Anti-noise performance of the pulse coupled neural network applied in discrimination of neutrons and gamma-rays

Anti-noise performance of the pulse coupled neural network applied in discrimination of neutrons and gamma-rays 1 2 first-author https://orcid.org/0000-0003-0729-4526 1 2 3 corresp zuozhuo01@gmail.com (Zhuo Zuo) zuozhuo01@gmail.com (Zhuo Zuo) zuozhuo01@gmail.com https://orcid.org/0000-0003-2341-177X 2 https://orcid.org/0000-0001-7424-6618 4 2 3 2 3 State Key Laboratory of Geohazrd Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China State Key Laboratory of Geohazrd Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China The Engineering & Technical College of Chengdu University of Technology, Leshan Sichuan 614000, China The Engineering & Technical College of Chengdu University of Technology, Leshan Sichuan 614000, China Southwestern Institute of Physics, Chengdu Sichuan 610225, China Southwestern Institute of Physics, Chengdu Sichuan 610225, China Chengdu University, Chengdu Sichuan 610106, China Chengdu University, Chengdu Sichuan 610106, China In this study, the anti-noise performance of a pulse-coupled neural network (PCNN) was investigated in the neutron and gamma-ray (n- <math id="M1"><mi>γ</mi></math> ) discrimination field. The experiments were conducted in two groups. In the first group, radiation pulse signals were pre-processed using a Fourier filter to reduce the original noise in the signals, whereas in the second group, the original noise was left untouched to simulate an extremely high-noise scenario. For each part, artificial Gaussian noise with different intensity levels was added to the signals prior to the discrimination process. In the aforementioned conditions, the performance of the PCNN was evaluated and compared with five other commonly used methods of n- <math id="M2"><mi>γ</mi></math> discrimination: (1) zero crossing, (2) charge comparison, (3) vector projection, (4) falling-edge percentage slope, and (5) frequency gradient analysis. The experimental results showed that the PCNN method significantly outperforms other methods with outstanding <math id="M3"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value at all noise levels. Furthermore, the fluctuations in <math id="M4"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value of PCNN were significantly better than those obtained via other methods at most noise levels and only slightly worse than those obtained via the charge comparison and zero-crossing methods under extreme noise conditions. Additionally, the changing patterns and fluctuations of the <math id="M5"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value were evaluated under different noise conditions. Hence, based on the results, the parameter selection strategy of the PCNN was presented. In conclusion, the PCNN method is suitable for use in high-noise application scenarios for n- <math id="M6"><mi>γ</mi></math> discrimination because of its stability and remarkable discrimination performance. It does not rely on strict parameter settings and can realize satisfactory performance over a wide parameter range. Pulse Coupled Neural Network Zero Crossing Frequency Gradient Analysis Vector Projection Charge Comparison n-γ discrimination Pulse Shape Discrimination Introduction 1 Given that the demand for neutron detection has significantly increased in recent years, neutron monitoring has become vital in numerous fields such as deep-space exploration [ 1 1 ], reactors [ 2 2 , 3 3 ], radiopharmaceuticals [ 4 4 ], geology [ 5 5 ], national security [ 6 6 , 7 7 ], and meteorology [ 8 8 ]. One of the essential challenges in neutron detection is the presence of accompanying gamma rays. They are generated by the interaction, i.e., inelastic scattering and radiation capture, between the neutrons and the surrounding environment. Hence, typically, an extensive gamma background is present wherever neutrons exist [ 9 9 ]. For most radiation detectors, the incident neutrons and gamma rays are simultaneously recognized, and it is difficult to differentiate the signal coming from neutrons or gamma rays. To distinguish these signals, pulse-shape discrimination (PSD) was developed [ 10 10 , 11 11 ], which is based on the differences in [ 12 12 ] the shapes of neutrons and gamma-rays pulse signals (n- <math id="M7"><mi>γ</mi></math> PSs). Therefore, discrimination techniques rely heavily on scintillator materials, and many scintillators with cutting-edge discrimination performance have been developed [ 13 13 - 15 15 13-15 ]. It should be noted that most organic scintillators exhibit similar decay characteristics, but other scintillators exhibit different characteristics such as CLYC. Among PSD-capable scintillators, plastic scintillators are more commercially friendly, convenient to transform, and adaptive to different work conditions; however, their discrimination capabilities are limited. Conversely, liquid and crystal [ 16 16 ] scintillators commonly outperform plastic scintillators in terms of discrimination, although they are more expensive. Furthermore, liquid scintillators are difficult to store and transport. Plastic scintillators are preferred in many scenarios due to their application requirements and cost control. However, many broadly used discrimination methods, such as charge comparison [ 17 17 ] and the zero-crossing method [ 18 18 ], perform poorly in plastic scintillators when compared to liquid scintillators. Consequently, a discrimination method capable of realizing a higher performance in plastic scintillators is required. In 2021, Liu et al. resolved this problem by proposing a pulse-coupled neuron network (PCNN)-based discrimination method [ 19 19 ], which exhibits a remarkable discrimination effect when applied to n- <math id="M8"><mi>γ</mi></math> PS data acquired for a plastic scintillator. This method can recognize dynamic information inside n- <math id="M9"><mi>γ</mi></math> PSs and has the advantage of no pre-training process requirements. Furthermore, the anti-noise ability was mentioned in their study, but they did not further validate the anti-noise performance of the PCNN applied in n- <math id="M10"><mi>γ</mi></math> discrimination. Although many studies have been performed to control the noise in various detection systems [ 20 20 - 22 22 20-22 ], it is still an inevitable problem in any radiation detection system; hence, it is crucial for discrimination methods to maintain a stable performance under the influence of noise. In the present study, experiments were conducted to investigate the noise immunity of the PCNN. The n- <math id="M11"><mi>γ</mi></math> mixed field, used in the study, was generated by a 4.5-MeV (mean energy) 241 Am-Be neutron source, and n- <math id="M12"><mi>γ</mi></math> PSs were measured from this field via an EJ299-33 plastic scintillator and a 9821B photomultiplier. Different levels of Gaussian noise were added to the pulse shapes prior to the discrimination process. The experimental results of the PCNN were compared with the other five commonly used discrimination methods under different noise conditions, namely, zero crossing (ZC), charge comparison (CC), vector projection (VP), falling edge percentage slope (FEPS), and frequency gradient analysis (FGA) methods. Additionally, the influence of the PCNN parameters on its anti-noise effect was elucidated. Based on this, a parameter decision strategy was presented for high-noise application scenarios. The remainder of this paper is organized as