Advertisement

Construction of a 3D Geometric Model of a Presynaptic Bouton for Use in Modeling of Neurotransmitter Flow

  • Andrzej Bielecki
  • Maciej Gierdziewicz
  • Piotr Kalita
  • Kamil Szostek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9972)

Abstract

This paper refers strongly to mathematical modeling of diffusive process in a presynaptic bouton. Creation of a robust three-dimensional model of the bouton geometry is the topic of the paper. Such a model is necessary for partial differential equations that describe the aforementioned flows. The proposed geometric model is based on ultrathin sections obtained by using electron microscopy. The data structure which describes the surface of the whole bouton as well as the surfaces of some internal organelles is created as the result of the modeling procedure.

Keywords

Presynaptic bouton Partial differential equations model Three-dimensional bouton geometry Finite elements method 

Notes

Acknowledgement

The work of Piotr Kalita has been supported by the National Science Center of Poland under the Maestro Advanced Project No. DEC-2012/06/A/ST1/00262.

References

  1. 1.
    AutoDesk 3DS max software. http://www.autodesk.pl/products/3ds-max/
  2. 2.
    Bielecki, A., Kalita, P.: Model of neurotransmitter fast transport in axon terminal of presynaptic neuron. J. Math. Biol. 56, 559–576 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bielecki, A., Kalita, P., Lewandowski, M., Siwek, B.: Numerical simulation for a neurotransmitter transport model in the axon terminal of a presynaptic neuron. Biol. Cybern. 102, 489–502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bielecki, A., Kalita, P., Lewandowski, M., Skomorowski, M.: Compartment model of neuropeptide synaptic transport with impulse control. Biol. Cybern. 99, 443–458 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bielecki, A., Kalita, P.: Dynamical properties of the reaction-diffusion type model of fast synaptic transport. J. Math. Anal. Appl. 393, 329–340 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bobrowski, A.: Boundary conditions in evolutionary equations in biology. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds.) Evolutionary Equations with Applications in Natural Sciences. LNM, vol. 2126, pp. 47–92. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-11322-7_2 Google Scholar
  7. 7.
    Bobrowski, A., Morawska, K.: From a PDE model to an ODE model of dynamics of synaptic depression. Discrete Continuous Dyn. Syst. Ser. B 17, 2313–2327 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bui, L., Glavinovic, M.: Synaptic activity slows vesicular replenishment at excitatory synapses of rat hippocampus. Cogn. Neurodyn. 7, 105–120 (2013)CrossRefGoogle Scholar
  9. 9.
    Bui, L., Glavinovic, M.: Is replenishment of the readily releasable pool associated with vesicular movement? Cogn. Neurodyn. 8, 99–110 (2014)CrossRefGoogle Scholar
  10. 10.
    Bui, L., Glavinovic, M.: Temperature dependence of vesicular dynamics at excitatory synapses of rat hippocampus. Cogn. Neurodyn. 8, 277–286 (2014)CrossRefGoogle Scholar
  11. 11.
    Burger, B., Bettinghausen, S., Hesser, R., Hesser, J.: Real-time GPU-based ultrasound simulation using deformable mesh models. IEEE Trans. Med. Imaging 32(3), 609–618 (2013)CrossRefGoogle Scholar
  12. 12.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems (Classics in Applied Mathematics). 2nd edn. Society for Industrial and Applied Mathematics (SIAM) (2002)Google Scholar
  13. 13.
    Derakhshani, D., Munn, R.L.: Introducing 3ds Max 9: 3D for Beginners. pp. 164–177 (2007). ISBN 9781118058541Google Scholar
  14. 14.
    Hang, S.: TetGen: a quality tetrahedral mesh generator and 3D delaunay triangulator, version 1.4 user manual. WIAS - Weierstrass Institute for Applied Analysis and Stochastics (WIAS) (2006)Google Scholar
  15. 15.
    Hang, S.: TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), (2015). Article 11Google Scholar
  16. 16.
    Hughes, J.F., van Dam, A., McGuire, M., Sklar, D.F., Foley, J.D., Feiner, S.K., Akeley, K.: Computer Graphics: Principles and Practice, 3rd edn. Addison-Wesley Professional, Boston (2013)Google Scholar
  17. 17.
    Knodel, M.M., Geiger, R., Ge, L., Bucher, D., Grillo, A., Wittum, G., Schuster, C., Queisser, G.: Synaptic bouton properties are tuned to best fit the prevailing firing pattern. Front. Comput. Neurosci. 8, Article 101 (2014)Google Scholar
  18. 18.
    Miller, G., Talmor, D., Teng, S.H., Walkington, N., Wang, H.: Control volume meshes using sphere packing: generation, refinement and coarsening. In: Proceedings of the Fifth International Meshing Roundtable, pp. 47–61 (1996)Google Scholar
  19. 19.
    Murray, J.D., Van Ryper, W.: Encyclopedia of Graphics File Formats, 2nd edn. O’Reilly Media, USA (1996)Google Scholar
  20. 20.
    Oñate, E., Rojek, J., Taylor, R., Zienkiewicz, O.: Finite calculus formulation for incompressible solids using linear triangles and tetrahedra. Int. J. Numer. Methods Eng. 59, 1473–1500 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saleewong, T., Srikiatkhachorn, A., Maneepark, M., Chonwerayuth, A., Bongsebandhu-Ghubhakdi, S.: Quantyfying altered long-term potential in the CA1 hippocampus. J. Integr. Neurosci. 11, 243–264 (2012)CrossRefGoogle Scholar
  22. 22.
    Trayanova, N.A.: Whole-heart modeling: applications to cardiac electrophysiology and electromechanics. Circ. Res. 108(1), 113–128 (2011)CrossRefGoogle Scholar
  23. 23.
    Wilhelm, B.G.: Stoichiometric biology of the synapse. Dissertation in partial fulfillment of the requirements for the degree “Doctor of Natural Sciences (Dr. rer. nat)” in the Neuroscience Program at the Georg August University Göttingen, Faculty of Biology, Göttingen, Germany (2013)Google Scholar
  24. 24.
    Wilhelm, B.G., Mandad, S., Truckenbrodt, S., Kröhnert, K., Schäfer, C., Rammner, B., Seong, J.K., Gala, A.C., Krauss, M., Haucke, V., Urlaub, H., Rizzoli, S.O.: Composition of isolated synaptic boutons reveals the amounts of vesicle trafficking proteins. Science 344, 1023–1028 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Andrzej Bielecki
    • 1
  • Maciej Gierdziewicz
    • 1
  • Piotr Kalita
    • 2
  • Kamil Szostek
    • 3
  1. 1.Chair of Applied Computer Science, Faculty of Electrical Engineering, Automation, Computer Science and Biomedical EngineeringAGH University of Science and TechnologyKrakówPoland
  2. 2.Chair of Optimization and Control Theory, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.Chair of Geoinformatics and Applied Computer Science, Faculty of Geology, Geophysics and Environmental ProtectionAGH University of Science and TechnologyKrakówPoland

Personalised recommendations