Astrocytic modulation on neuronal electric mode selection induced by magnetic field effect

Abstract

Astrocytes as well as electromagnetic induction have been primarily considered as main factors in regulating neuronal firing patterns in the recent decade. In this work, an improved neuron-astrocyte model in consideration of the modulation of astrocytes and the electromagnetic induction is employed to explore the extend to which both of the factors affect the firing modes of the neurons. The “alternation mode”, defined as the alternative of neural normal spiking mode with the high-frequency bursting-like mode, clearly shows the functions of astrocytes on neurons. Moreover, the firing pattern of the neuron becomes more abnormal when astrocytes are hyper-excitable, the reason why the abnormal coupling of the astrocyte leads to the “alternation mode” of the neuron have been studied. In addition, the effect of electromagnetic induction manifests nonlinear characteristic towards neurons, complex firing modes of neurons are observed in the weaker field and a switching mode consists with quiescent and spiking mode appears when there is a higher stronger field. This approved model can reveal the normal or abnormal electric activities of neuron considered electromagnetic induction induced by the degree of excitability of the astrocyte. These results can provide potential understanding about the effects of astrocyte on neuronal activity when the coupling of electromagnetic field is considered.

Introduction

It is ever believed that biological neurons can show a variety of electric activities and distinct firing modes transitions in the physiological or pathological conditions in brain. Hodgkin and Huxley initiated the dynamical research on neuron via creating HH neuron model by means of equivalent circuits for the first time(Hodgkin and Huxley 1952). Several simplifications of HH model,such as HR(Hindmarsh-Rose), FHN(FitzHughNagumo) and ML(Morris-Lecar) neuron model, have been proposed for clearer and more uncomplicated comprehension on neuron model from the viewpoint of dynamics for the last decades(Hindmarsh and Rose 1982; Fitzhugh 1960; Morris and Lecar 1981), and the dynamical properties of the neuron are clearly understood via bifurcation analysis(Zhan and Liu 2019; Wu et al. 2020; Guan et al. 2020). The dynamic analysis methods on neuron models reveals not only the dynamical properties of neurons but the biological features of the neurons existed in human brains(Mondal et al. 2019). Zhao studied the modulations of hyperpolarization-activated cation current($I_h$) on resonance in a bursting developed HH neuron model and acquired a novel condition for suprathreshold resonance and the underlying codimension-2 bifurcations(Guan et al. 2021). Fan dynamically explained the seizure abatements induced by single-pulse alternately resetting stimulation(SARS) via calculating the averaged mean firing rate (AMFR) of neurons and triggering averaged mean firing rates (TAMFRs) of 2Hz-4Hz spike and wave discharges(SWD)(Fan et al. 2020). Feng disclosed the temporal property in pattern formation of the ML neural network, which is characterized by the theory of critical slowing down(Feng et al. 2020). Wang observed that the activities of neurons based on the Hodgkin-Huxley neuron model have duality, that is, subthreshold neurons and suprathreshold neurons have energy expenditure exists, but only the suprathreshold ones have energy absorption(Wang et al. 2018). The duality of neuronal activity has also been found in Chay model(Zhu et al. 2019). And it has been proved in functional networks such as intellectual exploration(Wang et al. 2017).

Many physiological experimental evidence have verified that the other kind of numerous neuron cells named glias, especially, astrocytes, act firmly on supporting and nourishing neurons but also participating in the electric activities of them(Lines et al. 2020; Laura et al. 2019). Postnov proposed a tripartite synapse model coupled the presynaptic and postsynaptic neurons together with an astrocyte as the most classic dynamical astrocytic model(Postnov et al. 2008). Verisokin established a realistic data-driven spatial astrocytic network and built the mathematic model by local cell morphology(Verisokin et al. 2021). Yu investigated the effect of astrocytic feedback on the collective activities of neuronal network, especially in epilepsy(Yu et al. 2020). Du constructed a novel neuron-astrocyte coupling model in consideration of each module composed with a neuron and four surrounding astrocytes interacted separately and including extracellular potassium dynamics(Du et al. 2018). Li described an altered astrocytic glutamate decay rate or baseline value as a dysfunction on glutamate clearance from the extracellular space as a result of EAAT2 downregulation(Li et al. 2016). In summary, the regulation of astrocytes on neurons is no longer ignored in the cellular level researches of the brain activities.

