2.1.1 Origin and mathematical expression

The pulse-coupled neural network (PCNN) was introduced by Eckhorn and Reitboeck et al. in 1990 [20], which works similarly to real biological neurons, does not require cumbersome work to train the neural network before using it, and, hence, does not require the tested data to be highly similar to the trained data. Based on the neurophysiological findings of the cat’s primary visual cortex [21,22], they developed a neural network whose central novel principle was abandoning the conventional receptive field and instead introducing a secondary receptive field, the linking field. The integrated input of the aforementioned linking field has an internal cellular circuit, which is capable of modulating the primary feeding receptive field input, influencing the generation of pulse bursts. For a signal fed to the PCNN, which has multiple sampling points — usually, an image [23,24] — the unique character of the linking field makes it possible to capture the dynamic properties contained in the signal and retrieve the inherent relationships between neighboring sampling points, which is especially crucial for processing signals whose dynamic information is as important as, if not more important than, the information of each sampling point.

The model of the PCNN incorporates the accepted, modulation, and pulse generator domains. The accepted domain has two components: feedback input (FI) and link input (LI), which together determine the potential of a neuron. When the potential of a neuron exceeds its dynamic threshold, it is considered activated. The mathematical formulas of the PCNN are as follows [25]:

Where F_ij and L_ij denote the FI and LI of $\left(i,j\right)$ respectively and coordinate $\left(k,l\right)$ represents one of the neighboring neurons of $\left(i,j\right)$; n is the iteration count; α_F and α_L represent the decay times of FI and LI, respectively; V_F and V_L denote the amplification coefficients of FI and LI, respectively; M is the connection weight matrix, which represents the extent of influence that central neurons received from surrounding neurons; S_ij denotes the external input; β is a coefficient influencing the weight of FI and LI in internal activity threshold U_ij; θ_ij represents the dynamic threshold, and Y is the timing pulse sequence defined by U_ij and θ_ij; V_θ and α_θ denote the amplification coefficient and decay time of the dynamic threshold θ_ij, respectively. Note that the connections of the FI should be slower than those of the LI.

Intuitively, as shown in Figs. 1a and 1b, the internal activity threshold U_ij has a direct influence on the timing pulse sequence Y and Y can further influence the dynamic threshold θ_ij. Meanwhile, the change in the dynamic threshold θ_ij influences the state of the timing pulse sequence Y because Y is defined by U_ij and ${\theta }_{ij}$ together [26]. Regarding the ignition of a group of closely connected neurons, as shown in Fig. 1-c, the internal activity threshold U_ij (or could be seen as the total input) and the dynamic threshold θ_ij are simultaneously augmented by multiple pulses (external input S_ij). After the stimulation of multiple pulses, the growth rate of U_ij begins to slow down while that of θ_ij remains, which means that θ_ij will eventually exceed U_ij and reset occurs. For each neuron, the number of times U_ij is greater than θ_ij after stimulation by multiple pulses is defined as the ignition time, which is the value of the corresponding location of this neuron in the ignition map.

Fig. 1Full-screen View

(Color online) Schematic of the pulse coupled neural network. (a) The relationship between U_ij, Y, and θ_ij. The internal activity threshold U_ij influences the timing pulse sequence Y and Y can further influence the dynamic threshold θ_ij; in turn, the changing dynamic threshold θ_ij influences the state of the timing pulse sequence Y. (b) The basic PCNN model. (c) The ignition situation of a neuron stimulated by multiple pulses. The internal activity threshold U_ij (or could be seen as the total input) and the dynamic threshold θ_ij are simultaneously augmented by the multiple pulses (external input S_ij). After stimulation via multiple pulses, the growth rate of U_ij begins to slow while that of θ_ij remains, indicating that θ_ij will eventually exceed U_ij and reset occurs.

2.1.2 Implementation on n-γ discrimination

PCNN has a tremendous number of applications in image processing [26] but has never been used in the PSD field. Similar to the information contained in an image, the information contained in a pulse signal also has dynamic properties, which are essential to the discrimination process. The distinguished performance of the PCNN in processing dynamic information enables a computer to capture the dynamic characteristics of pulse signals and perform outstanding n-γ discrimination. Specifically, the major difference between neutron and gamma-ray signals is in the falling edge and delayed fluorescence and the detection process of this difference does not only concern the amplitude of each sampling point of the falling edge and afterglow effect peak but is also related to the changing rate of the amplitudes in these locations, which is the aforementioned dynamic information one wants to capture, used to discriminate neutron and gamma-ray signals.

