An alternating electric field can influence the membrane potential of neurons [14]. The relationship between the external alternating electric field $E$ and the cell membrane depolarization voltage $\mathrm{\Delta }V$ can be described as [15]: (1) $\frac{d\mathrm{\Delta }V}{\mathit{dt}}+\frac{\mathrm{\Delta }V}{\tau }=\frac{\lambda }{\tau }E$ where $\lambda$ represents the polarization length and $\tau$ represents Maxwell-Wagner time constant [16].

The alternating electric field $E$ can be described as:

(2) $E\left(t\right)=\frac{A}{\omega }\sin \left(\omega t\right)$ where $A$ represents amplitude and $\omega$ represents angular frequency [17]. According to equation (1) and equation (2), the cell membrane depolarization voltage $\mathrm{\Delta }V$ due to an alternating electric field $E$ can be described as:

(3) $\mathrm{\Delta }V\mathrm{(t)}=\lambda \frac{A}{\omega }\frac{\sin \left(\omega t\right)-2\pi f\tau \cos \left(\omega t\right)}{1+(\mathit{\omega \tau }{)}^2}\mathrm{.}$

In the model, the order of magnitude $\tau$ is generally $10^{-10}$ ms; the frequency $f$ is located in the range of extremely low frequencies [18]. Therefore, the cell membrane depolarization voltage $\mathrm{\Delta }V$ can be approximated described as follows:

(4) $\mathrm{\Delta }V\left(t\right)=\lambda \frac{A}{\omega }\sin \left(\omega t\right)$ where polarization length $\lambda =1\mathit{mm}$ ; angular frequency $\omega =0.1\pi$ rad/s [19]. Cell membrane depolarization voltage $\mathrm{\Delta }V$ can be seen as an external disturbance of the cell membrane voltage $v\left(t\right)$ . The cell membrane voltage $v\left(t\right)$ under the electric field can be described as [20]:

The Izhikevich neuron model under the alternating electric field can be obtained by introducing equation (5) into the mathematical model of the Izhikevich neuron described in [12].

The pulse time sequence $S\left(t\right)$ of neurons with regard to each action potential generated by a neuron has a time function similar to $\delta$ . It can be described as:

(6) $S\left(t\right)=\sum _m\mathrm{‍}\delta (t-t(m\left)\right)$ where $t\left(m\right)$ is the firing moment of neurons. The value of $S\left(t\right)$ in the firing moment is one, and at other times is set to zero. The pulse time sequence clearly shows both the firing moments and firing rate.

The inter-spike interval (ISI) is the difference between two adjacent firing moments of a neuron. It is described as:

The firing rate of a single neuron is evaluated by dividing the average value of the ISIs (ms) by 1000. The firing rate of the neural network is evaluated by the average firing rate values of all neurons. The smaller the relative variation rate of firing rate, the better the anti-interference ability of the deep spiking neural network. When the relative variation rate of firing rate is less than 10%, the influence of electric field on the neural network is relatively small.

To investigate the influence of different electric field amplitudes on pulse time sequence of a deep spiking neural network, an electric field interference with amplitudes of 2 mV , 10 mV , or 20 mV are added to the deep spiking neural network. The pulse time sequence diagrams of the output layer neuron under different electric field amplitudes are shown in Fig. 1.

Figure.1.

Pulse time sequence diagrams.

From the Fig. 1, the abscissa gives simulation time; the ordinate gives the time of the action potential. The firing rate is 11.1 Hz under non-interference, the firing rates are 11.2 Hz, 11.4 Hz, and 12.0 Hz under the electric field amplitudes of 2 mV, 10 mV, and 20 mV, respectively. The fluctuations generated by electric field interference are 0.61%, 2.69% and 8.30%, respectively. Simulation results show that with the increased of electric field amplitude, the fluctuations are getting larger. This proved that with the increased electric field amplitude, the influence on firing moment and firing rate of the output layer neuron are all increased.

To further investigate the influence of electric field interference on the firing rate, an electric field interference with different amplitudes was added to the deep spiking neural network. The range of the electric field amplitude A was set to [1, 20] mV, and the step length was set to 1 mV. Simulation results showed that the firing rate of each layer in the deep spiking neural network was within the range of 6.7 Hz to 12.0 Hz. Normalized firing rate of each layer in the deep spiking neural network for different electric field amplitudes A is shown in Fig. 2.

Figure.2.

The firing rate of each layer.

From Fig. 2, the abscissa gives the electric field amplitude, the ordinate gives the number of layers, and the color-map gives the firing rate after normalization. Red indicates a higher neuron firing rate. Blue indicates a lower neuron firing rate. The figure shows that the firing rate of each layer increases with increased electric field amplitude. When the electric field amplitude is constant, the firing rate shows a trend of fluctuation with an increased number of layers.

