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Nonlinear system dynamics of calcium and nitric oxide due to cell memory and superdiffusion in neurons

  • Anand Pawar , ,
  • Kamal Raj Pardasani
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  • Department of Mathematics, Bioinformatics and Computer Applications, Maulana Azad National Institute of Technology, Bhopal-462003, Madhya Pradesh, India

Author to whom any correspondence should be addressed.

Received date: 2023-10-03

  Revised date: 2024-02-04

  Accepted date: 2024-03-20

  Online published: 2024-04-17

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The integer-order interdependent calcium ([Ca2+]) and nitric oxide (NO) systems are unable to shed light on the influences of the superdiffusion and memory in triggering Brownian motion (BM) in neurons. Therefore, a mathematical model is constructed for the fractional-order nonlinear spatiotemporal systems of [Ca2+] and NO incorporating reaction-diffusion equations in neurons. The two-way feedback process between [Ca2+] and NO systems through calcium feedback on NO production and NO feedback on calcium through cyclic guanosine monophosphate (cGMP) with plasmalemmal [Ca2+]-ATPase (PMCA) was incorporated in the model. The Crank–Nicholson scheme (CNS) with Grunwald approximation along spatial derivatives and L1 scheme along temporal derivatives with Gauss–Seidel (GS) iterations were employed. The numerical outcomes were analyzed to get insights into superdiffusion, buffer, and memory exhibiting BM of [Ca2+] and NO systems. The conditions, events and mechanisms leading to dysfunctions in calcium and NO systems and causing different diseases like Parkinson’s were explored in neurons.

Cite this article

Anand Pawar , Kamal Raj Pardasani . Nonlinear system dynamics of calcium and nitric oxide due to cell memory and superdiffusion in neurons[J]. Communications in Theoretical Physics, 2024 , 76(5) : 055002 . DOI: 10.1088/1572-9494/ad35b4

Introduction

Diverse mechanisms like neurotransmitter release, learning, memory and neuronal excitability [13] are greatly influenced by nitric oxide signaling. Nitric oxide also controls the various physiological functions in neuromuscular [4] and immune [5] etc. The concentration levels of NO exert cytotoxic as well as cytoprotective impacts in a neuron cell [6]. It is noted in the experimental studies that the derivatives of nitric oxide such as nitroglycerine, guanylate cyclase, nitrates etc are beneficial for the smooth muscles of the heart. The excess production and accumulation of NO in neuron cells cause neurotoxicity, damage of DNA, protein modifications etc, which lead to neuronal injury and cell death [7, 8]. Garthwaite [9] investigated the function of NO signaling in the nervous system. Tsoukias [10] examined the mechanisms which influence nitric oxide availability in both normal and diseased conditions. The dysregulation in NO can lead to neuronal diseases like Parkinson’s [7], primary headaches [11], HIV-associated dementia [12], stroke [13] etc.
The mathematics of diffusion mechanism in a variety of media including sheets, cylinders, etc with varying diffusion coefficients has been studied by Crank [14]. The presynaptic diffusion of calcium ions with different regulatory processes including influx, SERCA pump and binding proteins in neurons has been examined by Fogelson et al [15]. Experimental studies suggest that the [Ca2+]-buffer causes a measurable reduction in the free [Ca2+] in neurons [16]. The [Ca2+] diffusion models involving and validating buffering process for [Ca2+] channels were framed by Smith et al [17, 18]. The ryanodine receptor (RyR) and IP3-receptor (IP3R) are the elementary events, which release [Ca2+] ions to the cytosol of cells [19]. The diffusion and geometrical arrangement of calcium ions can cause oscillations in calcium signaling and these oscillations in calcium were noticed because of [Ca2+] or IP3-induced calcium release [20] and excessive slow buffer concentrations in neurons [21]. The calcium diffusion models in recent years have been explored in various cells including astrocyte cells [22, 23], neurons [2427], acinar cells [2830], Oocytes [3134], hepatocyte [35, 36], fibroblast [3739], T-lymphocyte [40, 41], β-cell [42] and myocytes [4345] by employing different analytical and numerical procedures. Incorporating biophysical components such as L-type calcium channels, Na+/K+ channels, buffer, sodium–calcium exchanger, etc, Tewari and Pardasani [25] explored the Na+ pump effects on [Ca2+] oscillations in neurons by employing Galerkin’s method. Jha and Adlakha [27] extended the work of Tewari and Pardasani [25] for two-dimensional cases in neuronal cells utilizing the finite element technique. The dysregulation in [Ca2+] with various pathologies is associated with several illnesses such as Alzheimer’s and Parkinson’s [46].
The NO interaction with other secondary messengers including [Ca2+], proteins, O2, etc influences the regulatory processes in the numerous cells of the human body. The intracellular [Ca2+] release elevates calcium concentration, which causes NO generation in the cell. The glutamate-induced calcium influx leads to the NO formation in neurons [47, 48]. The [Ca2+]-dependent and [Ca2+]-independent NO generation influence the regulation of various biological functions in different tissues [49]. The elevated calcium concentration levels, which are induced by the voltage-dependent [Ca2+] channels bind the nitric oxide synthase (nNOS) that generates NO in neurons [6]. Numerous [Ca2+] channels including voltage-gated [50], [Ca2+]-dependent K+ channels [51], etc have notable functions in NO formation in neuron cells. Calcium and NO signaling significantly influence each other as calcium regulates nitric oxide appropriately to prevent a neuron cell from toxicity [52] and calcium signal is amplified by nitric oxide signaling in neurons [53]. The enhanced calcium levels in cells are regulated by potent vasodilators release including nitric oxide [54]. The elevated NO levels cause an elevation in cGMP concentration, which further causes reduction in the cytosolic [Ca2+] levels through the PMCA channel, and these reduced calcium levels lead to the decrease in the [Ca2+]-dependent NO generation in cells [55, 56]. The mathematical models of [Ca2+] and NO production in nerve cells were explored by Dormanns et al [57] and Plank et al [58]. Pawar and Pardasani [5961] explored the dysregulatory effects of [Ca2+] and IP3 mechanisms on formations of NO, β-amyloid (Aβ), adenosine triphosphate (ATP), etc in neurons. Also, the dependence of NO, Aβ and dopamine molecules on [Ca2+] signaling in neurons has been discussed by Pawar and Pardasani [6264] by utilizing a finite element procedure. But, all of the above-mentioned models are based on the integer-order systems.
The fractional calculus effectively characterizes the nature, behavior as well as physical and geometrical interpretation of complex dynamics that emerge in biological tissues [65]. Currently, fractional calculus is a robust method for solving the reaction-diffusion problem by combining memory phenomena [66]. The study of memory and heredity phenomena in various disciplines like biology, psychology, mechanics, etc is made possible by fractional-order derivatives. The application of fractional time derivatives governs the memory phenomena in mathematical modeling. The fractional time derivative has been identified as a measure of memory [67]. There is complete memory loss at integer-order, i.e. α = 1. When the fractional-order α is reduced from the integer-order, the memory trace exhibits a nonlinear growth and its dynamics are highly dependent on time. The memory effect highlights the disparity in the derivatives of fractional-order and integer-order systems [68]. The one-sided and two-sided fractional-order space derivatives for flow equations with variable coefficients have been discussed by Meerschaert and Tadjeran [69, 70] utilizing the Grunwald approximation. The matrix approximation has been analyzed by Podlubny et al [71] for handling fractional-order partial differential equations. In biological systems, with the help of the memory phenomenon, cells can learn from their experiences to control the dysregulated cellular mechanisms [72]. The effects of memory and Atangana–Baleanu–Caputo (ABC) operator on the neuronal [Ca2+] signaling were studied by Joshi and Jha [66]. In the Parkinson-affected neuron cells, Joshi and Jha [73] studied the calcium distribution involving calbindin-D28K buffer and voltage-gated calcium channels, with fraction derivatives. Superdiffusion is a rather uncommon mechanism that is noted in porous glassware, systems in biology, the movement of molecules, etc [74]. The superdiffusion phenomenon is an enhancement of the diffusion process that occurs when the fractional derivatives is replaced for the second-order derivatives in the transport model [75]. In the scenario where 1 < α < 2, a superdiffusive flow is observed, which is characterized by the rapid spreading of a cloud of diffusing particles at a rate above that predicted by the conventional diffusion model [76]. It is reported that the occurrence of superdiffusive behavior is contingent upon the absence of equilibrium between birth and death processes in tumor cells [77]. The influences of memory and superdiffusion on the systems of calcium and and calcium and IP3 were studied in neurons [78, 79].
The individual systems of [Ca2+] and NO offer very limited insights into the cellular mechanisms in neurons. The interactive system dynamics of [Ca2+] and NO provide additional information about different mechanisms in neuron cells. However, the aforementioned studies were based on integer-order dynamics, which cannot shed light on the superdiffusion mechanism as well as cell memory effects in nerve cells. The individual fractional-order dynamics of calcium signaling are noticed, which provides information on the memory in neurons. But, no research was conducted on fractional-order nonlinear interactive systems of [Ca2+] and NO incorporating two-way feedback mechanism between [Ca2+] and NO dynamics in neurons. Also, the functional and dysfunctional effects of the superdiffusion process, memory exhibiting BM, etc on the nonlinear systems of calcium and NO in neurons have not been explored earlier. The objective of studying fractional-order systems is in their ability to capture the memory and superdiffusion properties exhibited by biological systems. This enables us to create more accurate models for understanding the dynamics of neuronal diseases. Within fractional-order models, the memory feature enables the incorporation of a wider range of prior information, resulting in more precise prediction and translation of the models. Thus, the fractional-order interactive systems of [Ca2+] and NO in neurons have been explored using a mathematical model. The CNS with the Grunwald approximation for spatial derivatives and L1 method for temporal derivatives was employed and the resultant nonlinear equations system was solved utilizing the GS iterations. This paper examines the influences of superdiffusion, buffering process, memory with Brownian motion, etc on the interactive nonlinear dynamical systems of [Ca2+] and NO in neurons.