follows. In Sect. 2, several n- <math id="M13"><mi>γ</mi></math> discrimination methods are introduced. In Sect. 3, evaluation criteria for n- <math id="M14"><mi>γ</mi></math> discrimination are defined. In Sect. 4, details of the experimental design and results are presented, and the characteristics of the PCNN parameters are analyzed. Finally, conclusions of this study are presented in Sect. 5. Principles of discrimination methods 1 Zero Crossing 2 The zero-crossing (ZC) method is one of the most used methods in the n- <math id="M15"><mi>γ</mi></math> discrimination field based on a simple methodology, while it offers reliable discrimination results [ 18 18 , 23 23 ]. To discriminate the pulse signals using this method, n- <math id="M16"><mi>γ</mi></math> PS must be first transformed into a bipolar pulse signal. Second, the so-called zero-crossing time should be calculated by finding the time interval between the beginning of the n- <math id="M17"><mi>γ</mi></math> PS and zero-crossing point of the bipolar pulse signal, which are set as 10% of the pulse maximum and the first sample after the pulse's peak that crosses the baseline, respectively. Finally, the neutrons and gamma rays can be discriminated by comparing the zero-crossing times of different n- <math id="M18"><mi>γ</mi></math> PSs. Specifically, a digital <math id="M19"><mrow><mi>C</mi><mi>R</mi><mo>−</mo><mi>R</mi><msup><mi>C</mi><mn>2</mn></msup></mrow></math> filter was used for the transformation from the original n- <math id="M20"><mi>γ</mi></math> PS to the bipolar pulse signal, whose mathematical expression is as follows [ 24 24 ]: <math display="block" id="M21"><mtable><mtr><mtd><mi>y</mi><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mn>3</mn><mi>δ</mi><mi>y</mi><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>−</mo><mn>3</mn><msup><mi>δ</mi><mn>2</mn></msup><mi>y</mi><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>]</mo></mrow><mo>+</mo><msup><mi>δ</mi><mn>3</mn></msup><mi>y</mi><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mi>T</mi><mi>δ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ω</mi><mi>T</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mi>x</mi><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>−</mo><mi>T</mi><msup><mi>δ</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>ω</mi><mi>T</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mi>x</mi><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>]</mo></mrow><mo>,</mo></mtd></mtr></mtable></math> where <math id="M22"><mi>y</mi></math> denotes the bipolar pulse signal obtained by this filter process, <math id="M23"><mi>x</mi></math> denotes the original n- <math id="M24"><mi>γ</mi></math> PS, <math id="M25"><mi>n</mi></math> denotes the sample index, <math id="M26"><mi>δ</mi></math> and <math id="M27"><mi>ω</mi></math> denote constants defined by: <math display="block" id="M28"><mrow><mi>δ</mi><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac bevelled="true"><mi>T</mi><mi>τ</mi></mfrac></mrow></msup><mo>,</mo></mrow></math> <math display="block" id="M29"><mrow><mi>ω</mi><mo>=</mo><mfrac><mn>1</mn><mi>τ</mi></mfrac><mo>,</mo></mrow></math> where <math id="M30"><mrow><mi>τ</mi><mo>=</mo><mi>R</mi><mi>C</mi></mrow></math> denotes the shaping time and <math id="M31"><mi>T</mi></math> denotes the sampling interval of the pulse signal. Furthermore, <math id="M32"><mi>τ</mi></math> and <math id="M33"><mi>T</mi></math> were determined using the detection system and its settings. Given the slower decay speed of neutron pulse signals (organic scintillators), the zero-crossing time of neutrons is longer than that of gamma rays. Charge Comparison 2 As the charge comparison (CC) method exhibits outstanding discriminating efficiency and stability, it has been widely used in many areas that require the n- <math id="M34"><mi>γ</mi></math> discrimination technique [ 17 17 ]. A criterion termed as charge ratio <math id="M35"><mrow><msub><mi>R</mi><mtext>c</mtext></msub></mrow></math> is used to transmit the pulse shape information of neutrons and gamma rays, which can be calculated as follows [ 25 25 ]: <math display="block" id="M36"><mrow><msub><mi>R</mi><mtext>c</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>Q</mi><mtext>N</mtext></msub></mrow><mrow><msub><mi>Q</mi><mtext>M</mtext></msub></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M37"><mrow><msub><mi>Q</mi><mtext>N</mtext></msub></mrow></math> denotes the charge of the slow component, which can be calculated by integrating the amplitude of the falling edge and delayed fluorescence parts of a pulse signal, <math id="M38"><mrow><msub><mi>Q</mi><mtext>M</mtext></msub></mrow></math> denotes the charge of the entire signal, which is defined as the amplitude integration of all sampling points of a given pulse signal. Given that the decay speed of a pulse signal generated by a neutron is slower than that of a gamma-ray photon, along with the effect of the delayed fluorescence that is characteristic of neutron pulses, the slow component charge <math id="M39"><mrow><msub><mi>Q</mi><mtext>N</mtext></msub></mrow></math> due to neutrons is significantly larger than that of gamma-ray signals. Hence, the <math id="M40"><mrow><msub><mi>R</mi><mtext>c</mtext></msub></mrow></math> -value for the neutrons is higher than that for the gamma rays because of the differences between the n- <math id="M41"><mi>γ</mi></math> PSs. Vector Projection 2 In the vector projection (VP) method [ 26 26 , 27 27 ], the pulse signals of different particles are considered as vectors pointing in different directions in a vector space. First, n- <math id="M42"><mi>γ</mi></math> PSs were normalized. Second, two ideal pulse signals corresponding to neutron and gamma rays are defined. They are typically obtained by averaging a significant number of respective n- <math id="M43"><mi>γ</mi></math> PSs. Then, every single vector (i.e., n- <math id="M44"><mi>γ</mi></math> PS) is projected in the projection direction, which is defined as the difference between ideal pulses. The projection results of neutrons and gamma rays differ and act as the discrimination factor. The projection results of neutrons are generally larger or smaller than those of gamma rays based on whether the projection direction is positive or negative with respect to the vector space. In this study, the projection results of gamma rays are set as smaller than those of neutrons to maintain the same distribution of the neutron and gamma peaks in the histogram of counts (Section 3) as those for other discrimination methods used in this study (i.e., the neutron peak on the right side with larger discrimination factors and the gamma-ray peak on the left side with smaller discrimination factors). Falling Edge Percentage Slope 2 The falling edge percentage slope (FEPS) method [ 27 27 , 28 28 ] aims to realize fast real-time n- <math id="M45"><mi>γ</mi></math> discrimination. To realize discrimination using this method, a region of interest (ROI) must be selected, which is located in the region with the most significant differences between n- <math id="M46"><mi>γ</mi></math> PSs in the falling-edge area. The ROI is specified by setting two thresholds on the top and bottom, termed as Above A threshold (AAT) and Below A threshold (BAT), respectively. The analysis includes finding the intersections between a n- <math id="M47"><mi>γ</mi></math> PS and these two thresholds: intersection <math id="M48"><mtext>Φ</mtext></math> at coordinates <math