When the PCNN is applied in neutron and gamma-ray signal processing, the connection weight matrix M is a one-dimensional vector because the pulse signals of neutrons and gamma rays are one-dimensional. After implementing the PCNN on a pulse signal, an ignition map was obtained, i.e., a matrix with the same dimensions as the original signal. By summing up part of the ignition map that contains the information of a signal’s peak and falling edge, the neutrons and gamma rays can be discriminated. Notice that, unlike other commonly used neural networks, the PCNN requires no training process before the discrimination process, making its computational complexity and time consumption much less than other conventional neural networks.

As shown in Fig. 3, the difference in the falling edge of neutrons and gamma rays is well detected by PCNN, exhibiting significant differences in the ignition maps of the n-γ signals. For the pulse signal of the gamma ray, the ignition times in the part of the falling edge decrease very quickly, which is attributed to the fast decay time of gamma-ray pulse signals. Conversely, the declining speed of the ignition times in the part of the falling edge is noticeably slower for the neutron signals than that of the gamma-ray pulses and the ignition times surge again when it comes to the part of delayed fluorescence. Consequently, by summing the ignition times of the partial ignition map that incorporates the rising edge, falling edge, and delayed fluorescence, the n-γ pulse signals could be well discriminated, granted that the sum of ignition times of neutron signals is distinctly larger than that of gamma-ray signals.

Fig. 3Full-screen View

(Color online) Comparison of normalized n–$\gamma$ pulses and their ignition maps. (a) The pulse signals of neutron and gamma-ray. (b) The ignition maps of neutron and gamma-ray, which are obtained by implementing PCNN on n-γ pulse signals. The difference between neutron and gamma-ray in (b) is similar but more distinct than that in (a), demonstrating that the inherent differences between n-γ pulses are successfully detected by the PCNN.

2.2.1 Model and mathematical expression

The back-propagation neural network (BPNN) is a product of artificial intelligence research based on the back-propagation algorithm, which is capable of learning from the examples fed to it [27]. Specifically, adapting the connection weights of the hidden neurons of the BPNN can provide an accurate approximation of the relationship between the training set and corresponding results; sometimes, even the aforementioned relationship is nonlinear [28]. With regard to the BPNN, the mathematical relationships between the inputs and outputs are not specified and the learning process has two steps: forward propagation of the signal and backward propagation of error. Specifically, when the output of the BPNN differentiates from the expected output, defined by the training set, the backward propagation stage begins to adjust the inherent relationships of the hidden node. Mathematically, the hidden node output model and output node output model are denoted as follows:

Where X_i denotes the input of the BPNN, O_j represents the output of the hidden node, Y_k denotes the output of the BPNN, h is a nonlinear function, W_ij and T_ik are weight matrices that are adjusted during backward propagation, and q represents the neural unit threshold.

2.2.2 Implementation on n-γ discrimination

In recent decades, back-propagation neural networks have been fully developed to address many civil engineering problems. The final objective of applying n-γ discrimination [29,30] is to determine the difference between neutron and gamma-ray pulse signals, by which an inherent relationship between a pulse signal and its category can be generalized and further used to discriminate the n-γ pulse signals fed to the BPNN in the future. To realize the aforementioned aim, the BPNN needs to be trained first using two sets of already well-discriminated neutron and gamma-ray pulse signals. In this training step, the BPNN tries to categorize the received pulse signals (the input patterns), with the recursive adjustment aided by the category results of each pulse signal (the output patterns). When the neural network has completed the training step, it can be used to process the test set and discriminate the pulse signals.

However, there are a few drawbacks of n-γ discrimination based on BPNN. The training and test sets need to be highly similar and hence these sets cannot use different filtering methods and the pulse signals of these two sets need to be retrieved from the same radiation detection device and come from the same radiation source, with the same average energy. These disadvantages restrict the general application of BPNN for n-γ discrimination.

4.1.1 PCNN

The parameters of the PCNN are set as: n=180, α_F=0.32, α_L=0.356, α_θ=0.08, V_F=0.0005, V_L=0.0005, V_θ=15, and linking weight M=[0.1409,0,0.1409]. The n-γ signals processed by the PCNN using the aforementioned parameters are shown in Fig. 3b. The ignition maps of pulse signals between 10 ns before the peak and 20 ns after the peak were selected as the discrimination intervals.

4.1.2 Other methods

To train the BPNN, 11,454 well-discriminated and low-noise signals were selected as the training set. There are two hidden layers of the BPNN used in our work, each of which has nine neurons. The entire training process took approximately 0.2 s, which is included in the total time consumption of BPNN in Table 1. With regard to the fractal spectrum method, the pulse signals from 38 – 130 ns after the peak were chosen as the discrimination interval. In the charge comparison method, the total component is composed of pulse signals between 15 ns before the peak and 200 ns after the peak and the slow component incorporates pulse signals between T and 200 ns after the peak. T is delay time, i.e., a parameter between 155 ns and 185 ns that affects the FoM-value of the charge comparison method. The abovementioned parameters of these three methods were optimized in our previous work [17].