Trend of firing rates in output the layer neuron under different electric field amplitudes A is shown in Fig. 3.

Fig. 3.

Firing rate of output layer neuron.

In Fig. 3, the abscissa gives the electric field amplitude; the ordinate gives the firing rate of the output neuron. When the electric field amplitude is in the range 0 mV to 6 mV, the firing rate is in the range of 11.1 Hz to 11.2 Hz. The firing rate is typically stable. When the electric field amplitude is in the range of 6 mV to 16 mV, the firing rate is in the range of 11.1 Hz to 11.9 Hz. The firing rate shows a marginally increasing trend. When the electric field amplitude is in the range 16 mV to 20 mV, the firing rate is in the range of 11.9 Hz to 12.0 Hz and the firing rate is effectively stable.

The relative variation rate of firing rate in the output layer neuron under different electric field amplitudes A is shown in Fig. 4.

Fig. 4.

The relative variation rate of firing rate.

In Fig. 4, the abscissa gives the electric field amplitude and the ordinate gives the firing rate of the output neuron. When the electric field amplitude is in the range of 1 mV to 20 mV, the relative variation rate of firing rate is less than 8.53%. An electric field amplitude in the range of 1 mV to 20 mV has little influence on the firing rate of the deep spiking neural network.

Correlation between the membrane potential can reflect the similarity degree of neuron membrane potential before and after electric field interference. The greater the correlation value is, the higher the similarity degree is before and after interference. Correlation coefficient can describe the correlation between the membrane potential quantitatively. The correlation coefficient can be described as follows:

(8) $ \rho _{ij} (\tau )=\frac{\sum\limits _{t=t_{1} }^{t_{2} -\tau +1}x_{i} (t)x_{j} (t+\tau ) }{\sqrt{\sum\limits _{t=t_{1} }^{t_{2} -\tau +1}x_{i} {}^{2} (t)\sum\limits _{t=t_{1} }^{t_{2} -\tau +1}x_{j} {}^{2} (t+\tau ) } } $ where ${\rho }_{\mathit{ij}}\left(\tau \right)$ is correlation coefficient; $[t_1,t_2]$ is simulation duration; $x_i$ is neuron membrane potential before electric field interference; $x_j$ is neuron membrane potential after electric field interference. The larger the value of ${\rho }_{\mathit{ij}}\left(\tau \right)$ is, the better the anti-interference ability of the deep spiking neural network is. When the correlation between the membrane potential is greater than half, the influence of electric field on neural network is relatively small.

In order to investigate the influence of electric field interference on correlation between the membrane potential, the electric field interference with different amplitudes is added to deep spiking neural network. The range of the electric field amplitude A is set to be [1, 20] mV, the step length is set to be 1 mV. The correlation of the membrane potential of each layer in deep spiking neural network under different electric field amplitudes is shown in Fig. 5.

Fig. 5.

Correlation between membrane potential of each layer.

From the Fig. 5, the abscissa gives the electric field amplitude; the ordinate gives the number of layers; the color-map represents the correlation between the membrane potential of each layer. Red indicates a higher correlation between the membrane potential in each layer. Blue indicates lower correlation between the membrane potential in each layer. The figure shows that the correlation between the membrane potential in each layer decreases with increased electric field amplitude. When the electric field amplitude is in the range of 0 mV to 7 mV, the correlation between the membrane potential of each layer is relatively large. When the electric field amplitude is in the range of 7 mV to 20 mV, the correlation between the membrane potential of each layer is relatively small.

The correlation trend between the membrane potential in output layer and other layers for different electric field amplitudes A is shown in Fig. 6.

Fig. 6.

Correlation between the membrane potential of the output layer neuron and other layers.

From the Fig. 6, the abscissa gives electric field amplitude; the ordinate gives the correlation between the membrane potential of the output layer neuron and other layers. With increased electric field amplitude, the correlation between the membrane potential shows a decreasing trend of fluctuations. When the electric field amplitude is 1 mV, the correlation between the membrane potential and membrane potential in other layers attains a maximum value of 0.71. When the electric field amplitude was 20 mV, the correlation between the membrane potential between layers attained the minimum value of 0.14.

According to Fig. 6, the correspondence between membrane potential correlations and electric field amplitude A are shown in Table.1. From Table.1, with an increase in the range of electric field amplitude, the range of correlations between the membrane potential of the output layer neuron and other layers is decreased. When the electric field amplitude is in the range of 1 mV to 9 mV, the correlation between membrane potential of the different layers is relatively large, whereas, the influence of the electric field on the correlation between the membrane potential of different layers is relatively small; when the electric field amplitude is in the range of 10 mV to 20 mV, the value of correlation is relatively small, the influence of electric field on the correlation between the membrane potential is relatively large.