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Year 2020, Volume: 41 Issue: 1, 56 - 68, 22.03.2020

Mehmet Kocabıyık , Mevlüde Yakıt Ongun

https://doi.org/10.17776/csj.461819

Abstract

References

  • [1] Leffler S.R, Legue E., Aristizábal O., Joyner A. L., Peskin C. S. and Turnbull D. H., A Mathematical Model of Granule Cell Generation During Mouse Cerebellum Development. Bulletin of mathematical biology, 78(5) (2016) 859-878.
  • [2] Legue E., Elyn R. and Alexandra L. J., Clonal Analysis Reveals Granule Cell Behaviors and Compartmentalization that Determine the Folded Morphology of the Cerebellum. Development, 142 (9) (2015), 1661-1671.
  • [3] Mares V. and Lodin Z., The Cellular Kinetics of the Developing Mouse Cerebellum II. The Function of the External Granular Layer in the Process of Gyrification. Brain research, 23(3) (1970), 343-352.
  • [4] Espinosa J. S. and Liqun L., Timing Neurogenesis and Differentiation: Insights from Quantitative Clonal Analyses of Cerebellar Granule Cells. The Journal of Neuroscience, 28(10) (2008) 2301-2312.
  • [5] Fujita S., Quantitative Analysis of Cell Proliferation and Differentiation in the Cortex of the Postnatal Mouse Cerebellum. The Journal of Cell Biology, 32(2) (1967), 277-287.
  • [6] Goldowitz D., Cushing R. C., Laywell E., D’Arcangelo G., Sheldon M., Sweet H. O. and Curran T., Cerebellar Disorganization Characteristic of Reeler in Scrambler Mutant Mice Despite Presence of Reelin. The Journal of Neuroscience, 17(22) (1997) 8767-8777.
  • [7] Haddara M. A. and Nooreddin M. A., A Quantitative Study on the Postnatal Development of the Cerebellar Vermis of Mouse. Journal of Comparative Neurology, 128(2) (1966), 245-253.
  • [8] Hatten M. E., Rifkin D. B., Furie M. B., Mason C. A. and Liem R. K., Biochemistry of Granule Cell Migration in Developing Mouse Cerebellum. Progress in clinical and biological research, 85 (1981), 509-519.
  • [9] Seil F. J. and Robert M. H., Cerebellar Granule Cells in Vitro a Light and Electron Microscope Study. The Journal of cell biology, 45(2) (1970), 212-220.
  • [10] Szulc K. U., 4D MEMRI Atlas of Neonatal FVB/N Mouse Brain Development. Neuroimage, 118 (2015) 49-62.
  • [11] Atay M. T., Aytekin E. and Sure K., Magnus Series Expansion Method for Solving Nonhomogeneous Stiff Systems of Ordinary Differential Equations. Kuwait Journal of Science, 43(1) (2015).
  • [12] Bluman G. W. and Kumei S., Symmetries and Differential Equations. Springer, New York,1989.
  • [13] Bluman G., Invariant Solutions for Ordinary Differential Equations, SIAM Journal on Applied Mathematics, 50(6 ) (1990), 1706-1715.
  • [14] Cheviakov A. F., Gem Software Package for Computation of Symmetries and Conservation Laws of Differential Equations, Computer Physics Communications, 176(1) (2007), 48-61.
  • [15] Cheviakov A. F., Computation of fluxes of conservation laws. Journal of Engineering Mathematics, 66(1) (2010), 153-173.
  • [16] Cheviakov A. F., Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations. Mathematics in Computer Science, 4(2) (2010-2), 203-222.
  • [17] Cohen A., An Introduction to the Lie Theory of One-Parameter Groups with Applications to The Solutions of Differential Equations, D.C. Heath Co. Publishers,1911.
  • [18] Edwards M. and Nucci M. C., Application of Lie Group Analysis to a Core Group Model for Sexually Transmitted Diseases. Journal of Nonlinear Mathematical Physics, 13(2) (2006), 211-230.
  • [19] Hyden P.E., Symmetry Methods for Differential Equations (A Beginner’s Guide), Cambridge Texts in Applied Mathematics, 2000.
  • [20] Ibragimov N. H. and Nucci M.C., Integration of Third Order Ordinary Differential Equations by Lie’s Method: Equations Admitting Three-dimensional Lie algebras. Lie Groups and Their Applications, 1 (1994), 49-64.
  • [21] Ibragimov N. H., Selected Works. Vol. 1, 2. Karlskrona, Sweden: Alga Publications, Blekinge Institute of Technology, 2001.
  • [22] Kocabıyık M., Lie Symmetry Analysis for Differential Equations and Differential Equation Systems, Master Thesis, Suleyman Demirel University, 2017.
  • [23] Kule M., Controllability of Affine Control Systems on Graded Lie groups. Kuwait Journal of Science, 43(1) (2015).
  • [24] Oliver P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York,1986.
  • [25] Ongun M. Y. and Kocabıyık M., Lie Symmetry Analysis of the Hanta-Epidemic Systems. Journal of Mathematics and Computer Science-JMCS, 17(2) (2017), 332-344.
  • [26] Ovsiannikov L.V., Group Analysis of Differential Equations. Academic Press, New York,1982.
  • [27] Torrisi V. and Nucci M. C., Application of Lie Group Analysis to a Mathematical Model. The Geometrical Study of Differential Equations: NSF-CBMS Conference on the Geometrical Study of Differential Equations, June 20-25, 2000, Howard University, Washington. Vol. 285. American Mathematical Soc, 2000.

