1
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Staii C. Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns. Biomimetics (Basel) 2023; 8:267. [PMID: 37366862 DOI: 10.3390/biomimetics8020267] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2023] [Revised: 06/07/2023] [Accepted: 06/16/2023] [Indexed: 06/28/2023] Open
Abstract
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.
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Affiliation(s)
- Cristian Staii
- Department of Physics and Astronomy, Tufts University, Medford, MA 02155, 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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3
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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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Liebchen B, Mukhopadhyay AK. Interactions in active colloids. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:083002. [PMID: 34788232 DOI: 10.1088/1361-648x/ac3a86] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Accepted: 11/16/2021] [Indexed: 06/13/2023]
Abstract
The past two decades have seen a remarkable progress in the development of synthetic colloidal agents which are capable of creating directed motion in an unbiased environment at the microscale. These self-propelling particles are often praised for their enormous potential to self-organize into dynamic nonequilibrium structures such as living clusters, synchronized super-rotor structures or self-propelling molecules featuring a complexity which is rarely found outside of the living world. However, the precise mechanisms underlying the formation and dynamics of many of these structures are still barely understood, which is likely to hinge on the gaps in our understanding of how active colloids interact. In particular, besides showing comparatively short-ranged interactions which are well known from passive colloids (Van der Waals, electrostatic etc), active colloids show novel hydrodynamic interactions as well as phoretic and substrate-mediated 'osmotic' cross-interactions which hinge on the action of the phoretic field gradients which are induced by the colloids on other colloids in the system. The present article discusses the complexity and the intriguing properties of these interactions which in general are long-ranged, non-instantaneous, non-pairwise and non-reciprocal and which may serve as key ingredients for the design of future nonequilibrium colloidal materials. Besides providing a brief overview on the state of the art of our understanding of these interactions a key aim of this review is to emphasize open key questions and corresponding open challenges.
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Affiliation(s)
- Benno Liebchen
- Institute for Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
| | - Aritra K Mukhopadhyay
- Institute for Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
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5
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Razetti A, Medioni C, Malandain G, Besse F, Descombes X. A stochastic framework to model axon interactions within growing neuronal populations. PLoS Comput Biol 2018; 14:e1006627. [PMID: 30507939 PMCID: PMC6292646 DOI: 10.1371/journal.pcbi.1006627] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2018] [Revised: 12/13/2018] [Accepted: 11/09/2018] [Indexed: 12/16/2022] Open
Abstract
The confined and crowded environment of developing brains imposes spatial constraints on neuronal cells that have evolved individual and collective strategies to optimize their growth. These include organizing neurons into populations extending their axons to common target territories. How individual axons interact with each other within such populations to optimize innervation is currently unclear and difficult to analyze experimentally in vivo. Here, we developed a stochastic model of 3D axon growth that takes into account spatial environmental constraints, physical interactions between neighboring axons, and branch formation. This general, predictive and robust model, when fed with parameters estimated on real neurons from the Drosophila brain, enabled the study of the mechanistic principles underlying the growth of axonal populations. First, it provided a novel explanation for the diversity of growth and branching patterns observed in vivo within populations of genetically identical neurons. Second, it uncovered that axon branching could be a strategy optimizing the overall growth of axons competing with others in contexts of high axonal density. The flexibility of this framework will make it possible to investigate the rules underlying axon growth and regeneration in the context of various neuronal populations. Understanding how neuronal cells establish complex circuits with specific functions within a developing brain is a major current challenge. Over the last past years, enormous progress has been done to precisely resolve brain anatomy and to dissect the mechanisms controlling the establishment of precise neuronal networks. However, due to the extreme complexity of the brain, it is still experimentally difficult to investigate in vivo how neurons interact with each other and with their physical environments to innervate target territories during development. Here, we have developed a framework that integrates a dynamic 3D mathematical model of single axonal growth with parameters estimated from neurons grown in vivo and simulations of entire populations of growing axons. The emergent properties of our model enable the study of the mechanistic principles underlying the growth of axonal population in developing brains. Specifically, our results highlight the impact of mechanical interactions on both individual and collective axon growth, and uncover how branching regulate this process.