id="M49"><mrow><mrow><mo>(</mo><mrow><msub><mtext>Φ</mtext><mi>x</mi></msub><mo>,</mo><msub><mtext>Φ</mtext><mi>y</mi></msub></mrow><mo>)</mo></mrow></mrow></math> for AAT and intersection <math id="M50"><mrow><mtext>Ψ </mtext><mrow><mo>(</mo><mrow><msub><mtext>Ψ</mtext><mi>x</mi></msub><mo>,</mo><msub><mtext>Ψ</mtext><mi>y</mi></msub></mrow><mo>)</mo></mrow></mrow></math> for BAT. Finally, the discrimination factor, i.e., the <math id="M51"><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></math> , is calculated as follows: <math display="block" id="M52"><mrow><msub><mi>R</mi><mtext>s</mtext></msub><mo>=</mo><mfrac><mrow><msub><mtext>Ψ</mtext><mi>y</mi></msub><mo>−</mo><msub><mtext>Φ</mtext><mi>y</mi></msub></mrow><mrow><msub><mtext>Ψ</mtext><mi>x</mi></msub><mo>−</mo><msub><mtext>Φ</mtext><mi>x</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> Usually, the BAT is a constant and set as 10% of the maximum of a n- <math id="M53"><mi>γ</mi></math> PS, whereas the AAT is a manually determined parameter that varies from 30% to 90% of the maximum. As gamma-ray pulses exhibit a faster decline speed, the absolute value of their <math id="M54"><mrow><msub><mi>R</mi><mtext>s</mtext></msub></mrow></math> is considerably larger than those of neutrons. Additionally, the <math id="M55"><mrow><msub><mi>R</mi><mtext>s</mtext></msub></mrow></math> -values of n- <math id="M56"><mi>γ</mi></math> PSs are negative; hence, in the histogram of counts of n- <math id="M57"><mi>γ</mi></math> PSs mentioned in Sect. 3, the gamma-ray band is located on the left side of the neutron band. Frequency Gradient Analysis 2 The frequency gradient analysis (FGA) method is a frequency-domain-based method with theoretically specialized anti-noise ability as proposed in 2010 [ 29 29 ]. When applying it to n- <math id="M58"><mi>γ</mi></math> PSs, a radiation pulse signal must be transformed into the frequency domain using the Fourier transform. In contrast to the original time-domain signal, in which the most significant differences between n- <math id="M59"><mi>γ</mi></math> PSs are located at the falling-edge area, the frequency-domain signal exhibits distinct differences between n- <math id="M60"><mi>γ</mi></math> PSs at the beginning of the signal. The amplitudes of the transformed n-γ frequency-domain signals at zero frequency, equal to the average values of the entire time-domain pulse signals of the neutron and gamma ray, respectively, significantly differ from each other. Subsequently, two points in the early parts of a frequency-domain signal are selected to calculate the gradient value, R g , which is further used in the discrimination process. The mathematical formula for the discrimination factor, termed as the frequency gradient, is defined as follows: <math display="block" id="M61"><mrow><msub><mi>R</mi><mi>g</mi></msub><mo>=</mo><mfrac><mrow><mrow><mo>|</mo><mrow><mi>X</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>−</mo><mrow><mo>|</mo><mrow><mi>X</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mi>f</mi></mfrac><mo>,</mo></mrow></math> where <math id="M62"><mrow><mi>X</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></math> denotes the frequency-domain signal transformed from an original time-domain n- <math id="M63"><mi>γ</mi></math> PS using the Fourier transform, <math id="M64"><mi>f</mi></math> denotes the frequency, which is located at the initial parts of the frequency-domain signal mentioned earlier. The <math id="M65"><mrow><msub><mi>R</mi><mtext>g</mtext></msub></mrow></math> -value of neutrons is greater than that of gamma rays because of the faster decrease in the frequency domain signals of the neutrons. Pulse Coupled Neural Network 2 A pulse-coupled neural network (PCNN) was introduced into the field of n- <math id="M66"><mi>γ</mi></math> discrimination by Liu et al. in 2021 [ 19 19 ]. It has been shown to exhibit a significantly better performance than almost all the previous discrimination techniques. This is attributed to its ability to recognize and capture the dynamic information in n- <math id="M67"><mi>γ</mi></math> PSs, which is crucial, if not the most important, information for the discrimination process. This characteristic is inherited from the original structural design and application of the PCNN. Initially, the PCNN was inspired by the biological neuronal cortex of animals. The biological neurons of the visual cortex of animals receive stimulation from their eyes, leading to spike generation and transmission between cell assemblies in the cortex [ 30 30 , 31 31 ]. This interaction between cell assemblies can recognize and analyze the information contained in the stimuli [ 32 32 ] wherein the information is carried by the images that are observed by the eyes of the animals. Due to evolution, this working style of neurons is extremely effective in dealing with dynamic information in side images. Based on these neurological findings, Eckhorn et al. proposed an artificial cortical model [ 33 33 ] that enables computers to obtain parts of the dynamic information processing ability of biological neurons. A few years later, in 1994, Johnson et al. designed a PCNN model based on Eckhorn's original cortical model for the image-processing area [ 34 34 , 35 35 ]. Since then, PCNN has been widely used in many image processing applications such as image shadow removal [ 36 36 ], feature extraction [ 34 34 ], pattern recognition [ 32 32 ], image segmentation [ 35 35 ], and object recognition [ 37 37 , 38 38 ]. The structural design of the PCNN incorporates three closely connected parts: accepted, modulation, and pulse generator domains [ 39 39 ]. First, the link input (LI) and feedback input (FI) constitute the accepted domain, and they are both modulated by the surrounding neurons via weighting matrices <math id="M68"><mi>M</mi></math> and <math id="M69"><mi>W</mi></math> . The LI is responsible for the stimulus provided by the surrounding neurons, whereas the input signal's outer stimulus <math id="M70"><mi>S</mi></math> mainly affects the FI. Second, the modulation domain controls the relationship between the internal threshold and internal activity by changing the character of the dynamic threshold <math id="M71"><mi>θ</mi></math> . Finally, the pulse generator domain is responsible for the activation of a neuron by comparing the value of its internal activity <math id="M72"><mi>U</mi></math> and dynamic threshold <math id="M73"><mi>θ</mi></math> . If the activity exceeds the threshold, then the neuron is activated (or ignited), and consequently passes a stimulus to its neighboring neurons. The mathematical equations for these activities are as follows [ 40 40 ]: <math display="block" id="M74"><mrow><msub><mi>F</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>α</mi><mtext>F</mtext></msub></mrow></msup><msub><mi>F</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><msub><mi>V</mi><mtext>F</mtext></msub><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>k</mi><mi>l</mi></mrow></munder><mrow><msub><mi>M</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mi>Y</mi><mrow><mi>k</mi><mi>l</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><msub><mi>S</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mstyle><mo>,</mo></mrow></math> <math