Analysis of granule cell generation system by lie symmetry method

Year 2020, Volume: 41 Issue: 1, 56 - 68, 22.03.2020

Mehmet Kocabıyık , Mevlüde Yakıt Ongun

https://doi.org/10.17776/csj.461819

Abstract

In this paper, we study main headlines of brain development which is a major problem in neurobiology in present. Our aim here is to find the analytical solution of the equation that belongs to the brain development system. For this solution, the exchange of cerebellum granule cells in EGL (External granule layer of the cerebellum) is discussed. For this reason, Lie symmetry analysis is used. Obtaining solutions of this system means determining the behavior of the granular cell numbers at different stages. Knowing the behavior of these cells provides important information about the progression and development of diseases. Examples of these diseases are abnormal cerebellum development, cerebellum cancer. In this study, time dependent probability function related to division of two granule cells is examined. Then, analytical solutions are obtained for three different states of this function. Some tables and density graphics of these solutions are given.

Keywords

Lie symmetry Ordinary differantial equation Granule cell generation Cerebellum cancer.

References

  • [1] Leffler S.R, Legue E., Aristizábal O., Joyner A. L., Peskin C. S. and Turnbull D. H., A Mathematical Model of Granule Cell Generation During Mouse Cerebellum Development. Bulletin of mathematical biology, 78(5) (2016) 859-878.
  • [2] Legue E., Elyn R. and Alexandra L. J., Clonal Analysis Reveals Granule Cell Behaviors and Compartmentalization that Determine the Folded Morphology of the Cerebellum. Development, 142 (9) (2015), 1661-1671.
  • [3] Mares V. and Lodin Z., The Cellular Kinetics of the Developing Mouse Cerebellum II. The Function of the External Granular Layer in the Process of Gyrification. Brain research, 23(3) (1970), 343-352.
  • [4] Espinosa J. S. and Liqun L., Timing Neurogenesis and Differentiation: Insights from Quantitative Clonal Analyses of Cerebellar Granule Cells. The Journal of Neuroscience, 28(10) (2008) 2301-2312.
  • [5] Fujita S., Quantitative Analysis of Cell Proliferation and Differentiation in the Cortex of the Postnatal Mouse Cerebellum. The Journal of Cell Biology, 32(2) (1967), 277-287.
  • [6] Goldowitz D., Cushing R. C., Laywell E., D’Arcangelo G., Sheldon M., Sweet H. O. and Curran T., Cerebellar Disorganization Characteristic of Reeler in Scrambler Mutant Mice Despite Presence of Reelin. The Journal of Neuroscience, 17(22) (1997) 8767-8777.
  • [7] Haddara M. A. and Nooreddin M. A., A Quantitative Study on the Postnatal Development of the Cerebellar Vermis of Mouse. Journal of Comparative Neurology, 128(2) (1966), 245-253.
  • [8] Hatten M. E., Rifkin D. B., Furie M. B., Mason C. A. and Liem R. K., Biochemistry of Granule Cell Migration in Developing Mouse Cerebellum. Progress in clinical and biological research, 85 (1981), 509-519.
  • [9] Seil F. J. and Robert M. H., Cerebellar Granule Cells in Vitro a Light and Electron Microscope Study. The Journal of cell biology, 45(2) (1970), 212-220.