Besides astrocytes, another significant factor, magnetic induction, has been observed for decades in the biological experiment(Roth and Wikswo 1985; Barry et al. 2016) . Continuous fluctuation in neuronal membrane potential will generate a magnetic field around the neuron according to Maxwell electromagnetic induction theorem, the accompanying induction current can influence the firing modes of the neuron back. Memristor have been proposed for bridging the effect of electromagnetic induction and neurons via the method of the equivalent circuit(Ma and Tang 2017; Zhang and Liu 2019). Lv discovered the electromagnetic radiation can excite and inhibit the neuronal firing modes via an improved HR model by introducing the effect of electromagnetic induction(Lv et al. 2016). Liu studied the dynamical responses of a hybrid neuron model combined HR model with Wilson model in consideration of the effect of electromagnetic induction and observed that the frequency and amplitude of the induction influence the firing patterns of the neuron significantly respectively(Liu et al. 2019). Zhang proposed a simple neural circuit incorporated two physical electronic components and revealed that the electric activities of the neuron responded rapidly and sensitively to external magnetic field(Zhang et al. 2020). Therefore, the magnetic effect in neurons remains significant and urgent to pay attention.

However, there is few researches to focus on the neuronal electric activities modulated by astrocytes and the electromagnetic effect simultaneously. In this present work, we study the dynamic responses of an improved neuron-astrocyte coupling model, which we have used in the previous work(Yuan et al. 2020), in consideration of the electromagnetic induction. Our results reveal that the modulation of both of the astrocytes and the electromagnetic induction effect strongly affect the physiological or pathological firing modes and the patterns transition of the neuron.

Model

The improved Hodgkin-Huxley neuron model is displayed as:

$$\begin{array}{cc}\hfill C_m\frac{\mathit{dV}}{\mathit{dt}}& =-g_K{n}^4(V-V_K)-g_{\mathit{Na}}{m}^{3}h\left(V,-,V_{\mathit{Na}}\right)\hfill \\ \hfill & -g_L\left(V,-,V_L\right)+I_{\mathit{ext}}+k_a{I}_{\mathit{Astro}}+I_{\mathit{mag}},\hfill \end{array}$$ 1

$$\begin{array}{cc}\hfill \frac{\mathit{dm}}{\mathit{dt}}& ={\alpha }_m(1-m)-{\beta }_{m}m,\hfill \end{array}$$ 2

$$\begin{array}{cc}\hfill \frac{\mathit{dn}}{\mathit{dt}}& ={\alpha }_n(1-n)-{\beta }_{n}n,\hfill \end{array}$$ 3

$$\begin{array}{cc}\hfill \frac{\mathit{dh}}{\mathit{dt}}& ={\alpha }_h(1-h)-{\beta }_{h}h,\hfill \end{array}$$ 4

where V represents the membrane potential, and the probability of opening potassium ion ($K^{+}$) channel, sodium ion ($N{a}^{+}$) activation gate and sodium inactivation gate are $n^4$, $m^3$ and h severally. $g_K,g_{\mathit{Na}}$ and $g_L,$ $V_K,V_{\mathit{Na}}$ and $V_L$ are the maximal conductance and the reversal potentials of the potassium, sodium, and leakage system respectively for providing the ionic current. $C_m$ is the membrane capacitance. $k_a$ is the feedback intensity of the astrocyte current. The coefficients in Eq. (2) are given byNadkarni and Jung (2003):

$$\begin{array}{c}\hfill \begin{array}{cc}& {\alpha }_m=0.1\frac{25-V}{exp\left[\frac{25-V}{10}\right]-1};{\beta }_m=4exp\left[\frac{-V}{18}\right];\hfill \\ \hfill & {\alpha }_n=0.01\frac{10-V}{exp\left[\frac{10-V}{10}\right]-1};{\beta }_n=0.125exp\left[\frac{-V}{80}\right];\hfill \\ \hfill & {\alpha }_h=0.07exp\left[\frac{-V}{20}\right];{\beta }_h=\frac{1}{exp\left[\frac{30-V}{10}\right]+1}\hfill \\ \hfill \end{array}\end{array}$$ 5

