Estimating the Parameters of the Epileptor Model for Epileptic Seizure Suppression

Abstract

Epilepsy is one of the most common brain disorders worldwide, affecting millions of people every year. Given the partially successful existing treatments for epileptiform activity suppression, dynamic mathematical models have been proposed with the purpose of better understanding the factors that might trigger an epileptic seizure and how to mitigate it, among which Epileptor stands out, due to its relative simplicity and consistency with experimental observations. Recent studies using this model have provided evidence that establishing a feedback-based control approach is possible. However, for this strategy to work properly, Epileptor’s parameters, which describe the dynamic characteristics of a seizure, must be known beforehand. Therefore, this work proposes a methodology for estimating such parameters based on a successive optimization technique. The results show that it is feasible to approximate their values as they converge to reference values based on different initial conditions, which are modeled by an uncertainty factor or noise addition. Also, interictal (healthy) and ictal (ongoing seizure) conditions, as well as time resolution, must be taken into account for an appropriate estimation. At last, integrating such a parameter estimation approach with observers and controllers for purposes of seizure suppression is carried out, which might provide an interesting alternative for seizure suppression in practice in the future.

Fuzzy Takagi-Sugeno Modeling

Formulation for the Linear Sub-Systems of the Dynamic Matrix A

Consider the following nonlinear dynamic system:

$$\begin{aligned} \dot{\mathbf{x }}(t) = \mathbf{A} _{\text{nl}}{} \mathbf{x} (t) + \mathbf{B} _{\text{nl}}{} \mathbf{u} (t) \end{aligned}$$

(36)

For simplicity, if $\mathbf{u} (t)=0$, each element of $\dot{\mathbf{x }}$ can be expressed as:

$$\begin{aligned} x_{i}(t)=\sum _{j=1}^{N}f_{ij}(\mathbf{x} (t))x_{j}(t) \end{aligned}$$

(37)

where $f_{ij}(\mathbf{x} (t))$ represents any function at position (i, j), N is the dimension of the system, $f_{ij}(\mathbf{x} (t))=f_{k}(\mathbf{x} (t))$, $k=1,...,N_{\text{nl}}<N^{2}$, represents the $N_{\text{nl}}$ nonlinear functions present in the system. According to the Fuzzy Takagi-Sugeno modeling (FTSM), this system can be rewritten as Tanaka and Wang (2004):

$$\begin{aligned} \text{ Rule } \;\; i: {\left\{ \begin{array}{ll} \text{ If } \;\; z_{1}(t) \;\; \text{ is } \;\; M_{i1} \;\; \text{ and } \;\; ... \;\; \text{ and } \;\; z_{p}(t) \;\; \text{ is } \;\; M_{ip} \\ \text{ Then, } \;\;\;\;\; \dot{\mathbf{x }}(t)=\mathbf{A _{i}}\mathbf{x }(t), \;\; \;\; i=1,...,r \;\;\;\;\;\;\;\;\;\;\; \end{array}\right. } \end{aligned}$$

(38)

where $M_{ij}$ are called fuzzy sets, r is the total number of rules, $z_{1}(t),...,z_{p}(t)$ are known as premise variables, and $\mathbf{A} _{i}$ are linear sub-models that represent the original dynamics locally. To identify them, Taniguchi et al. (2001) proposed writting all of the nonlinear functions as linear combinations between their maximum and minimum values, which leads to their exact representation for a given time interval:

$$\begin{aligned} f_{k}(\mathbf{x} )=[g_{k}^{\text {min}}(\mathbf{x} )]\kern 0.1500em f_{k}^{\text {min}} + [g_{k}^{\text {max}}(\mathbf{x} )] \kern 0.1500em f_{k}^{\text {max}} \end{aligned}$$

(39)

where $f_{k}^{\text {max}}=$ max$[\kern 0.1500em f_{k}(\mathbf{x} (t))]$, $f_{k}^{\text {min}}=$ min$[\kern 0.1500em f_{k}(\mathbf{x} (t))]$, and:

$$\begin{aligned} \begin{array}{c} g_{k}^{\text {min}}(\mathbf{x} ) = \dfrac{[\kern 0.1500em f_{k}^{\text {max}}-f_{k}(\mathbf{x} )]}{f_{k}^{\text {max}}-f_{k}^{\text {min}}} \\ g_{k}^{\text {max}}(\mathbf{x} )=1-g_{k}^{\text {min}}(\mathbf{x} ) \end{array} \end{aligned}$$

(40)

known as membership functions (MF) Tanaka and Wang (2004), which must comply with the following restrictions: $g_{k}^{\text {min}}(\mathbf{x} ) \ge 0$, $g_{k}^{\text {max}}(\mathbf{x} ) \le 1$, and $g_{k}^{\text {min}}(\mathbf{x} ) + g_{k}^{\text {max}}(\mathbf{x} ) = 1$. Based on the latter condition ($g_{k}^{\text {min}}(\mathbf{x} ) + g_{k}^{\text {max}}(\mathbf{x} ) = 1$), Eq. (39) can be expressed as:

$$\begin{aligned} f_{k}(\mathbf{x} )= \prod _{i=1,i \ne k}^{N_{nl}} \{ g_{i}^{\text {min}}(\mathbf{x} ) + g_{i}^{\text {max}}(\mathbf{x} ) \}\{ [g_{k}^{\text {min}}(\mathbf{x} )]\kern 0.1500em f_{k}^{\text {min}} + [g_{k}^{\text {max}}(\mathbf{x} )]\kern 0.1500em f_{k}^{\text {max}} \} \end{aligned}$$

