Dynamical analysis of spontaneous Ca<sup>2+</sup> oscillations in astrocytes

Yapeng Zhang; Yu Chen; Quanbao Ji; Yapeng Zhang; Yu Chen; Quanbao Ji

doi:10.3934/era.2024020

Electronic Research Archive

2024, Volume 32, Issue 1: 405-417. doi: 10.3934/era.2024020

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Dynamical analysis of spontaneous Ca²⁺ oscillations in astrocytes

1.
School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China
2.
Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning 530006, China

Received: 27 September 2023 Revised: 08 December 2023 Accepted: 20 December 2023 Published: 28 December 2023

In this work, we focus on a nonlinear dynamical model proposed by Lavrentovich et al. to compute and simulate spontaneous Ca²⁺ oscillations evoked by calcium ion efflux in astrocytes. Selected parameters are chosen, with observation of periodic and chaotic Ca²⁺ oscillations in cytosol. The stability analysis of equilibrium is conducted using the center manifold theorem to investigate the dynamics underlying spontaneous Ca²⁺ oscillations in astrocytes. The results indicate that the Hopf bifurcation represents the dynamical changes in stability of spontaneous Ca²⁺ oscillations. In addition, numerical simulations are performed to further assess the validity of the aforementioned analysis.

Keywords:

Citation: Yapeng Zhang, Yu Chen, Quanbao Ji. Dynamical analysis of spontaneous Ca2+ oscillations in astrocytes[J]. Electronic Research Archive, 2024, 32(1): 405-417. doi: 10.3934/era.2024020

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Abstract

1. Introduction

Astrocytes have traditionally been auxiliary elements to central nervous system, thereby providing support and nutrients to neurons. Unlike neurons, astrocytes may exert an active role in neuronal firing activities and are extensively employed in research pertaining to diverse neurological disorders ^[1,2,3,4,5]. Astrocytes do not generate electrical signals themselves but participate in neuronal activity by regulating the release of "glial transmitters", such as glutamate and ATP, through intracellular Ca²⁺ oscillations ^[6,7]. In the brain, they occupy approximately 50% of the volume and can either be influenced by neurons or exhibit spontaneous Ca²⁺ oscillations ^[8]. Spontaneous Ca²⁺ oscillations typically encompass the following main processes: (i) channel dynamics; (ii) calcium-induced calcium release; and (iii) negative feedback regulation ^[9,10,11]. Many studies have unveiled the correlation between the onset and cessation of Ca²⁺ oscillations within the system ^{[12,13,14,15]}.

Intracellular Ca²⁺ oscillation in astrocytes is frequently triggered by an external stimulus, such as glutamate. Nevertheless, Aguado et al. ^[16] observed spontaneous activities in astrocytes, thereby suggesting the existence of potential bidirectional regulation. In fact, astrocytes not only actively participate in neuronal activity and regulate synaptic plasticity, but also contribute to neural repair ^[17,18]. Therefore, a dynamical analysis of spontaneous Ca²⁺ oscillations could contribute to comprehending the role of astrocytes in neural networks and provide valuable insights into the development of complex brain networks in future. Here, a local Hopf bifurcation of a mathematical model in astrocytes proposed by Lavrentovich et al. ^[19] is investigated. This model manifests the dynamical behaviors of astrocytes without external stimulation. The objective of this work is to investigate the stability and bifurcation to explore the effects of calcium release rates from the cytosol of astrocytes.

2. Models

The model in ^[20] considered intracellular Ca²⁺ oscillations triggered by external stimuli, the interplay between calcium induced calcium release and the degradation of inositol triphosphate (IP₃). In ^[21], the authors proposed a mathematical model that considered experimental data to predict the control and plasticity of intercellular Ca²⁺ waves. Subsequently, Lavrentovich et al. ^[19] simplified this model and provided an improved framework to evaluate spontaneous Ca²⁺ oscillations in astrocytes. The system is activated by the influx of extracellular Ca²⁺ into the cell and sustained through feedback mechanisms involving intracellular Ca²⁺ in the endoplasmic reticulum (ER) and IP₃. The equations for the temporal evolution of three variables are defined as follows:

$\left\{ {\begin{array}{*{20}{c}} {dC{a_{{\text{cyt}}}}/dt = {v_{{\text{in}}}} - {k_{{\text{out}}}}C{a_{{\text{cyt}}}} - {v_{{\text{serca}}}} + {v_{{\text{CICR}}}} + {k_{\text{f}}}\left( {C{a_{{\text{er}}}} - C{a_{{\text{cyt}}}}} \right){\kern 1pt} , } \\ {{\kern 1pt} {\kern 1pt} dC{a_{{\text{er}}}}/dt = {v_{{\text{serca}}}} - {v_{{\text{CICR}}}} - {k_{\text{f}}}\left( {C{a_{{\text{er}}}} - C{a_{{\text{cyt}}}}} \right){\kern 1pt} {\kern 1pt} , {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {dI{P_{\text{3}}}/dt = {v_{{\text{PLC}}}} - {k_{\deg }}I{P_3}{\kern 1pt} , {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array} } \right.$

(1)

