Fractional-order reaction–diffusion model to study the dysregulatory impacts of superdiffusion and memory on neuronal calcium and IP₃ dynamics

Abstract

The models for the dynamics of single systems like calcium ([Ca²⁺]) in a neuron cell are able to provide information about factors affecting the calcium regulation in the neuron cell. But in realistic conditions, the dynamics of calcium also depends on the dynamics of other chemical molecules like inositol 1, 4, 5-trisphophate (IP₃) in the nerve cell. Some research workers explored the interdependent [Ca²⁺] and IP₃ signaling systems in neurons, but that too for integer-order systems only. The studies on integer-order system dynamics are not able to generate insights about the superdiffusion mechanisms, cell memory causing Brownian motion of signaling molecules in neurons. Few attempts to investigate fractional reaction–diffusion models of individual [Ca²⁺] dynamics have been reported in neurons. However, no attempt is noticed in the literature to analyze the interdependent fractional processes of [Ca²⁺] and IP₃ dynamics in neurons. In the present paper, a mathematical model of nonlinear interdependent spatiotemporal [Ca²⁺] and IP₃ signaling systems is proposed incorporating fractional diffusion processes and cell memory of [Ca²⁺] and IP₃ causing Brownian motion effects in neurons. The numerical solution is obtained by the Crank–Nicolson scheme (CNS) with Grunwald estimation for fractional derivatives along spatial dimensions and L1 scheme for fractional derivatives along temporal dimensions with Gauss–Seidel (GS) iterations. The influences of the superdiffusion mechanism and cell memory on the interdependent [Ca²⁺] and IP₃ signaling systems in neurons are analyzed with the help of numerical results. A significant difference in results is observed between fractional-order and integer-order dynamics. It is concluded that the cell can use memory effects and superdiffusion mechanisms to regulate the calcium and IP₃ concentrations at appropriate levels. Further, the fractional-order processes like superdiffusion and cell memory can also be a cause of dysregulation of calcium and IP₃ dynamics leading to neurological disorders like Parkinson’s, etc.

Appendix: Model equations summary

For the solution of Eqs. (1) and (7), the CNS with Grunwald approximation to the fractional derivatives along space is expressed as [69]:

$$ \frac{{{\partial ^{{V_1}}}C}}{{\partial {x^{{{\rm{V}}_1}}}}} = \sum\limits_{k = 0}^{{\rm{i + 1}}} {\mkern 1mu} {\rm{ }}\left( {\frac{1}{{2{h^{{{\rm{V}}_1}}}{\mkern 1mu} }}g{1_{\rm{k}}}{C_{{\rm{i}} - {\rm{k}} + 1}}^{\rm{j}} + \frac{1}{{2{h^{{{\rm{V}}_1}}}{\mkern 1mu} }}g{1_{\rm{k}}}C_{{\rm{i - k + 1}}}^{{\rm{j + 1}}}{\rm{ }}} \right) $$

(18)

$$\frac{{{\partial ^{{V_2}}}P}}{{\partial {x^{{{\rm{V}}_2}}}}} = \sum\limits_{k = 0}^{{\rm{i + 1}}} {\left( {\frac{1}{{2{h^{{{\rm{V}}_2}}}{\mkern 1mu} }}g{2_{\rm{k}}}{P_{{\rm{i}} - {\rm{k}} + 1}}^{\rm{j}} + \frac{1}{{2{h^{{{\rm{V}}_2}}}{\mkern 1mu} }}g{2_{\rm{k}}}P_{{\rm{i - k + 1}}}^{{\rm{j + 1}}}} \right)} {\rm{ }} $$

(19)

Here, C = [Ca²⁺] and P = IP₃ and h represents step size along space, for i = 1, 2, 3, …, K-1. The normalized Grunwald weights g1_k and g2_k, which are dependent on the index k and order V₁ and V₂, sequentially can be represented as shown below:

$$ {{g1}}_{{\text{k}}} { = }\frac{{{{\Gamma (k - V}}_{{1}} {)}}}{{{{\Gamma ( - V}}_{{1}} {{)\Gamma (k + 1)}}}} $$

(20)

$$ {{g2}}_{{\text{k}}} { = }\frac{{{{\Gamma (k - V}}_{2} {)}}}{{{{\Gamma ( - V}}_{2} {{)\Gamma (k + 1)}}}} $$