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6
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Neural electrical activity and neural network growth. Neural Netw 2018; 101:15-24. [PMID: 29475142 DOI: 10.1016/j.neunet.2018.02.001] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2017] [Revised: 01/31/2018] [Accepted: 02/01/2018] [Indexed: 01/19/2023]
Abstract
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.
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7
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Paraskevov AV, Zendrikov DK. A spatially resolved network spike in model neuronal cultures reveals nucleation centers, circular traveling waves and drifting spiral waves. Phys Biol 2017; 14:026003. [PMID: 28333685 DOI: 10.1088/1478-3975/aa5fc3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
We show that in model neuronal cultures, where the probability of interneuronal connection formation decreases exponentially with increasing distance between the neurons, there exists a small number of spatial nucleation centers of a network spike, from where the synchronous spiking activity starts propagating in the network typically in the form of circular traveling waves. The number of nucleation centers and their spatial locations are unique and unchanged for a given realization of neuronal network but are different for different networks. In contrast, if the probability of interneuronal connection formation is independent of the distance between neurons, then the nucleation centers do not arise and the synchronization of spiking activity during a network spike occurs spatially uniform throughout the network. Therefore one can conclude that spatial proximity of connections between neurons is important for the formation of nucleation centers. It is also shown that fluctuations of the spatial density of neurons at their random homogeneous distribution typical for the experiments in vitro do not determine the locations of the nucleation centers. The simulation results are qualitatively consistent with the experimental observations.
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Affiliation(s)
- A V Paraskevov
- National Research Centre "Kurchatov Institute", 123182 Moscow, Russia. Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia
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8
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Sergi PN, Cavalcanti-Adam EA. Biomaterials and computation: a strategic alliance to investigate emergent responses of neural cells. Biomater Sci 2017; 5:648-657. [DOI: 10.1039/c6bm00871b] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Synergistic use of biomaterials and computation allows to identify and unravel neural cell responses.
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Affiliation(s)
- Pier Nicola Sergi
- The Biorobotics Institute
- Sant’ Anna Scuola Universitaria Superiore
- Pontedera
- 56025 Italy
| | - Elisabetta Ada Cavalcanti-Adam
- Max Planck Institute for Medical Research
- Dept Cellular Biophysics and Heidelberg University
- Dept Biophysical Chemistry
- Heidelberg
- Germany
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9
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Naoki H, Nishiyama M, Togashi K, Igarashi Y, Hong K, Ishii S. Multi-phasic bi-directional chemotactic responses of the growth cone. Sci Rep 2016; 6:36256. [PMID: 27808115 PMCID: PMC5093620 DOI: 10.1038/srep36256] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2015] [Accepted: 10/12/2016] [Indexed: 11/23/2022] Open
Abstract
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.
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Affiliation(s)
- Honda Naoki
- Graduate School of Medicine, Kyoto University, Sakyo, Kyoto, Japan.,Imaging Platform for Spatio-temporal Information, Kyoto University, Sakyo, Kyoto, Japan
| | - Makoto Nishiyama
- Department of Biochemistry, New York University School of Medicine, New York, USA.,Kasah Technology Inc. New York, New York, USA
| | - Kazunobu Togashi
- Department of Biochemistry, New York University School of Medicine, New York, USA
| | | | - Kyonsoo Hong
- Department of Biochemistry, New York University School of Medicine, New York, USA.,Kasah Technology Inc. New York, New York, USA
| | - Shin Ishii
- Imaging Platform for Spatio-temporal Information, Kyoto University, Sakyo, Kyoto, Japan.,Graduate School of Informatics, Kyoto University, Sakyo, Kyoto, Japan
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10
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Monteiro RLS, Carneiro TKG, Fontoura JRA, da Silva VL, Moret MA, Pereira HBDB. A Model for Improving the Learning Curves of Artificial Neural Networks. PLoS One 2016; 11:e0149874. [PMID: 26901646 PMCID: PMC4763452 DOI: 10.1371/journal.pone.0149874] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2015] [Accepted: 02/06/2016] [Indexed: 11/30/2022] Open
Abstract
In this article, the performance of a hybrid artificial neural network (i.e. scale-free and small-world) was analyzed and its learning curve compared to three other topologies: random, scale-free and small-world, as well as to the chemotaxis neural network of the nematode Caenorhabditis Elegans. One hundred equivalent networks (same number of vertices and average degree) for each topology were generated and each was trained for one thousand epochs. After comparing the mean learning curves of each network topology with the C. elegans neural network, we found that the networks that exhibited preferential attachment exhibited the best learning curves.
