Stochasticity and functionality of neural systems: Mathematical modelling of axon growth in the spinal cord of tadpole

The storage capacity of a directed graph and nodewise autonomous, ubiquitous learning

The brain, an exceedingly intricate information processing system, poses a constant challenge to memory research, particularly in comprehending how it encodes, stores, and retrieves information. Cognitive psychology studies memory mechanism from behavioral experiment level and fMRI level, and neurobiology studies memory mechanism from anatomy and electrophysiology level. Current research findings are insufficient to provide a comprehensive, detailed explanation of memory processes within the brain. Numerous unknown details must be addressed to establish a complete information processing mechanism connecting micro molecular cellular levels with macro cognitive behavioral levels. Key issues include characterizing and distributing content within biological neural networks, coexisting information with varying content, and sharing limited resources and storage capacity. Compared with the hard disk of computer mass storage, it is very clear from the polarity of magnetic particles in the bottom layer, the division of tracks and sectors in the middle layer, to the directory tree and file management system in the high layer, but the understanding of memory is not sufficient. Biological neural networks are abstracted as directed graphs, and the encoding, storage, and retrieval of information within directed graphs at the cellular level are explored. A memory computational model based on active directed graphs and node-adaptive learning is proposed. First, based on neuronal local perspectives, autonomous initiative, limited resource competition, and other neurobiological characteristics, a resource-based adaptive learning algorithm for directed graph nodes is designed. To minimize resource consumption of memory content in directed graphs, two resource-occupancy optimization strategies-lateral inhibition and path pruning-are proposed. Second, this paper introduces a novel memory mechanism grounded in graph theory, which considers connected subgraphs as the physical manifestation of memory content in directed graphs. The encoding, storage, consolidation, and retrieval of the brain's memory system correspond to specific operations such as forming subgraphs, accommodating multiple subgraphs, strengthening connections and connectivity of subgraphs, and activating subgraphs. Lastly, a series of experiments were designed to simulate cognitive processes and evaluate the performance of the directed graph model. Experimental results reveal that the proposed adaptive connectivity learning algorithm for directed graphs in this paper possesses the following four features: (1) Demonstrating distributed, self-organizing, and self-adaptive properties, the algorithm achieves global-level functions through local node interactions; (2) Enabling incremental storage and supporting continuous learning capabilities; (3) Displaying stable memory performance, it surpasses the Hopfield network in memory accuracy, capacity, and diversity, as demonstrated in experimental comparisons. Moreover, it maintains high memory performance with large-scale datasets; (4) Exhibiting a degree of generalization ability, the algorithm's macroscopic performance remains unaffected by the topological structure of the directed graph. Large-scale, decentralized, and node-autonomous directed graphs are suitable simulation methods. Examining storage problems within directed graphs can reveal the essence of phenomena and uncover fundamental storage rules hidden within complex neuronal mechanisms, such as synaptic plasticity, ion channels, neurotransmitters, and electrochemical activities.

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.

Connectomes: from a sparsity of networks to large-scale databases

The analysis of whole brain networks started in the 1980s when only a handful of connectomes were available. In these early days, information about the human connectome was absent and one could only dream about having information about connectivity in a single human subject. Thanks to non-invasive methods such as diffusion imaging, we now know about connectivity in many species and, for some species, in many individuals. To illustrate the rapid change in availability of connectome data, the UK Biobank is on track to record structural and functional connectivity in 100,000 human subjects. Moreover, connectome data from a range of species is now available: from Caenorhabditis elegans and the fruit fly to pigeons, rodents, cats, non-human primates, and humans. This review will give a brief overview of what structural connectivity data is now available, how connectomes are organized, and how their organization shows common features across species. Finally, I will outline some of the current challenges and potential future work in making use of connectome information.

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.

