Deterministic and Stochastic Elements of Axonal Guidance

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.

The adaptive stochasticity hypothesis: Modeling equifinality, multifinality, and adaptation to adversity

Neural phenotypes are the result of probabilistic developmental processes. This means that stochasticity is an intrinsic aspect of the brain as it self-organizes over a protracted period. In other words, while both genomic and environmental factors shape the developing nervous system, another significant-though often neglected-contributor is the randomness introduced by probability distributions. Using generative modeling of brain networks, we provide a framework for probing the contribution of stochasticity to neurodevelopmental diversity. To mimic the prenatal scaffold of brain structure set by activity-independent mechanisms, we start our simulations from the medio-posterior neonatal rich club (Developing Human Connectome Project, n = 630). From this initial starting point, models implementing Hebbian-like wiring processes generate variable yet consistently plausible brain network topologies. By analyzing repeated runs of the generative process (>107 simulations), we identify critical determinants and effects of stochasticity. Namely, we find that stochastic variation has a greater impact on brain organization when networks develop under weaker constraints. This heightened stochasticity makes brain networks more robust to random and targeted attacks, but more often results in non-normative phenotypic outcomes. To test our framework empirically, we evaluated whether stochasticity varies according to the experience of early-life deprivation using a cohort of neurodiverse children (Centre for Attention, Learning and Memory; n = 357). We show that low-socioeconomic status predicts more stochastic brain wiring. We conclude that stochasticity may be an unappreciated contributor to relevant developmental outcomes and make specific predictions for future research.

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.

Predicting the distribution of serotonergic axons: a supercomputing simulation of reflected fractional Brownian motion in a 3D-mouse brain model

The self-organization of the brain matrix of serotonergic axons (fibers) remains an unsolved problem in neuroscience. The regional densities of this matrix have major implications for neuroplasticity, tissue regeneration, and the understanding of mental disorders, but the trajectories of its fibers are strongly stochastic and require novel conceptual and analytical approaches. In a major extension to our previous studies, we used a supercomputing simulation to model around one thousand serotonergic fibers as paths of superdiffusive fractional Brownian motion (FBM), a continuous-time stochastic process. The fibers produced long walks in a complex, three-dimensional shape based on the mouse brain and reflected at the outer (pial) and inner (ventricular) boundaries. The resultant regional densities were compared to the actual fiber densities in the corresponding neuroanatomically-defined regions. The relative densities showed strong qualitative similarities in the forebrain and midbrain, demonstrating the predictive potential of stochastic modeling in this system. The current simulation does not respect tissue heterogeneities but can be further improved with novel models of multifractional FBM. The study demonstrates that serotonergic fiber densities can be strongly influenced by the geometry of the brain, with implications for brain development, plasticity, and evolution.

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.

Biological network growth in complex environments: A computational framework

Spatial biological networks are abundant on all scales of life, from single cells to ecosystems, and perform various important functions including signal transmission and nutrient transport. These biological functions depend on the architecture of the network, which emerges as the result of a dynamic, feedback-driven developmental process. While cell behavior during growth can be genetically encoded, the resulting network structure depends on spatial constraints and tissue architecture. Since network growth is often difficult to observe experimentally, computer simulations can help to understand how local cell behavior determines the resulting network architecture. We present here a computational framework based on directional statistics to model network formation in space and time under arbitrary spatial constraints. Growth is described as a biased correlated random walk where direction and branching depend on the local environmental conditions and constraints, which are presented as 3D multilayer grid. To demonstrate the application of our tool, we perform growth simulations of a dense network between cells and compare the results to experimental data from osteocyte networks in bone. Our generic framework might help to better understand how network patterns depend on spatial constraints, or to identify the biological cause of deviations from healthy network function.

A computational model of bidirectional axonal growth in micro-tissue engineered neuronal networks (micro-TENNs)

Micro-Tissue Engineered Neural Networks (Micro-TENNs) are living three-dimensional constructs designed to replicate the neuroanatomy of white matter pathways in the brain and are being developed as implantable micro-tissue for axon tract reconstruction, or as anatomically-relevant in vitro experimental platforms. Micro-TENNs are composed of discrete neuronal aggregates connected by bundles of long-projecting axonal tracts within miniature tubular hydrogels. In order to help design and optimize micro-TENN performance, we have created a new computational model including geometric and functional properties. The model is built upon the three-dimensional diffusion equation and incorporates large-scale uni- and bi-directional growth that simulates realistic neuron morphologies. The model captures unique features of 3D axonal tract development that are not apparent in planar outgrowth and may be insightful for how white matter pathways form during brain development. The processes of axonal outgrowth, branching, turning and aggregation/bundling from each neuron are described through functions built on concentration equations and growth time distributed across the growth segments. Once developed we conducted multiple parametric studies to explore the applicability of the method and conducted preliminary validation via comparisons to experimentally grown micro-TENNs for a range of growth conditions. Using this framework, the model can be applied to study micro-TENN growth processes and functional characteristics using spiking network or compartmental network modeling. This model may be applied to improve our understanding of axonal tract development and functionality, as well as to optimize the fabrication of implantable tissue engineered brain pathways for nervous system reconstruction and/or modulation.