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Affiliation(s)
- Roberto L. S. Monteiro
- Programa de Modelagem Computational, SENAI CIMATEC, Av. Orlando Gomes 1845, Salvador, 41.650-010, Brazil
- Universidade do Estado da Bahia, Salvador, Brasil
- * E-mail:
| | | | | | - Valéria L. da Silva
- Programa de Modelagem Computational, SENAI CIMATEC, Av. Orlando Gomes 1845, Salvador, 41.650-010, Brazil
| | - Marcelo A. Moret
- Programa de Modelagem Computational, SENAI CIMATEC, Av. Orlando Gomes 1845, Salvador, 41.650-010, Brazil
- Universidade do Estado da Bahia, Salvador, Brasil
| | - Hernane Borges de Barros Pereira
- Programa de Modelagem Computational, SENAI CIMATEC, Av. Orlando Gomes 1845, Salvador, 41.650-010, Brazil
- Universidade do Estado da Bahia, Salvador, Brasil
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11
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Roccasalvo IM, Sergi PN, Tonazzini I, Cecchini M, Micera S. Topographical strategies to control neural outgrowth. ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. ANNUAL INTERNATIONAL CONFERENCE 2016; 2015:7147-50. [PMID: 26737940 DOI: 10.1109/embc.2015.7320040] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
In this work a synergistic approach is used to investigate how directional anisotropic surfaces (i.e., nanogratings) control the alignment of PC12 neurites. Finite Element models were used to assess the distribution of stresses in non-spread growth cones and filopodia. The stress field was assumed to be the main triggering cause fostering the increase and stabilization of filopodia, so the local stress maxima were directly related to the neuritic orientation. Moreover, a computational framework was implemented within an open source 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 angle, p = 0.4685; tortuosity, p = 0.9075) with a standard level of confidence (95%).
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12
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Sergi PN, Marino A, Ciofani G. Deterministic control of mean alignment and elongation of neuron-like cells by grating geometry: a computational approach. Integr Biol (Camb) 2015; 7:1242-52. [PMID: 26114801 DOI: 10.1039/c5ib00045a] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Abstract
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.
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Affiliation(s)
- Pier Nicola Sergi
- The Biorobotics Institute, Scuola Superiore SantAnna, Viale Rinaldo Piaggio 34, Pontedera, 56025 Italy.
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13
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Roccasalvo IM, Micera S, Sergi PN. A hybrid computational model to predict chemotactic guidance of growth cones. Sci Rep 2015; 5:11340. [PMID: 26086936 PMCID: PMC4471899 DOI: 10.1038/srep11340] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2014] [Accepted: 05/15/2015] [Indexed: 11/08/2022] Open
Abstract
The overall strategy used by growing axons to find their correct paths during the nervous system development is not yet completely understood. Indeed, some emergent and counterintuitive phenomena were recently described during axon pathfinding in presence of chemical gradients. Here, a novel computational model is presented together with its ability to reproduce both regular and counterintuitive axonal behaviours. In this model, the key role of intracellular calcium was phenomenologically modelled through a non standard Gierer-Meinhardt system, as a crucial factor influencing the growth cone behaviour both in regular and complex conditions. This model was able to explicitly reproduce neuritic paths accounting for the complex interplay between extracellular and intracellular environments, through the sensing capability of the growth cone. The reliability of this approach was proven by using quantitative metrics, numerically supporting the similarity between in silico and biological results in regular conditions (control and attraction). Finally, the model was able to qualitatively predict emergent and counterintuitive phenomena resulting from complex boundary conditions.
