## 2 Methods  
For mathematical modeling of voltage-gated ion channels, the method of system identification known from control engineering was used. We considered the voltage-gated ion channel Kv10.1 as a system, object, or process in accordance with the control system theory. Kv10.1 was treated as a separate object (or system/process) within a space that can interact or connect with other elements. If a system is interpreted as a combination of elements that act together and perform specific objectives [22], then a single ion channel can be considered as a system or, in more precise terms, as an object in a control loop. As such, the channel can achieve the desired dynamics under nominal conditions and display acceptable behaviors under random conditions that deviate from the maximum prescribed boundary conditions. System identification is a scientific methodology that is used to develop mathematical models of a dynamical system. This method is based on observed or measured data of the system [23]. A block diagram of a general system (G(s)) represents a transfer function with a complex variable \(s\) in the Laplace domain, specified with an input \(c(t)\), output \(r(t)\), measured disturbances \(dc(t)\), and unmeasured disturbances \(d(t)\) (Figure 2a). An illustration of a voltage-gated ion channel as a system is shown in Figure 2b. The block diagram of the system shows its unilateral property, where system inputs and outputs, in general, are vectors. If there are multiple inputs and multiple outputs, then the system is called a MIMO system. In the case of a single input and a single output, however, the control system is called a SISO system. For example, if a voltage-gated ion channel has a voltage stimulus as input and a measured current as output, we assume that it can be described as a SISO system. Furthermore, we assume that the voltage-gated ion channel Kv10.1 is a linear time-invariant system based on the experimental results given in the following sections. A linear time-invariant system is defined by  
\[ \begin{array}{l}a_{n}r^{(n)}(t)+a_{n-1}r^{(n-1)}(t)+\ldots+a_{1}\dot{r}(t)+a_ {0}r(t)=\\ b_{0}c(t)+b_{1}\dot{c}(t)+\ldots+b_{n-1}c^{(n-1)}(t)+b_{n}c^{(n)}(t),\;\;\;n \geq m.\end{array}  \tag{1}\]  
where \(r(t)\) is the system's output and \(c(t)\) is the system's input. The transfer function, \(G(s)\), is defined as the ratio of the left Laplace-transformed output and the left Laplace-transformed input under zero initial conditions, where \(s\) is a complex variable in the Laplace (5) domain. Figure 2: Block diagram of a system (**a**); illustrated voltage-gated ion channel as a system (**b**). \[G(s)=\frac{\mathcal{L}\{r(t)\}}{\mathcal{L}\{c(t)\}}=\frac{R(s)}{ \mathcal{C}(s)} =\frac{b_{0}+b_{1}s+...+b_{n-1}s^{n-1}+b_{n}s^{n}}{a_{n}s^{n}+a_{n-1 }+...+a_{1}y(t)+a_{0}y(t)}= \tag{2}\] \[=\frac{\sum_{0}^{n}b_{n}s^{n}}{\sum_{0}^{n}a_{n}s^{n}}\]  
The application of the transfer function concept makes it possible to represent the system's dynamics in the form of algebraic equations. If the highest power of \(s\) in the denominator of a transfer function is equal to \(n\), the system is called an \(n\)th-order system [22]. The transfer function of a system is thus a mathematical model of this system. If the system's transfer function is unknown, we can estimate it based on known inputs and measured outputs for the system. This resulting transfer function provides a complete description of the system's dynamic behavior [22; 24; 25]. A transfer function can be determined experimentally based on measured input and output signals if a mathematical model cannot be developed using physical correlations and equations. Therefore, the process of developing a model from measured input and output data is known as system identification [23; 26]. The main elements in the system identification process cycle are experimental design, the experiment itself, data preprocessing, fitting the model to the data, testing the model structure, validating and auditing the model [26].