(21)

For Eqs. (1) and (7), the expressions for the fractional time derivative by the L1 formula are proposed by Oldham and Spanier [70] and shown as below:

$$ \frac{{\partial^{{{\text{U}}_{{1}} }} {{C}}}}{{\partial {{t}}^{{{\text{U}}_{{1}} }} }}{ = }\frac{{1}}{{{{\Gamma (2 - U}}_{{1}} {)}}}\sum\limits_{{{k = 0}}}^{{\text{j}}} {\frac{{{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j} + 1 - {\text{k}}}} {{ - C}}_{{\text{i }}}^{{\text{j} - {\text{k}}}} { )}}}{{{{(\Delta t)}}^{{{\text{U}}_{{1}} }} }}} $$

(22)

$$ \frac{{\partial^{{{\text{U}}_{2} }} {{P}}}}{{\partial {\text{t}}^{{{\text{U}}_{2} }} }}{ = }\frac{{1}}{{{{\Gamma (2 - U}}_{2} {)}}}\sum\limits_{{{k = 0}}}^{{\text{j}}} {\frac{{{{b2}}_{{\text{k}}} {{(P}}_{{\text{i }}}^{{\text{j} + 1 - {\text{k}}}} {{ - P}}_{{\text{i }}}^{{\text{j} - {\text{k}}}} { )}}}{{{{(\Delta t)}}^{{{\text{U}}_{2} }} }}} $$

(23)

where

$$b{1_{\rm{k}}} = {(k + 1)^{{\rm{1}} - {{\rm{U}}_1}}} - {(k)^{1 - {{\rm{U}}_1}}} $$

(24)

$$ {{b2}}_{{\text{k}}} ={{ (k + 1)}}^{{{{1 - \text{U}}}_{2} }} -{{ (k)}}^{{{{1 - \text{U}}}_{2} }} $$

(25)

For solving the acquired fractional-order equations, the CNS with Grunwald estimation along space derivative and the L1 formula along time derivative are utilized.

Utilizing Eqs. (18 & 22) in Eq. (1), we have

$$ \begin{aligned} \frac{{1}}{{{{\Gamma (2 - U}}_{{1}} {)}}}\sum\limits_{{\text{k = 0}}}^{{\text{j}}} {\frac{{{{b1}}_{{\text{k}}} {\text{(C}}_{{\text{i }}}^{{\text{j + 1} - \text{k}}} {{ - C}}_{{\text{i }}}^{{\text{j } - \text{k}}} { )}}}{{{{(\Delta t)}}^{{{\text{U}}_{{1}} }} }}}& = {{D}}_{{{\text{Ca}}}} \left( {\frac{{1}}{{{{2h}}^{{{\text{V}}_{{1}} }} \, }}\sum\limits_{{\text{k = 0}}}^{{\text{i + 1}}} {\frac{{{{\Gamma (k - V}}_{{1}} {)}}}{{{{\Gamma ( - V}}_{{1}} {{)\Gamma (k + 1)}}}} \, } {{C}}_{{\text{i} - \text{k + 1}}}^{{\text{j}}} + \frac{{1}}{{{{2h}}^{{{\text{V}}_{{1}} }} \, }}\sum\limits_{{{k = 0}}}^{{\text{i + 1}}} {\frac{{{{\Gamma (k - V}}_{{1}} {)}}}{{{{\Gamma ( - V}}_{{1}} {{)\Gamma (k + 1)}}}} \, } {{C}}_{{\text{i } - \text{ k + 1}}}^{{\text{j + 1}}} } \right) \\ \, & \quad + \frac{{1}}{{{{F}}_{{\text{C}}} }}\left( \begin{gathered} {{V}}_{{{\text{IPR}}}} {{m}}^{{3}} {{h}}^{{3}} \left( {{{C}}_{{{\text{ER}}}} - {{C}}_{{\text{i}}}^{{\text{j}}} } \right) - {{V}}_{{{\text{SERCA}}}} \left( {\frac{{\left( {{{C}}_{{\text{i}}}^{{\text{j}}} } \right)^{2} }}{{\left( {{{C}}_{{\text{i}}}^{{\text{j}}} } \right)^{2} + \left( {{{K}}_{{{\text{SERCA}}}} } \right)^{2} }}} \right) \hfill \\ + {{V}}_{{{\text{LEAK}}}} \left( {{{C}}_{{{\text{ER}}}} - {{C}}_{{\text{i}}}^{{\text{j}}} } \right) \hfill \\ \end{gathered} \right) \, - {{K}}^{ + } \left[ {\text{B}} \right]_{\infty } \left( {{{C}}_{{\text{i}}}^{{\text{j}}} - {{C}}_{\infty } } \right) \\ \end{aligned} $$

