Introduction

Based on the famous Hodgkin–Huxley (HH) neuronal model (Hodgkin and Huxley 1952), some simplified neural models were proposed to study the neural discharge behaviors (FitzHugh 1961; Hindmarsh and Rose 1982; Wilson and Cowan 1972). The effects of systemic parameters and external factors (such as the external current, time delay, synapses, electric and magnetic fields, patch temperature, various noises, etc.) on discharge behaviors of neural models have been widely investigated in previous works. For instance, different discharge patterns (e.g. period-1 bursting and period-3 bursting) are acquired by changing parameters of neuronal model (Liu et al. 2019). And the neural mode transition is also affected by the amplitude of stimulating current (Lu et al. 2017). Time delay are common in different physical and biological systems. For example, there exists the time delay in the transmission of signals between coupled neurons (Lu et al. 2019b; Mondal et al. 2019; Wang et al. 2019a). The time delay causes the adaptive regulation of neural system by proper autapse connection (Xu et al. 2017b). Time delay is adjusted to achieve system synchronization through the memristor coupling (Xu et al. 2017a). The reception of signal is dependent on synapses between neurons. For instance, both the speed of wave propagation and synchronization are affected by field coupling and chemical synapses (Ge et al. 2019). The synapse affects the transmission of information between other synapses (Maio et al. 2018; Guo et al. 2018). Most of the previous studies demonstrated that the electric and magnetic fields have great influences on neuronal discharge (Ge et al. 2018b; Xu et al. 2018b). For example, the various discharge patterns are caused by different electric field strengths (Ma et al. 2018). The firing patterns and energy are more complex under the electromagnetic radiation (Lu et al. 2019d; Jin et al. 2019). The transition between different states of neurons is various under the electromagnetic induction (Ge et al. 2018c). The synchronization is much more stable when the coupling strength of magnetic flux reaches a certain threshold (Ma et al. 2017). According to the similitude between synapse and memristor, the information was efficiently handled by an intelligent coding scheme (Zhu et al. 2018; Xu et al. 2018c; Ji et al. 2019). The energy method is another way to study the coding problems. It was proved that neurons are in different status of energy characteristics during excitement and subconscious activities (Wang et al. 2017). It is well known that temperature adjusts the dynamics of neural networks and hence the action potential is produced. The electrical activities of neurons can show various states under different temperatures (Lu et al. 2018; Xu et al. 2019c). It is important to study the influence of external factors on neurons, and the explore of thresholds is also very meaningful. The research has shown that subthreshold neurons consume energy, super-threshold neurons can both consume energy and absorb energy (Wang et al. 2018a).

Neuron is the basic element of biological neural network, the study of electrical activity of single neuron (Zhu et al. 2019) has been extended to the neural networks (Wang et al. 2018b; Wen et al. 2019a). For example, the feed-forward multilayer networks (Ge et al. 2018a; Lu et al. 2019c) were investigated. The dynamic behaviors, such as coherent resonance (Pikovsky and Kurths 1997; Xu et al. 2019b; Lu et al. 2019a), stochastic resonance (Gammaitoni et al. 1998; Guo et al. 2017; Lu et al. 2020) and vibrational resonance (Landa and Mc Clintock 2000; Yang et al. 2012; Ge et al. 2020b) were investigated in different neuron models, respectively. Further research on neural networks is an exploration of true neurophysiology. For example, long-term memory is induced by applying two stimuli in the bi-stable dynamic model through neural energy coding (Wang et al. 2019b). A study of the tactile neural network found that the firing rate of neurons are related to the number of connected receptors, the distance between neuron and force application point and the number of mechanoreceptors (Yao and Wang 2019). Visual attention increases firing rates of neurons and may adjust neuronal activity at the synaptic level in the Hodgkin–Huxley neural network (Zhang et al. 2019). A fuzzy method to improve learning rate is applied to memristor-based multi-layer neural network to accomplish complex tasks (Wen et al. 2019b). The study has shown that when neurons are excited, blood flow increases obviously, and the presence of negative energy in neurons at the sub-threshold state is the reason for the increased blood flow reaction time in the brain (Wang et al. 2015). When neurons are at an action potential in the brain, blood flow increases notably while oxygen consumption decreases. It explains that hemodynamics plays an important role in the use of energy (Zheng et al. 2014).