Mathematical formulation

The following assumptions have been made in order to have the major focus on the interdependence of calcium and nitric oxide dynamics in neuron cells.
a

(a) It is assumed that all other signaling systems are well regulated maintaining the homeostasis of the signaling molecules other than calcium and nitric oxide, which may have influence on [Ca2+] and NO dynamics in neuron cells. Therefore, it is justified to assume that other signaling systems like IP3, β-amyloid, ATP, dopamine, etc are in homeostasis and static mode and therefore can be assumed to be constant. Thus, the influences on [Ca2+] and NO dynamics will be static and constant.

b

(b) There are no external disturbances under normal conditions in neuronal cells.

c

(c) The parameters assumed to be vary in the normal ranges.

The fractional-order system of [Ca2+] dynamics incorporating EGTA buffer and ryanodine receptor with constant IP3 on the basis of integer-order Wagner et al [80] model in a neuron cell can be expressed as shown below,
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\partial }^{{U}_{1}}\left[{\mathrm{Ca}}^{2+}\right]}{\partial {t}^{{{\rm{U}}}_{1}}}={D}_{\mathrm{Ca}}\displaystyle \frac{{\partial }^{{V}_{1}}\left[{\mathrm{Ca}}^{2+}\right]}{\partial {x}^{{V}_{1}}}\\ \,+\,\left(\displaystyle \frac{{J}_{\mathrm{IPR}}-{J}_{\mathrm{SERCA}}{{+}{J}}_{\mathrm{LEAK}}{+J}_{\mathrm{RyR}}}{{F}_{{\rm{C}}}}\right)\\ \,-\,{{\rm{K}}}^{+}{\left[{\rm{B}}\right]}_{\infty }\left(\left[{\mathrm{Ca}}^{2+}\right]-{\left[{\mathrm{Ca}}^{2+}\right]}_{\infty }\right)-{J}_{\mathrm{PMCA}},\end{array}\end{eqnarray}$
where $0\lt {U}_{1}\leqslant 1,1\lt {V}_{1}\leqslant 2$ and the steady state levels of calcium and EGTA buffer are denoted sequentially by [Ca2+] and [B]. The [Ca2+] diffusion coefficient is expressed by DCa. K+ represents the rate of buffer association. t and x denote the time and site variables, sequentially.
Wagner et al [80] deduced the different terms of inflow and outflow for equation (1) as represented below,
$\begin{eqnarray}{J}_{\mathrm{IPR}}={V}_{\mathrm{IPR}}{m}^{3}{h}^{3}\left({\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{ER}}-\left[{\mathrm{Ca}}^{2+}\right]\right),\end{eqnarray}$
$\begin{eqnarray}{J}_{\mathrm{SERCA}}{=V}_{\mathrm{SERCA}}\left(\displaystyle \frac{{\left[{\mathrm{Ca}}^{2+}\right]}^{2}}{{\left[{\mathrm{Ca}}^{2+}\right]}^{2}{+{\rm{K}}}_{\mathrm{SERCA}}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{J}_{\mathrm{LEAK}}{=V}_{\mathrm{LEAK}}\left({\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{ER}}-\left[{\mathrm{Ca}}^{2+}\right]\right),\end{eqnarray}$
$\begin{eqnarray}{J}_{\mathrm{RyR}}={P}_{0}{V}_{\mathrm{RyR}}\left({\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{ER}}-\left[{\mathrm{Ca}}^{2+}\right]\right).\end{eqnarray}$
The fluxes of SERCA pump (JSERCA), leak (JLEAK), IP3R (JIPR) and ryanodine receptor (JRyR) are involved in the present model. The flux rate constants of IP3R, leak, SERCA and RyR are, respectively, denoted by VIPR, VLEAK, VSERCA, and VRyR. KSERCA denotes the Michaelis constant concerning SERCA pump.
Li and Rinzel [81] have given the m and h terms as follows,
$\begin{eqnarray}m=\left(\displaystyle \frac{\left[{\mathrm{IP}}_{3}\right]}{\left[{\mathrm{IP}}_{3}\right]+{{\rm{K}}}_{\mathrm{IP}3}}\right)\left(\displaystyle \frac{\left[{\mathrm{Ca}}^{2+}\right]}{\left[{\mathrm{Ca}}^{2+}\right]+{K}_{\mathrm{Ac}}}\right),\end{eqnarray}$
$\begin{eqnarray}h=\displaystyle \frac{{K}_{\mathrm{Inh}}}{{K}_{\mathrm{Inh}}+\left[{\mathrm{Ca}}^{2+}\right]}.\end{eqnarray}$
Here, the dissociation parameters of binding location of IP3 and [Ca2+] activation and [Ca2+] inhibition are denoted by KIP3, KAc and Kinh, correspondingly.
The NO dynamics with [Ca2+] signaling is given by Gibson et al [82] and can be depicted as follows,
$\begin{eqnarray}\displaystyle \frac{{\partial }^{{U}_{2}}\left[\mathrm{NO}\right]}{\partial {t}^{{U}_{2}}}={D}_{\mathrm{NO}}\displaystyle \frac{{\partial }^{{V}_{2}}\left[\mathrm{NO}\right]}{\partial {x}^{{V}_{2}}}+\left({J}_{\mathrm{production}}-{J}_{\mathrm{degradation}}\right),\end{eqnarray}$
where $0\lt {U}_{2}\leqslant 1,1\lt {V}_{2}\leqslant 2$ and DNO represent the NO transport coefficient. The [Ca2+]-dependent NO generation is depicted as shown below,
$\begin{eqnarray}{J}_{{\rm{production}}}={V}_{{\rm{NO}}}\left(\displaystyle \frac{\left[{{\rm{Ca}}}^{2+}\right]}{\left[{{\rm{Ca}}}^{2+}\right]+{{\rm{KL}}}_{{\rm{NO}}}}\right),\end{eqnarray}$
where VNO and KNO are the rate constants. The NO degradation flux is represented by Jdegradation is given by,
$\begin{eqnarray}{J}_{\mathrm{degradation}}{{=}{K}}_{1}\left[\mathrm{NO}\right].\end{eqnarray}$
The following represents the expression for [Ca2+]ER,
$\begin{eqnarray}{\left[{{\rm{Ca}}}^{2+}\right]}_{{\rm{T}}}={{\rm{F}}}_{{\rm{C}}}\,{\left[{{\rm{Ca}}}^{2+}\right]}_{{\rm{C}}}+{{\rm{F}}}_{{\rm{E}}}\,{\left[{{\rm{Ca}}}^{2+}\right]}_{{\rm{ER}}}.\end{eqnarray}$
The focus of the present study is to evaluate the superdiffusion and cell memory triggering Brownian motion of calcium and nitric oxide in neuronal cells. In view of the above, the integer-order cGMP is considered in the present study because the fractional model of cGMP will involve cell memory triggering the Brownian motion of cGMP, which might override the influence of superdiffusion and cell memory of NO molecules on both calcium and NO signaling systems in neuronal cells.
The NO/cGMP kinetics are expressed below as follows [83],
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial \left[\mathrm{cGMP}\right]}{\partial t}={V}_{\mathrm{cGMP}}\left(\displaystyle \frac{{g}_{0}+{g}_{1}[\mathrm{NO}]+{[\mathrm{NO}]}^{2}}{{a}_{0}+{a}_{1}[\mathrm{NO}]+{[\mathrm{NO}]}^{2}}\right)\\ \,-\,{X}_{\mathrm{cGMP}}\left(\displaystyle \frac{{\left[\mathrm{cGMP}\right]}^{2}}{{K}_{\mathrm{cGMP}}+\left[\mathrm{cGMP}\right]}\right),\end{array}\end{eqnarray}$
where the flux of cGMP generation is depicted as a function of NO distribution and g0, g1, a0, a1, VcGMP, XcGMP and KcGMP are constant and their values are acquired in the model by fitting NO/cGMP pathways to experiment data [83].
The expression of plasmalemmal [Ca2+]-ATPase (PMCA) in the presence of cGMP can be represented as follows,
$\begin{eqnarray}\begin{array}{l}{J}_{\mathrm{PMCA}}=\,{V}_{\mathrm{cGMP}}^{\mathrm{PMCA}}\\ \,\times \,\left(1+1.8\displaystyle \frac{2.\left[\mathrm{cGMP}\right]}{{K}_{\mathrm{cGMP}}^{\mathrm{PMCA}}+\left[\mathrm{cGMP}\right]}\right)\left(\displaystyle \frac{\left[{\mathrm{Ca}}^{2+}\right]}{{K}_{\mathrm{PMCA}}+\left[{\mathrm{Ca}}^{2+}\right]}\right).\end{array}\end{eqnarray}$
Here, ${V}_{\mathrm{cGMP}}^{\mathrm{PMCA}}\,\mathrm{and}\,{K}_{\mathrm{cGMP}}^{\mathrm{PMCA}}$ are respectively the cGMP levels at half-activation of PMCA, maximum current in cGMP concentration’s absence.