display="block" id="M75"><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>α</mi><mtext>L</mtext></msub></mrow></msup><msub><mi>L</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><msub><mi>V</mi><mtext>L</mtext></msub><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>k</mi><mi>l</mi></mrow></munder><mrow><msub><mi>W</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mi>Y</mi><mrow><mi>k</mi><mi>l</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow></mrow></mstyle><mo>,</mo></mrow></math> <math display="block" id="M76"><mrow><msub><mi>U</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><msub><mi>F</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mo>{</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><msub><mi>L</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mo>,</mo></mrow></math> <math display="block" id="M77"><mrow><mo> </mo><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>1</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>U</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>></mo><msub><mi>θ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext></mrow></mtd></mtr></mtable></mrow></mrow><mo>,</mo></mrow></math> <math display="block" id="M78"><mrow><msub><mi>θ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>α</mi><mi>θ</mi></msub></mrow></msup><msub><mi>θ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><msub><mi>V</mi><mi>θ</mi></msub><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where <math id="M79"><mi>F</mi></math> denotes the FI and <math id="M80"><mi>L</mi></math> denotes the LI; subscripts <math id="M81"><mrow><mi>i</mi><mi>j</mi></mrow></math> denote the location of a neuron at coordinate <math id="M82"><mrow><mrow><mo>(</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mo>)</mo></mrow></mrow></math> ; <math id="M83"><mi>n</mi></math> denotes the iteration count; <math id="M84"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> and <math id="M85"><mrow><msub><mi>α</mi><mtext>L</mtext></msub></mrow></math> denote the decay time constants of FI and LI, respectively; <math id="M86"><mrow><msub><mi>V</mi><mtext>F</mtext></msub></mrow></math> and <math id="M87"><mrow><msub><mi>V</mi><mtext>L</mtext></msub></mrow></math> denote the amplification coefficients of FI and LI, respectively; <math id="M88"><mi>M</mi></math> and <math id="M89"><mi>W</mi></math> symbolize the weighting matrices of FI and LI, respectively; <math id="M90"><mi>S</mi></math> denotes the input signal's outer stimulus; <math id="M91"><mi>U</mi></math> denotes the internal activity; <math id="M92"><mi>Y</mi></math> denotes the timing pulse sequence that determines whether a neuron located at <math id="M93"><mrow><mrow><mo>(</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mo>)</mo></mrow></mrow></math> should be fired ( <math id="M94"><mrow><msub><mi>U</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>></mo><msub><mi>θ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math> , <math id="M95"><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> ) or not fired ( <math id="M96"><mrow><msub><mi>U</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>≤</mo><msub><mi>θ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math> , <math id="M97"><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> ); <math id="M98"><mi>θ</mi></math> denotes the dynamic threshold; <math id="M99"><mi>β</mi></math> denotes the linking strength, which modulates the contribution from the FI and LI to the internal activity <math id="M100"><mi>U</mi></math> ; <math id="M101"><mrow><msub><mi>α</mi><mi>θ</mi></msub></mrow></math> and <math id="M102"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> denote the decay time constant and amplification coefficient of the dynamic threshold, respectively. To discriminate n- <math id="M103"><mi>γ</mi></math> PSs using the PCNN method [ 19 19 ], the neuron network should be first provided with n- <math id="M104"><mi>γ</mi></math> PSs by generating an ignition map (a vector of the same size as the n- <math id="M105"><mi>γ</mi></math> PS) that corresponds to each fed n- <math id="M106"><mi>γ</mi></math> PS. The pulse shape differences between neutrons and gamma rays are amplified and more discernible in the ignition maps of neutrons and gamma rays. Subsequently, the discrimination factor was obtained by integrating the ignition times (the number of times a neuron at one point is ignited) of parts of the ignition map that correspond to the parts incorporating the falling edge and delayed fluorescence in the original n- <math id="M107"><mi>γ</mi></math> PS. It should be noted that the discrimination factors of neutrons are greater than those of gamma rays because the ignition times of a point are more significant if the amplitude is higher. Evaluation criteria 1 In this study, Figure of merit ( <math id="M108"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) is used to objectively evaluate the discrimination performance of several methods under different conditions. After the discrimination process, the discrimination results, i.e., the discrimination factors of n- <math id="M109"><mi>γ</mi></math> PSs, were used to create a histogram. This histogram consists of two peaks: the gamma-ray peak on the left side and neutron peak on the right side. We used a Gaussian fitting function to fit each peak and calculated the <math id="M110"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value using the following equation [ 41 41 ]: <math display="block" id="M111"><mrow><mtext>FoM</mtext><mo>=</mo><mfrac><mi>S</mi><mrow><mtext>FWH</mtext><msub><mtext>M</mtext><mi>n</mi></msub><mo>+</mo><mtext>FWH</mtext><msub><mtext>M</mtext><mi>γ</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M112"><mi>S</mi></math> denotes the distance between these two peaks, <math id="M113"><mrow><mi>F</mi><mi>W</mi><mi>H</mi><msub><mi>M</mi><mi>n</mi></msub></mrow></math> and <math id="M114"><mrow><mi>F</mi><mi>W</mi><mi>H</mi><msub><mi>M</mi><mi>γ</mi></msub></mrow></math> denote the full width at half maximum of the gamma ray (on the left side) and neutron (on the right side) peaks, respectively. For an excellent discrimination result, the peaks in the histogram are discernibly separated from each other, while each peak maintains a Gaussian distribution with low variance. This leads to a larger <math id="M115"><mi>S</mi></math> -value and smaller <math id="M116"><mrow><mi>F</mi><mi>W</mi><mi>H</mi><msub><mi>M</mi><mi>n</mi></msub></mrow></math> and <math id="M117"><mrow><mi>F</mi><mi>W</mi><mi>H</mi><msub><mi>M</mi><mi>γ</mi></msub></mrow></math> values. This implies that as the <math id="M118"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> value increases the discrimination results become better. Experiment 1 Experimental setups and parameter settings 2 The 9414 n- <math id="M119"><mi>γ</mi></math> PSs used in the study were collected under excitation from a 241 Am-Be isotope. To acquire data, we used a TPS2000B oscilloscope (with 200 <math id="M120"><mrow><mo>−</mo><mtext>MHz</mtext></mrow></math> bandwidth, 1-GS/s sampling rate, and 8 bits of vertical resolution), 9821 B photomultiplier, and n- <math id="M121"><mi>γ</mi></math> discrimination capable plastic scintillator (EJ299-33). We followed the Shannon criteria [ 42 42 ], the pulse duration was set to 160 ns to prevent the bandwidth from suppressing the information. The trigger threshold was set to 500 mV, which corresponds to a 1.6-MeVee (electron equivalent, i.e., signal produced by 1.6-MeV gamma-ray) threshold. Typical pulse waveforms of neutrons and gamma rays are shown in Fig. 1 Fig. 1 . All parameters of the discrimination methods used in the study are optimized. For the ZC method, <math