On the one hand, neurons are considered signal processors, and different external stimulus are transmitted through the nervous system in the form of signals. The responses of neural system to multiple signals (high-low frequency signal and envelope modulation signal) were studied, it was demonstrated that the appropriate amplitude and frequency can maximize the signal response (Yang et al. 2012; Guo et al. 2017).

On the other hand, there are different sources of noise in the biological nervous system, such as external synaptic noise and internal channel noise (Lecar and Nossal 1971; White et al. 2000; Schmid et al. 2003). The external synaptic noise can promote the dynamic systems from chaos to order (Yao and Ma 2018). The effect of additive on random burst synchronization was studied by changing the noise intensity, and it is found that long-term enhancement of synaptic strength makes good burst synchronization better (Kim and Lim 2018). And there exists an optimal channel noise which allows neurons to fire most regularly (Schmid et al. 2001; Jung and Shuai 2001). In the presence of ion channel noise, the regularity of firing pattern is greatly influenced (Xu et al. 2019a). Noise also enhances the signal propagation across voltage-dependent ion channels (Levin and Miller 1996).

In addition, based on biological experimental investigations, it was found that the toxins (e.g. tetraethylammonium and tetrodotoxin) can reduce the number of potassium and sodium channels opening (Narahashi and Moore 1968; Hoyt and Strieb 1971). It was showed that the blocking potassium channel lets the spiral wave more stable, while the blocking sodium channel breaks the spiral wave (Xu et al. 2018a). Furthermore, as the number of channels is reduced, the component of channel noise is increased (Goldman et al. 2001).

Therefore, how the ion channel blockage affects the discharge behaviors of stochastic HH neuronal model driven by multiple oscillating signals is a very interesting question. The effects of ionic channel blockages and ionic channel noise on the discharge behaviors of HH model with multiple oscillatory signals are studied in this paper.

The paper is organized as follows. In “Model description” section, the Hodgkin–Huxley neuron model driven by ionic channel blockages, ionic channel noise and multiple oscillatory signals is introduced. In “Results and discussions” section, the influences of multiple oscillatory signals on discharge behaviors are firstly discussed without ionic channel noise. Secondly, the effects of ionic channel blockages on discharge behaviors are studied in presence of ionic channel noise. We conclude the results in Conclusions.

Model description

The Hodgkin–Huxley neuron model driven by both ionic channel blockages and multiple oscillatory signals is described by

$$C\frac{{{\text{d}}V}}{{{\text{d}}t}} = - G_{\text{K}} x_{\text{K}} n^{4} \left( {V - E_{\text{K}} } \right) - G_{\text{Na}} x_{\text{Na}} m^{3} h\left( {V - E_{\text{Na}} } \right) - G_{\text{L}} \left( {V - E_{\text{L}} } \right) + I_{ext} ,$$
(1)

where V represents the membrane potential of neuron, and C = 1 μF/cm2 is the capacity of the cell membrane. The reversal potentials for the potassium, sodium and leakage currents are EK = −77 mV, ENa = 50 mV, EL = −54.4 mV, respectively. GK =36 mS/cm2, GNa = 120 mS/cm2 and GL = 0.3 mS/cm2 denote the maximum conductance of potassium, sodium and leakage currents, separately.

Based on Refs.(Narahashi and Moore 1968; Hoyt and Strieb 1971), the tetraethylammonium and tetrodotoxin can reduce the number of potassium and sodium channels open. In Eq. (1), parameters xK and xNa represent the ratios of ionic channel blockages, which is proportion coefficients of non blocking ion channels to the total number of ion channels (Hille 2001). The smaller the ratio is, the greater the degree of blockage will be.

The gating variables n, m and h which characterize the average proportion of working channels opening (Hodgkin and Huxley 1952) are given by

$$\frac{dy}{dt} = \alpha_{y} \left( {1 - y} \right) - \beta_{y} y,\quad \left( {y = n,m,h} \right)$$
(2)

where αy and βy are the switch rates of ionic channels which depend on voltage and read as follows:

$$\left\{ {\begin{array}{*{20}l} {\alpha_{n} = \frac{{0.01\left( {V + 55} \right)}}{{1 - \exp \left[ { - {{\left( {V + 55} \right)} \mathord{\left/ {\vphantom {{\left( {V + 55} \right)} {10}}} \right. \kern-0pt} {10}}} \right]}},\quad \beta_{n} = 0.125\exp \left[ { - {{\left( {V + 65} \right)} \mathord{\left/ {\vphantom {{\left( {V + 65} \right)} {80}}} \right. \kern-0pt} {80}}} \right],} \hfill \\ {\alpha_{m} = \frac{{0.1\left( {V + 40} \right)}}{{1 - \exp \left[ { - {{\left( {V + 40} \right)} \mathord{\left/ {\vphantom {{\left( {V + 40} \right)} {10}}} \right. \kern-0pt} {10}}} \right]}},\quad \beta_{m} = 4\exp \left[ { - {{\left( {V + 65} \right)} \mathord{\left/ {\vphantom {{\left( {V + 65} \right)} {18}}} \right. \kern-0pt} {18}}} \right],} \hfill \\ {\alpha_{h} = 0.07\exp \left[ { - {{\left( {V + 65} \right)} \mathord{\left/ {\vphantom {{\left( {V + 65} \right)} {20}}} \right. \kern-0pt} {20}}} \right],\quad \beta_{h} = \left\{ {1 + \exp \left[ { - {{\left( {V + 35} \right)} \mathord{\left/ {\vphantom {{\left( {V + 35} \right)} {10}}} \right. \kern-0pt} {10}}} \right]} \right\}^{ - 1} .} \hfill \\ \end{array} } \right.$$
(3)

The external current stimulation Iext in Eq. (1) is superimposed by two oscillatory signals: a low-frequency (LF) and a high-frequency (HF), and it is written as

$$I_{ext} = A{ \cos }\left( {\omega t} \right) + B\cos \left( {N\omega t} \right),$$
(4)

where ω = 0.3 is the angular frequency, and A = 1.5 μA/cm2 is the amplitude of LF oscillatory signal. The term B expresses the amplitude of HF oscillatory signal, and N means the multiple of angular frequency. It is assumed that HF oscillatory signal is harmonics of LF oscillatory signal, but it can be checked that non-harmonic HF do not change qualitative the results (Yang et al. 2012).

Ionic channel noises and ionic channel blockages play more crucial part in the neural dynamics in Hodgkin–Huxley neuron model. Considering ionic channel noise ξy(t) (Schmid et al. 2004), the gate variables (Schmid et al. 2003) obey the following Langevin equation

$$\frac{dy}{dt} = \alpha_{y} \left( {1 - y} \right) - \beta_{y} y + \xi_{y} \left( t \right),\quad \left( {y = n,m,h} \right)$$
(5)

where αy and βy are the opening and closing rates of ionic channels, and random variables ξy(t) which are Gaussian white noise sources (Schmid et al. 2004; Xu et al. 2018a) satisfy

$$\left\{ {\begin{array}{*{20}l} {\left\langle {\xi_{m} \left( t \right)\xi_{m} \left( {t^{\prime}} \right)} \right\rangle = \frac{2}{{N_{\text{Na}} x_{\text{Na}} }}\frac{{\alpha_{m} \beta_{m} }}{{\left[ {\alpha_{m} + \beta_{m} } \right]}}\delta \left( {t - t^{\prime}} \right),} \hfill \\ {\left\langle {\xi_{h} \left( t \right)\xi_{h} \left( {t^{\prime}} \right)} \right\rangle = \frac{2}{{N_{\text{Na}} x_{\text{Na}} }}\frac{{\alpha_{h} \beta_{h} }}{{\left[ {\alpha_{h} + \beta_{h} } \right]}}\delta \left( {t - t^{\prime}} \right),} \hfill \\ {\left\langle {\xi_{n} \left( t \right)\xi_{n} \left( {t^{\prime}} \right)} \right\rangle = \frac{2}{{N_{\text{K}} x_{\text{K}} }}\frac{{\alpha_{n} \beta_{n} }}{{\left[ {\alpha_{n} + \beta_{n} } \right]}}\delta \left( {t - t^{\prime}} \right).} \hfill \\ \end{array} } \right.$$
(6)

In Eq. (6), the total numbers of ion channels are NNa = ρNa × S and NK = ρK × S, where S is the size of membrane patch area, ρNa and ρK are ion channel densities. From the perspective of statistical physics, the ionic channel noise ξy(t) is decreased with the increasing of S, thus, the intensity of ionic channel noise can be enhanced by changing the membrane patch area S. Assuming ion channel density is homogeneous, and setting ρNa = 60 μm−2 and ρK = 18 μm−2. In this section, the integral step length is 0.01, the initial values (V0, m0, n0, h0) = (−65.0, 0.1, 0.3, 0.6), and the fourth-order Runge–Kutta method is used.