4.2.1 Pulse signals of neutrons and gamma-rays

In this work, a ²⁴¹Am-Be isotope neutron source was used to generate the n-$\gamma$ superposed field, with an average energy of 4.5 $\text{MeV}$. The detection equipment was composed of two parts: an EJ299-33 plastic scintillator and a digital oscilloscope with a 500 $\text{mV}$ trigger threshold (which corresponds to an energy of approximately 1.6 $\text{MeVee}$, where 1 $\text{MeVee}$ (i.e., $\text{MeV}$ electron equivalent) is defined as the amount of energy converted to light induced by an electron, which deposits 1 $\text{MeV}$ in the scintillator), a 200 $\text{MHz}$ bandwidth, and a 1 $\text{GS}/\text{s}$ sampling rate. Based on this setup, 9414 pulse signals were retrieved. A comparison of the normalized neutron and gamma-ray signals that we retrieved is shown in Fig. 3a. It can be seen that the rise speeds of both signals are similar, while the luminous attenuation rate of the neutron is noticeably slower than that of gamma rays; this is attributed to its longer decay time and delayed fluorescence [42].

4.2.2 Discrimination results

The 9414 n-γ pulse signals were filtered by the Fourier transform [34] and then discriminated by the aforementioned methods (processed on an AMD R9-5900X CPU); the results are shown in Figs. 4 and 5.

Fig. 4Full-screen View

(Color online) Scatter plot of pulse signals discriminated by different methods. The dots above the crossing line are discriminated as neutron signals and the dots below the crossing line are discriminated as gamma-ray signals. Well-separated neutron dots and gamma-ray dots indicate a good discrimination effect.

Fig. 5Full-screen View

(Color online) Gauss fitting curves for the histogram of four discrimination methods. A longer distance between neutrons and gamma-ray counting peaks or narrower and higher n-γ counting peaks indicates a better discrimination performance.

Figure 4 shows the scatter plot of pulse signals discriminated by different methods, where the dots above the crossing line are discriminated as neutron signals and the dots below the crossing line are discriminated as gamma-ray signals. For a good discrimination result, the group of neutron dots and the group of gamma-ray dots should be separated and each group should be distributed as centrally as possible while maintaining a good Gaussian distribution at the same time. It can be clearly seen that the PCNN method noticeably outperforms the others, with a clear gap between the neutron and gamma-ray dots. Meanwhile, both groups were centrally distributed, with very few dots located between the two groups or discrete on the outer sides.

Figure 5 shows the Gaussian fitting of the histogram of the four methods. The discrimination effect is reflected in two aspects: (i) the length of the gap between the two peaks and (ii) the shape of the two peaks. A wider gap corresponds to better discrimination. The narrower and higher neutron peak (on the right side) and gamma peak (on the left side) represent better discrimination performance. As shown in Fig. 5, the fitting curve of the PCNN method exhibits better performance than that of the other methods, with a longer distance between neutrons and gamma-ray counting peaks and has narrower and higher n-γ counting peaks.

To quantitatively evaluate the performance of these methods, the FoM-value of their discrimination results were calculated, as presented in Table 1, along with the discrimination time (for the processing time of the 9414 pulse signals) of each method.

Table 1 demonstrates that our proposed method significantly outperforms the others, with a 26.49% improvement in the FoM-value compared to the charge comparison method, a 72.80% improvement compared to the BPNN, and a 66.24% improvement compared to the fractal spectrum method. This outstanding discrimination performance of the proposed method is because the PCNN is not based on every single point of a fixed vector of the input signal, as in other methods. In contrast, the PCNN not only considers the amplitude of a point but also the amplitudes of points before and after this location; i.e., the PCNN treats the input signal as a whole for analysis and processing, imparting an excellent anti-noise effect and the ability to process the dynamic information contained in a pulse signal. In addition, the discrimination time of our proposed method is quite satisfactory (2.22 s), taking slightly longer than the charge comparison method (1.96 s) and outperforming the BPNN (3.65 s) and fractal spectrum method (178.01 s). As a neural network, the PCNN takes such a short time compared to the BPNN and the fractal spectrum is attributed to its simple computational complexity, with fewer matrix operations compared to the fractal spectrum and without the need for a pre-training step compared to the BPNN. This experimental result shows that the PCNN has the potential to be implemented in real-time applications.