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Qian K, Pawar A, Liao A, Anitescu C, Webster-Wood V, Feinberg AW, Rabczuk T, Zhang YJ. Modeling neuron growth using isogeometric collocation based phase field method. Sci Rep 2022; 12:8120. [PMID: 35581253 PMCID: PMC9114374 DOI: 10.1038/s41598-022-12073-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 05/05/2022] [Indexed: 11/29/2022] Open
Abstract
We present a new computational framework of neuron growth based on the phase field method and develop an open-source software package called "NeuronGrowth_IGAcollocation". Neurons consist of a cell body, dendrites, and axons. Axons and dendrites are long processes extending from the cell body and enabling information transfer to and from other neurons. There is high variation in neuron morphology based on their location and function, thus increasing the complexity in mathematical modeling of neuron growth. In this paper, we propose a novel phase field model with isogeometric collocation to simulate different stages of neuron growth by considering the effect of tubulin. The stages modeled include lamellipodia formation, initial neurite outgrowth, axon differentiation, and dendrite formation considering the effect of intracellular transport of tubulin on neurite outgrowth. Through comparison with experimental observations, we can demonstrate qualitatively and quantitatively similar reproduction of neuron morphologies at different stages of growth and allow extension towards the formation of neurite networks.
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Affiliation(s)
- Kuanren Qian
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
| | - Aishwarya Pawar
- School of Mechanical Engineering, Purdue University, West Lafayette, 47907, USA
| | - Ashlee Liao
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
| | - Cosmin Anitescu
- Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423, Weimar, Germany
| | - Victoria Webster-Wood
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
- Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
| | - Adam W Feinberg
- Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
- Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA
| | - Timon Rabczuk
- Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423, Weimar, Germany
| | - Yongjie Jessica Zhang
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA.
- Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, 15213, USA.
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2
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Laranjeira S, Pellegrino G, Bhangra KS, Phillips JB, Shipley RJ. In silico framework to inform the design of repair constructs for peripheral nerve injury repair. J R Soc Interface 2022; 19:20210824. [PMID: 35232275 PMCID: PMC8889181 DOI: 10.1098/rsif.2021.0824] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Abstract
Peripheral nerve injuries affect millions of people per year and cause loss of sensation and muscle control alongside chronic pain. The most severe injuries are treated through a nerve autograft; however, donor site morbidity and poor outcomes mean alternatives are required. One option is to engineer nerve replacement tissues to provide a supportive microenvironment to encourage nerve regeneration as an alternative to nerve grafts. Currently, progress is hampered due to a lack of consensus on how to arrange materials and cells in space to maximize rate of regeneration. This is compounded by a reliance on experimental testing, which precludes extensive investigations of multiple parameters due to time and cost limitations. Here, a computational framework is proposed to simulate the growth of repairing neurites, captured using a random walk approach and parameterized against literature data. The framework is applied to a specific scenario where the engineered tissue comprises a collagen hydrogel with embedded biomaterial fibres. The size and number of fibres are optimized to maximize neurite regrowth, and the robustness of model predictions is tested through sensitivity analyses. The approach provides an in silico tool to inform the design of engineered replacement tissues, with the opportunity for further development to multi-cue environments.
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Affiliation(s)
- S. Laranjeira
- UCL Mechanical Engineering, London, UK,UCL Centre for Nerve Engineering, UK
| | | | - K. S. Bhangra
- Department of Pharmacology, UCL School of Pharmacy, London, UK,UCL Centre for Nerve Engineering, UK
| | - J. B. Phillips
- Department of Pharmacology, UCL School of Pharmacy, London, UK,UCL Centre for Nerve Engineering, UK
| | - R. J. Shipley
- UCL Mechanical Engineering, London, UK,UCL Centre for Nerve Engineering, UK
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Abstract
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.
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Affiliation(s)
- Hadrien Oliveri
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
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4
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Oliveri H, Franze K, Goriely A. Theory for Durotactic Axon Guidance. PHYSICAL REVIEW LETTERS 2021; 126:118101. [PMID: 33798338 DOI: 10.1103/physrevlett.126.118101] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 02/17/2021] [Indexed: 06/12/2023]
Abstract
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.
