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.
Similar content being viewed by others
References
Aletta, J. M., & Greene, L. A. (1998). Growth cone configuration and advance: a time lapse study using video-enhanced differential interference contrast microscopy. J. Neurosci., 8, 1425–1435.
Argiro, V., Bunge, M. B., & Johnson, M. I. (1984). Correlation between growth form and movement and their dependence on neuronal age. J. Neurosci., 4, 3051–3062.
Bastiani, M. J., Raper, J. A., & Goodman, C. S. (1984). Pathfinding by neuronal growth cones in grasshopper embryos. iii. Selective affinity of the g growth cone for the p cells within the a/p fascicle. J. Neurosci., 4, 2311–2328.
Betz, T., Lim, D., & Kas, J. A. (2006). Neuronal growth: a bistable stochastic process. Phys. Rev. Lett., 96, 098103.
Borisyuk, R., Cooke, T., & Roberts, A. (2008). Stochasticity and functionality of neural systems: Mathematical modelling of axon growth in the spinal cord of tadpole. BioSystems, 93, 101–114.
Brodel, P. (1992). The central nervous system: structure and function. New York: Oxford University Press.
Brown, A., Wang, L., & Jung, P. (2005). Stochastic simulation of neurofilament transport in axons: The “stop-and-go” hypothesis. Mol. Biol. Cell, 16, 4243–4255.
Buettner, H. M. (1996). Analysis of cell-target encounter by random filopodial projections. AlChE J., 42(4), 1127.
Chuckowree, J. A., Dickson, T. C., & Vickers, J. C. (2004). Intrinsic regenerative ability of mature cns neurons. Neuroscientist, 10, 280–285.
Craciun, G., Brown, A., & Friedman, A. (2005). A dynamical system model of neurofilament transport in axons. J. Theor. Biol., 237(3), 316–322.
de Curtis, I. (2007). Intracellular mechanisms for neuritogenesis. New York: Springer.
Dotti, C. G., Sullivan, C. A., & Banker, G. A. (1988). The establishment of polarity by hippocampal neurons in culture. J. Neurosci., 8, 1454–1468.
Dunn, G. A., & Brown, A. F. (1987). A unified approach to analysing cell motility. J. Cell Sci., Suppl., 8, 81–108.
Engle, E. C. (2010). Human genetic disorders of axon guidance. Cold Spring Harb. Perspect. Biol., 2(3), a001784.
Godement, P., Wang, L. C., & Mason, C. A. (1994). Retinal axon divergence in the optic chiasm: dynamics of growth cone behavior at the midline. J. Neurosci., 14, 7024–7039.
Goodhill, G. J. (1997). Diffusion in axon guidance. Eur. J. Neurosci., 9, 100–108.
Goodhill, G. J. (1998). Mathematical guidance for axons. Trends Neurosci., 21, 226–231.
Goodhill, G. J., Gu, M., & Urbacj, J. (2004). Predicting axonal response to molecular gradients with a computational model of filopodia dynamics. Neural Comput., 16, 2221–2243.
Goodhill, G. J., Mortimera, D., Feldnera, J., Vaughana, T., Vettera, I., Pujica, Z., Rosoffa, W. J., Burrageb, K., Dayand, P., & Richardsa, L. J. (2009). A Bayesian model predicts the response of axons to molecular gradients. Proc. Natl. Acad. Sci. USA, 106, 296–301.
Grilli, M., Ferrari, G. T., Uberti, D., Spano, P., & Memo, M. (2003). Alzheimer’s disease linking neurodegeneration with neurodevelopment. Funct. Neurol., 18, 145–148.
Harrison, R. (1907). Observations on the living developing nerve fiber. Anat. Rec., 1, 116–128.
Hentschel, H. G. E., & VanOoyen, A. (1999). Models of axon guidance and bundling during development. Proc. R. Soc. B, 266, 2231.
Huber, A. B., Kolodkin, A. L., Ginty, D. D., & Cloutier, J.-F. (2003). Signaling at the growth cone: ligand-receptor complexes and the control of axon growth and guidance. Annu. Rev. Neurosci., 26, 509–563.
Ionides, E. L., Fang, K. S., Isseroff, R. R., & Oster, G. F. (2004). Stochastic models for cell motion and taxis. J. Math. Biol., 48, 23–37.