Among the other additional currents in Equation (1), $I_{\mathit{ext}}$ determines an external stimulus current, and $I_{\mathit{mag}}$ denotes the electromagnetic feedback current induced from neuronal activity, given as follows(Lv et al. 2016):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_{\mathit{mag}}& =-k_1\rho (\varphi )V,\hfill \\ \hfill \frac{d\varphi }{\mathit{dt}}& =V-k_2\varphi ,\hfill \\ \hfill \rho (\varphi )& =\alpha +3\beta {\varphi }^2,\hfill \\ \hfill \end{array}\end{array}$$ 6

where feedback strength $k_1$ describes the negative feedback of the electromagnetic effect on neuronal firing modes, while the additive current generated by astrocytes is $I_{\mathit{astro}}$. Amounts of neurotransmitters have been released into the synaptic cleft after the neuron fires, it mainly binds to the postsynaptic membrane while a small part to the astrocyte, which can cause the astrocytic release of $I{P}_3$. The biological process is described as follows:

$$\begin{array}{c}\hfill \frac{d\left[I,P_3\right]}{\mathit{dt}}=\frac{1}{{\tau }_{I{P}_3}}\left({\left[I,P_3\right]}^{\ast },-,\left[I,P_3\right]\right)+r_{I{P}_3}\Theta \left(v,-,50,\mathrm{mV}\right),\end{array}$$ 7

where ${[I{P}_3]}^{\ast }$ is the equilibrated concentration of $I{P}_3$ and $r_{I{P}_3}$ defines the production efficiency of $I{P}_3$. These values were determined in Ref.Wang (1995). Once the neuron membrane potential exceeds $+50mV$, the second production term on the right side will be implemented because of the Heaviside function ($\Theta (x)$).

The increase of the $I{P}_3$ in the astrocytic cytoplasm activates the activities of calcium reservoir, primarily the endoplasmic reticulum (ER), to release calcium ions($C{a}^{2+}$). Li-Rinzel model(Nadkarni and Jung 2003) is adopt to describe the dynamics of $[C{a}^{2+}]$:

$$\begin{array}{cc}\hfill \frac{d\left[C,a^2,+\right]}{\mathit{dt}}=& -J_{\mathit{channel}}-J_{\mathit{pump}}-J_{\mathit{leak}},\hfill \end{array}$$ 8

$$\begin{array}{cc}\hfill \frac{\mathit{dq}}{\mathit{dt}}=& {\alpha }_q(1-q)-{\beta }_{q}q,\hfill \end{array}$$ 9

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{\mathit{channel}}& =c_1{v}_1{m}_{\infty }^3{n}_{\infty }^3{q}^3\left(\left[C,a^{2+}\right],-,{\left[C,a^{2+}\right]}_{\mathit{ER}}\right),\hfill \\ \hfill J_{\mathit{pump}}& =\frac{v_3{\left[C,a^{2+}\right]}^2}{k_3^2+{\left[C,a^{2+}\right]}^2},\hfill \\ \hfill J_{\mathit{leak}}& =c_1{v}_2\left(\left[C,a^{2+}\right],-,{\left[C,a^{2+}\right]}_{\mathit{ER}}\right),\hfill \end{array}\end{array}$$ 10

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill m_{\infty }& =\frac{\left[I,P_3\right]}{\left[I,P_3\right]+d_1},n_{\infty }=\frac{\left[C,a^{2+}\right]}{\left[C,a^{2+}\right]+d_5},\hfill \\ \hfill {\left[C,a^{2+}\right]}_{\mathit{ER}}& =\frac{c_0-\left[C,a^{2+}\right]}{c_1},\hfill \\ \hfill {\alpha }_q& =a_2{d}_2\frac{\left[I,P_3\right]+d_1}{\left[I,P_3\right]+d_3},{\beta }_q=a_2\left[C,a^{2+}\right]\hfill \end{array}\end{array}$$ 11

where q represents the activated fraction of $I{P}_{3}Rs$ ($I{P}_3$ receptors) on the ER surface, $J_{\mathit{channel}}$, $J_{\mathit{pump}}$ and $J_{\mathit{leak}}$ denotes the $[C{a}^{2+}]$ that moves from the ER to the cytoplasm through the $I{P}_{3}R$ channel, pumps from the cytoplasm into the ER and leaks from the ER to the inner space of the astrocyte, respectively.