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.

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.

Deterministic control of mean alignment and elongation of neuron-like cells by grating geometry: a computational approach

Neuron-like cells are driven by their surrounding environment through local topography. A causal mechanotransductive web of topography-force relationships influences and controls complex cellular phenomena such as growth and alignment. This work aimed to provide a computational framework able to model the behaviour of neuron-like (PC12) cells on gratings, accounting for the twofold ability of topographical cues to simultaneously align and enhance the growth of cells. In particular, starting from the mechanical behaviour of the growth cone and filopodia, the effect of grating geometry (e.g., the periodicity and the size of grooves and ridges) on the neuritic mean alignment angle and on the outgrowth rate of cells was explored through theoretical tools and combinatorial simulations, which were able to predict (R(2) > 0.9) experimental data in a time range of 72-120 hours.

Synergistic effects of 3D ECM and chemogradients on neurite outgrowth and guidance: a simple modeling and microfluidic framework

During nervous system development, numerous cues within the extracellular matrix microenvironment (ECM) guide the growing neurites along specific pathways to reach their intended targets. Neurite motility is controlled by extracellular signal sensing through the growth cone at the neurite tip, including chemoattractive and repulsive cues. However, it is difficult to regenerate and restore neurite tracts, lost or degraded due to an injury or disease, in the adult central nervous system. Thus, it is important to evaluate the dynamic interplay between ECM and the concentration gradients of these cues, which would elicit robust neuritogenesis. Such information is critical in understanding the processes involved in developmental biology, and in developing high-fidelity neurite regenerative strategies post-injury, and in drug discovery and targeted therapeutics for neurodegenerative conditions. Here, we quantitatively investigated this relationship using a combination of mathematical modeling and in vitro experiments, and determined the synergistic role of guidance cues and ECM on neurite outgrowth and turning. Using a biomimetic microfluidic system, we have shown that cortical neurite outgrowth and turning under chemogradients (IGF-1 or BDNF) within 3D scaffolds is highly regulated by the source concentration of the guidance cue and the physical characteristics of the scaffold. A mechanistic-driven partial differential equation model of neurite outgrowth has been proposed, which could also be used prospectively as a predictive tool. The parameters for the chemotaxis term in the model are determined from the experimental data using our microfluidic assay. Resulting model simulations demonstrate how neurite outgrowth was critically influenced by the experimental variables, which was further supported by experimental data on cell-surface-receptor expressions. The model results are in excellent agreement with the experimental findings. This integrated approach represents a framework for further elucidation of biological mechanisms underlying neuronal responses of specialized cell types, during various stages of development, and under healthy or diseased conditions.

The behavior of neuronal cells on tendon-derived collagen sheets as potential substrates for nerve regeneration

Peripheral nervous system injuries result in a decreased quality of life, and generally require surgical intervention for repair. Currently, the gold standard of nerve autografting, based on the use of host tissue such as sensory nerves is suboptimal as it results in donor-site loss of function and requires a secondary surgery. Nerve guidance conduits fabricated from natural polymers such as collagen are a common alternative to bridge nerve defects. In the present work, tendon sections derived through a process named bioskiving were studied for their potential for use as a substrate to fabricate nerve guidance conduits. We show that cells such as rat Schwann cells adhere, proliferate, and align along the fibrous tendon substrate which has been shown to result in a more mature phenotype. Additionally we demonstrate that chick dorsal root ganglia explants cultured on the tendon grow to similar lengths compared to dorsal root ganglia cultured on collagen gels, but also grow in a more oriented manner on the tendon sections. These results show that tendon sections produced through bioskiving can support directional nerve growth and may be of use as a substrate for the fabrication of nerve guidance conduits.

