Three-Dimensional Finite Element Model to Study Calcium Distribution in Astrocytes in Presence of VGCC and Excess Buffer

  • Brajesh Kumar JhaEmail author
  • Amrita Jha
  • Neeru Adlakha
Original Research


The role of astrocytes in physiological processes is always a matter of interest for biologists, mathematicians and computer scientists. Similar to neurons, astrocytes propagate Ca2+ over long distances in response to stimulation and release gliotransmitters in a Ca2+-dependent manner to modulate various important brain functions. There are various processes and parameters that affect the cytoplasmic calcium concentration level of astrocytes like calcium buffering, influx via calcium channels, etc. Buffers bind with calcium ion (Ca2+) and makes calcium bound buffers. Thus, it decreases the calcium concentration [Ca2+] level. Ca2+ enters into the cytosol through voltage gated calcium channel (VGCC) and thus it increases the concentration level. In view of above, a three-dimensional mathematical model is developed for combined study of the effect of buffer and VGCC on cytosolic calcium concentration in astrocytes. Finite element method is applied to find the solution using hexagonal elements. A computer programme is developed for entire problem to simulate the results. The obtained results show that high affinity buffer reveals the effect of VGCC and at low buffer concentration VGCC effects more significantly.


Ca2+ concentration Buffer Voltage gated calcium channel Finite element method 

Mathematics Subject Classification

65L60 62P10 92C35 



  1. 1.
    Deitmer, J.W., Verkhratsky, A.J., Lohr, C.: Calcium signalling in glial cells. Cell Calcium 24, 405–416 (1998)CrossRefGoogle Scholar
  2. 2.
    Verkhratsky, A., Butt, A.: Glial Neurobiology: A Textbook. Wiley, New York (2007)CrossRefGoogle Scholar
  3. 3.
    Fellin, T.: Communication between neuron and astrocytes: relevance to the modulation of synaptic and network activity. J. Neurochem. 108(3), 533–544 (2009)CrossRefGoogle Scholar
  4. 4.
    Nedergaard, M., Rodriguez, J.J., Verkhratsky, A.: Glial calcium and disease of the nervous system. Cell Calcium 47, 140–149 (2010)CrossRefGoogle Scholar
  5. 5.
    Liu, Q.S., Xu, Q., Kang, J., Nedergaard, M.: Astrocyte activation of presynaptic metabotropic glutamate receptors modulates hippocampal inhibitory synaptic transmission. Neuron Glia Biol. 1, 307–316 (2004)CrossRefGoogle Scholar
  6. 6.
    Fiacco, T.A., Agulhon, C., McCarthy, K.D.: Sorting out astrocyte physiology from pharmacology. Annu. Rev. Pharmacol. Toxicol. 49, 151–174 (2009)CrossRefGoogle Scholar
  7. 7.
    Cornell-Bell, A., Finkbeiner, S.M.: Ca2+ waves in astrocytes. Cell Calcium 12, 185–204 (1991)CrossRefGoogle Scholar
  8. 8.
    Zeng, S., Li, B., Zeng, S., Chen, S.: Simulation of spontaneous Ca2+ oscillations in astrocytes mediated by voltage-gated calcium channels. Biophys. J. 97, 2429–2437 (2009)CrossRefGoogle Scholar
  9. 9.
    Jha, A., Adlakha, N., Jha, B.: Finite element model to study effect of Na+–Ca2+ exchangers and source geometry on calcium dynamics in a neuron cell. J. Mech. Med. Biol. 16(2), 1–22 (2015)Google Scholar
  10. 10.
    Jha, A., Adlakha, N.: Finite element model to study the effect of exogenous buffer on calcium dynamics in dendritic spines. Int. J. Model. Simul. Sci. Comput. 5(2), 1–12 (2014)CrossRefGoogle Scholar
  11. 11.
    Jha, A., Adlakha, N.: Analytical solution of two dimensional unsteady state problem of calcium diffusion in a neuron cell. J. Med. Imaging Health Inform. 4(4), 547–553 (2014)CrossRefGoogle Scholar
  12. 12.