The regulation of an astrocyte couples with a feedback electric current(Nadkarni and Jung 2003):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_{\mathit{Astro}}& =2.11\Theta (lny)lny,\hfill \\ \hfill y& =\left[C,a^{2+}\right]/\mathrm{nM}-{\left[C,a^{2+}\right]}_{\mathit{th}},\hfill \end{array}\end{array}$$ 12

The values of these parameters are listed in Table 1.

Table 1.

Parameters and constants in the simulation

	Parameter	Value
$C_m$	1	$\mu \mathrm{F}/{\mathrm{cm}}^2$
$g_{\mathrm{K}}$	36.0	$mS/{\mathrm{cm}}^2$
$g_{\mathrm{Na}}$	120.0	$mS/{\mathrm{cm}}^2$
$g_{\mathrm{L}}$	0.3	$mS/{\mathrm{cm}}^2$
$V_{\mathrm{K}}$	$-12.0$	mV
$V_{\mathrm{Na}}$	115.0	mV
$V_{\mathrm{L}}$	10.6	mV
$I_{\mathrm{ext}}$	11	$\mu \mathrm{A}/{\mathrm{cm}}^2$
${[I{P}_3]}^{\ast }$	160.0	${\mathrm{nmolL}}^{-1}$
${\tau }_{I{P}_3}$	0.00014	$m{s}^{-1}$
$c_0$	2.0	$\mu {\mathrm{molL}}^{-1}$
$c_1$	0.185	–
$v_1$	6	${\mathrm{s}}^{-1}$
$v_2$	0.11	${\mathrm{s}}^{-1}$
$v_3$	0.9	$\mu {\mathrm{molL}}^{-1}$
$k_3$	0.1	$\mu {\mathrm{molL}}^{-1}$
$d_1$	0.13	$\mu {\mathrm{molL}}^{-1}$
$d_2$	1.049	$\mu {\mathrm{molL}}^{-1}$
$d_3$	0.9434	$\mu {\mathrm{molL}}^{-1}$
$d_5$	0.08234	$\mu {\mathrm{molL}}^{-1}$
$a_2$	0.2	$\mu {\mathrm{molL}}^{-1}{\mathrm{s}}^{-1}$
$\alpha$	0.1	–
$\beta$	0.02	–
$k_1$	0.01	–
$k_2$	0.5	–
$r_{I{P}_3}$	0.8	$\mu$M$s^{-1}$
${[C{a}^{2+}]}_{\mathit{th}}$	196.69	nM

Numerical results

In this section, the fourth-order Runge-Kutta algorithm is implemented to solve the ordinary differential equations above with a time step of 0.01 ms. The initial values of the membrane potential(V), gate variables(m,n,h), magnetic flux($\varphi$), inositol 1,4,5-triphosphate($I{P}_3$) and calcium ion($C{a}^{2+}$) are 0mV, 0.2, 0.8, 0.8, 0nWb, 200nM and 200nM respectively.

“Alternation mode” in the neuron-coupled astrocyte system without electromagnetic flux

Spiking neurons can release plenty of neurotransmitters into the synaptic cleft, some of them can bind to astrocytic membrane leading to the increase of $I{P}_3$ in astrocytes. The parameter $r_{I{P}_3}$, which is defined as the production efficiency of $I{P}_3$, bridges the modulation of the neuron on its coupled astrocyte. We first focus on the dynamic role of astrocytic $I{P}_3$ in the coupling system without taking electromagnetic effect into account($k_1=0$), and the feedback intensity of the astrocyte current is set as $k_a=1$. Several neuronal firing patterns with $C{a}^{2+}$ oscillation are plotted in Fig. 1. Electric activities of the neuron experiences from a normal spiking mode (Fig. 1 a) to the other spiking in a higher frequency (Fig. 1 c), an abnormal pattern arises (Fig. 1 b) between the two spiking modes. Here we define the abnormal neuron firing as “alternation mode” for there are two kinds of neuronal oscillations appear alternatively.

Fig. 1.

Neuronal membrane potential, astrocytic feedback current and astrocyte calcium concentration under different $r_{I{P}_3}$ without magnetic effect. Red dashed lines denote the moment when $C{a}^{2+}={[C{a}^{2+}]}_{\mathit{th}}$. (Color figure online)

$I_{\mathit{Astro}}$ remains zero and $C{a}^{2+}$ oscillates under the threshold (Fig. 1 a), which indicates that the astrocyte cannot provide enough stimulants back to affect the firing modes of the neuron. While the $C{a}^{2+}$ beyond the threshold modulates neuronal electric patterns all the time (Fig. 1 c), and the feedback current $I_{\mathit{Astro}}$ is also oscillating at a high level. “Alternation mode” appears when the $C{a}^{2+}$ passes through the threshold along with the $I_{\mathit{Astro}}$ in high-value or zero alternatively.