$\begin{array}{l} {v_{{\text{PLC}}}} = {v_{\text{p}}}\left( {\frac{{Ca_{{\text{cyt}}}^{\text{2}}}}{{Ca_{{\text{cyt}}}^{\text{2}} + k_{\text{p}}^2}}} \right), {v_{{\text{serca}}}} = {v_{{\text{M2}}}}\left( {\frac{{Ca_{{\text{cyt}}}^{\text{2}}}}{{Ca_{{\text{cyt}}}^2 + k_2^2}}} \right), \quad \hfill \\ {v_{{\text{CICR}}}} = 4{v_{{\text{M3}}}}\left( {\frac{{k_{{\text{CaA}}}^{\text{n}}Ca_{{\text{cyt}}}^{\text{n}}}}{{\left( {Ca_{{\text{cyt}}}^{\text{n}} + k_{{\text{CaA}}}^{\text{n}}} \right)\left( {Ca_{{\text{cyt}}}^{\text{n}} + k_{{\text{CaI}}}^{\text{n}}} \right)}}} \right) \times \left( {\frac{{IP_{\text{3}}^{\text{m}}}}{{IP_{\text{3}}^{\text{m}} + k_{{\text{ip3}}}^{\text{m}}}}} \right)\left( {C{a_{{\text{er}}}} - C{a_{{\text{cyt}}}}} \right). \hfill \\ \end{array}$

The three variables represent the concentration of Ca²⁺ within the cytosol (Ca_cyt), the concentration Ca²⁺ within the ER (Ca_er), and the IP₃ concentration (IP₃). The following equations describe the specific meanings of certain parameters, such as the rate of Ca²⁺ pumping into the ER by the reticulum's ATPase (v_serca), the rate of Ca²⁺ flux from the ER to the cytosol mediated by the IP₃ receptors (v_CICR) and IP₃ production (v_PLC). The specific meanings and values of other parameters can be referred to in ^[19,20,21].

Table 1. Details of the parameters in Lavrentovich model.

Parameter	Value and units	Description
v_M2	15.0 μM/s	maximum Ca²⁺ efflux from the pump
k_out	bifurcation parameter	The rate of calcium release from the cytosol
k_deg	0.08 s^-1	Rate constant of IP₃ degradation
k_CaA	0.27 μM	activating affinities
k_ip3	0.1 μM	apparent affinity for IP₃
v_M3	40.0 s^-1	maximum Ca²⁺ flux entering the cytosol
k₂	0.1 μM	threshold constants for pumping
k_CaI	0.27 μM	inhibiting affinities
m	2.2	Hill coefficient
n	2.02	Hill coefficient
k_f	0.5 s^-1	Leakage flux
v_p	0.05	rate of PLC (phospholipase C) activation
v_in	0.05 s^-1	Ca²⁺ influx across the plasma membrane
k_p	0.164 μM	Ca²⁺ activation threshold for PLC

| Show Table

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3. Stability and bifurcation analysis

k_out is selected as the bifurcation parameter. For convenience, we write $\alpha = C{a_{{\text{cyt}}}},$ $\; \beta = C{a_{{\text{er}}}}, \;$ $\gamma = I{P_3}$ and $\theta$ = k_out. The system dynamics are determined by the following form:

$\left\{ \begin{array}{l} \dot \alpha = 0.05 + 0.5{\mkern 1mu} \beta - 0.5{\mkern 1mu} \alpha - \frac{{15{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.01}} - {\mkern 1mu} \theta {\mkern 1mu} \alpha - \frac{{11.36{\mkern 1mu} {\alpha ^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\alpha - {\mkern 1mu} \beta } \right)}}{{{{\left( {{\alpha ^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\gamma ^{2.2}} + 0.00631} \right)}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ \dot \beta = 0.5{\mkern 1mu} \alpha - 0.5{\mkern 1mu} \beta + \frac{{15{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.01}} + \frac{{11.36{\mkern 1mu} {\alpha ^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\alpha - {\mkern 1mu} \beta } \right)}}{{{{\left( {{\alpha ^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\gamma ^{2.2}} + 0.00631} \right)}}, {\kern 1pt} \hfill \\ \dot \gamma = \frac{{0.05{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.0269}} - 0.08{\mkern 1mu} \gamma . \hfill \\ \end{array} \right.$

(2)

The equilibrium points satisfy the equations on the left side of system (2) when they equal zero. Subsequently, by calculating

$\frac{{0.05{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.0269}} - 0.08{\mkern 1mu} \gamma = 0,$

we can obtain

$\gamma = \frac{{0.05{\alpha ^2}}}{{0.08({\alpha ^2} + 0.0269)}}.$

Then,

$0.5{\mkern 1mu} \alpha - 0.5{\mkern 1mu} \beta + \frac{{15{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.01}} + \frac{{11.36{\mkern 1mu} {\alpha ^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\alpha - {\mkern 1mu} \beta } \right)}}{{{{\left( {{\alpha ^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\gamma ^{2.2}} + 0.00631} \right)}} = 0$

can be obtained by

$\begin{array}{l}\beta = \frac{0.5\alpha +\frac{15{\alpha }^{2}}{{\alpha }^{2}+0.01}+\frac{11.36{\alpha }^{3.02}{\sigma }_{1}}{\left({\sigma }_{1}+0.00631\right){\left({\alpha }^{2.02}+0.07102\right)}^{2}}}{\frac{11.36{\alpha }^{2.02}{\sigma }_{1}}{\left({\sigma }_{1}+0.00631\right){\left({\alpha }^{2.02}+0.07102\right)}^{2}}+0.5}, \\ {\sigma }_{1} = {\left(\frac{5{\alpha }^{2}}{8{\alpha }^{2}+0.2152}\right)}^{2.2}.\end{array}$