Initial conditions

Smith [18] and Liew and Raychaudhuri [84] framed the initial conditions for [Ca2+] and nitric oxide as shown below,
$\begin{eqnarray}{\left[{{\rm{Ca}}}^{2+}\right]}_{t=0}=0.1\,\mu {\rm{M}}.\end{eqnarray}$
$\begin{eqnarray}{\left[{\rm{NO}}\right]}_{t=0}=0\,\mu {\rm{M}}.\end{eqnarray}$

Boundary conditions

For [Ca2+], the required boundary conditions are framed by Smith [18] as represented below,
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{x\to 0}\left(-{D}_{\mathrm{Ca}}\displaystyle \frac{\partial \left[{\mathrm{Ca}}^{2+}\right]}{\partial x}\right)=\sigma ,\end{eqnarray}$
where σ denotes the [Ca2+] source inflow.
The [Ca2+] reaches the background levels at the opposite side from the source location, i.e.
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{x\to 5}\left[{{\rm{Ca}}}^{2+}\right]={\left[{{\rm{Ca}}}^{2+}\right]}_{\infty }=0.1\,\mu {\rm{M}}.\end{eqnarray}$
For NO dynamics, the boundary conditions framed by Kavdia et al [85] as represented below,
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{x\to 0}\left(\displaystyle \frac{\partial \left[\mathrm{NO}\right]}{\partial x}\right)=0,\end{eqnarray}$
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{x\to 5}\left(\displaystyle \frac{\partial \left[\mathrm{NO}\right]}{\partial x}\right)=0.\end{eqnarray}$