id="M122"><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow></math> ns and <math id="M123"><mrow><mi>T</mi><mo>=</mo><mn>72</mn></mrow></math> ns. For the CC method, we set the range <math id="M124"><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>−</mo><mn>10</mn><mo>,</mo><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>+</mo><mn>90</mn></mrow><mo>)</mo></mrow></mrow></math> ns as the total component, where the <math id="M125"><mover accent="true"><mi>P</mi><mo>^</mo></mover></math> denotes the time when a pulse reaches its maximum value, and the range <math id="M126"><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>+</mo><mn>27</mn><mo>,</mo><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>+</mo><mn>90</mn></mrow><mo>)</mo></mrow></mrow></math> ns denotes the slow component. For the FEPS method, the aforementioned threshold A is set to 60%. For the FGA method, <math id="M127"><mrow><mi>f</mi><mo>=</mo><mn>1</mn></mrow></math> . Finally, for the PCNN method, we set <math id="M128"><mrow><mi>n</mi><mo>=</mo><mn>180</mn></mrow></math> , <math id="M129"><mrow><msub><mi>α</mi><mtext>F</mtext></msub><mo>=</mo><mn>0.325</mn></mrow></math> , <math id="M130"><mrow><msub><mi>α</mi><mtext>L</mtext></msub><mo>=</mo><mn>0.356</mn></mrow></math> , <math id="M131"><mrow><msub><mi>α</mi><mi>θ</mi></msub><mo>=</mo><mn>0.081</mn></mrow></math> , <math id="M132"><mrow><msub><mi>V</mi><mtext>F</mtext></msub><mo>=</mo><mn>0.0005</mn></mrow></math> , <math id="M133"><mrow><msub><mi>V</mi><mtext>L</mtext></msub><mo>=</mo><mn>0.0005</mn></mrow></math> , <math id="M134"><mrow><msub><mi>V</mi><mi>θ</mi></msub><mo>=</mo><mn>16.8</mn></mrow></math> , <math id="M135"><mrow><mi>β</mi><mo>=</mo><mn>0.67</mn></mrow></math> , <math id="M136"><mrow><mi>M</mi><mo>=</mo><mi>N</mi><mo>=</mo><mtext> </mtext><mrow><mo>[</mo><mrow><mn>0.1509</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0.1509</mn></mrow><mo>]</mo></mrow></mrow></math> , and the integration range is set as <math id="M137"><mrow><mrow><mo>(</mo><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>−</mo><mn>7</mn><mo>,</mo><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>+</mo><mn>123</mn></mrow><mo>)</mo></mrow></mrow></math> . Discrimination results and analysis 2 Filtering is a standard process for most discrimination approaches, and it reduces the noise from the detection system (e.g., the noise of the photomultiplier tube or random voltage fluctuation). Zuo et al. demonstrated that different discrimination methods include an optimal filtering method, which can exhibit optimal discrimination performance [ 43 43 ]. In the study, the most commonly used filtering method, the Fourier filter, was applied to all discrimination methods to control variables such that the change in <math id="M138"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> is only affected by the type of discrimination method. Additionally, in the study, we focus on the effect of noise. Hence, evaluation was conducted twice as follows: using the raw n-γ PSs and using filtered n- <math id="M139"><mi>γ</mi></math> PSs. Although ZC, CC, VP, and PCNN methods can work adequately with or without the filter, the FEPS method is unable to correctly discriminate n- <math id="M140"><mi>γ</mi></math> PSs when the n- <math id="M141"><mi>γ</mi></math> PSs are not pre-processed by the filter. Conversely, there is a significant decrease in the performance of the FGA method when n- <math id="M142"><mi>γ</mi></math> PSs are pre-processed. Hence, the discrimination processes can be divided into two groups as follows: (i) five methods that benefit from filtering (ZC, CC, VP, FEPS, and PCNN) and (ii) five methods that do not benefit from the filtering process (ZC, CC, VP, FGA, and PCNN). The discrimination results are shown in Fig. 2 Fig. 2 and 3 3 as two-dimensional histograms. In the histograms, count refers to the number of n- <math id="M143"><mi>γ</mi></math> PSs with a specific range of normalized discrimination factors and normalized maximum pulse amplitudes. The band on the left side originates from gamma-ray pulses, and the band on the right side originates from neutron signals. As shown in Fig.2 Fig.2 , CC (a) and PCNN (b) methods significantly outperform other methods with band shapes that are consistent with a Gaussian distribution and a wide gap between the gamma ray and neutron bands. For ZC (c) and VP (d) methods, an excessively high number of n- <math id="M144"><mi>γ</mi></math> PS counts were located between the bands, and they were difficult to identify. The discrimination performance of the FEPS (e) method is poor given that both bands are smeared (larger FWHM), and thus they almost overlap with each other. The results of the raw data processing are shown in Fig. 3 Fig. 3 . In general, when raw data are used, the discrimination effects of ZC (a) CC (b) VP(c) and PCNN (d) methods are slightly degraded when compared with their performance with filter processing as shown in Fig. 2 Fig. 2 . The number of counts located between the n- <math id="M145"><mi>γ</mi></math> bands significantly increased when band widths (FWHM) increased. Evidently, the residual noise in n- <math id="M146"><mi>γ</mi></math> PS negatively affects the discrimination process. More experiments were performed to determine the specific impact of noise on the different methods,. Thus, different intensity levels of Gaussian noise were added to the n- <math id="M147"><mi>γ</mi></math> PSs before the discrimination process to evaluate the different methods. The Gaussian noise <math id="M148"><mrow><mover><mi>X</mi><mo>˘</mo></mover><mo> </mo></mrow></math> follows the distribution <math id="M149"><mrow><mover><mi>X</mi><mo>˘</mo></mover><mo> </mo><mo>~</mo><mi>N</mi><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mtext> </mtext><mn>0.2</mn></mrow><mo>)</mo></mrow></mrow></math> , and the intensity level is defined as a constant <math id="M150"><mi>c</mi></math> . The artificial noise added to the n- <math id="M151"><mi>γ</mi></math> PSs is <math id="M152"><mrow><mi>c</mi><mover><mi>X</mi><mo>˘</mo></mover></mrow></math> . However, it is not possible to represent the effectiveness of a method under a given noise level by the result of a single experiment because different random sequences <math id="M153"><mrow><mover><mi>X</mi><mo>˘</mo></mover></mrow></math> generated in every independent experiment can lead to different discrimination results. The average discrimination performance tended to be stable only when the number of independent experiments was sufficient. This average performance represents general effectiveness under specific noise conditions. The following steps were performed to determine the effectiveness of the method under a given noise level: • Independently repeat the discrimination experiment 4000 times. • A comparison of the averaged <math id="M154"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value of the first 2000 times and that of the rest ensures that the difference between them is sufficiently low (less than 0.001). • The average <math id="M155"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value of all 4000 experiments as the discrimination performance under a specific noise level. Hence, we avoided variation in the <math id="M156"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value due to random noise, and thus the average performance under a specific circumstance is distinct. The experiment was divided into two groups, similar to the former. First, the n- <math