## 2 Adaptive PCNN Combined with GWO and BEMD
### Implementing the PCNN Theory  
[PERSON] [33] proposed the pulse-coupled neural network, which is a third-generation artificial neural network with wide application potential. In 1999, [PERSON] [34] used circuit theory to introduce the PCNN model more specifically, which was subsequently widely applied in various image processing fields. Today, the continuous development of the PCNN model has given it incomparable superiority over the other current image denoising methods. There are two notable features of PCNNin the form of the global coupling and pulsed synchronization of the neurons [35]. In image processing, a PCNN is a single layer two-dimensional array of laterally linked neurons. Each neuron in the network corresponds strictly to one pixel only in an input image and receives the intensity of that corresponding pixel as an external stimulus. Each neuron also connects with its neighboring neurons and receives local stimuli from them. The PCNN model (Figure 1) is composed of three parts: the dendritic tree of the coupled linking subsystem, the modulation subsystem, and the pulse generator of the dynamic threshold subsystem and the firing subsystem. The surrounding outputs of the PCNN in the last iteration can be obtained using the linking compartment, while the initial output of the PCNN is set to zero [36]. For the input stimulus, the feeding compartment is applied at the corresponding location. Overall, the dendritic tree is defined mathematically as shown below:  
\[F_{ij}[c]=I_{ij} \tag{6}\]  
\[L_{ij}[c]=V_{L}\sum_{k,l}W_{ijkl}Y[c-1] \tag{7}\]  
where \(F_{ij}[c]\) represents the feeding input compartment of neuron \((i,j)\) and \(L_{ij}[c]\) is the linking compartment. \(I_{ij}\) is the external stimulus of neuron \((i,j)\) and \(V_{L}\) is a normalizing constant. \(w_{ijkl}\) denotes the weighted coefficient of neurons \((k,l)\) to \((i,j)\), which is the reciprocal of the Euclidean distance and is defined as shown in (8) [9]:  
\[W_{ijkl}=\frac{1}{\sqrt{\left(i-k\right)^{2}+(k-l)^{2}}} \tag{8}\]  
The stimuli from the dendritic tree are applied to the modulation subsystem, and the modulation process can be described as shown in (9) below. \[u_{ij}[c]=F_{ij}[c]\big{\{}1+\beta_{T}L_{ij}[c]\big{\}} \tag{9}\]  
where \(\beta_{T}\) is the linking strength factor between synapses, and \(u_{ij}[c]\) denotes the internal state of the neuron. Next, the threshold \(E_{ij}[c]\), which represents the dynamical threshold, is generally set to zero. In each iteration, the threshold \(E_{ij}[c]\) will be shown as follows:  
\[E_{ij}[c]=e^{-\alpha^{T}}E_{ij}[c-1]+V_{T}Y_{ij}[c] \tag{10}\]  
where \(\alpha_{T}\) is the exponential decay time constant, and \(\upsilon_{T}\) is the intrinsic voltage constant of \(E_{ij}[c]\). Figure 1: The structure of the simplified Pulse Coupled Neural Network. Finally, the internal state \(U_{ij}[c]\) must be compared with that of the dynamic threshold \(E_{ij}[c]\), and the pulse output \(Y_{ij}[c]\) can be obtained subsequently using the firming subsystem:  
\[Y_{ij}[c]=\left\{ \begin{array}{l}1,U_{ij}(c)>E[c-1],\\ 0,U_{ij}(c)\leq E[c-1],\end{array} \right. \tag{11}\]  
The neuron's pulse generator includes a step function generator and a threshold signal generator. The threshold input at each time step is updated using the form of an exponential decay. At each step, the neuron output \(Y_{ij}[c]\) is set as 1, which means it can be fired. The internal activity \(U_{ij}[c]\) is greater than the threshold function \(E_{ij}[c]\). The neuron output \(Y_{ij}[c]\) is then reset to zero when \(U_{ij}[c]\) is less than \(E_{ij}[c]\). Thus, in a single time step, the pulse generator produces a single pulse at its output whenever the values of \(U_{ij}[c]\) exceed those of \(E_{ij}[c]\). The linking parameter \(\beta_{T}\) is taken on as a global variable. \(\alpha_{T}\) and \(v_{T}\) are critical parameters that can easily affect the PCNN processing treatment.