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Affiliation(s)
| | - Silvestro Micera
- The BioRobotics Institute, Scuola Superiore Sant’Anna, Pisa, Italy
- Bertarelli Foundation Chair in Translational NeuroEngineering Laboratory, Institute of Bioengineering, School of Engineering, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland
- Center for Neuroprosthetics, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland
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14
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Mottini A, Descombes X, Besse F, Pechersky E. Discrete stochastic model for the generation of axonal trees. ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. ANNUAL INTERNATIONAL CONFERENCE 2015; 2014:6814-7. [PMID: 25571561 DOI: 10.1109/embc.2014.6945193] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
In this work we propose a 2D discrete stochastic model for the simulation of axonal biogenesis. The model is defined by a third order Markov Chain. The model considers two main processes: the growth process that models the elongation and shape of the neurites and the bifurcation process that models the generation of branches. The growth process depends, among other variables, on the external attraction field generated by a chemoattractant molecule secreted by the target area. We propose an estimation scheme of the involved parameters from real fluorescent confocal microscopy images of single neurons within intact adult Drosophila fly brains. Both normal neurons and neurons in which certain genes were inactivated have been considered (two mutations). In total, 53 images (18 normal, 21 type 1 mutant and 14 type 2 mutant) were used. The model parameters allow us to describe pathological characteristics of the mutated populations.
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15
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Meyer M, Schimansky-Geier L, Romanczuk P. Active Brownian agents with concentration-dependent chemotactic sensitivity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022711. [PMID: 25353513 DOI: 10.1103/physreve.89.022711] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2013] [Indexed: 06/04/2023]
Abstract
We study a biologically motivated model of overdamped, autochemotactic Brownian agents with concentration-dependent chemotactic sensitivity. The agents in our model move stochastically and produce a chemical ligand at their current position. The ligand concentration obeys a reaction-diffusion equation and acts as a chemoattractant for the agents, which bias their motion towards higher concentrations of the dynamically altered chemical field. We explore the impact of concentration-dependent response to chemoattractant gradients on large-scale pattern formation, by deriving a coarse-grained macroscopic description of the individual-based model, and compare the conditions for emergence of inhomogeneous solutions for different variants of the chemotactic sensitivity. We focus primarily on the so-called receptor-law sensitivity, which models a nonlinear decrease of chemotactic sensitivity with increasing ligand concentration. Our results reveal qualitative differences between the receptor law, the constant chemotactic response, and the so-called log law, with respect to stability of the homogeneous solution, as well as the emergence of different patterns (labyrinthine structures, clusters, and bubbles) via spinodal decomposition or nucleation. We discuss two limiting cases, where the model can be reduced to the dynamics of single species: (I) the agent density governed by a density-dependent effective diffusion coefficient and (II) the ligand field with an effective bistable, time-dependent reaction rate. In the end, we turn to single clusters of agents, studying domain growth and determining mean characteristics of the stationary inhomogeneous state. Analytical results are confirmed and extended by large-scale GPU simulations of the individual based model.
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Affiliation(s)
- Marcel Meyer
- Department of Physics, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Lutz Schimansky-Geier
- Department of Physics, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Pawel Romanczuk
- Physikalisch-Technische Bundesanstalt, Abbestraße 2-12, 10587 Berlin, Germany
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16
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Rizzo DJ, White JD, Spedden E, Wiens MR, Kaplan DL, Atherton TJ, Staii C. Neuronal growth as diffusion in an effective potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042707. [PMID: 24229213 DOI: 10.1103/physreve.88.042707] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2013] [Revised: 09/20/2013] [Indexed: 06/02/2023]
Abstract
Current understanding of neuronal growth is mostly qualitative, as the staggering number of physical and chemical guidance cues involved prohibit a fully quantitative description of axonal dynamics. We report on a general approach that describes axonal growth in vitro, on poly-D-lysine-coated glass substrates, as diffusion in an effective external potential, representing the collective contribution of all causal influences on the growth cone. We use this approach to obtain effective growth rules that reveal an emergent regulatory mechanism for axonal pathfinding on these substrates.