Two-photon polymerization of sub-micrometric patterned surfaces: investigation of cell-substrate interactions and improved differentiation of neuron-like cells

Direct Laser Writing (DLW) is an innovative tool that allows the photofabrication of high resolution 3D structures, which can be successfully exploited for the study of the physical interactions between cells and substrates. In this work, we focused our attention on the topographical effects of submicrometric patterned surfaces fabricated via DLW on neuronal cell behavior. In particular, we designed, prepared, and characterized substrates based on aligned ridges for the promotion of axonal outgrowth and guidance. We demonstrated that both rat PC12 neuron-like cells and human SH-SY5Y derived neurons differentiate on parallel 2.5 μm spaced submicrometric ridges, being characterized by strongly aligned and significantly longer neurites with respect to those differentiated on flat control substrates, or on more spaced (5 and 10 μm) ridges. Furthermore, we detected an increased molecular differentiation toward neurons of the SH-SY5Y cells when grown on the submicrometric patterned substrates. Finally, we observed that the axons can exert forces able of bending the ridges, and we indirectly estimated the order of magnitude of these forces thanks to scanning probe techniques. Collectively, we showed as submicrometric structures fabricated by DLW can be used as a useful tool for the study of the axon mechanobiology.

Cell guidance on nanogratings: a computational model of the interplay between PC12 growth cones and nanostructures

Background

Recently, the effects of nanogratings have been investigated on PC12 with respect to cell polarity, neuronal differentiation, migration, maturation of focal adhesions and alignment of neurites.

Methodology/Principal Findings

A synergistic procedure was used to study the mechanism of alignment of PC12 neurites with respect to the main direction of nanogratings. Finite Element simulations were used to qualitatively assess the distribution of stresses at the interface between non-spread growth cones and filopodia, and to study their dependence on filopodial length and orientation. After modelling all adhesions under non-spread growth cone and filopodial protrusions, the values of local stress maxima resulted from the length of filopodia. Since the stress was assumed to be the main triggering cause leading to the increase and stabilization of filopodia, the position of the local maxima was directly related to the orientation of neurites. An analytic closed form equation was then written to quantitatively assess the average ridge width needed to achieve a given neuritic alignment (R² = 0.96), and the alignment course, when the ridge depth varied (R² = 0.97). A computational framework was implemented within an improved free Java environment (CX3D) and in silico simulations were carried out to reproduce and predict biological experiments. No significant differences were found between biological experiments and in silico simulations (alignment, p = 0.3571; tortuosity, p = 0.2236) with a standard level of confidence (95%).

Conclusions/Significance

A mechanism involved in filopodial sensing of nanogratings is proposed and modelled through a synergistic use of FE models, theoretical equations and in silico simulations. This approach shows the importance of the neuritic terminal geometry, and the key role of the distribution of the adhesion constraints for the cell/substrate coupling process. Finally, the effects of the geometry of nanogratings were explicitly considered in cell/surface interactions thanks to the analytic framework presented in this work.

Fast rearrangement of the neuronal growth cone's actin cytoskeleton following VEGF stimulation

The neuronal growth cone plays a crucial role in the development of the nervous system. This highly motile structure leads the axon to its final destination by translating guidance cues into cytoskeletal rearrangements. Recently, vascular endothelial growth factor (VEGF), which is essential for angiogenesis and vascular sprouting, has been found to exert a trophic activity also on neurons, leading to an increased axonal outgrowth, similar to the well-known nerve growth factor (NGF). The neurotrophic properties of VEGF are likely to be promoted via the VEGF receptor 2 (VEGFR-2) and neuropilin-1 (NRP-1). In the long term, VEGF attracts and influences the growth cone velocity and leads to growth cone enlargement. The present study focuses on immediate VEGF effects using RFP-actin and GFP-NF-M microinjected chicken dorsal root ganglia for live cell imaging of the neuronal growth cone. We analyzed actin and neurofilament dynamics following VEGF and NGF treatment and compared the effects. Furthermore, key signaling pathways of VEGF were investigated by specific blocking of VEGFR-2 or NRP-1. With the aid of confocal laser scanning microscopy and stimulated emission depletion microscopy, we show for the first time that VEGF has a quick effect on the actin-cytoskeleton, since actin rearrangements were identifiable within a few minutes, leading to a dramatically increased motion. Moreover, these effects were strongly enhanced by adding both VEGF and NGF. Most notably, the effects were inhibited by blocking VEGFR-2, therefore we propose that the immediate effects of VEGF on the actin-cytoskeleton are mediated through VEGFR-2.