Results and discussion

The fractional-order systems of [Ca2+] and NO are depicted in the graphical form using acquired numerical findings. Table 1 illustrates the numeric data of the various parameters employed in the current examination.
Table 1. Numeric Data (Gibson et al [82] and Wagner et al [80]).
Parameters Values Parameters Values
VIPR 8.5 s−1 KAc 0.8 μM
Kserca 0.4 μM VLeak 0.01 s−1
Kinh 1.8 μM KIP3 0.15 μM
DCa 16 μm2 s−1 Vserca 0.65 μM s−1
FC 0.83 k+ 1.5 μM−1 s−1
KNO 0.09 μM s−1 DNO 3300 μm2 s−1
Kdeg 0.0145 s−1 FE 0.17
VNO 0.45 μM g0 4.8 nM2
g1 35.33 nM a0 1200.16 nM2
g2 37.33 nM VcGMP 1.260 μM s−1
XcGMP 0.0695 μM s−1 VcGMP 2 μM
${V}_{\mathrm{cGMP}}^{\mathrm{PMCA}}$ 2.9 pA ${K}_{\mathrm{cGMP}}^{\mathrm{PMCA}}$ 1 μM
Figure 1 shows the influence of superdiffusion of [Ca2+] ions on [Ca2+] concentration at 1.0 s and position 0 μm. The diffusion of calcium ions, binding of [Ca2+] ions by buffers and extrusion of cytosolic [Ca2+] ions to the ER via the pump (SERCA) lead to the decrease in the [Ca2+] concentration with locations far from the source up to the opposite end of neuronal cells. The spatial [Ca2+] attains the peak concentration nearer the source for different fractional-order space derivatives since [Ca2+] ions release from source channels and cytosolic [Ca2+] concentration elevates in cells. In figure 1(B), the [Ca2+] levels increase over time and approach the equilibrium at 0.5 sec in cells. When the spatial derivative’s order reduces, the spatiotemporal [Ca2+] concentration also reduces in neuronal cells. The decrease in [Ca2+] concentration is attributed to the acceleration of diffusion, specifically referred to as the superdiffusion process of [Ca2+] ions, leading to the expression of superdiffusive behavior. Thus, this indicates that the enhancement in the diffusion mechanism affects the [Ca2+] signaling in neuron cells.
Figure 1. [Ca2+] concentration at V2 = 2.0, U1 and U2 = 1.0 for time (1.0 s) and site (0 μm) concerning distinct [Ca2+] space fractional derivatives.
The superdiffusion mechanism of [Ca2+] ions influences the NO generations in neurons as exhibited in figure 2 at time (1.0 s) and location (0 μm). The [Ca2+] levels and NO formation flux profiles fall down with distant positions from the source site due to the enhanced transportation of neuronal [Ca2+] ions as illustrated in figures 1(A) and 2(A). The temporal [Ca2+] and NO generation flux rise over time growth and reach the equilibrium at the same period in cells. Thus, the superdiffusion of [Ca2+] ions influences NO production through [Ca2+] signaling in neuron cells.
Figure 2. NO generation flux at V2 = 2.0, U1 and U2 = 1.0 for time (1.0 s) and site (0 μm) concerning distinct [Ca2+] space fractional derivatives.
The superdiffusion [Ca2+] ions affect the NO levels in neuron cells as shown in figure 3 for time (0.03 s) and site (0 μm). The spatial NO levels fall down from position 0 to 5 μm and the temporal NO levels increase over time and reach the equilibrium at the same duration of 0.5 s concerning distinct fractional-order [Ca2+] space derivatives. When the order of [Ca2+] space derivatives decreases from 2 to 1.7, the spatiotemporal NO concentrations decrease due to the enhancement in the diffusion of neuronal calcium ions. Thus, this signifies that the NO concentration depends on the superdiffusion of [Ca2+] ions.
Figure 3. NO distribution at V2 = 2.0, U1 and U2 = 1.0 for time (0.03 s) and site (0 μm) concerning distinct [Ca2+] space fractional derivatives.
Figure 4 displays the influences of superdiffusion of NO molecules on the NO distribution for time (0.01 s) and site (0 μm). The reduction in the spatial derivative’s order causes a decrease in the spatial nitric oxide levels because of the enhanced diffusion of NO molecules in neuronal cells as illustrated in figure 4(A). The NO concentration attains the peak level nearer the source for the integer-order space derivative at time t = 0.01 s, but in the cases of fractional derivatives along space, the nitric oxide achieves a high concentration at the center of cells. The temporal NO concentration in figure 4(B) increases over time for different fractional-order space derivatives at location x = 0 μm in neurons. Thus, one may conclude that the superdiffusion of NO affects the nitric oxide signaling in neurons.
Figure 4. NO distribution at V1 = 2.0, U1 and U2 = 1.0 for time (0.1 s) and site (0 μm) concerning different NO space fractional derivatives.
The [Ca2+] memory exhibiting Brownian motion affects the [Ca2+] levels as depicted in figure 5 for time 0.1 s and site 2.5 μm. The [Ca2+] memory causes more enhancement in the spatial [Ca2+] profile for lower-order derivatives along time as exhibited in figure 5(A). The spatial [Ca2+] profile attains a high concentration close to the [Ca2+] source and reduces with distant site from source concerning different temporal derivatives in neuronal cells. In figure 5(B), the elevated temporal [Ca2+] concentration is noticed in the initial duration for lower-order temporal derivatives due to the [Ca2+] memory in neurons. The temporal [Ca2+] levels reach a steady state with the passage of time for various fractional time derivatives in cells. Thus, this signifies that [Ca2+] signaling is affected by the [Ca2+] cell memory in nerve cells.
Figure 5. [Ca2+] concentration at U2 = 1.0, V1 and V2 = 2.0 concerning distinct [Ca2+] time fractional derivatives (A) t = 0.1 s (B) x = 2.5 μm.
The [Ca2+] memory causing Brownian motion affects nitric oxide formation in neurons. In figure 6, the spatiotemporal NO formation flux profiles increase with the decrease in the temporal derivative’s order because of the [Ca2+] memory effects. In figures 5(A) and 6(A), the spatial [Ca2+] and NO generation flux become lower with distant sites from the source concerning different order of temporal derivatives in neurons. The [Ca2+] and NO formation flux profiles are more elevated along temporal dimensions in the initial periods for lower-order temporal derivatives because of the calcium memory effects in neurons and with the progress of time, calcium and NO production flux attain a steady state for various fractional-order time derivatives in the cell as illustrated in figures 5(B) and 6(B). Thus, it can be concluded that calcium memory and Brownian motion affect NO formation in neurons.
Figure 6. NO generation flux at U2 = 1.0, V1 and V2 = 2.0 concerning distinct [Ca2+] time fractional derivatives (A) t = 0.1 s (B) x = 2.5 μm.
Figure 7 shows that [Ca2+] memory exhibits BM effects on neuronal NO levels for time 0.01 s and site 2.5 μm. As the temporal derivative’s order reduces, the NO concentration elevates due to the [Ca2+] cell memory in neurons. The NO concentration decreases with distance and elevates with time for different orders of time derivatives in cells. The temporal NO concentration is more elevated in the initial time for lower-order temporal derivatives due to calcium memory in neuron cells. Thus, this indicates that the cell memory of calcium ions influences the nitric oxide concentration levels in a neuron cell.
Figure 7. NO concentration at U2 = 1.0, V1 and V2 = 2.0 for time (0.01 s) and site (2.5 μm) concerning distinct [Ca2+] time fractional derivatives.
The influences of the [Ca2+] memory exhibiting Brownian motion on the [Ca2+] distribution at location 0 μm for buffer value 75 μM as illustrated in figure 8. In the initial time, the oscillations in the [Ca2+] concentration enhance as the fractional temporal derivative’s order reduces because the higher buffer level tries to lower the [Ca2+] concentration and the calcium memory effects try to enhance [Ca2+] levels in the initial times in neurons. The mismatches among the buffer mechanism and calcium memory mechanism as well as other mechanisms which elevate the calcium concentration are responsible for the fluctuations in [Ca2+] profiles in cells. When the time derivative’s order elevates from U1 = 0.8 to 1.0, the oscillations in the [Ca2+] concentration in the initial period decrease due to the fading calcium memory in neuron cells.
Figure 8. [Ca2+] concentration with U2 = 1.0, V1 and V2 = 2.0, buffer 75 μM at x = 0 μm for distinct [Ca2+] time fractional derivatives.
The effects of [Ca2+] memory and higher EGTA buffer on NO generation are exhibited in figure 9 at site 0 μm in neurons. In figures 8 and 9, the oscillations in the [Ca2+] and NO formation flux increase in the initial time duration with the reduction in the order of temporal derivatives because of the mismatches among buffer, calcium memory mechanism and also other processes which increase calcium concentration in neurons. Thus, this signifies that the dysregulation in the processes of [Ca2+] signaling due to memory effects and higher buffer concentration may contribute to the dysregulation in NO formation in neuronal cells.
Figure 9. NO formation flux with U2 = 1.0, V1 and V2 = 2.0, buffer 75 μM at x = 0 μm for distinct [Ca2+] time fractional derivatives.
Figure 10 shows the NO distribution at position 0 μm for U1 = 0.8, 0.85, 0.9 and 1.0 at [B] = 75 μM in neurons. As the temporal derivative’s order decreases, the oscillations in the NO concentration profiles increase due to the mismatches in the various processes of calcium signaling such as higher buffer, the memory of [Ca2+] ions, etc. Thus, the memory effects and the buffering mechanism of calcium signaling affect NO concentration levels in neurons. Also, the effects of reducing the concentration of buffers on the signaling of calcium and nitric oxide in fractional time spaces are demonstrated in figures 5(B) and 7(B). These Figures examine the lower buffer concentration ([B] = 5 μM) in neuronal cells, with the fractional time derivatives ranging from U1 = 0.8 to 1.0. The buffering mechanism functions as a process to decrease concentration, while fractional time derivatives are linked to cell memory phenomena, which serve as a mechanism to increase concentration during the initial time in neuronal cells. Due to the lower buffer concentration, there is no mismatch between the buffering mechanism and cell memory mechanism in neuronal cells. At lower-order fractional derivatives, the cell memory phenomenon causes an earlier accomplishment of peak calcium concentration and more quickly approaching to equilibrium state compared to the higher-order fractional derivatives in neuronal cells. While higher buffer concentrations with cell memory causes disturbances in calcium and NO distribution in the form of oscillations in neuronal cells as illustrated in figures 8 and 10.
Figure 10. NO concentration with U2 = 1.0, V1 and V2 = 2.0, buffer 75 μM at x = 0 μm for distinct [Ca2+] time fractional derivatives.
The absolute relative approximate errors were calculated and exhibited in table 2 for calcium and NO concentrations by utilizing the GS iterations (Itr). After Itr = 500, the model’s accuracies for [Ca2+] and NO concentrations are, respectively, 100% and 99.85% at site 0, 0.5, and 1.0 μm. Therefore, concerning the proposed model, the lowest accuracy and maximum error are sequentially 99.85% and 0.15%.
Table 2. The analysis of absolute relative approximate errors for [Ca2+] and NO concentrations utilizing GS iterations at distinct positions.
Absolute relative approximate errors for [Ca2+] Absolute relative approximate errors for NO
Itr 0 μm 0.5 μm 1.0 μm 0 μm 0.25 μm 1.0 μm
10 and 11 0.092891% 0.1% 0.1% 9.45% 9.43% 9.48%
20 and 21 0.94357 $\times $ 10−5% 0.10199 $\times $ 10−4% 0.90226 $\times $ 10−4% 5.2% 5.26% 5.28%
100 and 101 0.0% 0.0% 0.0% 1.83% 1.83% 1.82%
500 and 501 0.0% 0.0% 0.0% 0.15% 0.15% 0.15%
The concentrations of [Ca2+] and NO at [B] = 0.16 μM, VSERCA= 0.7 μM s−1, σ = 4.0 pA, P0 = 0 and IP3 concentration as 3.0 μM are numerically obtained and compared with earlier findings [80] for duration of 50 s and the results are in agreement as depicted in table 3. The computed root mean square error for table 3 is 0.0232, which is very low. However, no theoretical or experimental findings are noticed for additional validation of [Ca2+] and nitric oxide mechanisms and the outcomes are in accordance with biological realities.
Table 3. [Ca2+] concentrations compared to Wagner et al [80] at 50 s.
Site (μm) [Ca2+] concentrations (Wagner et al  [80]) [Ca2+] concentrations (current findings)
0 1.350 000 000 000 00 1.349 532 787 423 240
0.5 1.232 315 089 570 071 1.247 780 196 638 780
1.0 1.115 735 369 664 420 1.132 627 546 781 410
2.0 0.878 398 878 928 465 0.904 465 212 671 969
3.0 0.628 405 963 029 872 0.661 742 116 024 014
4.0 0.366 529 266 967 666 0.391 925 819 057 789
5.0 0.100 000 000 000 000 0.100 000 000 000 000
To conduct sensitivity analysis, the nonlinear systems of calcium and NO are tested with varying maximum and minimum values of distinct parameters as illustrated in table 4. The minimum and maximum values provide the permissible range for the parameter values. These selected values are then used to conduct model simulations, resulting in the model outputs. The problem was run with absolute and relative errors set to 1.5 $\times $ 10−3, treating it as the exact solution to the system of partial differential equations of calcium and NO in neuronal cells. Table 4 exhibits the variations in the parameter values and their significant effects on the concentration levels of [Ca2+] and NO in neuronal cells.
Table 4. Minimum and maximum values of few parameters of [Ca2+] and NO dynamics for sensitivity analysis.
Parameters Minimum values Maximum values Sensitivity to [Ca2+] Sensitivity to NO
P0 0.0 1.0 68.14% 29.93%
DCa(μm2s−1) 16 32 32.89% 3.80%
[IP3] M) 0.16 3.0 20.44% 6.35%
VIPR (s−1) 0 8.5 23.82% 7.61%

