## 2 Adaptive PCNN Combined with GWO and BEMD
### GWO Algorithm  
Over recent decades, meta-heuristic optimization techniques, which are inspired by related physical phenomena, animal behavior or evolutionary concepts, have become increasingly popular [39]. Inspired by the behavior of grey wolves, the Grey Wolf Optimization (GWO) algorithm is a new meta-heuristics method. The tracking processes of grey wolves when encircling and attacking prey are the main steps in the numerical pattern of the GWO algorithm [40]. The GWO mimics the leadership hierarchy and the hunting mechanism of grey wolves in nature very well. The intelligent grey wolf optimizer was proposed by [PERSON] [41] and can perform appropriate trade-offs between the exploration and exploitation abilities of the algorithm. In solving complex benchmark functions, the GWO can be implemented effectively to address complex multivariate optimization problems. In the simulation of the leadership hierarchy, the grey wolves involve four types, designated \(\alpha\), \(\beta\), \(\delta\) and \(\omega\)[42]. The hierarchy and relationships of the grey wolves and their corresponding duties are described in detail in Figure 2.  
In the GWO, the optimal solution to the problem is considered to be the prey and first three best solutions are regarded as \(\alpha\), \(\beta\) and \(\delta\), respectively. Other than the three solutions mentioned above, \(\omega\) must follow the three dominant wolves and holds the rest of the candidate solutions [43]. The initial stage of the GWO algorithm in the hunting process is encircling the prey. The mathematical formulation used to mimic the encircling process is given as follows:  
\[\stackrel{{\rightarrow}}{{D}}=\left[\stackrel{{ \rightarrow}}{{\mathbf{C}}}\cdot\stackrel{{\rightarrow}}{{x}}_{p} (t)-\stackrel{{\rightarrow}}{{x}}(t)\right] \tag{12}\]  
\[\stackrel{{\rightarrow}}{{X}}(t+1)=\stackrel{{ \rightarrow}}{{x}}_{p}(t)-\stackrel{{\rightarrow}}{{A}}\cdot \stackrel{{\rightarrow}}{{D}} \tag{13}\]  
where \(t\) indicates the current iteration, \(\stackrel{{\rightarrow}}{{x}}_{p}\) and \(\stackrel{{\rightarrow}}{{x}}\) are the position vectors of the grey wolves and a grey wolf, respectively. The vectors \(\stackrel{{\rightarrow}}{{A}}\) and \(\stackrel{{\rightarrow}}{{C}}\) are calculated as follows:  
\[\stackrel{{\rightarrow}}{{A}}=2\cdot\stackrel{{ \rightarrow}}{{a}}\cdot\stackrel{{\rightarrow}}{{e}}_{1}- \stackrel{{\rightarrow}}{{a}} \tag{14}\]  
\[\stackrel{{\rightarrow}}{{C}}=2\cdot\stackrel{{ \rightarrow}}{{e}}_{2} \tag{15}\]  
where \(\stackrel{{\rightarrow}}{{e}}_{1}\) and \(\stackrel{{\rightarrow}}{{e}}_{2}\) are random vectors in [0, 1], and \(\stackrel{{\rightarrow}}{{a}}\) is a motion vector acting as a variable that decreases linearly from 2 to 0 over the iteration process. Depending on the random vectors \(\stackrel{{\rightarrow}}{{e}}_{1}\) and \(\stackrel{{\rightarrow}}{{e}}_{2}\) position updating can be performed randomly over the search domain. We assume that \(\alpha\), \(\beta\) and \(\delta\) all have better knowledge about the potential location of the prey. Therefore, the other grey wolves represented by \(\omega\) are obliged to update their positions based on the best three positions obtained to date [44]. The grey hunting characteristic can then be presented numerically as follows:  
\[\left\{\begin{array}{l}\stackrel{{\rightarrow}}{{D}}_{a}= \left|\stackrel{{\rightarrow}}{{\mathbf{C}}}_{1}\cdot\stackrel{{ \rightarrow}}{{X}}_{a}-\stackrel{{\rightarrow}}{{X}}\right|\\ \stackrel{{\rightarrow}}{{D}}_{\beta}=\left|\stackrel{{ \rightarrow}}{{\mathbf{C}}}_{2}\cdot\stackrel{{ \rightarrow}}{{X}}_{\beta}-  
The best three initial solutions are considered, and the residual solutions are eliminated, and thus the average measure of the three best solutions is provided by \(\overset{\rightarrow}{X}\). The hunting process is terminated only after the prey is attacked and the prey should pause its movement.