Subsequently, by substituting $\beta$ and $\gamma$ into the equation

$0.05 + 0.5{\mkern 1mu} \beta - 0.5{\mkern 1mu} \alpha - \frac{{15{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.01}} - {\mkern 1mu} \theta {\mkern 1mu} \alpha - \frac{{11.36{\mkern 1mu} {\alpha ^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\alpha - {\mkern 1mu} \beta } \right)}}{{{{\left( {{\alpha ^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\gamma ^{2.2}} + 0.00631} \right)}} = 0,$

we have the following:

$\left\{ \begin{array}{l} f(\alpha , \theta ) = 0.05 + 0.5{\mkern 1mu} \beta - 0.5{\mkern 1mu} \alpha - \frac{{15{\mkern 1mu} {\alpha ^2}}}{{{\alpha ^2} + 0.01}} - {\mkern 1mu} \theta {\mkern 1mu} \alpha - \frac{{11.36{\mkern 1mu} {\alpha ^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\alpha - {\mkern 1mu} \beta } \right)}}{{{{\left( {{\alpha ^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\gamma ^{2.2}} + 0.00631} \right)}} = 0 \hfill \\ \beta = \frac{{0.5\alpha + \frac{{15{\alpha ^2}}}{{{\alpha ^2} + 0.01}} + \frac{{11.3625{\alpha ^{3.02}}{\sigma _1}}}{{\left( {{\sigma _1} + 0.0063} \right){{\left( {{\alpha ^{2.02}} + 0.071} \right)}^2}}}}}{{\frac{{11.3625{\alpha ^{2.02}}{\sigma _1}}}{{\left( {{\sigma _1} + 0.0063} \right){{\left( {{\alpha ^{2.02}} + 0.071} \right)}^2}}} + 0.5}} \hfill \\ \gamma = \frac{{0.05{\alpha ^2}}}{{0.08({\alpha ^2} + 0.0269)}}. \hfill \\ \end{array} \right.$

(3)

Assuming that the equilibrium points are denoted as ${\alpha _0}, \; {\beta _0}, \; {\gamma _0},$ we have the following equations through the substitution j₁ = j $-$ j₀ ( $j = \alpha, \; \beta, \; \gamma$ ):

$\left\{ \begin{array}{l} {{\dot \alpha }_1} = 0.05 + 0.5{\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right) - 0.5{\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) - \frac{{15{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.01}} - {\mkern 1mu} \theta {\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) \hfill \\ - \frac{{11.36{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\left( {{\alpha _1} + {\alpha _0}} \right) - {\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right)} \right)}}{{{{\left( {{{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}} + 0.00631} \right)}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ {{\dot \beta }_1} = 0.5{\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) - 0.5{\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right) + \frac{{15{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.01}} \hfill \\ + \frac{{11.36{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}}{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}}{\mkern 1mu} \left( {\left( {{\alpha _1} + {\alpha _0}} \right) - {\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right)} \right)}}{{{{\left( {{{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}} + 0.00631} \right)}}, {\kern 1pt} \hfill \\ {{\dot \gamma }_1} = \frac{{0.05{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.0269}} - 0.08{\mkern 1mu} \left( {{\gamma _1} + {\gamma _0}} \right). \hfill \\ \end{array} \right.$

(4)

Systems (2) and (4) could exhibit identical characteristics with mutual equilibrium points (0, 0, 0). Therefore, we can readily compute the Jacobian matrix of the system as follows:

$J = \left( {\begin{array}{*{20}{c}} {{i_{11}}}&{{i_{12}}}&{i{}_{13}} \\ {{i_{21}}}&{{i_{22}}}&{{i_{23}}} \\ {{i_{31}}}&{{i_{32}}}&{{i_{33}}} \end{array}} \right)$

where

$\left( {\begin{array}{*{20}{c}} {{\sigma _1} - \theta - \frac{{30\alpha }}{{{\alpha ^2} + 0.01}} - {\sigma _2} - {\sigma _6} + {\sigma _5} - 0.5}&{{\sigma _2} + 0.5}&{{\sigma _3} - {\sigma _4}} \\ {\frac{{30\alpha }}{{{\alpha ^2} + 0.01}} - {\sigma _1} + {\sigma _2} + {\sigma _6} - {\sigma _5} + 0.5}&{ - {\sigma _2} - 0.5}&{{\sigma _4} - {\sigma _3}} \\ {\frac{{0.1\alpha }}{{{\alpha ^2} + 0.026896}} - \frac{{0.1{\alpha ^3}}}{{{{\left( {{\alpha ^2} + 0.026896} \right)}^2}}}}&0&{ - 0.08} \end{array}} \right)$

$\begin{array}{l}\text{ }{\sigma }_{1} = \frac{30{\alpha }^{3}}{{\left({\alpha }^{2}+0.01\right)}^{2}};\text{ }{\sigma }_{2} = \frac{11.363{\alpha }^{2.02}{\gamma }^{2.2}}{{\sigma }_{7}}\text{ };\text{ }{\sigma }_{3} = \frac{24.998{\alpha }^{2.02}{\gamma }^{3.4}\left(\alpha -\beta \right)}{{\left({\alpha }^{2.02}+0.071016\right)}^{2}{\sigma }_{8}{}^{2}};\\ \text{ }{\sigma }_{4} = \frac{24.998{\alpha }^{2.02}{\gamma }^{1.2}\left(\alpha -\beta \right)}{{\sigma }_{7}};\text{ }{\sigma }_{5} = \frac{45.905{\alpha }^{3.04}{\gamma }^{2.2}\left(\alpha -\beta \right)}{{\left({\alpha }^{2.02}+0.071016\right)}^{3}{\sigma }_{8}};\\ \text{ }{\sigma }_{6} = \frac{22.952{\alpha }^{1.02}{\gamma }^{2.2}\left(\alpha -\beta \right)}{{\sigma }_{7}};\text{ }{\sigma }_{7} = {\left({\alpha }^{2.02}+0.071016\right)}^{2}{\sigma }_{8};\\ \text{ }{\sigma }_{8} = {\gamma }^{2.2}+\mathrm{0.0063096.}\end{array}$