(26)

Utilizing the Eqs. (19 and 23) in Eq. (7), we obtain

$$ \begin{aligned} \frac{{1}}{{{{\Gamma (2 - U}}_{2} {)}}}\sum\limits_{{\text{k = 0}}}^{{\text{j}}} {\frac{{{{b2}}_{{\text{k}}} {{(P}}_{{\text{i }}}^{{{\text{j + 1}} - {{k}}}} \, - {{P}}_{{\text{i }}}^{{{\text{j}} - {{k}}}} { )}}}{{{{(\Delta t)}}^{{{\text{U}}_{2} }} }}} =\, & {{D}}_{{\text{a}}} \left( {\frac{{1}}{{{{2h}}^{{{\text{V}}_{{2}} }} \, }}\sum\limits_{{\text{k = 0}}}^{{\text{i + 1}}} {\frac{{{{\Gamma (k}} - {{V}}_{{2}} {)}}}{{{\Gamma (} - {{V}}_{{2}} {{)\Gamma (k + 1)}}}} \, } {{P}}_{{{\text{i}} - {{k + 1}}}}^{{\text{j}}} + \frac{{1}}{{{{2h}}^{{{\text{V}}_{{2}} }} \, }}\sum\limits_{{{k = 0}}}^{{\text{i + 1}}} {\frac{{{{\Gamma (k}} - {{V}}_{{2}} {)}}}{{{\Gamma (} - {{V}}_{{2}} {{)\Gamma (k + 1)}}}} \, } {{P}}_{{{{i}} - {{k + 1}}}}^{{\text{j + 1}}} } \right) \\ & \quad { + }\frac{1}{{{{F}}_{{{c}}} }}\left( \begin{gathered} {{V}}_{{{\text{production}}}} \left( {\frac{{\left( {{{C}}_{{\text{i}}}^{{\text{j}}} } \right)^{2} }}{{\left( {{{C}}_{{\text{i}}}^{{\text{j}}} } \right)^{2} + \left( {{{K}}_{{{\text{production}}}} } \right)^{2} }}} \right) - {\lambda V}_{{\text{a}}} \left( {{1} - \left( {\frac{{{{C}}_{{\text{i}}}^{{\text{j}}} }}{{{{C}}_{{\text{i}}}^{{\text{j}}} { + 0}{{.39}}}}} \right)} \right)\left( {\frac{{{{P}}_{{\text{i}}}^{{\text{j}}} }}{{{{P}}_{{\text{i}}}^{{\text{j}}} { + 2}{{.5}}}}} \right) \hfill \\ - {\lambda V}_{{\text{b}}} \left( {\frac{{{{C}}_{{\text{i}}}^{{\text{j}}} }}{{{{C}}_{{\text{i}}}^{{\text{j}}} { + 0}{{.39}}}}} \right)\left( {\frac{{{\text{P}}_{{\text{i}}}^{{\text{j}}} }}{{{{P}}_{{\text{i}}}^{{\text{j}}} { + 0}{{.5}}}}} \right) - {\lambda V}_{{{\text{ph}}}} \left( {\frac{{{{P}}_{{\text{i}}}^{{\text{j}}} }}{{{{P}}_{{\text{i}}}^{{\text{j}}} { + 30}}}} \right) \hfill \\ \end{gathered} \right) \\ \end{aligned} $$

(27)

