Hao-Ran Liu? Yu-Xin Cheng? Zhuo Zuo,2? Tian-Tian Sun ?Kai-Min Wang

Abstract Neutron and gamma ray pulse signal discrimination technology is an essential part of many modern scientific fields, such as biology, geology, radiation imaging, and nuclear medicine. Neutrons are always accompanied by gamma rays due to their unique penetration characteristic; thus, the development of n-γ discrimination methods is especially crucial. In the present study, a novel n-γdiscrimination method is proposed that implements a pulse-coupled neural network for n-γ discrimination. In addition, experiments were conducted on the pulse signals detected by an EJ299-33 plastic scintillator, which is especially suitable for n-γ discrimination. The proposed method was compared to three other discrimination methods, including the back-propagation neural network(BPNN), the fractal spectrum method, and the charge comparison method, with respect to two aspects: (i) the figure of merit (FoM) and (ii) discrimination time.

Keywords Pulse-coupled neural network ?Charge

1 Introduction

In the past 50 years, neutron detection technology has become highly crucial in many areas, such as irradiation facilities[1],reactors[2],national defense[3],biology[4],geology [5], medical science [6], and deep-space exploration [7]. Because of the unique interaction between neutrons and the surrounding environment, neutrons are commonly accompanied by gamma rays. However, most radiation detectors used to detect neutrons are also sensitive to gamma rays, which makes n-γ discrimination especially important. In 1979, Brooks et al. [8] found that neutrons and gamma rays generated different pulse shapes when they interacted with organic detectors. Based on this difference between neutrons and gamma rays, pulse shape discrimination(PSD)has been studied[9,10],many novel detectors with excellent performance have been developed[11, 12], and the feasibility of plastic scintillators for PSD applications has been studied [13–15]. Various discrimination methods and discussions of their performance have been introduced and Liu et al.presented a fractal spectrum method in 2016 [16], which has the advantages of antinoise and high discrimination but requires a very long discrimination time. In 2021, Zuo and Liu et al. compared the performances of three different discrimination methods implemented by four different filtering methods [17]; the intelligent discrimination method they used was the conventional back-propagation neural network(BPNN),which requires the training and prediction sets to be highly similar and processes the dynamic information poorly [18].

Generally, all of the commonly used n-γ discrimination methods mentioned above can be categorized into three categories:time-domain,frequency-domain,and intelligent methods. Although the discrimination time of the timedomain methods is relatively short, their discrimination performances are not satisfactory. In addition, regarding the frequency-domain and intelligent methods, although these discrimination methods generally tend to outperform time-domain methods,most require a significant amount of matrix operations,i.e.,the algorithm implementation could be relatively cumbersome and time-consuming, which is not conducive to real-time online analysis.These problems raise the question of whether it is possible to find an n-γ discrimination method that is capable of processing the dynamic information contained in the pulse signals,similar to the frequency-domain and intelligent methods while having low computational difficulty and maintaining its time consumption at the same level as the time-domain methods.

To solve this problem, we present a novel n-γ discrimination method, which is based on a pulse-coupled neural network (PCNN). Experiments were conducted to verify the performance of the proposed method, in which it was compared with the BPNN method, fractal spectrum method,and charge comparison method,with respect to the figure of merit (FoM) and discrimination time. To retrieve the data from the n-γ superposed field,a plastic scintillator(EJ299-33)and a digital oscilloscope were used,which was set with a sampling rate of 1 GS/s, a bandwidth of 200 MHz, and a trigger threshold of 500 mV. The pulse duration was 160 ns, respecting the Shanon criteria, and the bandwidth did not bury the useful signal [19]. The experimental results showed that the proposed method significantly outperformed the others, with an outstanding FoMvalue and short discrimination time consumption.

The remainder of this paper is organized as follows:the fundamentals of our proposed method and the other three discrimination methods are presented in Sect. 2. The evaluation criteria of n-γ discrimination performance are proposed in Sect. 3. The discrimination results of the different methods are compared and discussed in Sect. 4.The conclusions of the work are presented in Sect. 5.