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Affiliation(s)
- Daniel J Rizzo
- Department of Physics and Astronomy, Center for Nanoscopic Physics, Tufts University, Medford, Massachusetts 02155, USA
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17
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Suleymanov Y, Gafarov F, Khusnutdinov N. Modeling of interstitial branching of axonal networks. J Integr Neurosci 2013; 12:103-16. [DOI: 10.1142/s0219635213500064] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
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18
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Zemianek JM, Shultz AM, Lee S, Guaraldi M, Yanco HA, Shea TB. Transient epileptiform signaling during neuronal network development: regulation by external stimulation and bimodal GABAergic activity. Int J Dev Neurosci 2012; 31:131-7. [PMID: 23220177 DOI: 10.1016/j.ijdevneu.2012.11.003] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2012] [Revised: 11/09/2012] [Accepted: 11/10/2012] [Indexed: 01/25/2023] Open
Abstract
A predominance of excitatory activity, with protracted appearance of inhibitory activity, accompanies cortical neuronal development. It is unclear whether or not inhibitory neuronal activity is solicited exclusively by excitatory neurons or whether the transient excitatory activity displayed by developing GABAergic neurons contributes to an excitatory threshold that fosters their conversion to inhibitory activity. We addressed this possibility by culturing murine embryonic neurons on multi-electrode arrays. A wave of individual 0.2-0.4 mV signals ("spikes") appeared between approx. 20-30 days in culture, then declined. A transient wave of high amplitude (>0.5 mV) epileptiform activity coincided with the developmental decline in spikes. Bursts (clusters of ≥3 low-amplitude spikes within 0.7s prior to returning to baseline) persisted following this decline. Addition of the GABAergic antagonist bicuculline initially had no effect on signaling, consistent with delayed development of GABAergic synapses. This was followed by a period in which bicuculline inhibited overall signaling, confirming that GABAergic neurons initially display excitatory activity in ex vivo networks. Following the transient developmental wave of epileptiform signaling, bicuculline induced a resurgence of epileptiform signaling, indicating that GABAergic neurons at this point displayed inhibitory activity. The appearance of transition after the developmental and decline of epileptiform activity, rather than immediately after the developmental decline in lower-amplitude spikes, suggests that the initial excitatory activity of GABAergic neurons contributes to their transition into inhibitory neurons, and that inhibitory GABAergic activity is essential for network development. Prior studies indicate that a minority (25%) of neurons in these cultures were GABAergic, suggesting that inhibitory neurons regulate multiple excitatory neurons. A similar robust increase in signaling following cessation of inhibitory activity in an artificial neural network containing 20% inhibitory neurons supported this conclusion. Even a minor perturbation in GABAergic function may therefore foster initiation and/or amplification of seizure activity, as well as perturbations in long-term potentiation.