From Eq. (26), we obtain,

$$ \begin{aligned} & \frac{{1}}{{{{\Gamma (2 - U}}_{{1}} {)}}}\sum\limits_{{\text{k} = 0}}^{{\text{j}}} {\frac{{{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j} + 1 - \text{k}}} {{ - C}}_{{\text{i }}}^{{\text{j} - \text{k}}} { )}}}{{{{(\Delta t)}}^{{{\text{U}}_{{1}} }} }}} \\ & \quad = {{D}}_{{{\text{Ca}}}} \left( {\frac{{1}}{{{{2h}}^{{{\text{V}}_{{1}} }} \, }}\sum\limits_{{\text{k = 0}}}^{{\text{i} + 1}} {{{g1}}_{{\text{k}}} \, } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j}}} + \frac{{1}}{{{{2h}}^{{{\text{V}}_{{1}} }} \, }}\sum\limits_{{{k = 0}}}^{{\text{i} + 1}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j} + 1}} } \right){{ + d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)} + {{f}}_{\text{i}}^{\text{j} + 1} \end{aligned}$$

(28)

$$ \begin{aligned} \sum\limits_{{{k = 0}}}^{{\text{j}}} {{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j} + 1 - \text{k}}} {{ - C}}_{{\text{i }}}^{{\text{j} - \text{k}}} { )}} = & {{D}}_{{{\text{Ca}}}} \frac{{{{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} }}{{{{2h}}^{{{\text{V}}_{{1}} }} }}\left( {\sum\limits_{{\text{k = 0}}}^{{\text{i} + 1}} {{{g1}}_{{\text{k}}} \, } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j}}} + \sum\limits_{{{k = 0}}}^{{\text{i} + 1}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j} + 1}} } \right) \\ \, & \quad \, + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)} + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{f1}}_{{\text{i}}}^{{\text{j} + 1}} \\ \end{aligned} $$

(29)

Defining, ${{B1 = D}}_{{{\text{Ca}}}} \frac{{{{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} }}{{{{2h}}^{{{\text{V}}_{{1}} }} }}$, then Eq. (29) can be written as follows:

$$ \begin{aligned} \sum\limits_{{\text{k = 0}}}^{{\text{j}}} {{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j} + 1 - \text{k}}} {{ - C}}_{{\text{i }}}^{{\text{j} - \text{k}}} { )}} = & {{B1}}\left( {\sum\limits_{{\text{k = 0}}}^{{\text{i} + 1}} {{{g1}}_{{\text{k}}} \, } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j}}} + \sum\limits_{{{k = 0}}}^{{\text{i} + 1}} {{\text{g1}}_{{\text{k}}} } {{C}}_{{\text{i} - \text{k} + 1}}^{{\text{j} + 1}} } \right) \\ \, & \quad \, + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)} + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{f1}}_{{\text{i}}}^{{\text{j} + 1}} \\ \end{aligned} $$

(30)

$$ \begin{aligned} & {{b1}}_{0} {{(C}}_{{\text{i }}}^{{\text{j + 1 }}} {{ - C}}_{{\text{i }}}^{{\text{j }}} {)} + {{b1}}_{1} {{(C}}_{{\text{i }}}^{{\text{j }}} {{ - C}}_{{\text{i }}}^{{\text{j } - \text{ 1}}} {)} + \sum\limits_{{{k = 2}}}^{{\text{j}}} {{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j + 1 } - \text{ k}}} {{ - C}}_{{\text{i }}}^{{\text{j } - \text{ k}}} { )}} \\ & \quad = {{B1g1}}_{0} {{C}}_{{\text{i + 1 }}}^{{\text{j}}} + {{B1g1}}_{1} {{C}}_{{\text{i }}}^{{\text{j}}} + {{B1}}\left( {\sum\limits_{{\text{k = 2}}}^{{\text{i}}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i } - \text{ k}}}^{{\text{j}}} } \right) \\ & \quad \quad \, + {{B1g1}}_{0} {{C}}_{{\text{i + 1 }}}^{{\text{j + 1}}} + {{B1g1}}_{1} {{C}}_{{\text{i }}}^{{\text{j + 1}}} + {{B1g1}}_{2} {{C}}_{{\text{i } - \text{ 1 }}}^{{\text{j + 1}}} \\ \, & \quad \quad \, + {{B1}}\left( {\sum\limits_{{{k = 3}}}^{{\text{i}}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i } - \text{ k}}}^{{\text{j + 1}}} } \right) + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{f1}}_{{\text{i}}}^{{\text{j + 1}}} \\ \end{aligned} $$

