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

Borisyuk, Roman; Cooke, Tom; Roberts, Alan

doi:10.1016/j.biosystems.2008.03.012

Cited by 26 publications

(37 citation statements)

References 27 publications

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“…Motivations to model the growth cone cover formation of the extracellular gradient that the growth cone senses427273, axonal pathfinding by the gradient cone74757677787980, the growth cone movement in three-dimensional space25818283, axonal specification during neuronal polarization8485868788 and the gradient sensing based on intracellular reaction23257489 or on Bayesian information approach9091.…”

Section: Discussionmentioning

confidence: 99%

Multi-phasic bi-directional chemotactic responses of the growth cone

Naoki

Nishiyama

Togashi

et al. 2016

Sci Rep

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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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Section: Discussionmentioning

confidence: 99%

Multi-phasic bi-directional chemotactic responses of the growth cone

Naoki

Nishiyama

Togashi

et al. 2016

Sci Rep

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“…Similar to previous methods, the formulation of the model is guided by experimental data. This approach is useful because the model is simple enough to allow for the analytical calculations as a means to compare experimental findings through optimal parameter estimates (Borisyuk et al 2008). This type of model also allows for the simulation of axons in a network assigning synapses under strict probabilistic rules, ultimately leading to the reconstruction of various anatomical architectures such as the spinal cord.…”

Section: Introductionmentioning

confidence: 99%

“…This highlights the importance of using experimental systems that permit detailed examination of typical axon behaviors, such as saltatory growth (Odde et al 1996) characterized by alternating periods of extension and retraction, or axon turning in homogeneous environments and in response to localized guidance cues. Computational models based on axon behaviors in model systems can be extremely helpful in testing hypotheses regarding the rules that account for axonogenesis leading to functional neural assemblies Borisyuk et al 2008;Krottje and Ooyen 2007;Maskery and Shinbrot 2005).…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modeling of Axonal Formation Part I: Geometry

et al. 2011

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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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“…In another model that simulates axon growth in the tadpole spinal cord [126], the axon growth angle depends on the tendency to turn towards the gradient angle and noise. The position of the growth cone at timestep n is (x n , y n ) and the heading direction of the n th step is θ n .…”

Section: Random Walk Models Of Growth Cone Trajectoriesmentioning

confidence: 99%

“…Li et al simulated trajectories by assuming the turning angle of the growth cone is in proportion to the angle between the neurite and the resultant filopodial tension [131]. In [126], the axon growth angle depends on the tendency to turn towards the gradient angle and noise. The noise term is small (2-5 • ), leading to straight paths that resemble axon growth in the tadpole spinal cord.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modelling of axon guidance in chemical gradients

Nguyen¹

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Correct wiring is crucial to the proper functioning of the nervous system. Key signals guiding growth cones, the motile tips of developing axons, to their targets are molecular gradients. The receptors on the membrane of the growth cone bind stochastically with guidance molecules in the environment, giving an estimate of the direction of the gradient. The growth cone then executes biased random movements to navigate toward the chemoattractant source or away from the chemorepellent source. This thesis addresses two questions: 1) how the positioning and diffusion of receptors on the growth cone membrane affect the accuracy of gradient estimation, 2) how trajectories are influenced by chemical gradients.Membrane receptors can diffuse freely on the membrane, smearing out the directional information about the gradient. It was unknown how the positioning and diffusion of receptors affect the estimate of the gradient. To address question (1) above, we utilise an ideal-observer approach, assuming that the growth cone can perform maximum likelihood estimation of the gradient based on the stochastic binding patterns with ligand molecules. The performance of gradient sensing is measured by the Fisher Information between the binding pattern and the gradient direction. The quality of gradient sensing decreases with higher diffusion constant of the receptors and improves with higher concentration. We then extend to a two-dimensional model of an elliptical growth cone with a general prior distribution. With a random uniform distribution of receptors, the shape of the growth cone can introduce bias in the gradient estimate. This bias can be corrected by a non-uniform distribution of receptors with higher density near the minor axis of the growth cone.Besides stochastic gradient sensing, growth cones also trace highly stochastic trajectories, and it is unclear how molecular gradients bias their movement. We then introduce a mathematical model of a correlated random walk based on persistence, bias and noise to describe growth cone trajectories, constrained directly by measurements of the detailed statistics of growth cone movements in both attractive and repulsive gradients in a microfluidic device. This model explains the long-standing mystery that average axon turning angles in gradients in vitro plateau very rapidly with time at relatively small values of 10-20• . This work introduces the most accurate predictive model of growth cone trajectories to date, and calls into question the ability of molecular gradients alone to provide reliable guidance cues for growing axons.ii Declaration by AuthorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

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

Cited by 26 publications

References 27 publications

Multi-phasic bi-directional chemotactic responses of the growth cone

Multi-phasic bi-directional chemotactic responses of the growth cone

Mathematical Modeling of Axonal Formation Part I: Geometry

Mathematical modelling of axon guidance in chemical gradients

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