Inter-spike interval (ISI) diagram has been used for a holistic observation of the neuronal firing patterns transition (Fig. 2). Low frequency spiking remains until $r_{I{P}_3}$ greater than $r_{I{P}_{3}a}=0.307mM{s}^{-1}$, at which high frequency spiking appears and mixes with the lower. As a consequence, “alternation mode” appears. When $r_{I{P}_3}$ increases continually, the lower disappears completely at about $r_{I{P}_{3}b}=0.642mMs-1$. Actually, higher or lower frequency firings indicate the state of neuronal activities. A neuron can reach the electric activities of low-frequency or high-frequency, and even contain the two frequency, which manifests that the neuron experiences both of the states and the firing mode occurs alternatively. In summary, the $I{P}_3$ production efficiency $r_{I{P}_3}$ can enhance the $C{a}^{2+}$ activities in astrocyte via increasing the amount of the second messager(namely stimulant)$I{P}_3$. Moreover, low-frequency firing mode appears at the weaker $r_{I{P}_3}$ relatively, whereas the stronger one can produce high-frequency neuronal patterns, “alternation mode” appears while both of them coexist.

Fig. 2.

The ISI diagram respect to $r_{I{P}_3}$ without consideration of electromagnetic induction($k_1=0$)

Neuronal electric activities modulated by the effect of electromagnetic induction

The parameter $k_1$ reflects the intensity of the electromagnetic field induced by the fluctuation of neuronal membrane potential, so it is set to change different values for investigating the dynamical role of electromagnetic effect in neuronal electric activities. In this part, $r_{I{P}_3}$ is taken at 0.2, 0.5 and $0.8\mu M{s}^{-1}$ respectively for showing weak, moderate and strong effect of production efficiency of IP3 according to Section 3.1, and the feedback intensity of the astrocyte current is also fixed as $k_a=1$.

The neuronal firing patterns and astrocytic calcium concentration are shown in Fig. 3 with varying the induction intensity $k_1$ under the $0.2\mu M{s}^{-1}$ $r_{I{P}_3}$. The ISI diagram is also shown in Fig. 4. Obviously, increasing $k_1$ inhibits the spiking mode of the neuron and the $C{a}^{2+}$ activities in the astrocyte continually, and a completely inhibition appears when $k_1$ exceeds 0.0212.

Fig. 3.

Neuronal membrane potential and astrocyte calcium concentration under different $k_1$ while $r_{I{P}_3}=0.2mM{s}^{-1}$

Fig. 4.

The ISI diagram respect to $k_1$ with the condition of $r_{I{P}_3}=0.2\mu M{s}^{-1}$, the value when $k_1$ is greater than 0.0212 is none

Subsequently, $r_{I{P}_3}$ is fixed at $0.5\mu M{s}^{-1}$, the firing modes and the oscillation of $C{a}^{2+}$ are calculated in Fig. 5. The ISI is plotted in Fig. 6 as well. $k_1$ still plays the role of suppressing neuronal electric activities, the peak of the membrane potential decrease slowly while the $k_1$ increases, then the neuron goes back to the quiescent state when the $k_1$ is strong enough. There is a switching state of resting state and “alternation mode” when the $k_1$ exceeds 0.008 until 0.018. Neuron spiking mode is not changing continually, it returns to the quiescent state suddenly in several special intervals (none value indicates the quiescent state) and is completely suppressed at a much greater $k_1$.

Fig. 5.

Neuronal membrane potential, astrocytic feedback current and astrocyte calcium concentration under different $k_1$ while $r_{I{P}_3}=0.5\mu M{s}^{-1}$

Fig. 6.