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Affiliation(s)
- Hadrien Oliveri
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Kristian Franze
- Department of Physiology, Development, and Neuroscience, University of Cambridge, Cambridge CB2 3DY, United Kingdom
- Institute of Medical Physics and Micro-Tissue Engineering, Friedrich-Alexander University Erlangen-Nuremberg, Erlangen 91052, Germany
- Max-Planck-Zentrum für Physik und Medizin, Erlangen 91052, Germany
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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Yang X, Sun A, Ju BF, Xu S. A rotary scanning method to evaluate grooves and porosity for nerve guide conduits based on ultrasound microscopy. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2018; 89:073705. [PMID: 30068110 DOI: 10.1063/1.5004783] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2017] [Accepted: 06/21/2018] [Indexed: 06/08/2023]
Abstract
Grooved nerve guide conduits (NGCs) have been effective in the clinical treatment of peripheral nerve injury. They are generally fabricated from a micro-structured spinneret using a spinning process, which easily can cause a variety of pores and morphological deviation. The topography of internal grooves as well as the porosity can greatly influence the therapeutic effect. Traditional optical or scanning electron microscopy (SEM) methods can be used to image the grooves; however, these methods are destructive and require slicing NGCs to prepare specimens suitable for imaging. Moreover, lengthy experiments and large batches of NGCs are required to ensure reliable results from both in vitro experiments and clinical studies. In this paper, a non-destructive method for evaluating the grooves and porosity of NGCs is proposed using ultrasonic imaging combined with rotary scanning and an image analysis algorithm. Two ultrasonic methods were used: a 25-MHz point-focus ultrasonic transducer applied to observe axial cross sections of the conduits and a 100-MHz point-focus ultrasonic transducer to detect large pores caused by defects. Furthermore, a theoretical algorithm for detecting the local porosity of a conduit based on density is proposed. Herein, the proposed acoustic method and traditional optical methods are evaluated and compared. A parameter representing the specific surface area of the internal grooves is introduced and computed for both the optical and acoustic methods, and the relative errors of the computed parameter values for three different NGCs were 7.0%, 7.9%, and 15.3%. The detected location and shape of pores were consistent between the acoustic and optical methods, and greater porosity was observed in the middle of the conduit wall. In this paper, the results of the acoustic and optical methods are presented and the errors relating to the acoustic factors, device characteristics, and image processing method are further analyzed.
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Affiliation(s)
- Xiaoyu Yang
- State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, People's Republic of China
| | - Anyu Sun
- State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, People's Republic of China
| | - Bing-Feng Ju
- State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, People's Republic of China
| | - Shaoning Xu
- Zhejiang Information Institute of Machinery Industry, Hangzhou 310027, People's Republic of China
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Recho P, Jerusalem A, Goriely A. Growth, collapse, and stalling in a mechanical model for neurite motility. Phys Rev E 2016; 93:032410. [PMID: 27078393 DOI: 10.1103/physreve.93.032410] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2015] [Indexed: 06/05/2023]
Abstract
Neurites, the long cellular protrusions that form the routes of the neuronal network, are capable of actively extending during early morphogenesis or regenerating after trauma. To perform this task, they rely on their cytoskeleton for mechanical support. In this paper, we present a three-component active gel model that describes neurites in the three robust mechanical states observed experimentally: collapsed, static, and motile. These states arise from an interplay between the physical forces driven by the growth of the microtubule-rich inner core of the neurite and the acto-myosin contractility of its surrounding cortical membrane. In particular, static states appear as a mechanical balance between traction and compression of these two parallel structures. The model predicts how the response of a neurite to a towing force depends on the force magnitude and recovers the response of neurites to several drug treatments that modulate the cytoskeleton active and passive properties.
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Affiliation(s)
- Pierre Recho
- Mathematical Institute, University of Oxford, Oxford OX26GG, United Kingdom
| | - Antoine Jerusalem
- Department of Engineering Science, University of Oxford, Oxford OX13PJ, United Kingdom
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Oxford OX26GG, United Kingdom
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O'Toole M, Lamoureux P, Miller KE. Measurement of subcellular force generation in neurons. Biophys J 2016; 108:1027-37. [PMID: 25762315 DOI: 10.1016/j.bpj.2015.01.021] [Citation(s) in RCA: 48] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2014] [Revised: 01/19/2015] [Accepted: 01/23/2015] [Indexed: 11/24/2022] Open
Abstract
Forces are important for neuronal outgrowth during the initial wiring of the nervous system and after trauma, yet subcellular force generation over the microtubule-rich region at the rear of the growth cone and along the axon has never, to our knowledge, been directly measured. Because previous studies have indicated microtubule polymerization and the microtubule-associated proteins Kinesin-1 and dynein all generate forces that push microtubules forward, a major question is whether the net forces in these regions are contractile or expansive. A challenge in addressing this is that measuring local subcellular force generation is difficult. Here we develop an analytical mathematical model that describes the relationship between unequal subcellular forces arranged in series within the neuron and the net overall tension measured externally. Using force-calibrated towing needles to measure and apply forces, in combination with docked mitochondria to monitor subcellular strain, we then directly measure force generation over the rear of the growth cone and along the axon of chick sensory neurons. We find the rear of the growth cone generates 2.0 nN of contractile force, the axon generates 0.6 nN of contractile force, and that the net overall tension generated by the neuron is 1.3 nN. This work suggests that the forward bulk flow of the cytoskeletal framework that occurs during axonal elongation and growth-cone pauses arises because strong contractile forces in the rear of the growth cone pull material forward.
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Affiliation(s)
- Matthew O'Toole
- Department of Mathematics, Kettering University, Flint, Michigan
| | - Phillip Lamoureux
- Department of Zoology, Michigan State University, East Lansing, Michigan
| | - Kyle E Miller
- Department of Zoology, Michigan State University, East Lansing, Michigan.