(31)

$$ \begin{aligned} & {{ - B1g1}}_{0} {{C}}_{{\text{i + 1 }}}^{{\text{j + 1 }}} + \left( {{{b1}}_{{0}} {{ - B1g1}}_{1} } \right){{C}}_{{\text{i }}}^{{\text{j + 1}}} {{ - B1g1}}_{2} {{C}}_{{\text{i } - \text{ 1 }}}^{{\text{j + 1 }}} {{ - B1}}\left( {\sum\limits_{{\text{k = 3}}}^{{\text{i + 1}}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i } - \text{ k + 1}}}^{{\text{j + 1}}} } \right) \, \\ & \quad {{ = B1g1}}_{0} {{C}}_{{\text{i + 1 }}}^{{\text{j}}} + \left( {{{B1g1}}_{1} {{ + b1}}_{{0}} {{ - b1}}_{{1}} } \right){{C}}_{{\text{i }}}^{{\text{j}}} \\ \, & \quad \quad \, + {{b1}}_{1} {{C}}_{{\text{i }}}^{{\text{j } - \text{ 1}}} + {{B1}}\left( {\sum\limits_{{\text{k = 2}}}^{{\text{i + 1}}} {{{g1}}_{{\text{k}}} } {{C}}_{{\text{i } - \text{ k + 1}}}^{{\text{j}}} } \right) - \sum\limits_{{\text{k = 2}}}^{{\text{j}}} {{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j + 1 } - \text{ k}}} {{ - C}}_{{\text{i }}}^{{\text{j } - \text{ k}}} { )}} \\ \, & \quad \quad \, + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)} + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{f1}}_{{\text{i}}}^{{\text{j + 1}}} \\ \end{aligned} $$

(32)

where ${{d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)}$ depicts the nonlinear terms for Eq. (26).

Equation (32) expresses the nonlinear equations system, which is rewritten as,

$$ \begin{aligned} {\overline{{A}}}1{\overline{{C}}}^{{\text{j + 1}}} &= {{\overline{M}1 \overline{C}}}^{{\text{j}}} + ({{b1}}_{1} {{C}}_{{\text{i }}}^{{\text{j } - \text{ 1}}} - \sum\limits_{{{k = 2}}}^{{\text{j}}} {{{b1}}_{{\text{k}}} {{(C}}_{{\text{i }}}^{{\text{j + 1 } - \text{ k}}} {{ - C}}_{{\text{i }}}^{{\text{j } - \text{ k}}} { )}} ) \\ & \quad + {{\Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{\text{U}}_{{1}} }} {{d1(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {{) + \Gamma (2 - U}}_{{1}} {{)(\Delta t)}}^{{{{U}}_{{1}} }} {\overline{{F}}\text{1}}^{\text{j} + 1} \\ \end{aligned} $$

(33)

where

$$ {\overline{{C}}}^{{\text{j + 1}}} = \left[ {{{C}}_{{0}}^{{\text{j + 1}}} {{, C}}_{{1}}^{{\text{j + 1}}} {{, C}}_{{2}}^{{\text{j + 1}}} {, }...{{ , C}}_{{{K}}}^{{\text{j + 1}}} } \right]^{{\text{T}}} $$

(34)

$$ {\overline{{C}}}^{{\text{j}}} = \left[ {{{0, C}}_{{1}}^{{\text{j}}} {{, C}}_{{2}}^{{\text{j}}} {, }...{{ , C}}_{{\text{K } - \text{ 1}}}^{{\text{j}}} } \right]^{{\text{T}}} $$

(35)

$$ {\overline{{F}}}1^{{\text{j}}} = \left[ {{{0, f1}}_{{1}}^{{\text{j}}} {{, f1}}_{{2}}^{{\text{j}}} {, }...{{, f1}}_{{\text{K } - \text{ 1}}}^{{\text{j}}} } \right]^{{\text{T}}} $$

(36)

The coefficients matrix is expressed by ${{\overline{A}1 = [A1}}_{{\text{i,j}}} {] }$ for i, j = 1, 2, …, K-1 as,

$$A{1_{{\rm{i,j}}}} = \left\{ {\begin{array}{*{20}{l}}0&{{\mkern 1mu} {\rm{When }}j \ge i + 2}\\{ - g{1_0}B1}&{{\rm{When }}j = i + 1}\\{b{1_0} - g{1_1}B1}&{{\rm{When }}j = i{\rm{ }}}\\{ - g{1_2}B1}&{{\rm{When }}j = i - 1{\rm{ }}}\\{ - g{1_{{\rm{i - j + 1}}}}B1}&{{\rm{When }}j \le i - 1{\rm{ }}}\end{array}} \right\} $$

