Abstract
Ca2+ signaling is an essential function of neurons to control synaptic activity, memory formation, fertilization, proliferation, etc. Protein and voltage-dependent calcium channels (VDCCs) maintain an adequate level of calcium concentration ([Ca2+]). An alteration in [Ca2+] leads to the death of the neurons that start the primary symptoms of the disease. The present study deals with cell memory-based mathematical modeling of Ca2+ that is characterized by the presence of protein and VDCC. We developed a two-dimensional Ca2+ neuronal model to study the spatiotemporal behavior of the Ca2+ profile. All principal parameters like buffer concentration, diffusion coefficient, VDCC fluxes, etc. are incorporated in this model. Apposite initial and boundary conditions are applied to the physiology of the problem. We obtained an approximate Ca2+ profile by the fractional integral transform method. The application of obtained results is performed to provide its implications to estimate the [Ca2+] in neurodegenerative disease. It is observed that the protein and VDCC provide a significant impact in the presence of cell memory. The memory of cells shrinks the Ca2+ flow from elevation and provides better results to estimated Ca2+ flow in the disease state.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Abboubakar H, Kumar P, Erturk VS, Kumar A (2021) A mathematical study of a tuberculosis model with fractional derivatives. Int J Model Simul Sci Comput 12(4):2150037
Akgül EK, Akgül A, Yavuz M (2021) New illustrative applications of integral transforms to financial models with different fractional derivatives. Chaos Solitons Fractals 146:110877
Bonyah E, Yavuz M, Baleanu D, Kumar S (2022) A robust study on the listeriosis disease by adopting fractal-fractional operators. Alex Eng J 61(3):2016–2028
Borak S, Härdle W, Weron R (2005) Stable distributions. In: Statistical tools for finance and insurance, Springer, 2005, pp 21–44.
Brini M, Calì T, Ottolini D, Carafoli E (2014) Neuronal calcium signaling: function and dysfunction. Cell Mol Life Sci 71(15):2787–2814
Calì T, Ottolini D, Brini M (2014) Calcium signaling in Parkinson’s disease. Cell Tissue Res 357(2):439–454
Crank J (1975) The mathematics of diffusion, 2nd edn. Oxford University Press, Ely House, London W.I.
Crisanto-Neto JC, da Luz MGE, Raposo EP, Viswanathan GM (2018) An efficient series approximation for the Lévy α-stable symmetric distribution. Phys Lett Sect A Gen At Solid State Phys 382(35):2408–2413
Dargan SL, Parker I (2003) Buffer kinetics shape the spatiotemporal patterns of IP3-evoked Ca2+ signals. J Physiol 553(3):775–788
Dave DD, Jha BK (2021a) On finite element estimation of calcium advection diffusion in a multipolar neuron. J Eng Math 128(1):1–15
Dave DD, Jha BK (2021b) Mathematical modeling of calcium oscillatory patterns in a neuron. Interdiscip Sci Comput Life Sci 13(1):12–24
Hurley MJ, Brandon B, Gentleman SM, Dexter DT (2013) Parkinson’s disease is associated with altered expression of CaV1 channels and calcium-binding proteins. Brain 136(7):2077–2097
Iacopino AM, Christakos S (1990) Specific reduction of calcium-binding protein (28-kilodalton calbindin-D) gene expression in aging and neurodegenerative diseases. Proc Natl Acad Sci U S A 87(11):4078–4082
Jena RM, Chakraverty S, Yavuz M, Abdeljawad T (2021) A new modeling and existence-uniqueness analysis for Babesiosis disease of fractional order. Mod Phys Lett B 35(30):2150443
Jha A, Adlakha N (2015) Two-dimensional finite element model to study unsteady state Ca2+ diffusion in neuron involving ER LEAK and SERCA. Int J Biomath 8(1):1–14
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 Simul Sci Comput 4(2):1250030
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(3):1–11
Jha A, Adlakha N, Jha BK (2015) 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
Jha BK, Joshi H, Dave DD (2018) 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
Jha BK, Jha A, Adlakha N (2020) Three-dimensional finite element model to study calcium distribution in astrocytes in presence of VGCC and excess buffer. Differ Equ Dyn Syst 28(3):603–616
Joshi H, Jha BK (2021a) Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects. Int J Nonlinear Sci Numer Simul. https://doi.org/10.1515/ijnsns-2020-0254
Joshi H, Jha BK (2021b) On a reaction–diffusion model for calcium dynamics in neurons with Mittag-Leffler memory. Eur Phys J Plus 136(6):1–15
Joshi H, Jha BK (2021c) Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Math Model Numer Simul Appl 1(2):84–94
Joshi H, Jha BK (2022) Generalized diffusion characteristics of calcium model with concentration and memory of cells: a spatiotemporal approach. Iran J Sci Technol Trans A Sci 46(1):309–322
Keener J, Sneyd J (2009) Mathematical Physiology, Second. Interdisciplinary Applied Mathematics, Springer US
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, New York