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8
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Abstract
Chemotaxis plays a crucial role in many biological processes, including nervous system development. However, fundamental physical constraints limit the ability of a small sensing device such as a cell or growth cone to detect an external chemical gradient. One of these is the stochastic nature of receptor binding, leading to a constantly fluctuating binding pattern across the cell's array of receptors. This is analogous to the uncertainty in sensory information often encountered by the brain at the systems level. Here we derive analytically the Bayes-optimal strategy for combining information from a spatial array of receptors in both one and two dimensions to determine gradient direction. We also show how information from more than one receptor species can be optimally integrated, derive the gradient shapes that are optimal for guiding cells or growth cones over the longest possible distances, and illustrate that polarized cell behavior might arise as an adaptation to slowly varying environments. Together our results provide closed-form predictions for variations in chemotactic performance over a wide range of gradient conditions.
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Affiliation(s)
- Duncan Mortimer
- Queensland Brain Institute, University of Queensland, St. Lucia QLD 4072, Australia.
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9
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Chetta J, Kye C, Shah SB. Cytoskeletal dynamics in response to tensile loading of mammalian axons. Cytoskeleton (Hoboken) 2010; 67:650-65. [DOI: 10.1002/cm.20478] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Causin P, Facchetti G. Autocatalytic loop, amplification and diffusion: a mathematical and computational model of cell polarization in neural chemotaxis. PLoS Comput Biol 2009; 5:e1000479. [PMID: 19714204 PMCID: PMC2722090 DOI: 10.1371/journal.pcbi.1000479] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2008] [Accepted: 07/21/2009] [Indexed: 12/11/2022] Open
Abstract
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.
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Affiliation(s)
- Paola Causin
- Department of Mathematics F Enriques, Università degli Studi di Milano, Milano, Italy.
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11
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Abstract
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.
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Aletti G, Causin P. Mathematical characterisation of the transduction chain in growth cone pathfinding. IET Syst Biol 2008; 2:150-61. [PMID: 18537455 DOI: 10.1049/iet-syb:20070059] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022] Open
Abstract
Axon guidance by graded diffusible ligands plays an important role in the developing nervous system. Concentration gradients induce an asymmetric localisation of molecules in the axon tip, the growth cone, and the consequent internal polarised signalling pathway leads to rearrangement of the growth cone cytoskeleton and, ultimately, to motility. Here the authors provide a mathematical description of the growth cone transduction chain as a series of functional boxes characterised by input/output relations. The model relies on the assumption that the characteristic time of independent concentration measures by growth cone receptors, the characteristic time of growth cone internal reorganisation preceding motion and the characteristic time needed for a discernible axon turning belong to separated scales. The results give insight into the deterministic against stochastic regime of internal growth cone functions that are not readily accessible from experimental observations, pointing out a substantial equilibrium of the two contributions. The mathematical model predicts the decrease of the coefficient of variation of the signal moving down the functional chain leading to motion. Moreover, possible mechanisms that allow for buffering against noise are highlighted. These results have an interest also for the more experimentally minded reader, since they can be used to predict sample sizes for detecting significant differences in benchmark gradient assays.
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Affiliation(s)
- G Aletti
- Università degli Studi di Milano, Dipartimento di Matematica F. Enriques, Milano, Italy.
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13
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A physical model of axonal elongation: force, viscosity, and adhesions govern the mode of outgrowth. Biophys J 2008; 94:2610-20. [PMID: 18178646 DOI: 10.1529/biophysj.107.117424] [Citation(s) in RCA: 99] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Whether the axonal framework is stationary or moves is a central debate in cell biology. To better understand this problem, we developed a mathematical model that incorporates force generation at the growth cone, the viscoelastic properties of the axon, and adhesions between the axon and substrate. Using force-calibrated needles to apply and measure forces at the growth cone, we used docked mitochondria as markers to monitor movement of the axonal framework. We found coherent axonal transport that decreased away from the growth cone. Based on the velocity profiles of movement and the force applied at the growth cone, and by varying the attachment of the axonal shaft to the coverslip, we estimate values for the axial viscosity of the axon (3 x 10(6) +/- 2.4 x 10(6) Pa.s) and the friction coefficient for laminin/polyornithine-based adhesions along the axon (9.6 x 10(3) +/- 7.5 x 10(3) Pa.s). Our model suggests that whether axons elongate by tip growth or stretching depends on the level of force generation at the growth cone, the viscosity of the axon, and the level of adhesions along the axon.
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Graham BP, van Ooyen A. Mathematical modelling and numerical simulation of the morphological development of neurons. BMC Neurosci 2006; 7 Suppl 1:S9. [PMID: 17118163 PMCID: PMC1679805 DOI: 10.1186/1471-2202-7-s1-s9] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/02/2022] Open
Abstract
Background The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other. Methods A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described. Results Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development. Conclusion A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.
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Affiliation(s)
- Bruce P Graham
- Department of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, UK
| | - Arjen van Ooyen
- Department of Experimental Neurophysiology, Vrije Universiteit, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands
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