Kaethner, R. J., & Stuermer, R. J. (1992). Dynamics of terminal arbor formation and target approach of reinotectal axons in living zebrafish embryos: a time-lapse study of single axons. J. Neurosci., 12, 3257–3271.
Katz, M. J., George, E. B., & Gilbert, L. J. (1984). Axonal elongation as a stochastic walk. Cell Motil., 4, 351–370.
Kobayashi, T., Terjima, K., Nozumi, M., Igarashi, M., & Akazawa, K. (2010). A stochastic model of neuronal growth cone guidance regulated by multiple sensors. J. Theor. Biol., 266, 712–722.
Kramer, P. R. (2005). Brownian motion. In A. Scott (Ed.), Encyclopedia of nonlinear science. New York: Routledge.
Krottje, J. K., & Ooyen, A. V. (2007). A mathematical framework for modeling axon guidance. Bull. Math. Biol., 69, 3–31.
Letourneau, P. C. (1982). Nerve fiber growth and its regulation by extrinsic factors. In N. C. Spitzer (Ed.), Neuronal development. New York: Plenum.
Li, G. H., Qin, C. D., & Li, M. H. (1994). On the mechanisms of growth cone locomotion: modeling and computer simulation. J. Theor. Biol., 169, 355–362.
Lindsley, T. A., Kerlin, A. M., & Rising, L. J. (2003). Time-lapse analysis of ethanol’s effects on axon growth in vitro. Dev. Brain Res., 30, 191–199.
Maskery, S., & Shinbrot, T. (2005). Deterministic and stochastic elements of axonal guidance. Annu. Rev. Biomed. Eng., 7, 187–221.
Maskery, S., Buettner, H. M., & Shinbrot, T. (2004). Predicting axonal response to molecular gradients with a computational model of filopodia dynamics. BMC Neurosci., 5, 22.
Odde, D. J., & Buettner, H. M. (1998). Autocorrelation function and power spectrum of two-state random processes used in neurite guidance. Biophys. J., 75, 1189–1196.
Odde, D. J., Tanaka, E. M., Hawkins, S. S., & Buettner, H. M. (1996). Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol. Bioeng., 50, 452–461.
Ornstein, L. S. (1919). On the Brownian motion. Procesnieuws (Amst.), 21, 96–108.
Pearson, Y. (2009). Discrete and continuous stochastic models for neuromorphological data. Rensselaer Polytechnic Institute Library, Dissertation.
Pearson, Y., Drew, D., Castronovo, E., & Lindsley, T. (2011, in preparation). Mathematical modeling of axonal formation; part ii: Temporal Growth.
Segev, R., & Ben-Jacob, E. (2000). Generic modeling of chemotactic based self-wiring of neural networks. Neural Netw., 13, 185–199.
Selmeczi, D., Mosler, S., Hagedorn, P. H., Larsen, N. B., & Flyvbjerg, H. (2005). Cell motility as persistent random motion: theories from experiments. Biophys. J., 89, 912–931.
Shinbrot, T., Maskery, S. M., & Buettner, H. M. (2004). Growth cone pathfinding: a competition between deterministic and stochastic events. BMC Neurosci., 5, 22.
Siegman, A. E. (1979). Simplified derivation of the Fokker Planck equation. Am. J. Phys., 47, 545–547.
Song, H.-J., & Poo, M.-M. (2001). The cell biology of neuronal navigation. Nat. Cell Biol., 3, E81–E87.
VanDemark, K. L., Guizzetti, M., Giordano, G., & Costa, L. G. (2009). Ethanol inhibits muscarinic receptor induced axonal growth in rat hippocampal neurons. Alcohol. Clin. Exp. Res., 33, 1945–1955.
Wang, F.-S., Liu, C.-W., Diefenbach, T. J., & Jay, D. G. (2003). Modeling the role of myosin 1c in neuronal growth cone turning. Biophys. J., 85, 3319–3328.
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Pearson, Y.E., Castronovo, E., Lindsley, T.A. et al. Mathematical Modeling of Axonal Formation Part I: Geometry. Bull Math Biol 73, 2837–2864 (2011). https://doi.org/10.1007/s11538-011-9648-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-011-9648-2