    Tewari, S.G., Pardasani, K.R.: Modeling effect of sodium pump on calcium oscillations in neuron cells. J. Multiscale Model. 4(3), 1–16 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tewari, S.G., Pardasani, K.R.: Finite element model to study two dimensional unsteady state cytosolic calcium diffusion in presence of excess buffers. IAENG Int. J. Appl. Math. 40(3), 108–112 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jha, B.K., Adlakha, N., Mehta, M.N.: Two-dimensional finite element model to study calcium distribution in astrocytes in presence of excess buffer. Int. J. Biomath. 7(3), 1–11 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jha, B.K., Adlakha, N., Mehta, M.: Two-dimensional finite element model to study calcium distribution in astrocytes in presence of VGCC and excess buffer. Int. J. Model. Simul. Sci. Comput. 4(2), 1250030-1–1250030-15 (2016)Google Scholar
  16. 16.
    Naik, P.A., Pardasani, K.: Finite element model to study calcium distribution in oocytes involving voltage gated calcium channel, ryanodine receptor and buffers. Alex. J. Med. 52(1), 43–49 (2016)CrossRefGoogle Scholar
  17. 17.
    Naik, P.A., Pardasani, K.: 2D finite element analysis of calcium distribution in oocytes. Netw. Model. Anal. Health Inform. Bioinform. 7, 1–11 (2018)CrossRefGoogle Scholar
  18. 18.
    Kumar, H., Naik, P.A., Pardasani, K.: Finite element model to study calcium distribution in T lymphocyte involving buffers and ryanodine receptors. Proc. Natl. Acad. Sci. India Sect. A 88(4), 585–590 (2018)CrossRefGoogle Scholar
  19. 19.
    Naik, P.A., Pardasani, K.: Three dimensional finite element model to study effect of RyR calcium channel, ER leak and SERCA pump on calcium distribution in oocyte cell. Int. J. Comput. Methods 16(1), 1–19 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Dave, D.D., Jha, B.K.: Analytically depicting the calcium diffusion for Alzheimer’s affected cell. Int. J. Biomath. 11(7), 1–17 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jha, B.K., Joshi, H., Dave, D.D.: Portraying the effect of calcium-binding proteins on cytosolic calcium concentration distribution fractionally in nerve cells. Interdiscip. Sci. Comput. Life Sci. 10(4), 674–685 (2018)CrossRefGoogle Scholar
  22. 22.
    Adler, E., Augustine, G., Duffy, S., Charlton, M.: Alien intracellular calcium chelators attenuate neurotransmitter release at the squid giant synapse. J. Neurosci. 11(6), 1496–1507 (1991)CrossRefGoogle Scholar
  23. 23.
    Wang, Z., Tymianski, M., Jones, O.T., Nedergaard, M.: Impact of calcium buffering on the spatial and temporal characteristics of intercellular calcium signals in astrocytes. J. Neurosc. 17(19), 7359–7371 (1997)CrossRefGoogle Scholar
  24. 24.
    Smith, G.D., Dai, L., Miura, R.M., Sherman, A.: Asymptotic analysis of buffered calcium diffusion near a point source. SIAM J. Appl. Math. 61, 1816–1838 (2000)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Smith, G.D.: Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel. Biophys. J. 71, 3064–3072 (1996)CrossRefGoogle Scholar
  26. 26.
    Keener, J., Sneyd, J.: Mathematical physiology, vol. 8, pp. 53–56. Springer, Berlin (1998)CrossRefGoogle Scholar
  27. 27.
    Verkhratsky, A., Rodríguez, J.J., Parpura, V.: Molecular and cellular endocrinology calcium signalling in astroglia. Mol. Cell. Endocrinol. 353(1–2), 45–56 (2012)CrossRefGoogle Scholar
  28. 28.
    Hofmann, F., Biel, M., Flockerzi, V.: Molecular basis for Ca2+ channel diversity. Annu. Rev. Neurosci. 17, 399–418 (1994)CrossRefGoogle Scholar
  29. 29.
    Huguenard, J.R.: Low threshold calcium currents in central nervous system. Annu. Rev. Physiol. 58, 329–348 (1996)CrossRefGoogle Scholar
  30. 30.
    Macvicar, B.A.: Voltage-dependent calcium channels in glial cells. Science 226, 1345–1347 (1984)CrossRefGoogle Scholar
  31. 31.
    Rao, S.S.: Finite element method in engineering. Books. Elsevier Science and Technology, New York (2004)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of TechnologyPandit Deendayal Petroleum UniversityGandhinagarIndia
  2. 2.Department of Science and HumanitiesIndus UniversityAhmedabadIndia
  3. 3.Department of Applied MathematicsS. V. National Institute of TechnologySuratIndia

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