2 Fundamentals of discrimination methods

2.1 Pulse-coupled neural network

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 Fijand Lijdenote the FI and LI of (i,j ),respectively,and coordinate (k,l ) represents one of the neighboring neurons of (i,j ); n is the iteration count; αFand αLrepresent the decay times of FI and LI, respectively; VFand VLdenote 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; Sijdenotes the external input; β is a coefficient influencing the weight of FI and LI in internal activity threshold Uij;θijrepresents the dynamic threshold,and Y is the timing pulse sequence defined by Uijand θij;Vθand αθdenote the amplification coefficient and decay time of the dynamic threshold θij, respectively. Note that the connections of the FI should be slower than those of the LI.

Intuitively, as shown in Fig. 1a, b, the internal activity threshold Uijhas a direct influence on the timing pulse sequence Y and Y can further influence the dynamic threshold θij. Meanwhile, the change in the dynamic threshold θijinfluences the state of the timing pulse sequence Y because Y is defined by Uijand θijtogether[26]. Regarding the ignition of a group of closely connected neurons, as shown in Fig. 1c, the internal activity threshold Uij(or could be seen as the total input) and the dynamic threshold θijare simultaneously augmented by multiple pulses(external input Sij).After the stimulation of multiple pulses,the growth rate of Uijbegins to slow down while that of θijremains, which means that θijwill eventually exceed Uijand reset occurs. For each neuron, the number of times Uijis greater than θijafter 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.1 (Color online)Schematic of the pulse-coupled neural network.a The relationship between Uij, Y, and θij. The internal activity threshold Uij influences the timing pulse sequence Y and Y can further influence the dynamic threshold θij; in turn, the changing dynamic threshold θij 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 Uij (or could be seen as the total input) and the dynamic threshold θij are simultaneously augmented by the multiple pulses(external input Sij).After stimulation via multiple pulses,the growth rate of Uij begins to slow while that of θij remains, indicating that θij will eventually exceed Uij 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 onedimensional 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.

2.2 Back-propagation neural network

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 Xidenotes the input of the BPNN,Ojrepresents the output of the hidden node, Ykdenotes the output of the BPNN, h is a nonlinear function, Wijand Tjkare 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.

2.3 Fractal spectrum

The fractal spectrum method discriminates n-γ signals using fractal dimensions, which are the power spectra of signals[16].Specifically,the spectrum of a signal needs to first be calculated and then the frequency of this signal and its spectrum should be presented logarithmically,retrieving an approximate linear relationship between the aforementioned operation results. Finally, a linear regression analysis is implemented to obtain regression coefficients,which can be further used to discriminate the n-γ signals. The mathematical interpretation of the fractal spectrum method is defined as:where D is the input data,N is the calculation length of the Fourier transform,FDis the data processed by the discrete Fourier transform, and Pssrepresents the power spectrum.Based on Eqs.(8),(9),and(10),the power spectral density function can be calculated. Then, the logarithm of this function is taken and further fitted by formula (11), where G w( )denotes the power spectral density of the signal,w0is the reference space frequency, w represents the spatial frequency, and G(w0) denotes the signal conversion coefficient.Finally,the fractal dimension is defined as follows:Dim=2?5-α? (12)

2.4 Charge comparison

The charge comparison method [31]has become one of the most commonly used n-γ discrimination methods,which has a fine n-γ discrimination performance and consumes little time for simultaneous discrimination processing. With regard to its internal principle, owing to the different interaction characteristics between neutrons and gamma rays when they penetrate the plastic scintillator,the ratios of the slow component’s charge to the total charge are different for the pulse signals of neutrons and gamma rays.Based on the aforementioned characteristics,Hawkes et al. calculated the charge ratio R to discriminate n-γ signals [32], defined as follows:

where QNand QMrepresent the summations of the pulse voltage of the slow component and the whole signal,respectively. As a result of the longer decay time and delayed fluorescence of neutrons,the R-value is larger than that of gamma rays.

2.5 Filtering methods

Inevitably, the pulse signals detected by the radiation detector always incorporate noise, which makes the difference between the pulse signals of neutrons and gamma rays much less significant. Thus,the performance of direct n-γ discrimination is affected by the useless noise, making the preprocessing of the pulse signals especially important.There are many preprocessing methods,also called filtering methods, which are commonly used in radiation detection,such as the Fourier transform [33–35], wavelet transform[36–38],Kalman filter[39],and sliding average filter[40].A discrimination method shows different discrimination effects with different filtering methods; we have demonstrated in our previous work [17] that each discrimination method has the best-matched filtering method, with which it can achieve optimized performance.