id="M157"><mi>γ</mi></math> PSs were filtered via the Fourier filter before discrimination, thereby ensuring that the effect of noise mainly originates from the artificially added Gaussian noise with a controllable level. Second, raw n- <math id="M158"><mi>γ</mi></math> PSs were fed to the discrimination process, and thus the original noise from the radiation detection system remained unchanged and was further artificially increased by Gaussian noise to evaluate the performance of different methods. Their performance was quantified by the <math id="M159"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value and its fluctuation, which is defined as the absolute value of the difference between the <math id="M160"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value of n- <math id="M161"><mi>γ</mi></math> PSs without artificially added Gaussian noise and <math id="M162"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value under a given level of artificially added noise. For each noise level, the discrimination experiment for each method was performed multiple times (over 4000 times in this study) to obtain an average fluctuation in <math id="M163"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value, which denotes the general performance of the method under this specific noise scenario. The experimental results are shown in Fig. 4 Fig. 4 - 5 5 . In Fig. 4a Fig. 4a , in a manner similar to the intuitive results shown in Fig. 2 Fig. 2 , the discrimination methods are easily separated into two groups according to their performance. The <math id="M168"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values of the PCNN and CC methods fluctuate at approximately 1.7 while those of the other methods are roughly in the range of 1.0 to 1.2 and generally tend to decline as the noise level increases. As shown in Fig. 4b Fig. 4b , the FEPS method is the most sensitive to noise, and its fluctuations significantly increase when the noise level increases. This is because the FEPS method relies on the calculation of a certain area's slope of the falling edge, and this requires n- <math id="M169"><mi>γ</mi></math> PSs to be highly smooth. They are no longer smooth when Gaussian noise is added to the signals. Hence, the discrimination performance of the FEPS degrades significantly. Given this characteristic, the FEPS method cannot work properly without the filtering process, and it requires a filter to smooth n- <math id="M170"><mi>γ</mi></math> PSs. The VP, CC, and ZC methods exhibit a similar tendency of fluctuation in the <math id="M171"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value, which slowly increases with increases in the noise level. With respect to the PCNN method, its fluctuation initially shares the same pattern with other methods and then surprisingly decreases to zero. This is because the discrimination effect of the PCNN initially improved slightly at the low noise level (as shown in Fig. 4b Fig. 4b ) and then decreased when the noise level exceeded 0.01. This implies that the PCNN can realize a better discrimination performance under a low-noise scenario than that without-noise scenario. This property is realized by tuning the parameters of the PCNN, thereby making it more capable of anti-noise characteristics while sacrificing minimal discrimination performance. The parameter selection strategy is explicitly illustrated in Sect. 4.3. Figure 5a Figure 5a shows the experimental results without filtering process wherein it is observed that PCNN and CC methods still yield the optimal <math id="M172"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value and exhibit stable performance under high noise conditions. However, FGA and VP methods perform poorly under the high noise circumstances wherein the <math id="M173"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value fluctuates around 1 and 0.8 values, respectively, which are unacceptable in many n- <math id="M174"><mi>γ</mi></math> discrimination applications. Additionally, the effect of PCNN does not improve at the low artificial noise level as in the former experiment, thereby suggesting that raw n- <math id="M175"><mi>γ</mi></math> PSs already contain excessive noise. As shown in Fig. 5b Fig. 5b , the anti-noise ability of the VP method is unsatisfactory, and its fluctuations increase immediately when artificial noise is added. A comparison of the discrimination performance of the VP method with and without pre-processing indicated that its <math id="M176"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values significantly decrease from a normal level to a level that cannot be considered successful. The fluctuations in the FGA method initially increased and then decreased at a higher noise level. Given the frequency domain-based property of the FGA method, it can work well in such high noise condition, with its <math id="M177"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value fluctuating at approximately 1.03 and does not show a clear decline tendency. With respect to the CC and ZC methods, their fluctuations in <math id="M178"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value were the most stable and remained at a low level under all noise conditions. Finally, for the PCNN method, the change in the fluctuations in <math id="M179"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value are almost identical to those of the CC and ZC methods below 0.01 noise level. However, at higher levels, the fluctuations of the PCNN increase faster than those of the CC and ZC methods. The detailed discrimination performances of different methods under different situations are listed in Table 1 Table 1 . 1 1 Method/Noise level 1 1 0 1 1 0.5 1 1 1 1 1 1.5 1 1 2 6 1 Discrimination results under influence of noise with filtering process 1 1 ZC ( <math id="M180"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.097 1 1 1.094 1 1 1.091 1 1 1.084 1 1 1.075 1 1 CC ( <math id="M181"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.618 1 1 1.602 1 1 1.597 1 1 1.587 1 1 1.574 1 1 VP ( <math id="M182"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.024 1 1 1.011 1 1 1.008 1 1 1.006 1 1 1.001 1 1 FEPS ( <math id="M183"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.184 1 1 1.165 1 1 1.148 1 1 1.119 1 1 1.081 1 1 PCNN ( <math id="M184"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.720 1 1 1.741 1 1 1.740 1 1 1.731 1 1 1.721 1 1 Method/Noise level 1 1 0 1 1 0.5 1 1 1 1 1 1.5 1 1 2 1 1 ZC ( <math id="M185"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.003 1 1 0.005 1 1 0.011 1 1 0.020 1 1 CC ( <math id="M186"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.009 1 1 0.013 1 1 0.018 1 1 0.026 1 1 VP ( <math id="M187"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.011 1 1 0.015 1 1 0.017 1 1 0.021 1 1 FEPS ( <math id="M188"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.015 1 1 0.029 1 1 0.054 1 1 0.086 1 1 PCNN ( <math id="M189"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.012 1 1 0.011 1 1 0.006 1 1 0.000 6 1 Discrimination results under the effect of noise without filtering process 1 1 Method/Noise level 1 1 0 1 1 0.5 1 1 1 1 1 1.5 1 1 2 1 1 ZC ( <math id="M190"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.104 1 1 1.107 1 1 1.103 1 1 1.100 1 1 1.091 1 1 CC ( <math id="M191"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.563 1 1 1.572 1 1 1.566 1 1 1.557 1 1 1.546 1 1 VP ( <math id="M192"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 0.870 1 1 0.892 1 1 