Numerical study of axonal outgrowth in grooved nerve conduits

Nerve conduits with grooved inner texture, working as a topographical guidance cue, have been experimentally proved to play a significant role in axonal alignment. How grooved conduits guide axonal outgrowth is of particular interest for studying nerve regeneration. A viscoelastic model of axonal outgrowth in a conduit with a defined grooved geometry characterized by its width in the circumferential direction and its height in the radial direction is developed in this work. In this model, the axon is considered as an elastic beam and the axonal deformation and motion, including stretching, bending and torsion, are described using a Cosserat rod theory. The friction between axon and substrate is also considered in this model as well as the tip outgrowth. It is found that the directional outgrowth of the axon can be significantly improved by the grooved texture: when the groove width decreases or the groove height increases, the axonal elongation in the longitudinal direction of the conduit can be increased, which is in good agreement with experimental observations. This work is the first numerical model to study the effect of the substrate geometry on axonal outgrowth.

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

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

Membrane tension, myosin force, and actin turnover maintain actin treadmill in the nerve growth cone

A growth cone is a motile structure at the tips of axons that is driven by the actin network and guides axon extension. Low actin adhesion to the substrate creates a stationary actin treadmill that allows leading-edge protrusion when adhesion increases in response to guidance cues. We use experimental measurements in the Aplysia bag growth cone to develop and constrain a simple mechanical model of the actin treadmill. We show that actin retrograde flow is primarily generated by myosin contractile forces, but when myosin is inhibited, leading-edge membrane tension increases and drives the flow. By comparing predictions of the model with previous experimental measurements, we demonstrate that lamellipodial and filopodial filament breaking contribute equally to the resistance to the flow. The fully constrained model clarifies the role of actin turnover in the mechanical balance driving the actin treadmill and reproduces the recent experimental observation that inhibition of actin depolymerization causes retrograde flow to slow exponentially with time. We estimate forces in the actin treadmill, and we demonstrate that measured G-actin distributions are consistent with the existence of a forward-directed fluid flow that transports G-actin to the leading edge.

VEGF-induced growth cone enhancement is diminished by inhibiting tyrosine-residue 1214 of VEGFR-2

Axonal outgrowth is of paramount significance for establishing the intricate neuronal network both during embryogenesis and nerve regeneration. Vascular endothelial growth factor (VEGF), which is known for its essential role in vascular sprouting and its involvement in cancer, has recently been found to exert a trophic activity on neurons leading to an increased axonal outgrowth. Although two receptors, VEGFR-2 and neuropilin-1, were identified on neurons, the signaling pathways associated with them are not well understood. The aim of this study was to analyze the influence of VEGF on the growth cone morphology and motility of dorsal root ganglia (DRG) neurons. Moreover, we aimed for a deeper understanding of VEGFR-2 on growth cones that potentially mediates the stimulating and attractive effects. We cultivated chicken DRG in medium containing mouse VEGF and analyzed growth cone size. The data presented here show a positive effect of VEGF on growth cone size. Furthermore, we interrupted the activity of VEGFR-2 by either blocking the tyrosine residue 1214 (tyr1214) or by inhibiting the receptor phosphorylation with axitinib, a novel small molecule, which has recently entered phase III trials for cancer treatment. Disruption of the VEGFR-2 leads to a significantly diminished growth cone size. Based on these findings, we propose a positive effect of VEGF on peripheral nervous system growth cone size and show for the first time quantitative data to underline this hypothesis. Additionally, we propose that VEGFR-2 and especially the tyr1214-dependent pathway of VEGFR-2 are of importance in VEGF signaling in the growth cone of DRG neurons.

The limits of brain determinacy

The genes do not control everything that happens in a cell or an organism, because thermally induced molecular movements and conformation changes are beyond genetic control. The importance of uncontrolled events has been argued from the differences between isogenic organisms reared in virtually identical environments, but these might alternatively be attributed to subtle, undetected differences in the environment. The present review focuses on the uncontrolled events themselves in the context of the developing brain. These are considered at cellular and circuit levels because even if cellular physiology was perfectly controlled by the genes (which it is not), the interactions between different cells might still be uncoordinated. A further complication is that the brain contains mechanisms that buffer noise and others that amplify it. The final resultant of the battle between these contrary mechanisms is that developmental stochasticity is sufficiently low to make neurobehavioural defects uncommon, but a chance component of neural development remains. Thus, our brains and behaviour are not entirely determined by a combination of genes-plus-environment.