Theory for Durotactic Axon Guidance

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that, at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell's law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

Local CPG Self Growing Network Model with Multiple Physical Properties

Compared with traditional control methods, the advantage of CPG (Central Pattern Generator) network control is that it can significantly reduce the size of the control variable without losing the complexity of its motion mode output. Therefore, it has been widely used in the motion control of robots. To date, the research into CPG network has been polarized: one direction has focused on the function of CPG control rather than biological rationality, which leads to the poor functional adaptability of the control network and means that the control network can only be used under fixed conditions and cannot adapt to new control requirements. This is because, when there are new control requirements, it is difficult to develop a control network with poor biological rationality into a new, qualified network based on previous research; instead, it must be explored again from the basic link. The other direction has focused on the rationality of biology instead of the function of CPG control, which means that the form of the control network is only similar to a real neural network, without practical use. In this paper, we propose some physical characteristics (including axon resistance, capacitance, length and diameter, etc.) that can determine the corresponding parameters of the control model to combine the growth process and the function of the CPG control network. Universal gravitation is used to achieve the targeted guidance of axon growth, Brownian random motion is used to simulate the random turning of axon self-growth, and the signal of a single neuron is established by the Rall Cable Model that simplifies the axon membrane potential distribution. The transfer model, which makes the key parameters of the CPG control network—the delay time constant and the connection weight between the synapses—correspond to the axon length and axon diameter in the growth model and the growth and development of the neuron processes and control functions are combined. By coordinating the growth and development process and control function of neurons, we aim to realize the control function of the CPG network as much as possible under the conditions of biological reality. In this way, the complexity of the control model we develop will be close to that of a biological neural network, and the control network will have more control functions. Finally, the effectiveness of the established CPG self-growth control network is verified through the experiments of the simulation prototype and experimental prototype.

Neural electrical activity and neural network growth

The development of central and peripheral neural system depends in part on the emergence of the correct functional connectivity in its input and output pathways. Now it is generally accepted that molecular factors guide neurons to establish a primary scaffold that undergoes activity-dependent refinement for building a fully functional circuit. However, a number of experimental results obtained recently shows that the neuronal electrical activity plays an important role in the establishing of initial interneuronal connections. Nevertheless, these processes are rather difficult to study experimentally, due to the absence of theoretical description and quantitative parameters for estimation of the neuronal activity influence on growth in neural networks. In this work we propose a general framework for a theoretical description of the activity-dependent neural network growth. The theoretical description incorporates a closed-loop growth model in which the neural activity can affect neurite outgrowth, which in turn can affect neural activity. We carried out the detailed quantitative analysis of spatiotemporal activity patterns and studied the relationship between individual cells and the network as a whole to explore the relationship between developing connectivity and activity patterns. The model, developed in this work will allow us to develop new experimental techniques for studying and quantifying the influence of the neuronal activity on growth processes in neural networks and may lead to a novel techniques for constructing large-scale neural networks by self-organization.

Growth of adult spinal cord in knifefish: Development and parametrization of a distributed model

The study of indeterminate-growing organisms such as teleost fish presents a unique opportunity for improving our understanding of central nervous tissue growth during adulthood. Integrating the existing experimental data associated with this process into a theoretical framework through mathematical or computational modeling provides further research avenues through sensitivity analysis and optimization. While this type of approach has been used extensively in investigations of tumor growth, wound healing, and bone regeneration, the development of nervous tissue has been rarely studied within a modeling framework. To address this gap, the present work introduces a distributed model of spinal cord growth in the knifefish Apteronotus leptorhynchus, an established teleostean model of adult growth in the central nervous system. The proposed model incorporates two mechanisms, cell proliferation by active stem/progenitor cells and cell drift due to population pressure, both of which are subject to global constraints. A coupled reaction-diffusion equation approach was adopted to represent the densities of actively-proliferating and non-proliferating cells along the longitudinal axis of the spinal cord. Computer simulations using this model yielded biologically-feasible growth trajectories. Subsequent comparisons with whole-organism growth curves allowed the estimation of previously-unknown parameters, such as relative growth rates.

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.

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.

A mathematical model explains saturating axon guidance responses to molecular gradients

Correct wiring is crucial for the proper functioning of the nervous system. Molecular gradients provide critical signals to guide growth cones, which are the motile tips of developing axons, to their targets. However, in vitro, growth cones trace highly stochastic trajectories, and exactly how molecular gradients bias their movement is unclear. Here, we introduce a mathematical model based on persistence, bias, and noise to describe this behaviour, constrained directly by measurements of the detailed statistics of growth cone movements in both attractive and repulsive gradients in a microfluidic device. This model provides a mathematical explanation for why average axon turning angles in gradients in vitro saturate very rapidly with time at relatively small values. This work introduces the most accurate predictive model of growth cone trajectories to date, and deepens our understanding of axon guidance events both in vitro and in vivo.