0.892 1 1 0.892 1 1 0.892 1 1 FGA ( <math id="M193"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.034 1 1 1.072 1 1 1.065 1 1 1.058 1 1 1.052 1 1 PCNN ( <math id="M194"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> ) 1 1 1.744 1 1 1.736 1 1 1.734 1 1 1.723 1 1 1.698 1 1 Method/Noise level 1 1 0 1 1 0.5 1 1 1 1 1 1.5 1 1 2 1 1 ZC ( <math id="M195"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.002 1 1 0.001 1 1 0.003 1 1 0.011 1 1 CC ( <math id="M196"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.005 1 1 0.002 1 1 0.003 1 1 0.011 1 1 VP ( <math id="M197"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.025 1 1 0.025 1 1 0.024 1 1 0.024 1 1 FGA ( <math id="M198"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.037 1 1 0.030 1 1 0.023 1 1 0.017 1 1 PCNN ( <math id="M199"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> fluctuation) 1 1 0 1 1 0.004 1 1 0.005 1 1 0.012 1 1 0.026 In conclusion, as shown in Table 1 Table 1 , the PCNN method exhibits optimal discrimination performance with and without the filtering process, with <math id="M200"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values significantly exceeding those of all other methods. Additionally, the anti-noise ability of the PCNN exceeded that of other methods under filter conditions and reached the same level as the other fluctuation-stabilized methods (such as the CC and ZC methods) when the filtering process was removed. Although the anti-noise performance of ZC and FGA methods is good, their poor discrimination performance is a disadvantage that cannot be ignored. The CC method exhibited a discrimination performance second only to that of the PCNN and even better anti-noise performance under extreme noise conditions. Therefore, for high-noise applications, the PCNN method is recommended because it can remain stable under varying noise conditions and can exhibit an outstanding discrimination performance. However, if the noise conditions are extremely poor, then the use of the CC method is recommended to cross-check discrimination results of the PCNN method for ensuring the accuracy and effectiveness of discrimination results. Selection strategy of the parameters of PCNN 2 The discrimination performance and anti-noise capabilities are closely related to the parameters of the PCNN. Hence, it is important to determine the behavior of the PCNN when these parameters change. In this section, we evaluated the effect of the six main parameters of the PCNN. When a parameter is changed, the other parameters are fixed to the values mentioned in Sect. 4.1. For each parameter, experiments were conducted for different noise levels to determine its effect on anti-noise performance and under a zero artificial noise scenario to estimate its impact on discrimination performance. The experimental results are shown in Fig. 6 Fig. 6 . In the figure, the Y -axis on the left side denotes fluctuations in the <math id="M201"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> value measured for different noise levels, and the Y -axis on the right side represents the <math id="M202"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> value measured without artificial noise. As shown in Fig. 6a Fig. 6a , the <math id="M213"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value declines from approximately 1.85 to 1.65, with some periodical fluctuation when the value of <math id="M214"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> increases while the fluctuations in <math id="M215"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value stabilize at their lowest level when the value of <math id="M216"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> is approximately 0.325. The <math id="M217"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> is responsible for the decay rate of the FI. A larger <math id="M218"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> value indicates a faster return of activated neurons to the resting state. Hence, it increases the ignition frequency of the neuron. If the <math id="M219"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> -value is excessively low, then the ignition frequency of each neuron in the PCNN decreases, thereby adversely affecting the information recognition ability of the PCNN. Conversely, if the <math id="M220"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> -value is excessively high, it suppresses the effect on a neuron from its former iteration, and thus the FI is heavily reliant on the outer stimulus, which can deteriorate the discrimination and anti-noise performance of the PCNN. Hence, <math id="M221"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> should be selected as approximately 0.325. As shown in Fig. 6b Fig. 6b , a change in <math id="M222"><mrow><msub><mi>α</mi><mtext>L</mtext></msub></mrow></math> does not affect the discrimination or anti-noise capabilities over a wide range. This is because <math id="M223"><mrow><msub><mi>α</mi><mtext>L</mtext></msub></mrow></math> is responsible for the behavior of the LI, whereas the LI is not extremely important in n- <math id="M224"><mi>γ</mi></math> discrimination applications. The discrimination process mainly depends on the FI to extract information from n- <math id="M225"><mi>γ</mi></math> PSs, whereas the LI only slightly moderates the internal activity of neurons. Any value from 0.34 to 0.36 is acceptable for <math id="M226"><mrow><msub><mi>α</mi><mtext>L</mtext></msub></mrow></math> . As shown in Fig. 6c Fig. 6c , the <math id="M227"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value fluctuated at approximately 1.7 when the <math id="M228"><mi>M</mi></math> changes without an obvious pattern whereas the fluctuations in <math id="M229"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value tended to decrease when the value of <math id="M230"><mi>M</mi></math> increased. The <math id="M231"><mi>M</mi></math> affects the contribution of the surrounding neurons to the central neuron. Increases in the value make the connection between neighboring neurons closer. A stronger connection between neurons decreased the sensitivity of PCNN to the random fluctuation of the pulse signals, i.e., it improved interms of ignoring extraneous noise and focusing on the information carried by the signals. The recommended range of <math id="M232"><mi>M</mi></math> -value was 0.15 0.28. As shown in Fig. 6d Fig. 6d , the performance of the PCNN was steady for β values of <math id="M233"><mi>β</mi></math> lower than 0.72. For values exceeding this value, the performance of discrimination and noise immunity significantly decreased. This degradation originated from the relationship imbalance between FI and LI. As previously mentioned, FI is more important in n- <math id="M234"><mi>γ</mi></math> discrimination applications. When the <math id="M235"><mi>β</mi></math> value increased, the contribution of LI to the internal activity also increased, which directly decreased the contribution of FI's because the internal activity consisted of FI and LI. Thus, if the contribution of one increases, that of the other must proportionally decrease. Therefore, the <math id="M236"><mi>β</mi></math> values should be lower than 0.65. Fig. 6e and 6f Fig. 6e and 6f show the effect of the two parameters related to the dynamic threshold. The <math id="M237"><mrow><msub><mi>α</mi><mi>θ</mi></msub></mrow></math> , which is responsible for the decay speed of the dynamic threshold, exhibited an apparent optimal