics. Axons, being unique cellular processes in terms of high aspect ratios and often extreme lengths, also exhibit unique passive mechanical properties that might have evolved to stabilize them under mechanical stress. In this review, we summarize the experiments that have exposed some of these features. It is our view that axonal mechanics deserves much more attention not only due to its significance in the development and maintenance of the nervous system but also due to the susceptibility of axons to injury and neurodegeneration.

Biochemical Pathways of Cellular Mechanosensing/Mechanotransduction and Their Role in Neurodegenerative Diseases Pathogenesis

In this review, we shed light on recent advances regarding the characterization of biochemical pathways of cellular mechanosensing and mechanotransduction with particular attention to their role in neurodegenerative disease pathogenesis. While the mechanistic components of these pathways are mostly uncovered today, the crosstalk between mechanical forces and soluble intracellular signaling is still not fully elucidated. Here, we recapitulate the general concepts of mechanobiology and the mechanisms that govern the mechanosensing and mechanotransduction processes, and we examine the crosstalk between mechanical stimuli and intracellular biochemical response, highlighting their effect on cellular organelles' homeostasis and dysfunction. In particular, we discuss the current knowledge about the translation of mechanosignaling into biochemical signaling, focusing on those diseases that encompass metabolic accumulation of mutant proteins and have as primary characteristics the formation of pathological intracellular aggregates, such as Alzheimer's Disease, Huntington's Disease, Amyotrophic Lateral Sclerosis and Parkinson's Disease. Overall, recent findings elucidate how mechanosensing and mechanotransduction pathways may be crucial to understand the pathogenic mechanisms underlying neurodegenerative diseases and emphasize the importance of these pathways for identifying potential therapeutic targets.

Force-Regulated State Transitions of Growing Axons

Growing axons are one-dimensional active structures that are important for wiring the brain and repairing nerves. However, the biophysical mechanisms underlying the complex kinetics of growing axons remain elusive. Here, we develop a theoretical framework to recapitulate force-regulated states and their transitions in growing axons. We demonstrate a unique negative feedback mechanism that defines four distinct kinetic states in a growing axon, whose transitional boundaries depend on the interplay between cytoskeletal dynamics and axon-substrate adhesion. A phase diagram for axonal growth is formulated based on two dimensionless numbers.