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Affiliation(s)
- Jill M Zemianek
- Center for Cellular Neurobiology & Neurodegeneration Research, Department of Biological Sciences, University of Massachusetts Lowell, 01854, USA
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Gritsun TA, le Feber J, Rutten WLC. Growth dynamics explain the development of spatiotemporal burst activity of young cultured neuronal networks in detail. PLoS One 2012; 7:e43352. [PMID: 23028450 PMCID: PMC3447003 DOI: 10.1371/journal.pone.0043352] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2011] [Accepted: 07/23/2012] [Indexed: 11/28/2022] Open
Abstract
A typical property of isolated cultured neuronal networks of dissociated rat cortical cells is synchronized spiking, called bursting, starting about one week after plating, when the dissociated cells have sufficiently sent out their neurites and formed enough synaptic connections. This paper is the third in a series of three on simulation models of cultured networks. Our two previous studies [26], [27] have shown that random recurrent network activity models generate intra- and inter-bursting patterns similar to experimental data. The networks were noise or pacemaker-driven and had Izhikevich-neuronal elements with only short-term plastic (STP) synapses (so, no long-term potentiation, LTP, or depression, LTD, was included). However, elevated pre-phases (burst leaders) and after-phases of burst main shapes, that usually arise during the development of the network, were not yet simulated in sufficient detail. This lack of detail may be due to the fact that the random models completely missed network topology .and a growth model. Therefore, the present paper adds, for the first time, a growth model to the activity model, to give the network a time dependent topology and to explain burst shapes in more detail. Again, without LTP or LTD mechanisms. The integrated growth-activity model yielded realistic bursting patterns. The automatic adjustment of various mutually interdependent network parameters is one of the major advantages of our current approach. Spatio-temporal bursting activity was validated against experiment. Depending on network size, wave reverberation mechanisms were seen along the network boundaries, which may explain the generation of phases of elevated firing before and after the main phase of the burst shape.In summary, the results show that adding topology and growth explain burst shapes in great detail and suggest that young networks still lack/do not need LTP or LTD mechanisms.
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Affiliation(s)
- Taras A Gritsun
- Neural Engineering Department, Institute for Biomedical Engineering MIRA, University of Twente, Enschede, The Netherlands
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Pearson YE, Castronovo E, Lindsley TA, Drew DA. Mathematical modeling of axonal formation. Part I: Geometry. Bull Math Biol 2011; 73:2837-64. [PMID: 21390561 DOI: 10.1007/s11538-011-9648-2] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2010] [Accepted: 02/18/2011] [Indexed: 12/28/2022]
Abstract
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.
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Modeling of Neuronal Growth In Vitro: Comparison of Simulation Tools NETMORPH and CX3D. EURASIP JOURNAL ON BIOINFORMATICS & SYSTEMS BIOLOGY 2011; 2011:616382. [PMID: 21436988 DOI: 10.1155/2011/616382] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2010] [Accepted: 01/13/2011] [Indexed: 11/17/2022]
Abstract
We simulate the growth of neuronal networks using the two recently published tools, NETMORPH and CX3D. The goals of the work are (1) to examine and compare the simulation tools, (2) to construct a model of growth of neocortical cultures, and (3) to characterize the changes in network connectivity during growth, using standard graph theoretic methods. Parameters for the neocortical culture are chosen after consulting both the experimental and the computational work presented in the literature. The first (three) weeks in culture are known to be a time of development of extensive dendritic and axonal arbors and establishment of synaptic connections between the neurons. We simulate the growth of networks from day 1 to day 21. It is shown that for the properly selected parameters, the simulators can reproduce the experimentally obtained connectivity. The selected graph theoretic methods can capture the structural changes during growth.
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22
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A hybrid approach for the control of axonal outgrowth: preliminary simulation results. Med Biol Eng Comput 2010; 49:163-70. [DOI: 10.1007/s11517-010-0687-x] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2009] [Accepted: 09/15/2010] [Indexed: 12/17/2022]
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Koene RA, Tijms B, van Hees P, Postma F, de Ridder A, Ramakers GJA, van Pelt J, van Ooyen A. NETMORPH: A Framework for the Stochastic Generation of Large Scale Neuronal Networks With Realistic Neuron Morphologies. Neuroinformatics 2009; 7:195-210. [PMID: 19672726 DOI: 10.1007/s12021-009-9052-3] [Citation(s) in RCA: 95] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2008] [Accepted: 06/16/2009] [Indexed: 10/20/2022]
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GAFAROV F, KHUSNUTDINOV N, GALIMYANOV F. MORPHOLESS NEURONS COMPROMISE THE DEVELOPMENT OF CORTICAL CONNECTIVITY. J Integr Neurosci 2009; 8:35-48. [DOI: 10.1142/s0219635209002058] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2008] [Revised: 02/05/2009] [Indexed: 01/16/2023] Open
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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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26
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Gafarov FM. SELF-WIRING IN NEURAL NETS OF POINT-LIKE CORTICAL NEURONS FAILS TO REPRODUCE CYTOARCHITECTURAL DIFFERENCES. J Integr Neurosci 2006; 5:159-69. [PMID: 16783866 DOI: 10.1142/s0219635206001136] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2006] [Accepted: 04/03/2006] [Indexed: 11/18/2022] Open
Abstract
We propose a model for description of activity-dependent evolution and self-wiring between binary neurons. Specifically, this model can be used for investigation of growth of neuronal connectivity in the developing neocortex. By using computational simulations with appropriate training pattern sequences, we show that long-term memory can be encoded in neuronal connectivity and that the external stimulations form part of the functioning neocortical circuit. It is proposed that such binary neuron representations of point-like cortical neurons fail to reproduce cytoarchitectural differences of the neocortical organization, which has implications for inadequacies of compartmental models.
