Fractional Reaction Diffusion Model for Parkinson’s Disease

  • Hardik JoshiEmail author
  • Brajesh Kumar Jha
Conference paper
Part of the Lecture Notes in Computational Vision and Biomechanics book series (LNCVB, volume 30)


Calcium (Ca2+) ion known as a second messenger, involve in variety of signalling process, and directly link with the intracellular calcium concentration ([Ca2+]) that are continuously remodelled for the survival of the nerve cell. Buffer, also refer as a protein, react with Ca2+ and significantly lower down the intracellular [Ca2+] in nerve cell. There are numerous signalling processes in mammalian brain which can initiate at the high level of intracellular [Ca2+]. Voltage gated calcium channel (VGCC), and ryanodine receptor (RyR) are work as an outward source of Ca2+ which initiate, and sustain the signalling process for smooth functioning of the cells. Parkinson’s disease (PD) is a brain disorder of the central nervous system accompanied with the alteration of the signalling process. In present paper, a one dimensional fractional reaction diffusion model is consider to understand the physiological role of buffer, VGCC, and RyR in view of the PD.


Fractional approach Buffer Voltage gated calcium channel Ryanodine receptor Calcium concentration Parkinson’s disease 



We are very grateful to the reviewers for their fruitful comments and suggestions for making updation in the research article.


  1. 1.
    Surmeier DJ, Schumacker PT, Guzman JD, Ilijic E, Yang B, Zampese E (2017) Calcium and parkinson’s disease. Biochem Biophys Res Commun. 483:1013–1019CrossRefGoogle Scholar
  2. 2.
    Jha BK, Joshi H, Dave DD (2016) Portraying the effect of calcium-binding proteins on cytosolic calcium concentration distribution fractionally in nerve cells. Interdiscip SciGoogle Scholar
  3. 3.
    Zaichick SV, McGrath KM, Caraveo G (2017) The role of Ca2+ signaling in parkinson’s disease. Dis Model Mech 10:519–535CrossRefGoogle Scholar
  4. 4.
    Jha BK, Adlakha N, Mehta MN (2014) Two-dimensional finite element model to study calcium distribution in astrocytes in presence of excess buffer. Int J Biomath 7:1–11MathSciNetCrossRefGoogle Scholar
  5. 5.
    Schmidt H (2012) Three functional facets of calbindin D-28k. Front Mol Neurosci 5:25CrossRefGoogle Scholar
  6. 6.
    Jha BK, Adlakha N, Mehta MN (2013) Two-dimensional finite element model to study calcium distribution in astrocytes in presence of VGCC and excess buffer. Int J Model Simulation, Sci Comput 4Google Scholar
  7. 7.
    Jha A, Adlakha N (2014) Analytical solution of two dimensional unsteady state problem of calcium diffusion in a neuron cell. J Med Imaging Heal Informatics 4:547–553CrossRefGoogle Scholar
  8. 8.
    Naik PA, Pardasani KR (2015) One dimensional finite element model to study calcium distribution in oocytes in presence of VGCC, RyR and buffers. J Med Imaging Heal Informatics 5:471–476CrossRefGoogle Scholar
  9. 9.
    Tewari SG, Pardasani KR (2010) Finite element model to study two dimensional unsteady state cytosolic calcium diffusion in presence of excess buffers. IAENG Int J Appl Mathe 40:40-3-01Google Scholar
  10. 10.
    Panday S, Pardasani KR (2014) Finite element model to study the mechanics of calcium regulation in Oocyte. J Mech Med Biol 14:1450022-1-16 (2014)CrossRefGoogle Scholar
  11. 11.
    Kotwani M, Adlakha N, Mehta MN (2014) Finite element model to study the effect of buffers, source amplitude and source geometry on spatio-temporal calcium distribution in fibroblast cell. J Med Imaging Health Inform. 4:840–847CrossRefGoogle Scholar
  12. 12.
    Pathak K, Adlakha N (2016) Finite element model to study two dimensional unsteady state calcium distribution in cardiac myocytes. Alexandria J Med 52:261–268CrossRefGoogle Scholar
  13. 13.
    Dave DD, Jha BK (2018) Delineation of calcium diffusion in Alzheimeric brain. J Mech Med Biol 18:1850028-1-15CrossRefGoogle Scholar
  14. 14.
    Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59:1586–1593MathSciNetCrossRefGoogle Scholar
  15. 15.
    Agarwal R, Jain S, Agarwal RP (2018) Mathematical modeling and analysis of dynamics of cytosolic calcium ion in astrocytes using fractional calculus. J Fract Calc Appl 9:1–12MathSciNetGoogle Scholar
  16. 16.
    Smith GD (1996) Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel. Biophys J 71:3064–3072CrossRefGoogle Scholar
  17. 17.
    Podlubny I (1999) Fractional differential equations. Academic Press, New York, vol 198, pp 1–366Google Scholar
  18. 18.
    Borak S, Härdle W, Weron R (2005) Stable distributions. In: Statistical tools for finance and insurance. Diss Paper Springer pp. 21–44Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of TechnologyPandit Deendayal Petroleum UniversityGujaratIndia

Personalised recommendations