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.

Homeostatic structural plasticity increases the efficiency of small-world networks

In networks with small-world topology, which are characterized by a high clustering coefficient and a short characteristic path length, information can be transmitted efficiently and at relatively low costs. The brain is composed of small-world networks, and evolution may have optimized brain connectivity for efficient information processing. Despite many studies on the impact of topology on information processing in neuronal networks, little is known about the development of network topology and the emergence of efficient small-world networks. We investigated how a simple growth process that favors short-range connections over long-range connections in combination with a synapse formation rule that generates homeostasis in post-synaptic firing rates shapes neuronal network topology. Interestingly, we found that small-world networks benefited from homeostasis by an increase in efficiency, defined as the averaged inverse of the shortest paths through the network. Efficiency particularly increased as small-world networks approached the desired level of electrical activity. Ultimately, homeostatic small-world networks became almost as efficient as random networks. The increase in efficiency was caused by the emergent property of the homeostatic growth process that neurons started forming more long-range connections, albeit at a low rate, when their electrical activity was close to the homeostatic set-point. Although global network topology continued to change when neuronal activities were around the homeostatic equilibrium, the small-world property of the network was maintained over the entire course of development. Our results may help understand how complex systems such as the brain could set up an efficient network topology in a self-organizing manner. Insights from our work may also lead to novel techniques for constructing large-scale neuronal networks by self-organization.

Can simple rules control development of a pioneer vertebrate neuronal network generating behavior?

How do the pioneer networks in the axial core of the vertebrate nervous system first develop? Fundamental to understanding any full-scale neuronal network is knowledge of the constituent neurons, their properties, synaptic interconnections, and normal activity. Our novel strategy uses basic developmental rules to generate model networks that retain individual neuron and synapse resolution and are capable of reproducing correct, whole animal responses. We apply our developmental strategy to young Xenopus tadpoles, whose brainstem and spinal cord share a core vertebrate plan, but at a tractable complexity. Following detailed anatomical and physiological measurements to complete a descriptive library of each type of spinal neuron, we build models of their axon growth controlled by simple chemical gradients and physical barriers. By adding dendrites and allowing probabilistic formation of synaptic connections, we reconstruct network connectivity among up to 2000 neurons. When the resulting "network" is populated by model neurons and synapses, with properties based on physiology, it can respond to sensory stimulation by mimicking tadpole swimming behavior. This functioning model represents the most complete reconstruction of a vertebrate neuronal network that can reproduce the complex, rhythmic behavior of a whole animal. The findings validate our novel developmental strategy for generating realistic networks with individual neuron- and synapse-level resolution. We use it to demonstrate how early functional neuronal connectivity and behavior may in life result from simple developmental "rules," which lay out a scaffold for the vertebrate CNS without specific neuron-to-neuron recognition.

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

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

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.

Modeling the connectome of a simple spinal cord

In this paper we develop a computational model of the anatomy of a spinal cord. We address a long-standing ambition of neuroscience to understand the structure–function problem by modeling the complete spinal cord connectome map in the 2-day old hatchling Xenopus tadpole. Our approach to modeling neuronal connectivity is based on developmental processes of axon growth. A simple mathematical model of axon growth allows us to reconstruct a biologically realistic connectome of the tadpole spinal cord based on neurobiological data. In our model we distribute neuron cell bodies and dendrites on both sides of the body based on experimental measurements. If growing axons cross the dendrite of another neuron, they make a synaptic contact with a defined probability. The total neuronal network contains ∼1,500 neurons of six cell-types with a total of ∼120,000 connections. The anatomical model contains random components so each repetition of the connectome reconstruction procedure generates a different neuronal network, though all share consistent features such as distributions of cell bodies, dendrites, and axon lengths. Our study reveals a complex structure for the connectome with many interesting specific features including contrasting distributions of connection length distributions. The connectome also shows some similarities to connectivity graphs for other animals such as the global neuronal network of C. elegans. In addition to the interesting intrinsic properties of the connectome, we expect the ability to grow and analyze a biologically realistic spinal cord connectome will provide valuable insights into the properties of the real neuronal networks underlying simple behavior.

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.