Conclusion

A mathematical model employing the CNS with Grunwald estimation along fractional spatial derivatives and L1 method along fractional temporal derivatives with GS iterations has been framed for spatiotemporal nonlinear interactive dynamics of [Ca2+] and NO in neurons. The utilization of the L1 method along temporal dimensions incorporates a cellular memory trace capable of capturing and integrating the entire history of neural cell activity. The influences of buffer mechanism, superdiffusion, memory triggering BM, etc on the nonlinear dynamical systems of [Ca2+] and NO were explored in neuronal cells. One of our findings confirms that elevated calcium and nitric oxide concentrations are associated with a variety of neurodegenerative illnesses like Parkinson’s [7], Alzheimer’s [46], ischemia [86], etc as reported by the researchers. Further, the consequences of the superdiffusion mechanism, cellular memory triggering BM, etc of the cooperative calcium and nitric oxide systems in neurons have not been reported earlier by research workers. Few investigations were reported on the integer-order interactive dynamics of calcium and NO concerning the one-way feedback process in neuron cells. In the present model, the two-way feedback between calcium and NO through calcium feedback on NO production and NO feedback on calcium through NO/cGMP pathways with plasmalemmal [Ca2+]-ATPase PMCA was incorporated. The novel conclusion is the constitute processes and events that cause the elevation in [Ca2+] and NO concentration levels in neurons. Our numerical results confirm that utilizing fractional reaction-diffusion equations is a suitable method for investigating the superdiffusion and cell memory mechanisms of complex spatiotemporal [Ca2+] and nitric oxide dynamics. Thus, the findings of the present research provide novel insights into the superdiffusion, memory exhibiting Brownian motion, etc along different crucial cell mechanisms like buffer process on the nonlinear spatiotemporal interactive dynamics of [Ca2+] and nitric oxide in neurons.
The following conclusions can be drawn based on the numerical outcomes,
i

(i) The superdiffusion of calcium ions decreases calcium levels, which further reduces the NO production in neurons. The decrease in the formation of NO causes a decrease in nitric oxide levels in neurons. The occurrence of superdiffusive behavior might be observed when the concentrations of calcium and NO are not at appropriate levels within neuronal cells. The numerical results indicate that calcium ions and NO molecules exhibit a superdiffusion process in neuronal cells under neurotoxic conditions, leading to a decrease in their concentration levels. The transition from normal diffusion to superdiffusion occurs in all these cases as a result of cell development and death. Thus, the disturbances in the superdiffusion of calcium ions process can lead to dysregulation in the [Ca2+] and NO signaling by elevating [Ca2+] and NO levels, which are associated to several neuronal illnesses like Parkinson’s and Alzheimer’s.

ii

(ii) The cell memory effect of calcium ions enhances calcium levels, which further increases nitric oxide production as well as NO concentration in neurons. The elevated [Ca2+] and nitric oxide levels may result in neurotoxicity in the form of Alzheimer’s, ischemia, and other neurodegenerative diseases.

iii

(iii) The higher buffer levels with cell memory leads to the dysregulation in the [Ca2+], NO production and NO concentration in the form of fluctuations. These disturbances in the [Ca2+] and NO levels due to the dysregulation in the buffer capacity are linked to various neurological diseases such as Alzheimer’s.