The resulting characteristic equation can be easily obtained using the following:

${\lambda ^3} + {Q_3}{\lambda ^2} + {Q_2}\lambda + {Q_1} = 0,$

where

$\begin{array}{l} {Q_1} = - ({i_{11}} + {i_{22}} + {i_{33}}), \hfill \\ {Q_2} = {i_{11}}{i_{22}} + {i_{11}}{i_{33}} + {i_{22}}{i_{33}} - {i_{13}}{i_{31}} - {i_{12}}{i_{21}} - {i_{32}}{i_{23}}, \hfill \\ {Q_3} = i{}_{31}i{}_{13}{i_{22}} + {i_{12}}{i_{21}}{i_{33}} + {i_{32}}i{}_{23}{i_{11}} - {i_{11}}{i_{22}}{i_{33}} - {i_{12}}{i_{23}}{i_{31}} - {i_{13}}{i_{21}}{i_{32}}. \hfill \\ \end{array}$

The characteristic polynomial can be obtained and the Hurwitz matrix with Q_l (l = 1, 2, 3) coefficients is as follows:

${H_1} = \left( {{Q_1}} \right), \;{H_2} = \left( {\begin{array}{*{20}{c}} {{Q_1}}&1 \\ {{Q_3}}&{{Q_2}} \end{array}} \right), \;{H_3} = \left( {\begin{array}{*{20}{c}} {{Q_1}}&1&0 \\ {{Q_3}}&{{Q_2}}&1 \\ 0&0&{{Q_3}} \end{array}} \right) .$

The stability of the system can be determined by calculating the sign of H_l (l = 1, 2, 3). The dynamical behaviors of the system (4) can be obtained as the parameter k_out varies with use of the Routh-Hurwitz criteria ^[12].

When

1) $\theta$ < 0.421, there is a stable node;

2) $\theta$ = 0.421, there is a Hopf bifurcation point O₁ = (0.1183, 0.5907, 0.2146);

3) 0.421 < $\theta$ < 1.267, there exists an equilibrium;

4) $\theta$ = 1.267, there is a Hopf bifurcation point O₂ = (0.0345, 2.6574, 0.0246);

5) 1.267 < $\theta$ ≤ 1.284, there exists an equilibrium; and

6) $\theta$ > 1.284, there is a stable node.

Given j₁ = j $-$ j₀ ( $j = \alpha, \; \beta, \; \gamma, \; \theta$ ), the equilibrium of system (4) is ( ${\alpha _0}$ , ${\beta _0}$ , ${\gamma _0}$ ). We introduce a new parameter ${\theta _1}$ for ${{{\text{d}}{\theta _1}} \mathord{\left/ {\vphantom {{{\text{d}}{\theta _1}} {dt}}} \right. } {dt}} = 0$ .

$\left\{ \begin{array}{l} {{\dot \alpha }_1} = 0.05 + 0.5{\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right) - 0.5{\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) - \frac{{15.0{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.01}} - \left( {{\theta _1} + {\theta _0}} \right){\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) \hfill \\ - \frac{{11.36{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}}{\mkern 1mu} {\gamma ^{2.2}}{\mkern 1mu} \left( {\left( {{\alpha _1} + {\alpha _0}} \right) - \left( {{\beta _1} + {\beta _0}} \right)} \right)}}{{{{\left( {{{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}} + 0.00631} \right)}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ {{\dot \beta }_1} = 0.5{\mkern 1mu} \left( {{\alpha _1} + {\alpha _0}} \right) - 0.5{\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right) + \frac{{15{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.01}} \hfill \\ + \frac{{11.36{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}}{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}}{\mkern 1mu} \left( {\left( {{\alpha _1} + {\alpha _0}} \right) - {\mkern 1mu} \left( {{\beta _1} + {\beta _0}} \right)} \right)}}{{{{\left( {{{\left( {{\alpha _1} + {\alpha _0}} \right)}^{2.02}} + 0.07102} \right)}^2}{\mkern 1mu} \left( {{\mkern 1mu} {{\left( {{\gamma _1} + {\gamma _0}} \right)}^{2.2}} + 0.00631} \right)}}, {\kern 1pt} \hfill \\ {{\dot \gamma }_1} = \frac{{0.05{\mkern 1mu} {{\left( {{\alpha _1} + {\alpha _0}} \right)}^2}}}{{{{\left( {{\alpha _1} + {\alpha _0}} \right)}^2} + 0.0269}} - 0.08{\mkern 1mu} \left( {{\gamma _1} + {\gamma _0}} \right), \hfill \\ {{\dot \theta }_1} = 0. \hfill \\ \end{array} \right.$

(5)

Systems (2) and (5) have the same dynamics with mutual equilibrium points O ( ${\alpha _1}$ , ${\beta _1}$ , ${\gamma _1}$ , ${\theta _1}$ ) = (0, 0, 0, 0). For ${\theta _0}$ = 0.421, we calculate the eigenvalues at the equilibrium point: ξ₁ = -0.1200, ξ₂ = 2.2814i, ξ₃ = -2.2814i and ξ₄ = 0. The eigenvectors conform to the ensuing matrix:

