# Self-Organized Fuzzy Neural Network Nonlinear System Modeling Method Based on Clustering Algorithm
## 3 Implementation of Modeling Methods
### Self-Organizing Mechanism
#### 3.1.2 Adjustment of Neuron Width  
Neuronal activity is an important factor in the construction of neural network structures, and their local field potential (LFP) is used as the indicators for the construction of new neurons. In [20; 23], the mathematical model of LFP is:  
\[V(r)=\frac{1}{4\pi\tau}\times\frac{I_{o}}{|r-r_{o}|}. \tag{17}\]  
where \(V(r)\) is the extracellular potential at position \(r\) in the extracellular space, \(I_{o}\) is the current source, \(\tau\) is a constant value representing the conductivity, and \(|r-r_{o}|\) is the absolute distance between r and the current source position \(r_{o}\).  
From Equation (17), it is known that when the sample is far from the neuron, the local field potential value of the sample is smaller, so the activation strength (AS) of the sample on the neuron is defined as  
\[AS_{i}=\frac{\left(x_{i}-c_{i,l}\right)^{2}}{\sigma_{i,l}^{2}},i=1,2,\ldots, \tag{18}\]  
where \(i\) is the dimension of the input sample, \(j\) is the number of center columns closest to the sample, and \(\sigma_{i,l}\) is the width of the neuron.  
According to Figure 2, the activation strength of neuron decays as the distance of the sample from the center increases. When the value of \(AS_{i}\) as in Equation (19) is less than the minimum activation strength we set, and, here, the activation strength threshold is 0.2,  
\[AS_{i}<AS_{th}. \tag{19}\]  
The center is not well-activated by the sample, and the \(i\)th dimensional generalization performance of the neuron is not good. Therefore, we expand the neuron width of the current center to improve its generalization, and the width growth parameter \(\varepsilon\) is set to 0.01  
\[\sigma_{i,l}=(1+\varepsilon)\times\sigma_{i}. \tag{20}\]  
A large number of samples are gathered around an effective center. As the training samples enter, there is an overlap of samples [29], as shown in Figure 3. There is an overlap of samples attributed to centers A and B. Ref. [8] showed that the width of the MF neurons has a significant effect on the generalization performance of the system. If the width is too large, the MF neurons will have a large amount of overlap and will give a value close to  
Figure 2: Neuron activation strength.  
1 for the different input systems. Since the number of samples around neurons A and B differs, the activation strength of each sample for a neuron is also inversely proportional to the distance between the neurons. To evaluate the generalization properties of the MF neurons, the neuron density potential is defined as:  
\[density\_pot_{J}=\frac{\sum\limits_{k=1}^{p}\sqrt[]{\prod\limits_{i=1,2,...,m }AS_{i}}}{P}, \tag{21}\]  
where the density potential energy increases or decreases with the number of neurons, and \(J\) denotes the nearest neuron marker to the sample.  
If the density potential of a neuron is greater than the threshold we set, it means that the width of that central neuron can be appropriately converged in that neuron sample set to reduce redundancy. The density threshold is set according to the experiment, and the width convergence parameter \(\delta\) is set to 0.01  
\[density\_pot_{J}\geq den\_th, \tag{22}\]  
\[\sigma_{J}=(1-\delta)\times\sigma\_or_{J}, \tag{23}\]  
where \(\sigma\_or_{J}\) is the original width.