Thus, one may conclude that the superdiffusion, memory effects causing Brownian motion, higher buffer levels, etc can significantly affect the calcium, NO production and NO levels in neurons. Any alterations in these processes can cause alterations in [Ca2+] and NO mechanisms in neurons in the form of several neurological illnesses. The proposed mathematical model employing the CNS with the Grunwald scheme for spatial derivatives and the L1 method for temporal derivatives incorporating the GS iterations is highly effective at generating novel insights into the alterations in the cooperative nonlinear systems of [Ca2+] and NO in neurons.
The proposed model will be valid for the conditions assumed during the formulation like the other signaling systems are in a static mode with constant feedback in the proposed model. But, if the other signaling systems like IP3, β-amyloid, ATP, dopamine, etc are not in homeostasis, then their feedback will be dynamic, and therefore the present model is required to be modified by coupling with another dynamic system. Such models will be developed in future research to study the impacts of other signaling systems on [Ca2+] and NO dynamics in neuron cells.
The proposed model is quite effective in understating the sensitivity of interdependent [Ca2+] and NO dynamics with respect to other neurological conditions like Alzheimer’s and Parkinson’s diseases. The model is proposed for a general neuron cell and can be customized for different types of neurons and neurological conditions by incorporating appropriate data specific to neuron types and neurological disorders. Thus, the findings of the proposed model can be generalized for different types of neurons and neurological conditions. However, further generalization will require modeling of more than two signaling systems involving [Ca2+] and NO, which is the future scope of the development of this area.
Various cellular mechanisms and parameters influence the cooperative dynamics of [Ca2+] and nitric oxide in neurons. Thus, it is imperative to comprehend the impact of these parameters on the calcium and NO systems in neurons. The utilization of mathematical models and numerical simulation tools is able to predict the functioning of calcium and NO signaling systems in neuronal cells based on specific input parameters. The present study lays the groundwork for future research on the additional investigation and experimental validation of simulation predictions required to assess the applicability of the numerical models and tools. Also, this research holds particular significance for the scientific community’s understanding of numerous neurological illnesses. Biomedical scientists can utilize this study to develop strategies for diagnosing, preventing, and treating the aforementioned neurological illnesses.

Author contributions

In this paper, the authors contributed equally concerning problem formulation, solutions, review of literature, data correction, and findings interpretations. The Matlab code was created by Author (1).

Conflict of interest

Both authors of the current study state that there are no conflicts of interest.