$\left( {\begin{array}{*{20}{c}} { - 0.1825}&{ - 0.6869 - 0.1263i}&{ - 0.6869 + 0.1263i}&{ - 0.1920} \\ { - 0.4592}&{0.7134}&{0.7134}&{0.5352} \\ {0.8694}&{ - 0.0139 + 0.0565i}&{ - 0.0139 - 0.0565i}&{ - 0.4566} \\ 0&0&0&{0.6843} \end{array}} \right).$

Suppose

$\left( \begin{array}{l} {\alpha _1} \hfill \\ {\beta _1} \hfill \\ {\gamma _1} \hfill \\ {\theta _1} \hfill \\ \end{array} \right) = U\left( \begin{array}{l} {\text{x}} \hfill \\ y \hfill \\ z \hfill \\ s \hfill \\ \end{array} \right), U = \left( {\begin{array}{*{20}{c}} { - 0.1825}&{ - 0.6869}&{0.1263}&{ - 0.1920} \\ { - 0.4592}&{0.7134}&0&{0.5352} \\ {0.8694}&{ - 0.0139}&{ - 0.0565}&{ - 0.4566} \\ 0&0&0&{0.6843} \end{array}} \right) .$

System (5) can be replaced by

$\left( {\begin{array}{*{20}{c}} {\dot x} \\ {\dot y} \\ {\dot z} \\ {\dot s} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 0.1200}&0&0&0 \\ 0&0&{ - 2.2814}&0 \\ 0&{2.2814}&0&0 \\ 0&0&0&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ z \\ s \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {{g_1}} \\ {{g_2}} \\ {{g_3}} \\ {{g_4}} \end{array}} \right),$

(6)

and

$\begin{array}{l} \left( \begin{array}{l} {{\dot \alpha }_1} \hfill \\ {{\dot \beta }_1} \hfill \\ {{\dot \gamma }_1} \hfill \\ {{\dot \theta }_1} \hfill \\ \end{array} \right) = U\left( {\begin{array}{*{20}{c}} {\dot x} \\ {\dot y} \\ {\dot z} \\ {\dot s} \end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}} {\dot x} \\ {\dot y} \\ {\dot z} \\ {\dot s} \end{array}} \right) = {U^{ - 1}}\left( \begin{array}{l} {{\dot \alpha }_1} \hfill \\ {{\dot \beta }_1} \hfill \\ {{\dot \gamma }_1} \hfill \\ {{\dot \theta }_1} \hfill \\ \end{array} \right) = {U^{ - 1}}\left( \begin{array}{l} {f_1} \hfill \\ {f_2} \hfill \\ {f_3} \hfill \\ 0 \hfill \\ \end{array} \right) \hfill \\ \left( \begin{array}{l} {g_1} \hfill \\ {g_2} \hfill \\ {g_3} \hfill \\ {g_4} \hfill \\ \end{array} \right) = {U^{ - 1}}\left( \begin{array}{l} {f_1} \hfill \\ {f_2} \hfill \\ {f_3} \hfill \\ 0 \hfill \\ \end{array} \right) - \left( {\begin{array}{*{20}{c}} { - 0.1200}&0&0&0 \\ 0&0&{ - 2.2814}&0 \\ 0&{2.2814}&0&0 \\ 0&0&0&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ z \\ s \end{array}} \right), \hfill \\ \end{array}$

where

$\begin{array}{l} {g_1} = - 0.7699{f_1} - 0.7749{f_2} + 1.7211{f_3} - 0.12x, \hfill \\ {g_2} = 0.4956{f_1} + 1.9005{f_2} + 1.1078{{\text{f}}_3} - 2.2814z, \hfill \\ {g_3} = 11.7255{f_1} + 11.4558{f_2} + 8.5121{f_3} + 2.2814y, \hfill \\ {g_4} = 0, \hfill \\ {f_1} = 0.3636s - 0.1383x + 0.7001y - 0.06315z - {\sigma _2} + \left( {0.6843s + 0.4217} \right){\sigma _5} + {\sigma _1} + 0.2862, \hfill \\ {f_2} = 0.1383x - 0.3636s - 0.7001y + 0.06315z + {\sigma _2} - {\sigma _1} - 0.2362, \hfill \\ {f_3} = 0.03653s - 0.06955x + 0.001112y + 0.00452z + \frac{{0.05{\sigma _5}^2}}{{{\sigma _5}^2 + 0.0269}} - 0.01717, \hfill \\ {f_4} = 0, \hfill \\ \;\;{\sigma _1} = \frac{{11.36\left( {0.7272s - 0.2767x + 1.4y - 0.1263z + 0.4724} \right){\sigma _3}{\sigma _4}}}{{\left( {{\sigma _4} + 0.00631} \right){{\left( {{\sigma _3} + 0.07102} \right)}^2}}}, \hfill \\ \;\;{\sigma _2} = \frac{{15.0{\sigma _5}^2}}{{{\sigma _5}^2 + 0.01}}, \hfill \\ \;\;{\sigma _3} = {\left( {0.1263z - 0.1825x - 0.6869y - 0.192s + 0.1183} \right)^{2.02}}, \hfill \\ \;\;{\sigma _4} = {\left( {0.8694x - 0.4566s - 0.0139y - 0.0565z + 0.2146} \right)^{2.2}}, \hfill \\ \;\;{\sigma _5} = 0.192s + 0.1825x + 0.6869y - 0.1263z - 0.1183. \hfill \\ \end{array}$