Kumar H, Naik PA, Pardasani KR (2018) Finite element model to study calcium distribution in T lymphocyte involving buffers and ryanodine receptors. Proc Natl Acad Sci India Sect A Phys Sci 88(4):585–590
Kumar P, Suat Ertürk V, Nisar KS (2021) Fractional dynamics of huanglongbing transmission within a citrus tree. Math Methods Appl Sci 44(14):11404–11424
Magin RL (2006) Fractional calculus in bioengineering. Begell House
Manhas N, Pardasani KR (2014) Modelling mechanism of calcium oscillations in pancreatic acinar cells. J Bioenerg Biomembr 46(5):403–420
Manhas N, Sneyd J, Pardasani KR (2014) Modelling the transition from simple to complex Ca2+ oscillations in pancreatic acinar cells. J Biosci 39(3):463–484
McMahon A, Wong BS, Iacopino AM, Ng MC, Chi S, German DC (1998) Calbindin-D28k buffers intracellular calcium and promotes resistance to degeneration in PC12 cells. Mol Brain Res 54(1):56–63
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
Naik PA (2020) Modeling the mechanics of calcium regulation in T lymphocyte: a finite element method approach. Int J Biomath 13(5):2050038
Naik PA, Pardasani KR (2016) Finite element model to study calcium distribution in oocytes involving voltage gated Ca2+ channel, ryanodine receptor and buffers. Alex J Med 52(March):43–49
Naik PA, Zu J (2020) Modeling and simulation of spatial-temporal calcium distribution in T lymphocyte cell by using a reaction-diffusion equation. J Bioinform Comput Biol 18(2):2050013
Naik PA, Owolabi KM, Yavuz M, Zu J (2020) Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos Solitons Fractals 140:110272
Navarro-López EM, Çelikok U, Şengör NS (2021) A dynamical model for the basal ganglia-thalamo-cortical oscillatory activity and its implications in Parkinson’s disease. Cogn Neurodyn 15(4):693–720
Özköse F, Şenel MT, Habbireeh R (2021) Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Math Model Numer Simul Appl 1(2):67–83
Pak S (2009) Solitary wave solutions for the RLW equation by he’s semi inverse method. Int J Nonlinear Sci Numer Simul 10(4):505–508
Panday S, Pardasani KR (2014) Finite element model to study the mechanics of calcium regulation in oocyte. J Mech Med Biol 14(2):1–16
Paradisi P, Cesari R, Mainardi F, Tampieri F (2001) Fractional Fick’s law for non-local transport processes. Phys A Stat Mech Appl 293(1–2):130–142
Parkinson disease, World health organisation. [Online]. Available https://www.who.int/news-room/fact-sheets/detail/parkinson-disease. Accessed 16 Jul 2022
Pathak K, Adlakha N (2016) Finite element model to study two dimensional unsteady state calcium distribution in cardiac myocytes. Alex J Med 52(3):261–268
Pawar A, Pardasani KR (2022a) Effect of disturbances in neuronal calcium and IP3 dynamics on β-amyloid production and degradation. Cogn Neurodyn. https://doi.org/10.1007/s11571-022-09815-0
Pawar A, Pardasani KR (2022b) Effects of disorders in interdependent calcium and IP3 dynamics on nitric oxide production in a neuron cell. Eur Phys J Plus 137(5):1–19
Pawar A, Pardasani KR (2022c) Simulation of disturbances in interdependent calcium and β-amyloid dynamics in the nerve cell. Eur Phys J Plus 137(8):1–23
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their, 1st Editio. Academic Press, Elsevier
Podlubny I (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal 5(4):367–386
Schmidt H (2012) Three functional facets of calbindin D-28k. Front Mol Neurosci 5:25
Smith GD, Dai L, Miura RM, Sherman A (2001) Asymptotic analysis of buffered calcium diffusion near a point source. SIAM J Appl Math 61(5):1816–1838
Tewari SG, Pardasani KR (2012) Modeling effect of sodium pump on calcium oscillations in neuron cells. J Multiscale Model 04(03):1250010
Tewari V, Tewari S, Pardasani KR (2011) A model to study the effect of excess buffers and Na+ ions on Ca2+ diffusion in neuron cell. World Acad Sci Eng Technol 76(4):41–46
Veeresha P (2021) A numerical approach to the coupled atmospheric ocean model using a fractional operator. Math Model Numer Simul Appl 1(1):1–10
Yavuz M, Bonyah E (2019) New approaches to the fractional dynamics of schistosomiasis disease model. Phys A Stat Mech Appl 525:373–393
Yu Y, Han F, Wang Q, Wang Q (2022) Model-based optogenetic stimulation to regulate beta oscillations in Parkinsonian neural networks. Cogn Neurodyn 16(3):667–681
Yuan H-H, Chen R-J, Zhu Y-H, Peng C-L, Zhu X-R (2012) The neuroprotective effect of overexpression of calbindin-D28k in an animal model of Parkinson’s disease. Mol Neurobiol 47:117–122
Acknowledgements
No funding to declare. The authors are thankful to the editor and anonymous reviewers for their constructive comments and suggestions to improve the quality and presentation of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Joshi, H., Jha, B.K. 2D dynamic analysis of the disturbances in the calcium neuronal model and its implications in neurodegenerative disease. Cogn Neurodyn 17, 1637–1648 (2023). https://doi.org/10.1007/s11571-022-09903-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11571-022-09903-1