Appendix

Model equations summary

To solve the equations (1) and (7), the CNS with Grunwald procedure is employed along fractional space derivative [76] as depicted below,
$\begin{eqnarray}\displaystyle \frac{{\partial }^{{V}_{1}}C}{\partial {x}^{{V}_{1}}}=\displaystyle \sum _{k=0}^{i+{\rm{1}}}\left(\displaystyle \frac{1}{2{h}^{{V}_{1}}}g{1}_{k}{{\rm{C}}}_{i-k+{\rm{1}}}^{j}+\displaystyle \frac{1}{2{h}^{{V}_{1}}}g{1}_{k}{{\rm{C}}}_{i-k+{\rm{1}}}^{j+1}\right),\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\partial }^{{V}_{2}}P}{\partial {x}^{{V}_{2}}}=\displaystyle \sum _{k=0}^{i+{\rm{1}}}\left(\displaystyle \frac{1}{2{h}^{{V}_{2}}}g{2}_{k}{P}_{i-k+{\rm{1}}}^{j}+\displaystyle \frac{1}{2{h}^{{V}_{2}}}g{2}_{k}{P}_{i-k+{\rm{1}}}^{j+1}\right),\end{eqnarray}$
where C represents the [Ca2+] and P represents the NO concentration and the step size along space is denoted by h concerning i = 1, 2, 3…, K−1. The normalized Grunwald weights g1k and g2k are associated with the order V1 and V2 and index k are expressed as below,
$\begin{eqnarray}g{1}_{k}=\displaystyle \frac{{\rm{\Gamma }}\left(k-{V}_{1}\right)}{{\rm{\Gamma }}\left(-{V}_{1}\right){\rm{\Gamma }}\left(k+1\right)},\end{eqnarray}$
$\begin{eqnarray}{g}{2}_{{k}}=\displaystyle \frac{{\rm{\Gamma }}\left({k}-{{V}}_{2}\right)}{{\rm{\Gamma }}\left(-{{V}}_{2}\right){\rm{\Gamma }}\left({k}+1\right)}.\end{eqnarray}$
The L1 formula along temporal derivatives for equations (1) and (7) is reported as follows [87],
$\begin{eqnarray}\displaystyle \frac{{\partial }^{{{\rm{U}}}_{1}}C}{\partial {t}^{{U}_{1}}}=\displaystyle \frac{1}{{\rm{\Gamma }}\left(2-{U}_{1}\right)}\displaystyle \sum _{k=0}^{j}\displaystyle \frac{b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)}{{\left({\rm{\Delta }}t\right)}^{{U}_{1}}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\partial }^{{U}_{2}}P}{\partial {t}^{{U}_{2}}}=\displaystyle \frac{1}{{\rm{\Gamma }}\left(2-{U}_{2}\right)}\displaystyle \sum _{k=0}^{j}\displaystyle \frac{b{2}_{k}\left({P}_{i}^{j+1-k}-{P}_{i}^{j-k}\right)}{{\left({\rm{\Delta }}t\right)}^{{U}_{2}}},\end{eqnarray}$
where
$\begin{eqnarray}b{1}_{{k}}={\left(k+1\right)}^{1-{U}_{1}}-{\left(k\right)}^{1-{U}_{1}},\end{eqnarray}$
$\begin{eqnarray}b{2}_{k}={\left(k+1\right)}^{1-{U}_{2}}-{\left(k\right)}^{1-{U}_{2}}.\end{eqnarray}$
Utilizing equations (20) and (24) for equation (1), we get,
$\begin{eqnarray}\begin{array}{c}\frac{1}{{\rm{\Gamma }}\left(2-{{U}}_{1}\right)}\displaystyle \sum _{{k}=0}^{{j}}\frac{{{\rm{b1}}}_{{k}}\left({{C}}_{{i}}^{{j}+1-{k}}-{{C}}_{{i}}^{{j}-{k}}\right)}{{({\rm{\Delta }}{t})}^{{{U}}_{1}}}\\ \,=\,{{D}}_{{\rm{Ca}}}\left(\frac{1}{2{h}^{{{V}}_{1}}}\displaystyle \sum _{{k}=0}^{{i}+1}\frac{{\rm{\Gamma }}\left({k}-{{V}}_{1}\right)}{{\rm{\Gamma }}\left(-{{V}}_{1}\right){\rm{\Gamma }}\left({k}+1\right)}{{C}}_{{i}-{k}+1}^{{j}}\right.\\ \,\left.+\,\frac{1}{2{h}^{{{V}}_{1}}}\displaystyle \sum _{{k}=0}^{{i}+1}\frac{{\rm{\Gamma }}\left({k}-{{V}}_{1}\right)}{{\rm{\Gamma }}\left(-{{V}}_{1}\right){\rm{\Gamma }}\left({k}+1\right)}{{C}}_{{i}-{k}+1}^{{j}+1}\right)\\ \,+\,\frac{1}{{{F}}_{{C}}}\left(\begin{array}{c}{{V}}_{{\rm{IPR}}}{m}^{3}{h}^{3}\left({{C}}_{{\rm{ER}}}-{{C}}_{{i}}^{{j}}\right)-{{V}}_{{\rm{SERCA}}}\left(\frac{{\left({{C}}_{{i}}^{{j}}\right)}^{2}}{{\left({{C}}_{{i}}^{{j}}\right)}^{2}+{\left({{K}}_{{\rm{SERCA}}}\right)}^{2}}\right)\\ +{{V}}_{{\rm{LEAK}}}\left({{C}}_{{\rm{ER}}}-{{C}}_{{i}}^{{j}}\right)+{{V}}_{{\rm{RyR}}}{{P}}_{0}\left({{C}}_{{\rm{ER}}}-{{C}}_{{i}}^{{j}}\right)\end{array}\right)\\ \,-\,{{K}}^{+}{\left[{B}\right]}_{\infty }\left(\left[{{\rm{Ca}}}^{2+}\right]-{\left[{{\rm{Ca}}}^{2+}\right]}_{\infty }\right).\end{array}\end{eqnarray}$
Utilizing equations (21) and (25) for equation (7), we obtain,
$\begin{eqnarray}\begin{array}{l}\frac{1}{{\rm{\Gamma }}\left(2-{U}_{2}\right)}\displaystyle \sum _{k=0}^{j}\frac{b{2}_{k}\left({P}_{i}^{j+1-k}-{P}_{i}^{j-k}\right)}{{\left({\rm{\Delta }}t\right)}^{{U}_{2}}}\\ \,=\,{D}_{\mathrm{NO}}\left(\frac{1}{2{h}^{{V}_{2}}}\displaystyle \sum _{k=0}^{i+1}\frac{{\rm{\Gamma }}\left(k-{V}_{2}\right)}{{\rm{\Gamma }}\left(-{V}_{2}\right){\rm{\Gamma }}\left(k+1\right)}{P}_{i-k+1}^{j}\right.\\ \left.\,+\,\frac{1}{2{h}^{{V}_{2}}}\displaystyle \sum _{k=0}^{i+1}\frac{{\rm{\Gamma }}\left(k-{V}_{2}\right)}{{\rm{\Gamma }}\left(-{V}_{2}\right){\rm{\Gamma }}\left(k+1\right)}{P}_{i-k+1}^{j+1}\right)\\ \,+\,\left({V}_{\mathrm{NO}}\left(\frac{\left({C}_{i}^{j}\right)}{\left({C}_{i}^{j}\right)+\left({K}_{\mathrm{NO}}\right)}\right)-{K}_{1}{P}_{i}^{j}\right).\end{array}\end{eqnarray}$
Equation (28) can be re-written as follows,
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{{\rm{\Gamma }}\left(2-{U}_{1}\right)}\displaystyle \sum _{k=0}^{j}\displaystyle \frac{b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)}{{\left({\rm{\Delta }}t\right)}^{{U}_{1}}}\\ \,=\,{D}_{\mathrm{Ca}}\left(\displaystyle \frac{1}{2{h}^{{V}_{1}}}\displaystyle \sum _{k=0}^{i+1}g{1}_{k}{C}_{i-k+{\rm{1}}}^{j}+\displaystyle \frac{1}{2{h}^{{V}_{1}}}\displaystyle \sum _{k=0}^{i+{\rm{1}}}g{1}_{k}{C}_{i-k+1}^{j+1}\right)\\ \,+\,d1\left({C}_{i}^{j}{,P}_{i}^{j}\right)+{f}_{i}^{j+1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{k=0}^{j}{\mathrm{b1}}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)\\ \,=\,{D}_{\mathrm{Ca}}\displaystyle \frac{{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}}{2{h}^{{V}_{1}}}\left(\displaystyle \sum _{k=0}^{i+1}g{1}_{k}{C}_{i-k+1}^{j}+\displaystyle \sum _{k=0}^{i+1}g{1}_{k}{C}_{i-k+1}^{j+1}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}d1\left({C}_{i}^{j},{P}_{i}^{j}\right)+{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}f{1}_{i}^{j+1}.\end{array}\end{eqnarray}$
Defining, $B1={D}_{\mathrm{Ca}}\tfrac{{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}}{2{h}^{{V}_{1}}},$ then system depicted by (31) can be re-written as,
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{k=0}^{j}b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)\\ \,=\,B1\left(\displaystyle \sum _{k=0}^{i+1}g{1}_{k}{C}_{i-k+1}^{j}+\displaystyle \sum _{k=0}^{i+1}g{1}_{k}{C}_{i-k+1}^{j+1}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}d1\left({C}_{i}^{j},{P}_{i}^{j}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}f{1}_{i}^{j+1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\mathrm{b1}}_{0}\left({C}_{i}^{j+{\rm{1}}}-{C}_{i}^{j}\right)+b{1}_{1}\left({C}_{i}^{j}-{C}_{i}^{j-1}\right)\\ \,+\,\displaystyle \sum _{k=2}^{j}b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)\\ \,=\,B1g{1}_{0}{C}_{i+1}^{j}+B1g{1}_{1}{C}_{i}^{j}+B1\left(\displaystyle \sum _{k=2}^{i}g{1}_{k}{C}_{i-k}^{j}\right)\\ \,+\,B1g{1}_{0}{C}_{i+1}^{j+1}+B1g{1}_{1}{C}_{i}^{j+1}+B1g{1}_{2}{C}_{i-1}^{j+1}\\ \,+\,B1\left(\displaystyle \sum _{k=3}^{i}g{1}_{k}{C}_{i-k}^{j+1}\right)+{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}f{1}_{i}^{j+1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-B1g{1}_{0}{C}_{i+1}^{j+1}+\left(b{1}_{0}-B1g{1}_{1}\right){C}_{i}^{j+1}-B1g{1}_{2}{C}_{i-{\rm{1}}}^{j+1}\\ \,-\,B1\left(\displaystyle \sum _{k=3}^{i+1}g{1}_{k}{C}_{i-k+1}^{j+1}\right)=B1g{1}_{0}{C}_{i+1}^{j}\\ \,+\,\left(B1g{1}_{1}+b{1}_{0}-b{1}_{1}\right){C}_{i}^{j}\\ \,+\,b{1}_{1}{C}_{i}^{j-1}+B1\left(\displaystyle \sum _{k=2}^{i+1}g{1}_{k}{C}_{i-k+1}^{j}\right)\\ \,-\,\displaystyle \sum _{k=2}^{j}b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}d1\left({C}_{i}^{j}{,P}_{i}^{j}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}f{1}_{i}^{j+1},\end{array}\end{eqnarray}$