value range of approximately 0.083–0.088. In this range, discrimination and anti-noise capabilities were good. With respect to the amplification coefficient <math id="M238"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> , the <math id="M239"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value decreased from 1.9 to 1.5 when <math id="M240"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> increased. The fluctuations in <math id="M241"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value initially ameliorated when the value of <math id="M242"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> increased from 15 to 17. However, the fluctuations significantly increased when <math id="M243"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> exceeded 17. The <math id="M244"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> affects the amplification speed of the dynamic threshold. A larger <math id="M245"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> -value leads to a faster amplification speed, thereby curtailing the ignition process of the neurons and decreasing the ignition frequency. This negatively impacts the information extraction ability of the PCNN and results in a decrease in the <math id="M246"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value as shown in Fig. 6f Fig. 6f . Meanwhile, the considerable amplification speed makes it difficult for the dynamic threshold to be exceed by the stimulus of noise, which explains increases in the PCNN's noise immunity in the middle range of <math id="M247"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> value. The value of <math id="M248"><mrow><msub><mi>V</mi><mi>θ</mi></msub></mrow></math> should be in the 16–16.5 range. In general, all parameters of the PCNN can be selected from a wide range and still achieve an acceptable discrimination performance (with <math id="M249"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value from 1.6 to 1.9) and anti-noise performance (with fluctuations in <math id="M250"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value under 0.04). The result indicates that the PCNN method is not heavily constrained by its parameters when applied to n- <math id="M251"><mi>γ</mi></math> discrimination. In addition, the effect of many parameters on discrimination and anti-noise performance steadily fluctuated when the values of the parameters were within a reasonable range whereas the performance significantly decreased when the values were selected at extremes. The reason for this phenomenon was that the PCNN was composed of several closely connected parts. Hence, when the parameter responsible for one part exhibited significant changes, the other parts modulated it to make it work accurately. For example, if the <math id="M252"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> was excessively low, the decay speed of the FI significantly decreased, and the FI increased accordingly and provided a stronger stimulus to the internal activity. Nevertheless, the internal activity did not make the neurons stay activated forever. The dynamic threshold was amplified more times than in the usual <math id="M253"><mrow><msub><mi>α</mi><mtext>F</mtext></msub></mrow></math> -value scenario such that the neurons returned to the initial state. However, the modulation ability exhibited certain limitations. Modulation failed if the selected parameter was excessively radical to maintain the connection between different parts of the PCNN, thereby decreasing the performance of the PCNN. Conclusion 1 In the study, the anti-noise performance of the PCNN method for n- <math id="M254"><mi>γ</mi></math> discrimination was evaluated. The n- <math id="M255"><mi>γ</mi></math> pulses used in the study were generated via a plastic scintillator (EJ299-33) under 241 Am-Be excitation. A 9821 B photomultiplier and an oscilloscope with 200-MHz bandwidth, 1-GS/s sampling rate, and 8-bit vertical resolution were used to collect data. It is noted that the sampling rate of the oscilloscope influences the discrimination performance. The experiments were divided into two runs as follows: in the first run, pulses were pre-processed using the Fourier filter to reduce original noise in the signals, and in the second run, original raw signals were used to simulate an extremely high-noise scenario. For each run, artificial Gaussian noise at different levels was added to the signals before the discrimination process. Under these circumstances, the performance of the PCNN was evaluated and quantified via <math id="M256"><mrow><mo> </mo><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values and their fluctuations. The performance of the PCNN was compared with the other five commonly used methods, namely zero crossing, charge comparison, vector projection, falling edge percentage slope, and frequency gradient analysis. The experimental results indicated that the PCNN method outperforms most other methods (CC is close) in <math id="M257"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -value under all noise conditions. Furthermore, the fluctuations in <math id="M258"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values were lower for the PCNN than for the other methods when the pulse signals are pre-filtered under most conditions. Only for the additional artificial noise at high levels, the fluctuations in the <math id="M259"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> values of the PCNN exceeded those of the CC and ZC methods. The results demonstrated that the PCNN method exhibits outstanding anti-noise capability and can be applied to high-noise applications. Additionally, experiments were conducted to evaluate the effect of PCNN parameters. Variations in <math id="M260"><mrow><mi>F</mi><mi>o</mi><mi>M</mi></mrow></math> -values and their fluctuations were observed under different noise conditions. The experimental results suggested that PCNN does not rely on strict parameter settings and can realize satisfactory performance over a wide parameter range. In conclusion, the PCNN method is suitable for use in high-noise scenarios due to its stability and excellent discrimination performance. 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Swiderski et al. , Comparative study of large samples (2" × 2") plastic scintillators and EJ309 liquid with pulse shape discrimination (PSD) capabilities . J. Instrument. 9 , P06014 ( 2014 ) doi: 10.1088/1748-0221/9/06/p06014 http://doi.org/10.1088/1748-0221/9/06/p06014 Comparative study of large samples (2" × 2") plastic scintillators and EJ309 liquid with pulse shape discrimination (PSD) capabilities Z. Zuo , H.R. Liu , Y.C. Yan et al. , Adaptability of n–γ discrimination and filtering methods based on plastic scintillation . Nucl. Sci. Tech. 32 , 28 ( 2021 ). doi: 10.1007/s41365-021-00865-3 http://doi.org/10.1007/s41365-021-00865-3 Adaptability of n–γ discrimination and filtering methods based on plastic scintillation All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Hao-Ran Liu, Zhuo Zuo and Peng Li. 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. 10.1007/s41365-022-01054-6 This work was supported by the National Natural Science Foundation of China (Nos. 4210040255, U19A2086) and the Sichuan Science and Technology Program (No. 2021JDRC0108) . 2022-02-17 2022-04-19 2022-06-22 2022-07-08 2022-07-26 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01054-6 http://dx.doi.org/10.1007/s41365-022-01054-6 . 2022 Nuclear Science and Techniques 06 33