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Affiliation(s)
- Fail M Gafarov
- Department of Theoretical Physics, Tatar State University of Humanity and Pedagogic, 420021 Kazan, Mezhlauk Street, 1, Russia.
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Chao ZC, Bakkum DJ, Wagenaar DA, Potter SM. Effects of random external background stimulation on network synaptic stability after tetanization: a modeling study. Neuroinformatics 2006; 3:263-80. [PMID: 16077162 PMCID: PMC2584804 DOI: 10.1385/ni:3:3:263] [Citation(s) in RCA: 42] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
We constructed a simulated spiking neural network model to investigate the effects of random background stimulation on the dynamics of network activity patterns and tetanus induced network plasticity. The simulated model was a "leaky integrate-and-fire" (LIF) neural model with spike-timing-dependent plasticity (STDP) and frequency-dependent synaptic depression. Spontaneous and evoked activity patterns were compared with those of living neuronal networks cultured on multi-electrode arrays. To help visualize activity patterns and plasticity in our simulated model, we introduced new population measures called Center of Activity (CA) and Center of Weights (CW) to describe the spatio-temporal dynamics of network-wide firing activity and network-wide synaptic strength, respectively. Without random background stimulation, the network synaptic weights were unstable and often drifted after tetanization. In contrast, with random background stimulation, the network synaptic weights remained close to their values immediately after tetanization. The simulation suggests that the effects of tetanization on network synaptic weights were difficult to control because of ongoing synchronized spontaneous bursts of action potentials, or "barrages." Random background stimulation helped maintain network synaptic stability after tetanization by reducing the number and thus the influence of spontaneous barrages. We used our simulated network to model the interaction between ongoing neural activity, external stimulation and plasticity, and to guide our choice of sensory-motor mappings for adaptive behavior in hybrid neural-robotic systems or "hybrots."
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Affiliation(s)
- Zenas C. Chao
- Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0535, USA
| | - Douglas J. Bakkum
- Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0535, USA
| | - Daniel A. Wagenaar
- Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
| | - Steve M. Potter
- Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0535, USA
- Author to whom all correspondence and reprint requests should be addressed. E-mail:
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Abstract
An enormous literature has been developed on investigations of the growth and guidance of axons during development and after injury. In this review, we provide a guide to this literature as a resource for biomedical investigators. We first review briefly the molecular biology that is known to regulate migration of the growth cone and branching of axonal arbors. We then outline some important fundamental considerations that are important to the modeling of the phenomenology of these guidance effects and of what is known of their underlying internal mechanisms. We conclude by providing some thoughts on the outlook for future biomedical modeling in the field.
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Affiliation(s)
- Susan Maskery
- Biomedical Informatics, Windber Research Institute, Windber, PA 15963, USA.