An instruction language for self-construction in the context of neural networks

Biological systems are based on an entirely different concept of construction than human artifacts. They construct themselves by a process of self-organization that is a systematic spatio-temporal generation of, and interaction between, various specialized cell types. We propose a framework for designing gene-like codes for guiding the self-construction of neural networks. The description of neural development is formalized by defining a set of primitive actions taken locally by neural precursors during corticogenesis. These primitives can be combined into networks of instructions similar to biochemical pathways, capable of reproducing complex developmental sequences in a biologically plausible way. Moreover, the conditional activation and deactivation of these instruction networks can also be controlled by these primitives, allowing for the design of a "genetic code" containing both coding and regulating elements. We demonstrate in a simulation of physical cell development how this code can be incorporated into a single progenitor, which then by replication and differentiation, reproduces important aspects of corticogenesis.

Model of fasciculation and sorting in mixed populations of axons

We extend a recently proposed model [Chaudhuri et al., Europhys. Lett. 87, 20003 (2009)] aiming to describe the formation of fascicles of axons during neural development. The growing axons are represented as paths of interacting directed random walkers in two spatial dimensions. To mimic turnover of axons, whole paths are removed and new walkers are injected with specified rates. In the simplest version of the model, we use strongly adhesive short-range inter-axon interactions that are identical for all pairs of axons. We generalize the model to adhesive interactions of finite strengths and to multiple types of axons with type-specific interactions. The dynamic steady state is characterized by the position-dependent distribution of fascicle size and fascicle composition. With distance in the direction of axon growth, the mean fascicle size and emergent time scales grow monotonically, while the degree of sorting of fascicles by axon type has a maximum at a finite distance. To understand the emergence of slow time scales, we develop an analytical framework to analyze the interaction between neighboring fascicles.

Abnormal language pathway in children with Angelman syndrome

Angelman syndrome is a genetic disorder characterized by pervasive developmental disability with failure to develop speech. We examined the basis for severe language delay in patients with Angelman syndrome by diffusion tensor imaging. Magnetic resonance imaging/diffusion tensor imaging was performed in 7 children with genetically confirmed Angelman syndrome (age 70 ± 26 months, 5 boys) and 4 age-matched control children to investigate the microstructural integrity of arcuate fasciculus and other major association tracts. Six of 7 children with Angelman syndrome had unidentifiable left arcuate fasciculus, while all control children had identifiable arcuate fasciculus. The right arcuate fasciculus was absent in 6 of 7 children with Angelman syndrome and 1 of 4 control children. Diffusion tensor imaging color mapping suggested aberrant morphology of the arcuate fasciculus region. Other association tracts, including uncinate fasciculus, inferior fronto-occipital fasciculus, inferior longitudinal fasciculus, and corticospinal tract, were identifiable but manifested decreased fractional anisotropy in children with Angelman syndrome. Increased apparent diffusion coefficient was seen in all tracts except uncinate fasciculus when compared to control children. Patients with Angelman syndrome have global impairment of white matter integrity in association tracts, particularly the arcuate fasciculus, which reveals severe morphologic changes. This finding could be the result of a potential problem with axon guidance during brain development, possibly due to loss of UBE3A gene expression.

Engineered in vitro/in silico models to examine neurite target preference

Research on spinal cord injury (SCI) repair focuses on developing mechanisms to allow neurites to grow past an injury site. In this article, we observe that numerous divergent paths (i.e., spinal roots) are present along the spinal column, and hence guidance strategies must be devised to ensure that regrowing neurites reach viable targets. Therefore, we have engineered an in vitro micropatterned model in which cultured E7 dorsal root ganglia (DRG) explants may enter alternate pathways (?roots?) along a branching micropattern. Alongside this in vitro model, we have developed an in silico simulation that we validate by comparison with independent experiments. We find in both in silico and in vitro models that the probability of a neurite entering a given root decreases exponentially with respect to the number of roots away from the DRG; consequently, the likelihood of neurites reaching a distant root can be vanishingly small. This result represents a starting point for future strategies to optimize the likelihood that neurites will reach appropriate targets in the regenerating nervous system, and provides a new computational tool to evaluate the feasibility and expected success of neurite guidance in complex geometries.

