Epileptogenic zone localization and seizure control in coupled neural mass models

Abstract

Exact localization of the epileptogenic zone (EZ) is the first priority for ensuring epilepsy treatments and reducing side effects. The results of traditional visual methods for localizing the origin of seizures are far from satisfactory in some cases. Signal processing methods could extract substantial information that may complement visual inspection of EEG signals. In this study, EZ localization is changed into a driver identification problem, and a nonlinear interdependence measure, the weighted rank interdependence, is proposed and used as a driver indicator because it can detect coupling information, especially directionality, from EEG signals. A proportional integral derivative (PID) controller is then explored, using simulations, to establish its suitability for seizure control. The seizure control we propose rests on identifying the EZ using nonlinear interdependence measures of directed functional connectivity. Two directionally coupled neural mass models are employed for simulation investigation. Two parameters can adjust the sensitivity and completeness of the weighted rank interdependence for different applications, and their effect is discussed in the context of neural mass models. Simulation results demonstrate that use of the weighted rank interdependence for EZ identification can be applied to different EZ types, and the approach achieves an overall identification rate of 98.84 % for several EZ types. Simulations also indicate that PID control can effectively regulate synchronization between neural masses.

Appendix

Proposition

$H^{(k,a)}(X|Y)\in (0,1]$.

Proof

By definition, both $G_n^{(k,a)} (X)$ and $G_n^{(k,a)} (X|Y)$ are greater than 0, so $H^{(k,a)}(X|Y)$ is greater than 0. If X is consistent with Y, that is, $G_n^{(k,a)} (X)=G_n^{(k,a)} (X|Y)$ $(n=1,2,\ldots ,N)$, then $H^{(k,a)}(X|Y)=1$.

If X is not consistent with Y, two situations occur: The k nearest neighbors of ${\mathbf {y}}_n$ are the k nearest neighbors of ${\mathbf {x}}_n$, or the k nearest neighbors of ${\mathbf {y}}_n$ are not the k nearest neighbors of ${\mathbf {x}}_n$.

If the k nearest neighbors of ${\mathbf {y}}_n$ are the k nearest neighbors of ${\mathbf {x}}_n$, that is, $g_{nr_{n,j} }^X =j$ for $j=1,2,\ldots ,k$ and $g_{ns_{n,j} }^X \in \{1,2,\ldots ,k\}$, but $g_{ns_{n,j} }^X$ is not necessarily equal to j. $g_{ns_{n,j} }^X$ can become $g_{nr_{n,j} }^X$ after a sorting process realized via a sorting algorithm such as bubble sorting. In the bubble sorting algorithm, two adjacent elements are compared; if the former is greater than the latter, they switch positions, and otherwise they maintain their positions. This task is repeated for every pair of adjacent elements. After a comparison, if the positions remain unchanged, then $G_n^{(k,a)} (X|Y)$ is unchanged; if the elements are switched (e.g., if $g_{ns_{{n,j}_1}}^X$ and $g_{ns_{{n,j}_2}}^X$ are exchanged), then the change in $G_n^{(k,a)} (X|Y)$ is $-(g_{ns_{{n,j} _1}}^X -g_{ns_{{n,j}_2}}^X )$*$(a^{n_{j_1} }\hbox {-}a^{n_{j_1} +1})<$0, and therefore $G_n^{(k,a)} (X|Y)$ will be unchanged or will become smaller. After $\frac{k(k-1)}{2}$ comparisons, $G_n^{(k,a){\prime }} (X|Y)=G_n^{(k,a)} (X)<G_n^{(k,a)} (X|Y)$, and therefore $H^{(k,a)}(X|Y)<1$.

If the k nearest neighbors of ${\mathbf {y}}_n$ are not the k nearest neighbors of ${\mathbf {x}}_n$, a series of switches can yield the situation discussed above, whereby the k nearest neighbors of ${\mathbf {y}}_n$ are the k nearest neighbors of ${\mathbf {x}}_n$. Each switch is performed between $g_{ns_{{n,j}_3}}^X >k$ and $g_{ns_{{n,j}_4}}^X <k$, and the change in $G_n^{(k,a)} (X|Y)$ is $-(g_{ns_{{n,j}_3}}^X -g_{ns_{{n,j}_4} }^X )$*$a^{n_{j_3}}$. In this case, we can still obtain $G_n^{(k,a)^{\prime }} (X|Y)=G_n^{(k,a)} (X)<G_n^{(k,a)} (X|Y)$, and therefore $H^{(k,a)}(X|Y)<1$. $\square$