Fractional-order mathematical model for calcium distribution in nerve cells

Joshi, Hardik; Jha, Brajesh Kumar

doi:10.1007/s40314-020-1082-3

Cited by 32 publications

(16 citation statements)

References 58 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…13,14 The fractional calculus is not limited to any particular branch but is used in chemistry, biochemistry, biology, engineering, physics, hydrology, and medicine. [15][16][17][18][19][20][21][22][23][24][25][26][27][28] Yavuz and Bonyah, 29 in their paper, examined the fractional-order schistosomiasis model via exponential law kernel and Mittag-Leffler kernel in Liouville-Caputo sense. The iterative scheme through Laplace transform and Sumudu-Picard integration technique was applied for the solution of their model.…”

Section: Introductionmentioning

confidence: 99%

“…Presently, it finds a great application in real‐life modeling problems, especially biological systems 13,14 . The fractional calculus is not limited to any particular branch but is used in chemistry, biochemistry, biology, engineering, physics, hydrology, and medicine 15–28 . Yavuz and Bonyah, 29 in their paper, examined the fractional‐order schistosomiasis model via exponential law kernel and Mittag–Leffler kernel in Liouville–Caputo sense.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Ghori

Naik

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The present paper studies a fractional‐order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type‐II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely, basic reproduction number scriptR0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease‐free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity scriptR0. If scriptR0<1, then DFE is globally stable, leading to eradicating the population's disease. If scriptR0>1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters in disease transmission. The numerical solution of the model is obtained by the generalized Adams–Bashforth–Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results. From the obtained results, it is concluded that the order of the fractional derivative plays a significant role in the dynamic process. Also, from the sensitivity analysis results, it is seen that the parameter δ ( infection rate) is positively correlated to scriptR0, while the parameter μ (inhibition rate) is negatively correlated to scriptR0, which means these parameters are more sensitive than others in disease dynamics.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Ghori

Naik

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Also, there have been several experimental attempts that were performed in the past to identify the role and physiological impact of sodium calcium exchangers on various cells [13,14,15,16]. Beside this a researches has explored the role of parameters of calcium toolkits on astrocytes [17,18,19,20,21,22], neuron [23,24,25,26,27,28,29,30], oocytes [31,32], myocytes [33,34,35], hepatocytes [36], and T lymphocytes cells [37,38]. Thus a very little amount of work has attempted to study parameters of calcium toolkits by using the fractional calculus approach.…”

Section: Introductionmentioning

confidence: 99%

Chaos of calcium diffusion in Parkinson s infectious disease model and treatment mechanism via Hilfer fractional derivative

Joshi¹,

Jha²

2021

mmnsa

Self Cite

View full text Add to dashboard Cite

Calcium is a vital element in our body and plays a crucial role to moderate the calcium signalling process. Calcium-dependent protein and flux through the sodium-calcium exchanger are also involved in signalling process to perform and execute necessary cellular activities. The loss or alteration in this cellular activity starts the early progress of Parkinson's disease. A mathematical calcium model is developed in the form of the Hilfer fractional reaction-diffusion equation to examine the calcium diffusion in the cells. The effect of calcium-dependent protein and flux through the sodium-calcium exchanger is incorporated in the model. The solution of the Hilfer fractional calcium model is obtained by using the Sumudu transform technique in the form of the Wright function and Mittag-Leffler function. The graphical results are obtained for the different amounts of proteins, presence, and absence of sodium-calcium exchanger, and various orders of Hilfer derivative. The obtained results show that the modified calcium model is a function of time, position, and Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a rich physical interpretation of a calcium model as compared to the classical calcium model.

show abstract

“…Hence, complex systems can be modeled using multiple temporal and spatial scales without segmenting the given problem into a lot of smaller ones. Recently, numerous fractional calculus applications in neuroscience field have been explored [10][11][12][13][14][15][16]. The researchers [10][11][12] showed that the membrane ion channels, conductances, spiny dendrites and the spiking activities of neurons have the feature of fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, numerous fractional calculus applications in neuroscience field have been explored [10][11][12][13][14][15][16]. The researchers [10][11][12] showed that the membrane ion channels, conductances, spiny dendrites and the spiking activities of neurons have the feature of fractional calculus. Lundstrom et al [13] showed that the neocortical pyramidal neurons have fractional order adaptation that depends on the changes of fractional order stimulus.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and coupling of fractional-order models of the motor cortex and central pattern generators

2020

J. Neural Eng.

View full text Add to dashboard Cite

Objective. Fractional calculus plays a key role in the analysis of neural dynamics. In particular, fractional calculus has been recently exploited for analyzing complex biological systems and capturing intrinsic phenomena. Also, artificial neural networks have been shown to have complex neuronal dynamics and characteristics that can be modeled by fractional calculus. Moreover, for a neural microcircuit placed on the spinal cord, fractional calculus can be employed to model the central pattern generator (CPG). However, the relation between the CPG and the motor cortex is still unclear. Approach. In this paper, fractional-order models of the CPG and the motor cortex are built on the Van der Pol oscillator and the neural mass model (NMM), respectively. A self-consistent mean field approximation is used to construct the potential landscape of the Van der Pol oscillator. This landscape provides a useful tool to observe the 3D dynamics of the oscillator. To infer the relation of the motor cortex and CPG, the coupling model between the fractional-order Van der Pol oscillator and the NMM is built. As well, the influence of the coupling parameters on the CPG and the motor cortex is assessed. Main results. Fractional-order NMM and coupling model of the motor cortex and the CPG are first established. The potential landscape is used to show 3D probabilistic evolution of the Van der Pol oscillator states. Detailed observations of the evolution of the system states can be made with fractional calculus. In particular, fractional calculus enables the observation of the creation of stable modes and switching between them. Significance. The results confirm that the motor cortex and CPG have associated modes or states that can be switched based on changes in the fractional order and the time delay. Fractional calculus and the potential landscape are helpful methods for better understanding of the working principles of locomotion systems.

show abstract

Fractional-order mathematical model for calcium distribution in nerve cells

Cited by 32 publications

References 58 publications

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Chaos of calcium diffusion in Parkinson s infectious disease model and treatment mechanism via Hilfer fractional derivative

Dynamics and coupling of fractional-order models of the motor cortex and central pattern generators

Contact Info

Product

Resources

About