Mathematical modeling of axonal formation. Part I: Geometry

A stochastic model is proposed for the position of the tip of an axon. Parameters in the model are determined from laboratory data. The first step is the reduction of inherent error in the laboratory data, followed by estimating parameters and fitting a mathematical model to this data. Several axonogenesis aspects have been investigated, particularly how positive axon elongation and growth cone kinematics are coupled processes but require very different theoretical descriptions. Preliminary results have been obtained through a series of experiments aimed at isolating the response of axons to controlled gradient exposures to guidance cues and the effects of ethanol and similar substances. We show results based on the following tasks; (A) development of a novel filtering strategy to obtain data sets truly representative of the axon trail formation; (B) creation of a coarse graining method which establishes (C) an optimal parameter estimation technique, and (D) derivation of a mathematical model which is stochastic in nature, parameterized by arc length. The framework and the resulting model allow for the comparison of experimental and theoretical mean square displacement (MSD) of the developing axon. Current results are focused on uncovering the geometric characteristics of the axons and MSD through analytical solutions and numerical simulations parameterized by arc length, thus ignoring the temporal growth processes. Future developments will capture the dynamic growth cone and how it behaves as a function of time. Qualitative and quantitative predictions of the model at specific length scales capture the experimental behavior well.

Fasciculation and defasciculation of neurite bundles on micropatterned substrates

We describe experiments of fasciculation, i.e., bundling, of chick sensory neurites on 2D striped substrates. By Fourier decomposition, we separate left-going and right-going neurite components from in vitro images, and we find first that neurite bundles orient toward preferred angles with respect to the stripe direction, and second that in vitro bundles travel in leftward and rightward directions nearly uninterrupted by crossings of bundles traveling in the opposing direction. We explore mechanisms that lead to these behaviors, and summarize implications for future models for neurite outgrowth and guidance.

A stochastic model of neuronal growth cone guidance regulated by multiple sensors

Neuronal growth cones migrate directionally under the control of axon guidance molecules, thereby forming synapses in the developing brain. The signal transduction system by which a growth cone detects surrounding guidance molecules, analyzes the detected signals, and then determines the overall behavior remains undetermined. In this study, we describe a novel stochastic model of this behavior that utilizes multiple sensors on filopodia to respond to guidance molecules. Overall growth cone behavior is determined by using only the concentration gradients of guidance molecules in the immediate vicinity of each sensor. The detected signal at each sensor, which is treated as a vector quantity, is sent to the growth cone center and then integrated to determine axonal growth in the next step by means of a simple vector operation. We compared the results of computer simulations of axonal growth with observations of actual axonal growth from co-culture experiments using olfactory bulb and septum. The probabilistic distributions of axonal growth generated by the computer simulation were consistent with those obtained from the culture experiments, indicating that our model accurately simulates growth cone behavior. We believe that this model will be useful for elucidating the as yet unknown mechanisms responsible for axonal growth in vivo.

3D timelapse analysis of muscle satellite cell motility

Skeletal muscle repair and regeneration requires the activity of satellite cells, a population of myogenic stem cells scattered throughout the tissue and activated to proliferate and differentiate in response to myotrauma or disease. While it seems likely that satellite cells would need to navigate local muscle tissue to reach damaged areas, relatively little data on such motility exist, and most studies have been with immortalized cell lines. We find that primary satellite cells are significantly more motile than myoblast cell lines, and that adhesion to laminin promotes primary cell motility more than fourfold over other substrates. Using timelapse videomicroscopy to assess satellite cell motility on single living myofibers, we have identified a requirement for the laminin-binding integrin α7β1 in satellite cell motility, as well as a role for hepatocyte growth factor in promoting directional persistence. The extensive migratory behavior of satellite cells resident on muscle fibers suggests caution when determining, based on fixed specimens, whether adjacent cells are daughters from the same mother cell. We also observed more persistent long-term contact between individual satellite cells than has been previously supposed, potential cell-cell attractive and repulsive interactions, and migration between host myofibers. Based on such activity, we assayed for expression of “pathfinding” cues, and found that satellite cells express multiple guidance ligands and receptors. Together, these data suggest that satellite cell migration in vivo may be more extensive than currently thought, and could be regulated by combinations of signals, including adhesive haptotaxis, soluble factors, and guidance cues. Stem Cells2009;27:2527–2538

A framework for modeling the growth and development of neurons and networks

The development of neural tissue is a complex organizing process, in which it is difficult to grasp how the various localized interactions between dividing cells leads relentlessly to global network organization. Simulation is a useful tool for exploring such complex processes because it permits rigorous analysis of observed global behavior in terms of the mechanistic axioms declared in the simulated model. We describe a novel simulation tool, CX3D, for modeling the development of large realistic neural networks such as the neocortex, in a physical 3D space. In CX3D, as in biology, neurons arise by the replication and migration of precursors, which mature into cells able to extend axons and dendrites. Individual neurons are discretized into spherical (for the soma) and cylindrical (for neurites) elements that have appropriate mechanical properties. The growth functions of each neuron are encapsulated in set of pre-defined modules that are automatically distributed across its segments during growth. The extracellular space is also discretized, and allows for the diffusion of extracellular signaling molecules, as well as the physical interactions of the many developing neurons. We demonstrate the utility of CX3D by simulating three interesting developmental processes: neocortical lamination based on mechanical properties of tissues; a growth model of a neocortical pyramidal cell based on layer-specific guidance cues; and the formation of a neural network in vitro by employing neurite fasciculation. We also provide some examples in which previous models from the literature are re-implemented in CX3D. Our results suggest that CX3D is a powerful tool for understanding neural development.