In the present work, the discussion mainly regards the performance of different discrimination methods; hence,only one is uniformly used as the filtering method for all discrimination methods.In this work,the Fourier transform is selected as the filtering method due to its generally good performance when coupled with most discrimination methods [17] and its common application in radiation detection.

After implementing the Fourier transform, the pulse signal is transformed from the time-domain to the frequency-domain, where the low-amplitude frequency-domain information is the noise component. Filtering is realized by removing the noise component in the frequency-domain and further taking the inverse Fourier transform to obtain the pulse signal back to the time-domain. As a result of the discrete sampling process being required when the pulse signals are retrieved from the n-γ superposed field, the pulse signals are discrete; hence, a discrete Fourier transform(DFT)is required to filter the n-γ pulse signals. The mathematical transform formulas are as follows:

where Eq. (14) is the positive transform formula for the DFT, Eq. (15) is the inverse transform formula for the DFT, ωN=e (-2πi)/N denotes the frequency, G(u ) represents the frequency-domain signal, g(x ) denotes the timedomain signal, and N is the length of a signal.

3 Evaluation criteria

To evaluate the performance of different n-γ discrimination methods, objective evaluation criteria need to be used, which are the discrimination time (the CPU time required to process all given pulse signals)and the figure of merit (FoM) in this work. Regarding calculation of the FoM-value, the histogram of counts of neutrons and gamma rays must be drawn after the discrimination process, as shown in Fig. 2. Then, the mathematical formula of FoM [41] is given as follows:

Fig. 2 (Color online) n-γ discrimination evaluation criteria. The evaluation criteria FoM-value is defined by Eq. (14), incorporating three parts that are marked in this figure(S,FWHMn,and FWHMγ).A greater FoM-value corresponds to a better discrimination effect

where S denotes the distance between the neutron count and gamma ray count peaks,FWHMnand FWHMγare the half-height widths of the neutron counting peak and gamma ray counting peak, respectively, estimated by Gaussian fitting. A larger FoM-value corresponds to a better discrimination effect.

4 Experiment

4.1 Parameter settings

4.1.1 PCNN

4.1.2 Other methods

To train the BPNN,11,454 well-discriminated and lownoise 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 to 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 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].

Table 1 Discrimination time and FoM value of different discrimination methods

4.2 Experimental results and discussions

4.2.1 Pulse signals of neutrons and gamma rays

In this work, a241Am-Be isotope neutron source was used to generate the n-γ superposed field, with an average energy of 4.5 MeV. The detection equipment was composed of two parts: an EJ299-33 plastic scintillator and a digital oscilloscope with a 500 mV trigger threshold(which corresponds to an energy of approximately 1.6 MeVee,where 1 MeVee (i.e., MeV electron equivalent) is defined as the amount of energy converted to light induced by an electron, which deposits 1 MeV in the scintillator), a 200 MHz bandwidth,and a 1 GS/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, 5.

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.

Fig. 4 (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. 5 (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

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.

5 Conclusion

In this work, a PCNN-based n-γ discrimination method is proposed and compared with the other three methods,including the BPNN, fractal spectrum method, and charge comparison method.Our experiment was conducted on the data of an EJ299-33 plastic scintillator, by which 9414 n-γ pulse signals were retrieved and further processed on an AMD R9-5900X CPU. The experimental results show that our proposed method significantly outperforms the others,with the best FoM-value and short discrimination time.Specifically, the PCNN demonstrates a 26.49% improvement in the FoM-value compared to the charge comparison method, a 72.80% improvement compared to the BPNN,a 66.24% improvement compared to the fractal spectrum method, and the second-fastest discrimination time of 2.22 s.This outstanding performance is because 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.This short discrimination time of the PCNN is due to its simple computational complexity, with few matrix operations and without the need for a pre-training step. The distinguished n-γ discrimination effect of the PCNN enables broad application prospects in the radiation detection field and its relatively short discrimination time makes it potentially useful for real-time discrimination.To the best of our knowledge, this is the first report on applying a pulse-coupled neural network for n-γ discrimination. In our future work, we will verify the feasibility of other improved PCNN models, aiming to achieve an even better discrimination effect and shorter discrimination time.

Author contributionsAll authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Hao-Ran Liu, Yu-Xin Cheng, Zhuo Zuo, Tian-Tian Sun, and Kai-Min Wang. The first draft of the manuscript was written by Hao-Ran Liu, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.