System (5) has a center manifold with the following form:

$W_{{\text{loc}}}^{\text{c}}\left( {{O_1}} \right) = \left\{ {({\text{x}}, {\text{y}}, z, s) \in {R^4}|x = {h^*}(y, z, s), {h^*}(0, 0, 0) = 0, D{h^*}(0, 0, 0) = 0} \right\}.$

Substituting the equation into (6) yields the following equations:

$\left( {\begin{array}{*{20}{c}} {{{\dot h}^*}(y, z, s)} \\ {\dot y} \\ {\dot z} \\ {\dot s} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 0.1200}&0&0&0 \\ 0&0&{ - 2.2814}&0 \\ 0&{2.2814}&0&0 \\ 0&0&0&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{h^*}(y, z, s)} \\ y \\ z \\ s \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {{g_1}} \\ {{g_2}} \\ {{g_3}} \\ {{g_4}} \end{array}} \right).$

Let h (y, z, s) = ay² + bz² + cs² + dyz + eys + fzs +…; we have the following:

$N(h) = Dh \cdot \left[ {\begin{array}{*{20}{c}} {\dot y} \\ {\dot z} \\ {\dot s} \end{array}} \right] + 0.12h - {g_1} \equiv 0,$

(7)

where a = -0.7494, b = -0.7334, c = -2.1352, d = -0.0752, e = -0.0506, f = -0.2795. The system is described as follows:

$\left( {\begin{array}{*{20}{c}} {\dot y} \\ {\dot z} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&{ - 2.2814} \\ {2.2814}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} y \\ z \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {f\left( {y, z} \right)} \\ {{\text{g}}\left( {y, z} \right)} \end{array}} \right),$

(8)

where

$\begin{array}{l} f(y, z) = 2.375z - 0.9824y - 0.4704s - 0.005937sy - 0.0328sz - 0.008823yz \cdots , \hfill \\ g(y, z) = 0.02144z - 2.083y + 0.409s + 0.03185sy + 0.1759sz + 0.04733yz + 1.344{s^2} + \cdots . \hfill \\ \end{array}$

Hence, it is easy to obtain the following:

$\begin{array}{l} a = \frac{1}{{16}}{\left. {[f_{{\text{yyy}}}^{\text{1}} + f_{{\text{yzz}}}^{\text{1}} + f_{{\text{yyz}}}^{\text{2}} + f_{{\text{zzz}}}^{\text{2}}]} \right|_{(0, 0)}} + \frac{1}{{16 \times 2.2814}}[f_{{\text{yz}}}^1(f_{{\text{yy}}}^{\text{1}} + f_{{\text{zz}}}^{\text{1}}) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left. { - f_{{\text{yz}}}^2(f_{{\text{yy}}}^{\text{2}} + f_{{\text{zz}}}^2) - f_{{\text{yy}}}^1f_{{\text{yy}}}^{\text{2}} + f_{{\text{zz}}}^{\text{1}}f_{{\text{zz}}}^{\text{2}})]} \right|_{({\text{y}} = 0, {\text{z}} = 0, {\text{s}} = 0)}} = - 270.7099 < 0, \hfill \\ d = {\left. {\frac{{d({Re} (\xi (s))}}{{ds}}} \right|_{({\text{y}} = 0, {\text{z}} = 0, {\text{s}} = 0)}} = - 0.3697 < 0. \hfill \\ \end{array}$

(9)

Based on the aforementioned computation, we derive the following conclusions.

Conclusion 1: A subcritical Hopf bifurcation is observed as the parameter $\theta$ traverses the critical value of ${\theta _0}$ = 0.421. Below this threshold ( $\theta < {\theta _0}$ ), the equilibrium O₁ is always locally stable. For $\theta > {\theta _0}$ , the equilibrium O₁ turns to be unstable. and the system (2) begins to oscillate.

For ${\theta _0}$ = 1.267, the eigenvalues at the equilibrium point can be computed as follows: ξ₁ = -69.3501, ξ₂ = 0.0157i, ξ₃ = -0.0157i and ξ₄ = 0.

$\left( {\begin{array}{*{20}{c}} {\dot y} \\ {\dot z} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&{ - 0.0157} \\ {0.0157}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} y \\ z \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {f\left( {y, z} \right)} \\ {{\text{g}}\left( {y, z} \right)} \end{array}} \right),$

(10)

where

$\begin{array}{l} f(y, z) = 0.006927s - 0.008238y + 0.01683z + 0.0006307sy + 0.0005685sz - 0.0007773yz + \cdots , \hfill \\ g(y, z) = 0.02448s - 0.05694y - 0.08124z - 0.001969sy - 0.001775sz + 0.002427yz + \cdots . \hfill \\ \end{array}$

Similar to the previous computations, we have a = -0.12945 < 0 and d = -0.0113 < 0.

Conclusion 2: system (2) undergoes a supcritical Hopf bifurcation at ${\theta _0}$ = 1.267. When $\theta < {\theta _0}$ , the equilibrium O₂ turns to be unstable.

4. Numerical simulations

Next, we present the bifurcation diagrams by varying the values of parameter k_out, as illustrated in Figure 1a, b. As k_out varies from 0.2 to 1.5, the system undergoes bifurcations around 0.421 and 1.267. Figure 1a illustrates the bifurcation in terms of both the period and the amplitude. When the parameter value is between 0.42 and 0.49, the model exhibits a simple oscillation with a consistent amplitude. As k_out is further increased, the model displays complex Ca²⁺ oscillations with varying amplitudes and periods. The period gradually decreases near 1.28. In Figure 1b, the continuous line represents the equilibrium state. HB1 and HB2 correspond to two bifurcation points. Figure 1c displays the interspike interval (ISI) bifurcation. As the parameter increases, the value of ISI becomes larger, thereby indicating a decrease in frequency. Figure 1d depicts the corresponding Lyapunov exponent diagram.