Stochastic actin polymerization and steady retrograde flow determine growth cone advancement

Neuronal growth is an extremely complex yet reliable process that is directed by a dynamic lamellipodial structure at the tip of every growing neurite, called the growth cone. Lamellipodial edge fluctuations are controlled by the interplay between actin polymerization pushing the edge forward and molecular motor driven retrograde actin flow retracting the actin network. The leading edge switches randomly between extension and retraction processes. We identify switching of "on/off" states in actin polymerization as the main determinant of lamellipodial advancement. Our analysis of motility statistics allows for a prediction of growth direction. This was used in simulations explaining the amazing signal detection capabilities of neuronal growth by the experimentally found biased stochastic processes. Our measurements show that the intensity of stochastic fluctuations depend on changes in the underlying active intracellular processes and we find a power law eta = a*x(alpha) with exponent alpha = 2.63 +/- 0.12 between noise intensity eta and growth cone activity x, defined as the sum of protrusion and retraction velocity. Differences in the lamellipodial dynamics between primary neurons and a neuronal cell line further suggests that active processes tune the observed stochastic fluctuations. This hints at a possible role of noise intensity in determining signal detection sensitivity.

Cellular morphogenesis in silico

We describe a model that simulates spherical cells of different types that can migrate and interact either attractively or repulsively. We find that both expected morphologies and previously unreported patterns spontaneously self-assemble. Among the newly discovered patterns are a segmented state of alternating discs, and a "shish-kebab" state, in which one cell type forms a ring around a second type. We show that these unique states result from cellular attraction that increases with distance (e.g., as membranes stretch viscoelastically), and would not be seen in traditional, e.g., molecular, potentials that diminish with distance. Most of the states found computationally have been observed in vitro, and it remains to be established what role these self-assembled states may play in in vivo morphogenesis.

Autocatalytic loop, amplification and diffusion: a mathematical and computational model of cell polarization in neural chemotaxis

The chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.

Theoretical models of neural circuit development

Proper wiring up of the nervous system is critical to the development of organisms capable of complex and adaptable behaviors. Besides the many experimental advances in determining the cellular and molecular machinery that carries out this remarkable task precisely and robustly, theoretical approaches have also proven to be useful tools in analyzing this machinery. A quantitative understanding of these processes can allow us to make predictions, test hypotheses, and appraise established concepts in a new light. Three areas that have been fruitful in this regard are axon guidance, retinotectal mapping, and activity-dependent development. This chapter reviews some of the contributions made by mathematical modeling in these areas, illustrated by important examples of models in each section. For axon guidance, we discuss models of how growth cones respond to their environment, and how this environment can place constraints on growth cone behavior. Retinotectal mapping looks at computational models for how topography can be generated in populations of neurons based on molecular gradients and other mechanisms such as competition. In activity-dependent development, we discuss theoretical approaches largely based on Hebbian synaptic plasticity rules, and how they can generate maps in the visual cortex very similar to those seen in vivo. We show how theoretical approaches have substantially contributed to the advancement of developmental neuroscience, and discuss future directions for mathematical modeling in the field.

Protein cooperation: from neurons to networks

A constant pattern through the development of cellular life is that not only cells but also subcellular components such as proteins, either being enzymes, receptors, signaling or structural proteins, strictly cooperate. Discerning how protein cooperation originated and propagates over evolutionary time, how proteins work together to a shared outcome far beyond mere interaction, thus represents a theoretical and experimental challenge for evolutionary, molecular, and computational biology, and a timely fruition also for biotechnology. In this review, we describe some basic principles sustaining not only cellular but especially protein cooperative behavior, with particular emphasis on neurobiological systems. We illustrate experimental results and numerical models substantiating that bench research, as well as computer analysis, indeed concurs in recognizing the natural propensity of proteins to cooperate. At the cellular level, we exemplify network connectivity in the thalamus, hippocampus and basal ganglia. At the protein level, we depict numerical models about the receptosome, the protein machinery connecting neurotransmitters or growth factors to specific, unique downstream effector proteins. We primarily focus on the purinergic P2/P1 receptor systems for extracellular purine and pyrimidine nucleotides/nucleosides. By spanning concepts such as single-molecule biology to membrane computing, we seek to stimulate a scientific debate on the implications of protein cooperation in neurobiological systems.