(37)

Similarly, utilizing the L1 scheme along temporal dimensions and the CNS with Grunwald estimation along spatial dimensions for Eq. (27), we obtain the nonlinear equations system as,

$$ \begin{aligned} {{\overline{A}2 \overline{P}}}^{{\text{j} + 1}} = & {{ \overline{M}2 \overline{C}}}^{{\text{j}}} + ({{b2}}_{1} {{P}}_{{\text{i }}}^{{\text{j} - 1}} - \sum\limits_{{\text{k = 2}}}^{{\text{j}}} {{{b2}}_{{\text{k}}} {{(P}}_{{\text{i }}}^{{\text{j} + 1 - \text{k}}} {{ - P}}_{{\text{i }}}^{{\text{j} - \text{k}}} { )}} ) \\ \, & \quad \, + {{\Gamma (2 - U}}_{2} {{)(\Delta t)}}^{{{\text{U}}_{2} }} {{d2(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {{) + \Gamma (2 - U}}_{2} {{)(\Delta t)}}^{{{\text{U}}_{2} }} {\overline{{F}}\text{2}}^{{\text{j + 1}}} \\ \end{aligned} $$

(38)

where ${{d2(C}}_{{\text{i}}}^{{\text{j}}} {{,P}}_{{\text{i}}}^{{\text{j}}} {)}$ depicts the nonlinear terms of Eq. (27), where

$$ {\overline{{P}}}^{{\text{j + 1}}} = \left[ {{{P}}_{{0}}^{{\text{j} + 1}} {{, P}}_{{1}}^{{\text{j} + 1}} {{, P}}_{{2}}^{{\text{j} + 1}} {, }...{{ , P}}_{{\text{K}}}^{{\text{j} + 1}} } \right]^{{\text{T}}} $$

(39)

$$ {\overline{{P}}}^{{\text{j}}} =\left[ {{{0, P}}_{{1}}^{{\text{j}}} {{, P}}_{{2}}^{{\text{j}}} {, }...{{ , P}}_{{\text{K} - 1}}^{{\text{j}}} } \right]^{{\text{T}}} $$

(40)

$$ {\overline{{F}}\text{2}}^{{\text{j}}} = \left[ {{{0, f2}}_{{1}}^{{\text{j}}} {{, f2}}_{{2}}^{{\text{j}}} {, }...{{, f2}}_{{\text{K} - 1}}^{{\text{j}}} } \right]^{{\text{T}}} $$

(41)

The coefficients matrix is expressed by ${{\overline{A}2 = [A2}}_{{\text{i,j}}} {] }$ for i, j = 1, 2, 3, …, K-1 as follows:

$$A{2_{{\rm{i,j}}}} = \left\{ {\begin{array}{*{20}{l}}0&{{\mkern 1mu} {\rm{When }}j \ge i + 2}\\{ - g{2_0}B2}&{{\rm{When }}j = i + 1}\\{b{2_0} - g{2_1}B2}&{{\rm{When }}j = i{\rm{ }}}\\{ - g{2_2}B2}&{{\rm{When }}j = i - 1{\rm{ }}}\\{ - g{2_{{\rm{i}} - {\rm{j}} + 1}}B2}&{{\rm{When }}j \le i - 1{\rm{ }}}\end{array}} \right\}$$

(42)

where${{B2 = D}}_{{\text{i}}} \frac{{{{\Gamma (2 - U}}_{2} {{)(\Delta t)}}^{{{\text{U}}_{2} }} }}{{{{2h}}^{{{\text{V}}_{2} }} }}$.