Figure 1. (a) The interspike interval bifurcation diagram with k_out. (b) Bifurcation diagram in (k_out, Ca_cyt) plane, HB represents Hopf bifurcation points. (c) The interspike interval bifurcation diagram with k_out. (d) Lyapunov exponent with parameter k_out.

DownLoad: Full-Size Img PowerPoint

In Figure 2, we present the time course for different values of the parameter k_out. The left column displays the time series for different parameter values (Figure 2a1-f1), the middle column shows the corresponding phase portraits (Figure 2a2-f2) and the right column illustrates the variations in power and frequency for the corresponding time series (Figure 2a3-f3). For example, Figure 2a1 represents the time course of Ca²⁺ evolution for k_out = 1.2. To eliminate the initial value interference, the time evolution for the first 200 s is excluded, thus allowing only one peak to appear in the graph. In Figure 2a2, a periodic orbit is evident. Figure 2a3 illustrates its frequency variation.

Figure 2. Illustrations for spontaneous Ca²⁺ oscillations. The left side displays the time series for Ca_cyt. The middle graphs are the associated phase portraits. The right panels display the variations in power and frequency. Different values of k_out in each figure are: (a1-a3) k_out = 1.2 s^-1; (b1-b3) k_out = 0.7 s^-1; (c1-c3) k_out = 0.65 s^-1; (d1-d3) k_out = 0.6 s^-1; (e1-e3) k_out = 0.5 s^-1 and (f1-f3) k_out = 0.4966 s^-1.

DownLoad: Full-Size Img PowerPoint

In Figure 2b1, we present spontaneous bursting Ca²⁺ oscillations for k_out = 0.7. In Figure 2c1-e1, an increasing number of small spikes become apparent. The corresponding phase portrait diagrams are illustrated in Figure 2b2-e2. Figure 2f1 corresponds to the case with k_out = 0.4966 in the bifurcation diagram. The time series illustrates the phenomenon of bursting chaos. For a clearer representation, we have zoomed in on the time range of 10, 000 s. The upper right corner depicts a schematic of the local magnification.

5. Discussion

In this paper, we investigated the stability and bifurcation of spontaneous Ca²⁺ oscillations in astrocytes using a well-established mathematical model which measures the rate of calcium release from the cytosol as a controlling parameter. Within a specific range, we identified two Hopf bifurcation points. The stability analysis revealed their close association with spontaneous Ca²⁺ oscillations. To validate the theoretical predictions, numerical simulations were conducted to demonstrate the consistency with computations. When the parameters were varied, the stability of the system exhibited diverse dynamic behaviors.

We analyzed the spontaneous Ca²⁺ oscillations evoked by calcium ion efflux in astrocytes in the same model as compared with previous studies ^[12,19], and obtained more complex dynamical behaviors. For instance, as the rate of calcium release from the cytosol decreased, this model exhibited the gradual emergence of multiple peaks simultaneously, accompanied by an increasing number of smaller peaks, before culminating in irregular chaotic states. Time-frequency diagrams were presented to show more intuitive depictions of frequency changes. The complexity arose from bidirectional communication between neurons and astrocytes and significantly increased the richness of their dynamical behavior. Future research is needed to examine the potential dynamical mechanisms for bidirectional communication in detail.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 12062004), the Natural Science Foundation of Guangxi Minzu University (No. 2022KJQD01), the Guangxi Natural Science Foundation (2020GXNSFAA297240) and Guangxi Science and Technology Program (Grant No. AD23023001) and Xiangsi Lake Young Scholars Innovation Team of Guangxi Minzu University (No. 2021RSCXSHQN05).

Conflict of Interests

The authors declare there is no conflict of interest.