Cofilin/ADF is required for retinal elongation and morphogenesis of the Drosophila rhabdomere

Drosophila photoreceptors undergo marked changes in their morphology during pupal development. These changes include a five-fold elongation of the retinal cell body and the morphogenesis of the rhabdomere, the light sensing structure of the cell. Here we show that twinstar (tsr), which encodes Drosophila cofilin/ADF (actin-depolymerizing factor), is required for both of these processes. In tsr mutants, the retina is shorter than normal, the result of a lack of retinal elongation. In addition, in a strong tsr mutant, the rhabdomere structure is disorganized and the microvilli are short and occasionally unraveled. In an intermediate tsr mutant, the rhabdomeres are not disorganized but have a wider than normal structure. The adherens junctions connecting photoreceptor cells to each other are also found to be wider than normal. We propose, and provide data supporting, that these wide rhabdomeres and adherens junctions are secondary events caused by the inhibition of retinal elongation. These results provide insight into the functions of the actin cytoskeleton during morphogenesis of the Drosophila eye.

The genetics of brain wiring: from molecule to mind

What makes some people neurotic or schizophrenic or right-handed or fearless? The challenge in answering this is to map from genotype to anatomical and physiological phenotypes and beyond to behavior and cognition.

A drug delivery system based on alginate microspheres: mass-transport test and in vitro validation

This paper presents a drug delivery system based on alginate microspheres. The biocompatibility, the flexibility in size and shape, the ability to entrap biomolecules as well as cells make alginate based systems ideal for in vivo drug delivery. Specifically, considering the target application of neural regeneration and the issue of neuroprotection for the development of innovative neuroprostheses, the authors describe a system for controlled release of Netrin-1, an axonal guidance protein. Microspheres dimensioning (based on specifications of drug release time and release modality), microspheres realization, and mass transport tests are described. The release efficiency is finally assessed by in vitro experiments of axonal guidance performed on embryonic neuronal cells. Preliminary results show that neuronal axons grow approaching the Netrin-1 source, thus indicating an efficient entrapment and release of the protein in the microspheres, in agreement with the microsphere modelling described before.

Simulated morphogenesis of developmental folds due to proliferative pressure

We present a simulation that models individual cells as spherical particles that can migrate, interact, divide and differentiate. We simulate the evolution of a progenitor layer of cells that reproduce, leading either to more progenitors or to differentiated daughters. We find that this simplified model produces spontaneous folds whose lengths depend linearly on the ratio of rates of production of progenitors to differentiated daughters. We also find that folds grow approximately exponentially in time, and that larger folds can be placed via patterning events that perturb the positions of selected progenitor cells early in the developmental process.

Antagonistic forces generated by cytoplasmic dynein and myosin-II during growth cone turning and axonal retraction

Cytoplasmic dynein transports short microtubules down the axon in part by pushing against the actin cytoskeleton. Recent studies have suggested that comparable dynein-driven forces may impinge upon the longer microtubules within the axon. Here, we examined a potential role for these forces on axonal retraction and growth cone turning in neurons partially depleted of dynein heavy chain (DHC) by small interfering RNA. While DHC-depleted axons grew at normal rates, they retracted far more robustly in response to donors of nitric oxide than control axons, and their growth cones failed to efficiently turn in response to substrate borders. Live cell imaging of dynamic microtubule tips showed that microtubules in DHC-depleted growth cones were largely confined to the central zone, with very few extending into filopodia. Even under conditions of suppressed microtubule dynamics, DHC depletion impaired the capacity of microtubules to advance into the peripheral zone of the growth cone, indicating a direct role for dynein-driven forces on the distribution of the microtubules. These effects were all reversed by inhibition of myosin-II forces, which are known to underlie the retrograde flow of actin in the growth cone and the contractility of the cortical actin during axonal retraction. Our results are consistent with a model whereby dynein-driven forces enable microtubules to overcome myosin-II-driven forces, both in the axonal shaft and within the growth cone. These dynein-driven forces oppose the tendency of the axon to retract and permit microtubules to advance into the peripheral zone of the growth cone so that they can invade filopodia.