Let λ₁ is an eigenvalue of matrix A1, such that A1X = λ₁X for some non-zero vector X. Consider $\left| {{{x}}_{{\text{i}}} } \right|{{ = max}}\left\{ {\left| {{{x}}_{{\text{j}}} } \right|:{{ j = 0, 1, 2, }}...{{, K}}} \right\}$, then $\sum\nolimits_{{\text{j = 0}}}^{{\text{K}}} {{{A1}}_{{\text{i,j}}} } {{ x}}_{{\text{j}}} {{ = \lambda }}_{{1}} {{x}}_{{\text{i}}}$, this implies

$$ {\uplambda }_{{1}} {{ = A1}}_{{\text{i,i}}} + \sum\limits_{{{{j = 0, j}} \ne {{i}}}}^{{\text{K}}} {{{A1}}_{{\text{i,j}}} } \frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }} $$

(43)

Putting the A_i,j values in Eq. (43), we obtain

$$ {\uplambda }_{{1}} {{ = b1}}_{{0}} {{ - B1g1}}_{1} {{ - B1g1}}_{0} \frac{{{{x}}_{{\text{i} + 1}} }}{{{{x}}_{{\text{i}}} }}{{ - B1g1}}_{2} \frac{{{{x}}_{{\text{i} - 1}} }}{{{{x}}_{{\text{i}}} }}{{ - B1}}\left( {\sum\limits_{{\text{j = 0}}}^{{\text{i - 2}}} {{{g1}}_{{\text{i} - \text{j} + 1}} } \frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right) $$

(44)

$$ {\uplambda }_{{1}} {{ = b1}}_{{0}} {{ - B1}}\left( {g1_{1} + \sum\limits_{{{\text{j = 0, j}} \ne {{i}}}}^{{\text{i + 1}}} {{{g1}}_{{\text{i - j + 1}}} } \frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right) $$

(45)

Since $\left( {\sum\nolimits_{{{K = 0}}}^{\infty } {{{g1}}_{{\text{K}}} } = 0} \right)$ and g1₁ is the only expression in Grunwald weights sequence that is negative, and ${{g1}}_{{1}} { = } - {{V}}_{{1}}$ and thus for ${{1 < V}}_{{1}} \le {2 }$

$$ {{ - g1}}_{{1}} \ge \sum\limits_{{{\text{K = 0,K}} \ne {1}}}^{{{j}}} {{{g1}}_{{\text{K}}} } {{ for j = 0, 1, 2, }}... $$

(46)

Since $\left| {\frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right| \le 1{{ and g1}}_{{\text{j}}} \ge 0,{{ for j = 0, 2, 3, }}...{, }$

$$ \sum\limits_{{{{j = 0, j}} \ne {{i}}}}^{{\text{i} + 1}} {{{g1}}_{{\text{i} - \text{j} + 1}} } \left| {\frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right| \le \sum\limits_{{{\text{j} = 0, \text{j}} \ne {{i}}}}^{{\text{i} + 1}} {{{g1}}_{{\text{i} - \text{j} + 1}} } \le {{g1}}_{{1}} $$

(47)

$$ {{g1}}_{{1}} + \sum\limits_{{{\text{j} = 0, \text{j}} \ne {{i}}}}^{{\text{i} + 1}} {{{g1}}_{{\text{i} -\text{ j} + 1}} } \left| {\frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right| \le 0 $$

(48)

Similarly, for the matrix A2,

$$ {{g2}}_{{1}} + \sum\limits_{{{\text{j} = 0, \text{j}} \ne {{i}}}}^{{\text{i}+ 1}} {{{g2}}_{{\text{i} - \text{j} + 1}} } \left| {\frac{{{{x}}_{{\text{j}}} }}{{{{x}}_{{\text{i}}} }}} \right| \le 0 $$

(49)

For interdependent [Ca²⁺] and IP₃ signaling systems, we get the matrix representation as follows:

$${{{[A]}}_{{{2K + 1 \times 2K + 1}}}} = {\left[ {\begin{array}{*{20}{c}}{{{{{[A1]}}}_{{{K \times K}}}}}&0\\0&{{{{{[A2]}}}_{{{K \times K}}}}}\end{array}} \right]_{{{2K + 1 \times 2K + 1}}}}$$

(50)

Because, B1 and B2 are non-negative real numbers, all the eigenvalues of A1, A2 and A fulfill the condition $\left| {\uplambda } \right| \ge 1$. Then, matrix A is invertible, and each eigenvalue of A⁻¹ satisfies $\left| {\upeta } \right| \le 1$. Therefore, the spectral radius of matrix is ${{\rho (A}}^{{ - 1}} {)} \le 1$ and suggests that the employed scheme is unconditionally stable.