  • [10] Szulc K. U., 4D MEMRI Atlas of Neonatal FVB/N Mouse Brain Development. Neuroimage, 118 (2015) 49-62.
  • [11] Atay M. T., Aytekin E. and Sure K., Magnus Series Expansion Method for Solving Nonhomogeneous Stiff Systems of Ordinary Differential Equations. Kuwait Journal of Science, 43(1) (2015).
  • [12] Bluman G. W. and Kumei S., Symmetries and Differential Equations. Springer, New York,1989.
  • [13] Bluman G., Invariant Solutions for Ordinary Differential Equations, SIAM Journal on Applied Mathematics, 50(6 ) (1990), 1706-1715.
  • [14] Cheviakov A. F., Gem Software Package for Computation of Symmetries and Conservation Laws of Differential Equations, Computer Physics Communications, 176(1) (2007), 48-61.
  • [15] Cheviakov A. F., Computation of fluxes of conservation laws. Journal of Engineering Mathematics, 66(1) (2010), 153-173.
  • [16] Cheviakov A. F., Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations. Mathematics in Computer Science, 4(2) (2010-2), 203-222.
  • [17] Cohen A., An Introduction to the Lie Theory of One-Parameter Groups with Applications to The Solutions of Differential Equations, D.C. Heath Co. Publishers,1911.
  • [18] Edwards M. and Nucci M. C., Application of Lie Group Analysis to a Core Group Model for Sexually Transmitted Diseases. Journal of Nonlinear Mathematical Physics, 13(2) (2006), 211-230.
  • [19] Hyden P.E., Symmetry Methods for Differential Equations (A Beginner’s Guide), Cambridge Texts in Applied Mathematics, 2000.
  • [20] Ibragimov N. H. and Nucci M.C., Integration of Third Order Ordinary Differential Equations by Lie’s Method: Equations Admitting Three-dimensional Lie algebras. Lie Groups and Their Applications, 1 (1994), 49-64.
  • [21] Ibragimov N. H., Selected Works. Vol. 1, 2. Karlskrona, Sweden: Alga Publications, Blekinge Institute of Technology, 2001.
  • [22] Kocabıyık M., Lie Symmetry Analysis for Differential Equations and Differential Equation Systems, Master Thesis, Suleyman Demirel University, 2017.
  • [23] Kule M., Controllability of Affine Control Systems on Graded Lie groups. Kuwait Journal of Science, 43(1) (2015).
  • [24] Oliver P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York,1986.
  • [25] Ongun M. Y. and Kocabıyık M., Lie Symmetry Analysis of the Hanta-Epidemic Systems. Journal of Mathematics and Computer Science-JMCS, 17(2) (2017), 332-344.
  • [26] Ovsiannikov L.V., Group Analysis of Differential Equations. Academic Press, New York,1982.
  • [27] Torrisi V. and Nucci M. C., Application of Lie Group Analysis to a Mathematical Model. The Geometrical Study of Differential Equations: NSF-CBMS Conference on the Geometrical Study of Differential Equations, June 20-25, 2000, Howard University, Washington. Vol. 285. American Mathematical Soc, 2000.
There are 27 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Mehmet Kocabıyık

Mevlüde Yakıt Ongun

Publication Date March 22, 2020
Submission Date September 20, 2018
Acceptance Date February 24, 2020
Published in Issue Year 2020Volume: 41 Issue: 1

Cite

APA Kocabıyık, M., & Yakıt Ongun, M. (2020). Analysis of granule cell generation system by lie symmetry method. Cumhuriyet Science Journal, 41(1), 56-68. https://doi.org/10.17776/csj.461819