Results and discussions

Effects of multiple oscillatory signals on bifurcation points

Before discussing the effects of ionic channel noise on the spontaneous spiking activity, it is necessary to study the influences of amplitude of multiple oscillatory signals on the bifurcation points of HH neuron model with the ionic channel blockages, where the ionic channel noises are not considered.

Setting the multiple of angular frequency N = 30 and the amplitude of HF oscillatory signal B = 1.0 in Fig. 1. It exhibits that the bifurcation diagram for HH neuron model driven by multiple oscillatory signals is plotted under the ratio of potassium channel blockage xK. The previous studies (Schmid et al. 2003; Xu et al. 2018a) demonstrated that sodium channel blockage cannot induce spiking modes, while it is opposite for potassium channel blocking.

Fig. 1
figure 1

The bifurcation diagram for HH neuron model driven by multiple oscillatory signals is plotted versus the ratio of potassium channel blockage xK (from zero to one) at B = 1.0, N = 30, xNa = 1.0. The four critical points x1, x2, x3 and x4 are four bifurcation points. Stable solutions (red lines), unstable solutions (blue lines and black line), maximum and minimum membrane potentials (green lines). (Color figure online)

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As the ratio of potassium channel blockage is increased, the stable solutions (red lines) become unstable, and the unstable solutions (blue lines and black line) appear. And as xK continues to increase, the stable solutions emerge again. The green lines correspond to the maximum and minimum values of neural membrane potentials. When the ratio xK is between x1 and x2 or x3 and x4, the stable solutions and unstable solutions are coexisting.

Previous studies (Schmid et al. 2003) showed that coexistence interval of stable solutions and unstable solutions is between 0.09 and 0.11 or 0.55 and 0.64 in the original Hodgkin–Huxley model. The changes of four critical points with the amplitude of HF oscillatory signal in HH neuron model are shown in Table 1. As the increasing of B, the former two critical points alter slightly while the latter two critical points increase significantly. When B reaches a certain value, there exist action potentials under the completely open potassium channels. Table 1 indicates that the amplitude of HF oscillatory signal can change coexistence interval of stable solutions and unstable solutions.

Table 1 Statistics of four critical points (in Fig. 1) with the different amplitudes B of HF oscillatory signal, and N =30, ω = 0.3
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The firing sequences of neuron are hardly changed by altering the frequency of HF oscillatory signal under greater potassium channel blockage. While the ratio of potassium channel blockage is fixed at a large value (xK = 0.95) and setting the amplitude of HF oscillatory signal B = 10.0, the discharge mode is changed.

As shown in Fig. 2a, the interspike interval is roughly concentrated in two ranges—around 25 and 40. It is found that when the multiple N of frequency is small (Fig. 2b), the discharge pattern is changed specifically. There is only one spike interval which is about 20 when N is around 10. While there are two spike intervals (about 20 and 40) when N is around 30. The temporal evolution of membrane potentials for different multiples is illustrated in Fig. 2c. The period-1 firing, the period-4 firing and the period-3 firing at N = 10, 20, 30 are shown, respectively. It indicates that the converting in neuron firing patterns depends on the change of frequency in the case of less potassium channel blockage.

Fig. 2
figure 2

a Bifurcation diagram with the increasing of frequency multiple (N); b Enlarged view with small region N. The parameters: B = 10.0, xK = 0.95, and xNa = 1.0. c Temporal evolution of membrane potentials for different frequency multiples N = 10, 20, 30

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Effects of ionic channel blockages and ionic channel noise on discharge behaviors

Numerical simulations is carried out to comprehend the influences of ionic channel noise on the firing frequency. The time series of membrane potentials under potassium channel blockage and sodium channel blockage are represented in Fig. 3a, b at different membrane patch sizes, respectively. No matter which ion channel is blocked, the spike number reduces with the increase of membrane patch size S. The phenomenon is more obvious when sodium channel is blocked.