The ISI diagram respect to $k_1$ with the condition of $r_{I{P}_3}=0.5\mu M{s}^{-1}$. The red arrows indicate that there is no ISI at these $k_1$, and the firing patterns of the neuron are in resting state. (Color figure online)

The electric activities of the neuron and its ISI graph is plotted in Figs. 7 and 8 severally under the condition of $r_{I{P}_3}=0.8\mu M{s}^{-1}$. The role of $k_1$ in regulating neuronal firing has not change, the neuron experiences the high-frequency spiking and “alternation mode” then restores into quiescent state with the varying $k_1$. The switching states also appears along the $k_1$ changing from 0.008 to 0.0145.

Fig. 7.

Neuronal membrane potential and astrocyte calcium concentration under different $k_1$ while $r_{I{P}_3}=0.8\mu M{s}^{-1}$

Fig. 8.

The ISI diagram respect to $k_1$ with the condition of $r_{I{P}_3}=0.8\mu M{s}^{-1}$. The red arrows indicate that there is no ISI at these $k_1$, and the firing patterns of the neuron are in resting state. (Color figure online)

In summary, the effect of the electromagnetic induction will inhibit normal neuronal firing activities until quiescent state. Moreover, neuron can not spiking normally in specific intervals in the condition of relatively stronger induction field. The switching of spiking and resting state is worthy studying further deeply.

Astrocytic regulation on neuronal electric activities with electromagnetic induction

In addition to the effect of the electromagnetic induction, the astrocyte is another important factor in modulating neuronal firing modes. In this section, the intensity of magnetic effect $k_1$ is fixed at 0.01. Astrocytic coupling intensity $k_a$ is set to change for investigating its dynamical role in this system. $r_{I{P}_3}$ is the same as those in the previous section at 0.2, the other two values 0.5 and $0.8\mu M{s}^{-1}$ is considered later.

Actually, in the condition of $r_{I{P}_3}=0.2$, astrocyte can not act efficiently on neuronal firing modes. Fig. 9a shows the membrane potential of the neuron, Fig. 9b explains the reason why there is only one diagram plotted. With the change of $k_a$, the neuron electric activities is not changing at all.

Fig. 9.

a Neuronal membrane potential and astrocyte calcium concentration under $r_{I{P}_3}=0.2\mu M{s}^{-1}$. b The ISI diagram respect to $k_a$

Figure 10 shows neuronal electric activities under different $k_a$ when $r_{I{P}_3}$ is increased to $0.5\mu M{s}^{-1}$. Apparently, neuronal firing patterns are “alternation mode” all and can be divided into two parts: one is normal periodical spiking mode when astrocytic $C{a}^{2+}$ is below the ${[C{a}^{2+}]}_{\mathit{th}}$, and it is not changing obviously with the increasing $k_a$, the other is the abnormal state which becomes more abnormal when the more participation astrocyte provided. The abnormal firing mode remains normal relatively under the condition of a weaker coupling of the astrocyte(see Fig. 10a), the spike value of the abnormal state does not change significantly from the normal and its time span is also comparable to that of the normal situation. With $k_a$ increasing, the abnormal firing mode becomes more pathological and the interval it lasts shortens, the pathological state here means the electric activities of the neuron deviate from the normal physiological firing modes. Once there is a $k_a$ greater than 40, the oscillation of the membrane potential in abnormal state disappears completely and is substituted by the “high-threshold resting” state, which is a extreme pathological state for its high energy-consumption even there is no obvious change of the membrane potential. As to the normal firing mode, there is no apparent change under different $k_a$ except the increasing time span. Actually, the time span of the abnormal firing mode is also changing. Therefore, the time spans of the normal and abnormal electric activities needs investigating.

Fig. 10.

Neuronal membrane potential and astrocyte calcium concentration under different $k_a$ while $r_{I{P}_3}=0.5\mu M{s}^{-1}$

As plotted in Fig. 11, the time span of the abnormal state increases at the beginning and decreases after the maximal value point when $k_a=2$, and the parameter $k_a=2$ is also the minimal value point of that of the normal state, whereas, the maximal value arises at $k_a=6.1$. Holistically, the span of the normal does not change apparently even it increases or decreases along the varying parameter, while the span of the abnormal decreases observably.

Fig. 11.