References

[1]	J. L. Stobart, K. D. Ferrari, M. J. Barrett, Long-term in vivo calcium imaging of astrocytes reveals distinct cellular compartment responses to sensory stimulation, Cereb. Cortex, 28 (2018), 184–198. https://doi.org/10.1093/cercor/bhw366 doi: 10.1093/cercor/bhw366
[2]	G. Perea, A. Araque, Astrocytes potentiate transmitter release at single hippocampal synapses, Science, 317 (2007), 1083–1086. https://doi.org/10.1126/science.1144640 doi: 10.1126/science.1144640
[3]	A. Covelo, A. Araque, Lateral regulation of synaptic transmission by astrocytes, Neuroscience, 323 (2016), 62–66. https://doi.org/10.1016/j.neuroscience.2015.02.036 doi: 10.1016/j.neuroscience.2015.02.036
[4]	Z. Cao, L. Du, H. Zhang, L. Qiu, L. Yan, Z. Deng, Firing activities and magnetic stimulation effects in a Cortico-basal ganglia-thalamus neural network, Electron. Res. Arch., 30 (2022), 2054–2074. https://doi.org/10.3934/era.2022104 doi: 10.3934/era.2022104
[5]	Z. Wang, Y. Yang, L. Duan, Dynamic mechanism of epileptic seizures induced by excitatory pyramidal neuronal population, Electron. Res. Arch., 31 (2023), 4427–4442. https://doi.org/10.3934/era.2023226 doi: 10.3934/era.2023226
[6]	G. Wallach, J. Lallouette, N. Herzog, M. De Pittà, E. B. Jacob, H. Berry, et al., Glutamate mediated astrocytic filtering of neuronal activity, PLoS Comput. Biol., 10 (2014), e1003964. https://doi.org/10.1371/journal.pcbi.1003964 doi: 10.1371/journal.pcbi.1003964
[7]	A. Volterra, J. Meldolesi, Astrocytes, from brain glue to communication elements: the revolution continues, Nat. Rev. Neurosci., 6 (2005), 626–640. https://doi.org/10.1038/nrn1722 doi: 10.1038/nrn1722
[8]	E. V. Pankratova, M. S. Sinitsina, S. Gordleeva, V. B. Kazantsev, Bistability and chaos emergence in spontaneous dynamics of astrocytic calcium concentration, Mathematics, 10 (2022), 1337. https://doi.org/10.3390/math10081337 doi: 10.3390/math10081337
[9]	K. A. Woll, F. V. Petegem, Calcium-release channels: structure and function of IP₃ receptors and ryanodine receptors, Physiol. Rev., 102 (2022), 209–268. https://doi.org/10.1152/physrev.00033.2020 doi: 10.1152/physrev.00033.2020
[10]	S. Zeng, B. Li, S. Zeng, S. Chen, Simulation of spontaneous Ca²⁺ oscillations in astrocytes mediated by voltage-gated calcium channels, Biophys. J., 97 (2009), 2429–2437. https://doi.org/10.1016/j.bpj.2009.08.030 doi: 10.1016/j.bpj.2009.08.030
[11]	Y. Wang, A. K. Fu, N. Y. Ip, Instructive roles of astrocytes in hippocampal synaptic plasticity: neuronal activity‐dependent regulatory mechanisms, FEBS J., 289 (2022), 2202–2218. https://doi.org/10.1111/febs.15878 doi: 10.1111/febs.15878
[12]	H. Zuo, M. Ye, Bifurcation and numerical simulations of Ca²⁺ oscillatory behavior in astrocytes, Front. Phys., 8 (2020), 258. https://doi.org/10.3389/fphy.2020.00258 doi: 10.3389/fphy.2020.00258
[13]	Q. Ji, Y. Zhou, Z. Yang, X. Meng, Evaluation of bifurcation phenomena in a modified Shen–Larter model for intracellular Ca²⁺ bursting oscillations, Nonlinear Dyn., 84 (2016), 1281–1288. https://doi.org/10.1007/s11071-015-2566-3 doi: 10.1007/s11071-015-2566-3
[14]	X. Qie, Q. Ji, Computational analysis and bifurcation of regular and chaotic Ca²⁺ oscillations, Mathematics, 9 (2021), 3324. https://doi.org/10.3390/math9243324 doi: 10.3390/math9243324
[15]	L. Li, Z. Zhao, Inhibitory autapse with time delay induces mixed-mode oscillations related to unstable dynamical behaviors near subcritical Hopf bifurcation, Electron. Res. Arch., 30 (2022), 1898–1917. https://doi.org/10.3934/era.2022096 doi: 10.3934/era.2022096
[16]	F. Aguado, J. F. Espinosa-Parrilla, M. A. Carmona, E. Soriano, Neuronal activity regulates correlated network properties of spontaneous calcium transients in astrocytes in situ, J. Neurosci., 22 (2002), 9430–9444. https://doi.org/10.1523/JNEUROSCI.22-21-09430.2002 doi: 10.1523/JNEUROSCI.22-21-09430.2002
[17]	A. J. Barker, E. M. Ullian, New roles for astrocytes in developing synaptic circuits, Commun. Integr. Biol., 1 (2008), 207–211. https://doi.org/10.4161/cib.1.2.7284 doi: 10.4161/cib.1.2.7284
[18]	R. A. Chiareli, G. A. Carvalho, B. L. Marques, L. S. Mota, O. C. Oliveira-Lima, R. M. Gomes, et al., The role of astrocytes in the neurorepair process, Front. Cell Dev. Biol., 9 (2021), 665795. https://doi.org/10.3389/fcell.2021.665795 doi: 10.3389/fcell.2021.665795
[19]	M. Lavrentovich, S. Hemkin, A mathematical model of spontaneous calcium (Ⅱ) oscillations in astrocytes, J. Theor. Biol., 251 (2008), 553–560. https://doi.org/10.1016/j.jtbi.2007.12.011 doi: 10.1016/j.jtbi.2007.12.011
[20]	G. Houart, G. Dupont, A. Goldbeter, Bursting, chaos and birhythmicity originating from self-modulation of the inositol 1, 4, 5-trisphosphate signal in a model for intracellular Ca²⁺ oscillations, Bull. Math. Biol., 61 (1999), 507–530. https://doi.org/10.1006/bulm.1999.0095 doi: 10.1006/bulm.1999.0095
[21]	T. Höfer, L. Venance, C. Giaume, Control and plasticity of intercellular calcium waves in astrocytes: a modeling approach, J. Neurosci., 22 (2002), 4850–4859. https://doi.org/10.1523/JNEUROSCI.22-12-04850.2002 doi: 10.1523/JNEUROSCI.22-12-04850.2002

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Dynamical analysis of spontaneous Ca²⁺ oscillations in astrocytes

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Dynamical analysis of spontaneous Ca2+ oscillations in astrocytes

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Dynamical analysis of spontaneous Ca²⁺ oscillations in astrocytes