## 3 Methodology
### Novel and Reasonableness Hypothesis for the Accurately-Solved Nonlinear Problem in EEG Signals  
In order to accurately solve the nonlinear problem in EEG signals, it is important to understand the work and generation mechanism of EEG signals. Currently, there are no definitive theories or mathematical models to describe the process of EEG production. However, some basic facts have been discovered, which are simplified and described in Figure 3. EEG signals are first produced by the nerve cells in the brain, as shown in Figure 3a. Usually, the characteristics of electrical signals are decided by two elements: the structure of brain cells, and motivation signals, as shown in Figure 3b. Neurons, connected by synapses, can be considered a type of cell that has a specific function of discharging electricity. The purpose of this electricity is to store and transmit information among different neurons. Neurons are often considered as the basic signal-processing unit of the human brain, whose main function is to receive information and transmit it to other cells, as shown in Figure 3a. Figure 3: Hypothesis on the work mechanism of brain activity, (**a**) the nerve cells in the brain, (**b**) the structure of brain cells and motivation signals, and (**c**) collected electroencephalographic (EEG) signals. Figure 3b demonstrates the hypothesis on the work and generation mechanism of EEG signals, which can be divided into three parts: motivation signals denoted by \(x_{j,m}\), the working principle of neurons denoted by \(g(\cdot)\), and the functional electrical signal output denoted by \(y_{j,m}\). Clearly, the most important part is the working principle of neurons \(g(\cdot)\), and, generally, each neuron is not isolated; they all connect with each other. Moreover, the connections are denoted by \(\omega_{j,i}\), and the discharge function can be considered as two types of neurons: summation neuron \(\sum\) and multiplication neuron \(\prod\). Usually, summation neuron \(\sum\) refers to emphasizing signals according to linear functions, whereas multiplication neuron \(\prod\) generates the new signals or enhances signal energy in accordance with some nonlinear pattern, which results in the EEG signals being nonlinear. Generally, the neuron design is a simple but extremely efficient architecture, which makes it extremely responsive to any minor electrical signals, and microscopically, it is non-stationary. From the hypothesis on the construction of neurons, as shown in Figure 3b, functional electrical signal output can be expressed as follows:  
\[ \begin{split} y_{j,m}&=\prod_{l}g(w_{1}\cdot x_{1, n})\oplus\prod g(w_{2}\cdot x_{2,n})\oplus...\oplus\prod g(w_{L}\cdot x_{j,n})\\ &=\sum_{l=1}^{L}(\prod g(w_{l}\cdot x_{j,n})),\end{split}  \tag{12}\]  
where \(x_{j,n},y_{j,m}\) denote the motivation signals and function electrical signal output, respectively, and \(g(\cdot)\) is the working principle of neurons. It should be noted that the summation neuron \(\sum\) was not considered because EEG signals are often non-stationary and nonlinear signals, which are generally caused by multiplication neurons \(\prod\). Therefore, it was reasonable and suitable for the following research work to analyze EEG signals for automated epileptic seizure detection. From the hypothesis, the following were assumed to be true: EEG signals are quite different in different individuals due to differences in the construction of brain cells and the characteristics of EEG signals recorded in different brain activities, which can be discriminated by the features denoted by the coefficient \(\omega_{j,i}\). The frequency spectra of EEG signals are always aliasing, and generally cannot be obtained by a series of simple filters. Deconvolution may be an efficient method to separate EEG signals into various components.