The time span of the two firing modes respect to $k_a$ while $r_{I{P}_3}=0.5\mu M{s}^{-1}$

Subsequently, $r_{I{P}_3}$ is fixed at $0.8\mu M{s}^{-1}$. Research method is the same as that in the previous part and Fig. 12 shows neuronal electric activities and astrocytic $C{a}^{2+}$. The neuron experiences the transitions of spiking(see Fig. 12c and e), quiescent state (see Fig. 12b and d) and “alternation mode”(see Fig. 12a and f) when $k_a$ is lower than 5, neuronal activities transform frequently in this parameter interval. The variation tendency of the abnormal state with a $k_a$ greater than 5 is no less than that in the condition of $r_{I{P}_3}=0.5\mu M{s}^{-1}$.

Fig. 12.

Neuronal membrane potential and astrocyte calcium concentration under different $k_a$ while $r_{I{P}_3}=0.8\mu M{s}^{-1}$

The time spans of the normal and abnormal state are also calculated in Fig. 13, they are increasing or decreasing monotonously in two intervals: $[00.3]$ and $[3.7150]$. However, there is no values from 0.3 to 3.71, which means the neuron cannot fire normally or only one of the two states(normal and abnormal firing states) exists. Astrocytic calcium concentration is selected for elucidating the detailed dynamical behaviors in the “none value” range if there is only one parameter required in describing the neuronal electric activities. Actually, the electric activities of the neuron is corresponding with the $C{a}^{2+}$ in astrocyte, neuronal quiescent state indicates low-value unchanging $C{a}^{2+}$ (see Fig. 12b and d) while spiking state implies the high-value low-amplitude $C{a}^{2+}$ oscillation (see Fig. 12c and e). Moreover, the oscillation of $C{a}^{2+}$ in high-amplitude means the neuron is experiencing the “alternation mode”, high-amplitude here manifests the state that calcium concentration has penetrated the threshold $C{a}^{2+}=196.69nM$, that’s why the neuronal activities in Fig. 12a is recognized as “alternation mode”.

Fig. 13.

The time span of the two firing modes respect to $k_a$ while $r_{I{P}_3}=0.8\mu M{s}^{-1}$

After the transient state, maximal and minimal $C{a}^{2+}$ in astrocyte are chose for displaying states of the whole system (see Fig. 14). Obviously, the mode of $C{a}^{2+}$ activities is changing frequently when $k_a$ is less than 5 (Fig. 14b). High-amplitude oscillation of the $C{a}^{2+}$ arises at the beginning, and then enters into a switching mode combined the high-value oscillation with the quiescent state when $k_a$ is greater than $k_{a1}=0.3$, lower calcium concentration here indicates the quiescent state of the whole system and disappears while $k_a$ exceeds $k_{a2}=1.54$. However, the oscillation of $C{a}^{2+}$ comes back into a high-value state in the case of a greater $k_a$ than $k_{a3}=3.71$.

Fig. 14.

Stable maximal and minimal calcium concentration under different $k_a$ while $r_{I{P}_3}=0.8\mu M{s}^{-1}$

Conclusion and discussion

In this paper, we numerically explore the dynamic role of the astrocytes and electromagnetic effect in the approved neuron model, neuronal firing modes mainly experience among quiescent state, low-frequency spiking, high-frequency spiking and “alternation mode”, which is a switch state between high-frequency and low-frequency spiking and it only occurs when the $C{a}^{2+}$ penetrates through the threshold.

Apparently, electromagnetic induction brings inhibition effect into this coupling system, each case of the neuronal firing modes mentioned above returns to quiescent state when the induction effect intensity is strong enough. In some special intervals, the electric activities of the neuron can switch between the quiescent state with the spiking or “alternation mode”, this kind of modes selection corresponding to the induction indicates the sensitivity of the neuronal activities on the effect intensity, only in the specific ranges can the neuron fire periodically or remain quiet state.

The dynamical role of astrocyte in this approved model seems more diverse. The firing modes of the neuron is changing frequently under a weaker coupling with the astrocyte, high-value or high-amplitude oscillation even the quiescent state can be reached in the specific intervals. Additionally, there are two states appeared in the $[0.31.54]$ interval in Fig. 14(b), it is another kind of a bistable state, further study can disclose its dynamic characteristic deeply.

Based on the approved neuron-astrocyte coupling system in consideration of the electromagnetic induction, this paper study the firing modes selection of the neuron under these two factors. Different roles of them reflect various dynamical properties in this system, the neuron can manifest specific electric activities under the control of the factors above. These results can provide some insights into the effects of astrocytes and electromagnetic induction on neuronal activity.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.