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Ben-Jacob E. Bacterial self-organization: co-enhancement of complexification and adaptability in a dynamic environment. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2003; 361:1283-1312. [PMID: 12816612 DOI: 10.1098/rsta.2003.1199] [Citation(s) in RCA: 54] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
During colonial development, bacteria generate a wealth of patterns, some of which are reminiscent of those occurring in abiotic systems. They can exhibit rich behaviour, reflecting informative communication capabilities that include exchange of genetic materials and the fact that the colony's building blocks are biotic. Each has internal degrees of freedom, informatic capabilities and freedom to respond by altering itself and others via emission of signals in a self-regulated manner. To unravel the special secrets of bacterial self-organization, we conducted an integrative (experimental and theoretical) study of abiotic and biotic systems. Guided by the notion of general biotic motives and principles, I propose that the informative communication between individuals makes possible the enhancement of the individuals' regulated freedom, while increasing their cooperation. This process is accomplished via cooperative complexification of the colony through self-organization of hierarchical spatio-temporal patterning. The colonial higher complexity provides the degree of plasticity and flexibility required for better colonial adaptability and endurability in a dynamic environment. The biotic system can modify the environment and obtain environmental information for further self-improvement. I reflect on the potential applications of the new understanding on 'engineered self-organization of systems too complex to design' and other issues.
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Affiliation(s)
- Eshel Ben-Jacob
- School of Physics and Astronomy, Tel-Aviv University, 69978 Tel-Aviv, Israel
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Segev R, Benveniste M, Shapira Y, Ben-Jacob E. Formation of electrically active clusterized neural networks. PHYSICAL REVIEW LETTERS 2003; 90:168101. [PMID: 12732015 DOI: 10.1103/physrevlett.90.168101] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2002] [Indexed: 05/24/2023]
Abstract
Ordinarily, in vitro neurons self-organize into homogeneous networks of single neurons linked by dendrites and axons. We show that under special conditions they can also self-organize into neuronal clusters, which are linked by bundles of axons. Multielectrode array measurement reveals that the clusterized networks are also electrically active and exhibit synchronized bursting events similar to those observed in the homogeneous networks. From time-lapse recording, we deduced the features required for the neuronal clusterized versus homogeneous self-organization and developed a simple model for testing their validity.
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Affiliation(s)
- Ronen Segev
- School of Physics and Astronomy, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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Schweitzer F, Tilch B. Self-assembling of networks in an agent-based model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026113. [PMID: 12241243 DOI: 10.1103/physreve.66.026113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2002] [Indexed: 05/23/2023]
Abstract
We propose a model to show the self-assembling of networklike structures between a set of nodes without using preexisting positional information or long-range attraction of the nodes. The model is based on Brownian agents that are capable of producing different local (chemical) information and respond to it in a nonlinear manner. They solve two tasks in parallel: (i) the detection of the appropriate nodes, and (ii) the establishment of stable links between them. We present results of computer simulations that demonstrate the emergence of robust network structures and investigate the connectivity of the network by means of both analytical estimations and computer simulations.
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Affiliation(s)
- Frank Schweitzer
- Fraunhofer Institute for Autonomous Intelligent Systems, Schloss Birlinghoven, 53754 Sankt Augustin, Germany.
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Shefi O, Golding I, Segev R, Ben-Jacob E, Ayali A. Morphological characterization of in vitro neuronal networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:021905. [PMID: 12241212 DOI: 10.1103/physreve.66.021905] [Citation(s) in RCA: 58] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2002] [Revised: 05/20/2002] [Indexed: 05/23/2023]
Abstract
We use in vitro neuronal networks as a model system for studying self-organization processes in the nervous system. We follow the neuronal growth process, from isolated neurons to fully connected two-dimensional networks. The mature networks are mapped into connected graphs and their morphological characteristics are measured. The distributions of segment lengths, node connectivity, and path length between nodes, and the clustering coefficient of the networks are used to characterize network morphology and to demonstrate that our networks fall into the category of small-world networks.
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Affiliation(s)
- Orit Shefi
- School of Physics and Astronomy, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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Segev R, Ben Jacob E. Chemical waves and internal energy during cooperative self-wiring of neural nets. Neurocomputing 2001. [DOI: 10.1016/s0925-2312(01)00369-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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