Fig. 3
figure 3

Sampled time series under different membrane patch sizes at B = 3.0, N = 30. The membrane patch sizes are selected as a1, b1S = 0.5, a2, b2S = 2.0 and a3, b3S = 6.0. The ratios of ionic channels blockages are a1, a2, a3xK = 0.5, xNa = 1.0; b1, b2, b3xK = 1.0, xNa = 0.5. And n is the number of spikes in 1000 time units

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To accurately examine the effects of ionic channel noise on spontaneous action potentials, the mean interspike interval of membrane potential (Schmid et al. 2001) is introduced:

$$\left\langle T \right\rangle = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {t_{i} - t_{i - 1} } \right)} ,$$
(7)

where ti is the time of the i-th spike in the time series of membrane potential, and n represents the total numbers of spikes. The smaller 〈Tñ means the higher frequency of neuron firing. The total duration for calculating is 10,000 time units.

The growth of the mean interspike interval with the membrane patch size under different blocking degrees of ionic channels is plotted in Fig. 4. The mean interspike interval increases with the membrane patch size increases, and tends to stabilize (the same with larger membrane area) in Fig. 4a. Furthermore, the mean interspike interval of smaller ratio xK is always lower than that of larger ratio. It is similar to Fig. 4a about the change trend of 〈Tñ with S in Fig. 4b. While the difference is that the 〈Tñ of smaller ratio xNa is invariably higher than that of larger ratio. Figure 4 illustrates that potassium channel blockage and ionic channel noise promote neural firing, while sodium channel blockage inhibits neural firing.

Fig. 4
figure 4

The dependence between the mean interspike interval and ionic channel noise. The mean interspike interval under the four different ratios of axK = 1.0, 0.7, 0.4, 0.1, and xNa = 1.0; bxNa = 1.0, 0.7, 0.5, 0.3, and xk = 0.5. Here, B = 3.0, N = 30

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The relationship related to the number of neuron spikes between the membrane patch size and the blocking degree of ionic channels is known more clearly in Fig. 5. The vertical axis represents the membrane area. The horizontal axes represent the blocking degree of potassium and sodium ionic channels, respectively. Different colors are used in the right-most vertical bar graph to distinguish different spike number intervals (the number sequence on the right is the number of spikes). The color in Fig. 5a, b represents the corresponding spike number intervals. The frequency of spontaneous discharge is measured by counting the spike number. It is shown from Fig. 5a that the number of spikes adds with the intensification of potassium channel blocking. When S is extremely small, the number of spikes is decreased at the smaller ratio of potassium channel blockage and is increased at the larger ratio of potassium channel. It conveys that appropriate potassium channel blocking can promote neural discharge behaviors, while excessive blocking will reduce the number of spikes when the channel noise is highly large. For the sodium channels, the excitability of neuron is inhibited through the increase of ionic channel blocking (Fig. 5b), and the neuron firing is induced when the channel noise is large enough (i.e., quite small membrane area). The distribution trend of spike numbers is similar to Fig. 5 at larger value of the membrane patch size.

Fig. 5
figure 5

Distribution of the spike number with different membrane patch sizes and blocking degree of potassium and sodium channels at B = 3.0, N = 30. The ordinate axis S is the membrane patch size for axNa = 1.0; bxK = 1.0

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Time series diagram under different ratios of xK and same channel noise is depicted in Fig. 6a. The same conclusion as Fig. 4a is acquired, that is, the spike interval increases as the proportion of open potassium ion channels is enlarged. And it is found that the neuron firing becomes increasingly irregular. The fast Fourier transform is used to describe the power spectrum at three ratios of xK in Fig. 6b. The spectrum peaks are getting higher and sharper as the ratio xK decreases, which indicates that the discharge behaviors become more regular.

Fig. 6
figure 6

Sampled time series for membrane area with S = 15.0 under three different ratios of a1xK = 0.35, a2xK = 0.65, a3xK = 1.0. b Power spectrum of the time series under different ratios of potassium channel blockage xK. With parameters: xNa = 1.0, B = 3.0, N = 30

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In order to precisely characterize the regularity of neural pulse signals, an appropriate measure of consistency, the coefficient of variation (Schmid et al. 2001), is introduced, which is expressed as the ratio of standard deviation to average value

$$CV = \frac{{\sqrt {\left\langle {T^{2} } \right\rangle - \left\langle T \right\rangle^{2} } }}{\left\langle T \right\rangle },$$
(8)

the smaller the value CV is, the more regular the neural spiking will be.