(41)

or, alternatively:

$$\begin{aligned} f_{k}(\mathbf{x} )= \sum _{i=1}^{r_{1}}N_{i}(\mathbf{x} )f_{k}^{\text {min}} + \sum _{i=r_{1}+1}^{ (2^{N_{\text{nl}}}) }N_{i}(\mathbf{x} )f_{k}^{\text {max}} \end{aligned}$$

(42)

where $r_{1}=2^{(N_{\text{nl}}-1)}$, and $N_{i}(\mathbf{x} ) = \prod _{k=1}^{N_{\text{nl}}}g_{k}^{(.)}(\mathbf{x} )$,

which must meet the requirement: $\sum _{i=1}^{2^{N_{\text{nl}}}}N_{i} (\mathbf {x})=1$. The superscript (.) indicates a combination between the maximum and minimum values for the kth nonlinear function $g_{k}^{(.)} (\mathbf{x} )$. After substituting each $f_{k}$ according to Eq. (42) into the matrix $\mathbf {A}_{nl}$, the system in Eq. (36) is rewritten by

$$\begin{aligned} \dot{\mathbf {x}}(t) = \sum _{i=1}^{2^{N_{\text{nl}}}}N_{i} (\mathbf {x}) \left[ \mathbf {A}_{j} \mathbf {x}(t) \right] \end{aligned}$$

(43)

Each $\mathbf{A} _{j}$ is, thus, a linear sub-model. If matrix $\mathbf{B} _{\text{nl}}$ and the input are considered back into the formulation:

$$\begin{aligned} \dot{\mathbf {x}}(t) = \sum _{i=1}^{2^{N_{\text{nl}}}}N_{i} (\mathbf {x}) \left[ \mathbf {A}_{i} \mathbf {x}(t) \right] + \sum _{i=1}^{2^{N_{\text{nl}}}}N_{i} (\mathbf {x}) \left[ \mathbf {B}_{i} \mathbf {u}(t) \right] \end{aligned}$$

(44)

or, alternatively:

$$\begin{aligned} \dot{\mathbf {x}}(t) = \sum _{i=1}^{2^{N_{\text{nl}}}}N_{i} (\mathbf {x}) \left[ \mathbf {A}_{i} \mathbf {x}(t) + \mathbf {B}_{i} \mathbf {u}(t) \right] \end{aligned}$$

(45)

For the sake of understanding, Fig. 16 presents an example of the exact reconstruction of a nonlinear function using the membership functions and linear submodels. In this case: $f(t)=e^{-2t}\text {sin}(2\pi t)$, that is, $N_{\text{nl}}=1$. The membership functions $g_{1}^{\text{max}}(x(t))$ and $g_{1}^{\text{min}}(x(t))$ weigh the maximum and minimum values of f(t) ($f_{1}^{\text{max}}$ and $f_{1}^{\text{min}}$, respectively) over time such that its exact representation is obtained as $f_{1}(x(t))$.

Fig. 16

Concept of sector nonlinearity to reconstruct $f_{\text {sin}}(t)=e^{-2t}\text {sin}(2\pi t)$ (–), using the exact solution Tanaka and Wang (2004); Taniguchi et al. (2001) based on the maximum and minimum values of $f_{\text {sin}}(t)$, expressed by $f_{1}^{\text {max}}$ and $f_{1}^{\text {min}}$ ( ), and the membership functions: $g_{1}^{\text {min}}(x(t))$ ( ) and $g_{1}^{\text {max}}(x(t))$ ( ). The reconstructed model is represented by $f_{1}(x(t))$ ( )

Formulation for the Linear Sub-Systems of the Error Dynamics e(t)

To obtain the membership functions (MFs) used to model the error dynamics, the concept of sector nonlinearity is applied once again Tanaka and Wang (2004); Ichalal et al. (2011). Each partial derivative can be rewritten as:

$$\begin{aligned} a_{ij}\le \frac{\partial f_{i}(z^{i})}{\partial z_{j}} \le b_{ij} \end{aligned}$$

(46)

assuming it is differentiable on the interval $[a_{ij},b_{ij}]$, where $a_{ij}$ and $b_{ij}$ are individual maximum and minimum values for each i and j, respectively. Therefore, the above equation can be conveniently rewritten as:

$$\begin{aligned} \frac{\partial f_{i}(z^{i})}{\partial z_{j}}=\sum _{k=1}^{2}v_{ij}^{k}(z^{i})\tilde{a}_{ijk} \end{aligned}$$

(47)

where:

$$\begin{aligned} v_{ij}^{1}(z^{i})=\frac{ \frac{\partial f_{i}(z^{i})}{\partial z_{j}}-a_{ij} }{ b_{ij}-a_{ij} }, \;\;\; v_{ij}^{2}(z^{i})= 1-v_{ij}^{1}(z^{i}) \end{aligned}$$

(48)

which must meet the following requirements of convexity Ichalal et al. (2011):

$$\begin{aligned} \sum _{k=1}^{2}v_{ij}^{k}(z^{i})=1, \;\;\; 0\le v_{ij}^{k}(z^{i})\le 1, \;\;\; k=1,2 \end{aligned}$$

(49)