## 2 Materials and Methods  
### Modeled Signal  
The conducted research included modeling of the neural structure by the Jansen-Rit single column model [10]. It includes a population of pyramidal structures receiving stimulatory and inhibitory impulses from interneurons and an input fed with noise simulating the impact of other neural structures. Pyramidal cells of the cerebral cortex, due to their characteristic location in the cerebral cortex structure, are considered to be the main source of the electroencephalic signal from the human brain. The Jansen-Rit model's block diagram is shown in Figure 1.  
The \(P_{1}\), \(P_{2}\), and \(P_{3}\) blocks can be presented using the equations:  
\[h_{e}(t) =\begin{cases}Aate^{-at}&t\geq 0\\ 0&t<0\end{cases} \tag{1}\] \[h_{i}(t) =\begin{cases}Bbte^{-bt}&t\geq 0\\ 0&t<0\end{cases} \tag{2}\]where: \(A\) and \(B\) are the maximal post-synaptic potential amplitudes, \(a\) and \(b\) are membrane time-constants.  
Equation (1) can also be presented in the form of two first order differential equations:  
\[\left\{\begin{array}{ll}\dot{y}(t)=&z(t)\\ \dot{z}(t)=&Aax(t)-2 az(t)-a^{2}y(t)\end{array}\right. \tag{3}\]  
Equation (2) also can be expressed this way, by changing \"\(A\)\" and \"\(a\)\" terms to \"\(B\)\" and \"\(b\)\". The model can be therefore expressed using six first-order equations [10]:  
\[\left\{\begin{array}{ll}\dot{y}_{0}(t)=&y_{3}(t)\\ \dot{y}_{1}(t)=&y_{4}(t)\\ \dot{y}_{2}(t)=&y_{5}(t)\\ \dot{y}_{3}(t)=&A\cdot a\cdot\text{sigm}(y_{1}(t)-y_{2}(t))-2\cdot a\cdot y_{ 3}(t)-a^{2}\cdot y_{0}(t)\\ \dot{y}_{4}(t)=&A\cdot a\cdot(p(t)+C_{2}\cdot\text{sigm}(C_{1}\cdot y_{0}(t) ))-2\cdot a\cdot y_{4}(t)-a^{2}\cdot y_{1}(t)\\ \dot{y}_{5}(t)=&B\cdot b\cdot C_{4}\cdot\text{sigm}(C_{3}\cdot y_{0}(t))-2 \cdot b\cdot y_{5}(t)-b^{2}\cdot y_{2}(t)\end{array}\right. \tag{4}\]  
where _sigm_ is the sigmoidal function, responsible for the neural structure activation rate, dependent on the potential (graphical representation of this function was presented in Figure 2):  
\[sigm(v)=\frac{2e_{0}}{1+e^{(v_{0}-v)}}, \tag{5}\]  
\(p(t)\) is the signal delivered to the model, made of white noise, originally in the range of 120-320 pulses per second, \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are numbers of synapses established between particular neuron populations, bounded by equation:  
\[\left\{\begin{array}{ll}C_{2}=&0.80\cdot C_{1}\\ C_{3}=&0.25\cdot C_{1}\\ C_{4}=&0.25\cdot C_{1}\end{array}\right. \tag{6}\]  
Output of the model, or the modeled EEG signal, is denoted as \(y\) and can be obtained by simple subtraction:  
\[y(t)=y_{1}(t)-y_{2}(t). \tag{7}\]  
In the sigmoidal function the parameters \(e_{0}\) and \(v_{0}\) are directly responsible for the activation function's shape, i.e., the inflection point. The parameter \(r\) is responsible for  
Figure 1: The one-column Jansen–[PERSON] model. The \(P_{1}\) and \(P_{2}\) blocks are the functions (he) presented in Equation (1), \(P_{3}\) is the function (hi) presented in Equation (2). The \(S_{1}\), \(S_{2}\), and \(S_{3}\) (sigm) blocks are the sigmoidal functions presented in Equation (5), p is noise, while the \(C_{1}\)–\(C_{4}\) are the coefficients of connections between particular neural structures.  
steepness (slope). These parameters were being modified in a course of this study, since the postsynaptic potential transforms into a firing rate not exactly in the same way from a population to a population [10; 26].  
The model was simulated with frequency of 1 kHz, using mid-point numerical method of solving differential equations.