In Fig. 7, the coefficient of variation about the ratio xK is plotted under four different patch sizes. The CV values are relatively large in Fig. 7a, b, it expresses the neural pulse signal is more chaotic when the channel noise is louder. The channel noise is appropriately reduced, the curves in Fig. 7c, d decrease first and then increase (anti-unimodal) (Ghitany et al. 2019). The optimal ratio xK corresponds to the minimum value. Furthermore, the coefficient of variation is much smaller, the pulse is most stable and regular at this point. When the membrane patch size reaches larger value, there is still an optimal solution for the variation coefficient curve.

Fig. 7
figure 7

The dependence between the coefficient of variation and the ratio of potassium channel blockage under four different membrane patch sizes with aS = 0.3; bS = 2.5; cS = 6; dS = 15. The parameters: xNa = 1.0, B = 3.0, N = 30

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The stability of neuron pulse signals under sodium ion channel blockage is discussed in Fig. 8. The opposite of Fig. 6a is that the spike interval decreases with the adding of the proportion of open sodium ion channels (Fig. 8a). Accordingly, the values of power spectrum which correspond to these three ratios of sodium channel blockage are represented in Fig. 8b. Spectrum peaks gradually become higher as the ratio xNa increases, which is contrary to Fig. 6b. It is suggested that sodium channel blockage causes instability of neural pulses and makes them tend to chaos.

Fig. 8
figure 8

Sampled time series for membrane area with S = 15.0 under three different ratios of a1xNa = 0.45, a2xNa = 0.7, a3xNa = 1.0. b Power spectrum of the time series under different ratios of sodium channel blockage xNa. With parameters: xK = 0.5, B = 3.0, N = 30

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The relationship between the coefficient of variation and sodium channel blockage under four different membrane areas is discussed in Fig. 9. The coefficient of variation represented by the ordinate axis is higher when the membrane area S is quite small, and it becomes lower when S is increased. In “Model description” section, it is known that an increase in membrane area leads to a reduction in ionic channel noise. The CV value is very large when ionic channel noise is strong (Fig. 9a), and it is reduced when the noise is appropriately decreased (Fig. 9b–d). It means that appropriate ionic channel noise intensity makes neural impulses orderly. For sodium ion channel, it can be clearly seen that the neural pulse sequences become more chaotic when the blocking is enhanced (decrease of xNa) from Fig. 9c, d. The alike phenomenon occurs with the smaller ionic channel noise. It implies that the mechanisms of sodium and potassium channel blockages on discharge behaviors are quite different.

Fig. 9
figure 9

The dependence between the coefficient of variation and the ratio of sodium channel blockage under four different membrane patch sizes with aS = 0.3; bS = 2.5; cS = 6; dS = 15. The parameters: xK = 0.5, B = 3.0, N = 30

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Conclusions

In summary, the influences of multiple oscillatory signals without ionic channel noise firstly and ionic channel blockages in presence of ionic channel noise secondly on discharge behaviors are investigated. The characteristics of discharge behaviors are found by discussing the time evolution of membrane potential and the statistic parameters such as the interspike interval, number of spikes, coefficient of variation.

In the absence of ionic channel noise, it is found that the amplitude and frequency of the multiple oscillatory signals have significant influences on discharge behaviors: (1) Amplitude of multiple oscillatory signals changes the coexistence interval of stable solutions and unstable solutions. (2) The frequency can transform the firing pattern, such as different firing cycle (the period-1 firing, the period-4 firing and the period-3 firing). (3) In the presence of ionic channel noise, the potassium channel blockage promotes spontaneous spike discharging activities, while the sodium channel blockage restrains spontaneous spike discharging activities. (4) The coefficient variation appears decreasing-increasing (anti-unimodal) with the ratio of potassium channel blockage xK increasing, which indicates that there exists an optimal potassium channel blockage at which discharge behaviors are most regular. While the coefficient variation decreases monotonously with the ratio of sodium channel blockage xNa increases, which indicates that sodium channel blockage disrupts the neural systems’ order. (5) The discharge frequency becomes faster as the ionic channel noise amplifies, but the system is more chaotic under the considerable noise.

These mechanisms help to understand the dynamics of nonlinear systems and the biological significance (Ma et al. 2012; Yang et al. 2012; Xu et al. 2019a). Our research is expected to extend from the single neuron to complex neural networks, and it gives possible directions for discussing the enhancement of weak signal transmission ability in the feed-forward neural networks (Ge et al. 2020a).