where $d1\left({C}_{i}^{j},{P}_{i}^{j}\right)$ denotes the nonlinear expressions of equation (28).
Equation (34) depicts the nonlinear equations system and represents as follows,
$\begin{eqnarray}\begin{array}{l}\bar{A}1{}^{j+1}=\bar{M}1\,{\bar{{\rm{C}}}}^{j}\\ \,+\,\left(b{1}_{1}{C}_{i}^{j-1}-\displaystyle \sum _{k=2}^{j}b{1}_{k}\left({C}_{i}^{j+1-k}-{C}_{i}^{j-k}\right)\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{U}_{1}}d1\left({C}_{i}^{j},{P}_{{\rm{i}}}^{{\rm{j}}}\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{1}\right){\left({\rm{\Delta }}t\right)}^{{{\rm{U}}}_{1}}\,\bar{F}{1}^{j+1},\end{array}\end{eqnarray}$
where
$\begin{eqnarray}{\bar{{C}}}^{{j}+1}={\left[{{{C}}_{0}}^{{j}+1},\,{{{C}}_{1}}^{{j}+1},\,{{{C}}_{2}}^{{j}+1},\mathrm{..}.,\,{{{C}}_{{K}}}^{{j}+1}\right]}^{{\rm{T}}},\end{eqnarray}$
$\begin{eqnarray}{\bar{{C}}}^{{j}}={\left[0,\,{{{C}}_{1}}^{{j}},\,{{{C}}_{2}}^{{j}},\,\mathrm{..}.\,,\,{{{C}}_{{K}-1}}^{{j}}\right]}^{{\rm{T}}},\end{eqnarray}$
$\begin{eqnarray}\bar{{F}}{1}^{{j}}={\left[0,\,{f}{1}_{1}^{{j}},\,{f}{1}_{2}^{{j}},\mathrm{..}.,\,{f}{1}_{{K}-1}^{{j}}\right]}^{{\rm{T}}}.\end{eqnarray}$
The expression for the coefficients matrix $\bar{A}1=\left[A{1}_{i,j}\right]$ for i, j = 1, 2, …, K$-$1 is represented as follows,
$\begin{eqnarray}{A}{1}_{{i},{j}}=\left\{\begin{array}{cc}0 & {\rm{When}}\,{j}\geqslant {i}+2\\ -{g}{1}_{0}{B}1 & {\rm{When}}\,{j}={i}+1\\ {b}{1}_{0}-{g}{1}_{1}{B}1 & {\rm{When}}\,{j}={i}\\ -{g}{1}_{2}{B}1 & {\rm{When}}\,{j}={i}-1\\ -{g}{1}_{{i}-{j}+1}{B}1 & {\rm{When}}\,{j}\leqslant {i}-1\end{array}\right\}.\end{eqnarray}$
The similar procedure has been employed for equation (29), we acquired the nonlinear equations system as follows,
$\begin{eqnarray}\begin{array}{l}\bar{A}2{\bar{P}}^{j+1}=\bar{M}2{\bar{C}}^{j}+\left(b{2}_{1}{P}_{i}^{j-1}-\displaystyle \sum _{k=2}^{j}b{2}_{k}\left({P}_{i}^{j+1-k}-{P}_{i}^{j-k}\right)\right)\\ \,+\,{\rm{\Gamma }}\left(2-{U}_{2}\right){\left({\rm{\Delta }}t\right)}^{{U}_{2}}d2\left({C}_{i}^{j},{P}_{i}^{j}\right)+{\rm{\Gamma }}\left(2-{U}_{2}\right){\left({\rm{\Delta }}t\right)}^{{U}_{2}}\,\bar{F}{2}^{j+1},\end{array}\end{eqnarray}$
where $d2\left({C}_{i}^{j},{P}_{i}^{j}\right)$ denotes the nonlinear expressions of equation (31).
The expression for the coefficients matrix $\bar{{A}}2=\left[{A}{2}_{{i},{j}}\right]$ for i, j = 1, 2, …, K−1 is represented as follows,
$\begin{eqnarray}{A}{2}_{{i},{j}}=\left\{\begin{array}{cc}0 & {\rm{When}}\,{j}\geqslant {i}+2\\ -{g}{2}_{0}{B}2 & {\rm{When}}\,{j}={i}+1\\ {b}{2}_{0}-{g}{2}_{1}{B}2 & {\rm{When}}\,{j}={i}\\ -{g}{2}_{2}{B}2 & {\rm{When}}\,{j}={i}-1\\ -{g}{2}_{{i}-{j}+1}{B}2 & {\rm{When}}\,{j}\leqslant {i}-1\end{array}\right\}.\end{eqnarray}$
Here, $B2={D}_{{\rm{N}}{\rm{O}}}\tfrac{{\rm{\Gamma }}\left(2-{U}_{2}\right){\left({\rm{\Delta }}t\right)}^{{U}_{2}}}{2{h}^{{V}_{2}}}$
Let an eigenvalue of matrix A1 be λ1, such that A1X = λ1X concerning some non-zero vector X.
Let $\left|{{x}}_{{i}}\right|=\,{\rm{\max }}\left\{\left|{{x}}_{{j}}\right|:{j}=0,\,1,\,\mathrm{..}.,{K}\right\},$ then ${\sum }_{j=0}^{K}A{1}_{i,j}{x}_{j}={\lambda }_{1}{x}_{i},$ thus
$\begin{eqnarray}{\lambda }_{1}={A}{1}_{{i},{i}}+\displaystyle \sum _{{j}=0,\,{j}\ne {i}}^{{K}}{A}{1}_{{i},{j}}\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}.\end{eqnarray}$
Putting the values of A1i,j in equation (42),
$\begin{eqnarray}\begin{array}{l}{\lambda }_{1}=b{1}_{0}-B1g{1}_{1}-B1g{1}_{0}\displaystyle \frac{{x}_{i+1}}{{x}_{i}}\\ \,-\,B1g{1}_{2}\displaystyle \frac{{x}_{i-1}}{{x}_{i}}-B1\left(\displaystyle \sum _{j=0}^{i-2}g{1}_{i-j+1}\displaystyle \frac{{x}_{j}}{{x}_{i}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}{\lambda }_{1}={b}{1}_{0}-{B}1\left(g{1}_{1}+\displaystyle \sum _{{j}=0,{j}\ne {i}}^{{i}+1}{\rm{g}}{1}_{{i}-{j}+1}\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}\right).\end{eqnarray}$
Since, $\left(\displaystyle {\sum }_{{K}=0}^{\infty }{g}{1}_{{K}}=0\right)$ and g11 are the only expression in Grunwald weights sequence that is negative and $g{1}_{1}=-\alpha ,$ and for $1\lt \alpha \leqslant 2$
$\begin{eqnarray}-{g}{1}_{1}\geqslant \displaystyle \sum _{{K}=0,\,{K}\ne 1}^{{j}}{g}{1}_{{K}}\,{\rm{for}}\,{j}\,=0,\,1,\,2,\mathrm{..}.,\end{eqnarray}$
Since $\left|\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}\right|\leqslant 1,\,{\rm{and}}\,{g}{1}_{{j}}\geqslant 0,\,{\rm{for}}\,{j}=0,\,2,\,3,\,\mathrm{..}.,$
$\begin{eqnarray}\displaystyle \sum _{{j}=0,\,{j}\ne {i}}^{{i}+1}{g}{1}_{{i}-{j}+1}\left|\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}\right|\leqslant \displaystyle \sum _{{j}=0,\,{j}\ne {i}}^{{i}+1}{g}{1}_{{i}-{j}+1}\leqslant {g}{1}_{1},\end{eqnarray}$
$\begin{eqnarray}{g}{1}_{1}+\displaystyle \sum _{{j}=0,\,{j}\ne {i}}^{{i}+1}{g}{1}_{{i}-{j}+1}\left|\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}\right|\leqslant 0.\end{eqnarray}$
Similarly, for matrix A2,
$\begin{eqnarray}{g}{2}_{1}+\displaystyle \sum _{{j}=0,\,{j}\ne {i}}^{{i}+1}{g}{2}_{{i}-{j}+1}\left|\displaystyle \frac{{{x}}_{{j}}}{{{x}}_{{i}}}\right|\leqslant 0.\end{eqnarray}$
For calcium and NO systems, we get the matrix expression as follows,
$\begin{eqnarray}{\left[A\right]}_{2K+1\times 2K+1}={\left[\begin{array}{cc}{\left[A1\right]}_{K\times K} & 0\\ 0 & {\left[A2\right]}_{K\times K}\end{array}\right]}_{2K+1\times 2K+1}.\end{eqnarray}$
The representation of equation (49) is used regarding the stability analysis of the employed numerical scheme by computing spectral radius (SR). As B1 and B2 are non-negative real numbers, thus every eigenvalue of A1, A2 and A fulfills $\left|\lambda \right|\geqslant 1.$ Then matrix A is invertible, thus every eigenvalue of A−1 fulfills $\left|\eta \right|\leqslant 1.$ This implies that the SR of matrix $\rho \left({A}^{-1}\right)\leqslant 1.$ Thus this suggests that the utilized numerical technique is unconditionally stable.
The Crank–Nicholson scheme, an implicit method is the favored technique for discretizing classical PDEs because of its unconditional stability, which allows for unrestricted time step sizes. Nevertheless, the aforementioned outcome indicates that stability deteriorates in the fractional scenario when employing the conventional Grunwald procedures to estimate the derivatives. To address this scenario, a modified version of the Grunwald formula can be employed, wherein the function evaluations are shifted one grid point to the right. The alteration renders the Crank–Nicholson approach and Grunwald formula both consistent and unconditionally stable. According to the lax equivalence theorem [88], the finite difference solution converges to the true solution as Δt → 0 and Δx → 0.
Further, the stability and error analysis has been performed to ensure that the computational approach used is appropriate, which is evident by its performance and effectiveness in this study.
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