Complex network based models of ECoG signals for detection of induced epileptic seizures in rats

Abstract

The automatic detection of seizures bears a considerable significance in epileptic diagnosis as it can efficiently lead to a considerable reduction of the workload of the medical staff. The present study aims at automatic detecting epileptic seizures in epileptic rats. To this end, seizures were induced in rats implementing the pentylenetetrazole model, with the electrocorticogram (ECoG) signals during, before and after the seizure periods being recorded. For this purpose, five algorithms for transforming time series into complex networks based on visibility graph (VG) algorithm were used. In this study, VG based methods were used for the first time to analyze ECoG signals in rats. Afterward, Standard measures in network science (graph properties) were made to examine the topological structure of these networks produced on the basis of ECoG signals. Then these measures were given to a classifier as input features so that the ECoG signals could be classified into seizure periods and seizure-free periods. Artificial Neural Network, considered a popular classifier, was used in this work. The experimental results showed that the method managed to detect epileptic seizure in rats with a high accuracy of 92.13%. Our proposed method was also applied to the recorded EEG signals from Bonn database to show the efficiency of the proposed method for human seizure detection.

Appendix

Fractal dimension, largest Lyapunove exponent and Bispectrum

In this section, three methods used in work are briefly introduced.

1. FD: fractal systems have self-similarity characteristic. Self-similarity can be measured by the number of basic building units that form a pattern, and this measure is defined as the FD. Several algorithms have been proposed for FD estimation (Lacasa et al. 2009; Higuchi 1988; Katz 1988; Asvestas et al. 1999), and in this work Higuchi (Higuchi 1988) method were used.

2. LLE: LE is a measure of the exponential divergence/convergence of initially nearby trajectories in the phase space (Abarbanel et al. 1993). Since there was only time series, a pseudo phase space or Reconstructed Phase Space (RPS) of the system is constructed using Time Delay Embedding (TDE) method.

Suppose {x_i} represents the time series. The RPS is created with a time delay $\tau$ and an embedding dimension m. The RPS matrix is formed as follows (Nasrolahzadeh et al. 2015):

$$\left[ {\begin{array}{*{20}l} {x_{0} } \hfill & {x_{\tau } } \hfill & \cdots \hfill & {x_{{\left( {n - 1} \right)\tau }} } \hfill \\ {x_{1} } \hfill & {x_{1 + \tau } } \hfill & \cdots \hfill & {x_{{1 + \left( {n - 1} \right)\tau }} } \hfill \\ {x_{2} } \hfill & {x_{2 + \tau } } \hfill & \cdots \hfill & {x_{{2 + \left( {n - 1} \right)\tau }} } \hfill \\ \vdots \hfill & \vdots \hfill & \vdots \hfill & \vdots \hfill \\ \end{array} } \right]$$

Parameter $\tau$ can be obtained through a number of different methods. In this study, “finding of the mutual information function” method was used for estimation of $\tau$.

After the optimal lag has been selected, the dimension (m) is estimated by Cao’s method (Nasrolahzadeh et al. 2015).The number of Lyapunov exponents is equal to (that of) the embedding dimension of the attractor. For a system to have at least one positive LE (which implies that the largest Lyapanove exponent (LLE) is greater than zero) leads to be chaotic.

Consider two nearest neighboring points in the phase space at time 0 and t, the distances of the points in the ith direction from these points are shown by ${\delta x}_{\text{i}} (0)$ and ${\delta x}_{\text{i}} (\text{t})$, respectively. The Lyapunov exponent is defined by the mean growth rate $\uplambda_{\text{i}}$ of the initial distance;

$$\frac{{\delta x_{i} (t)}}{{\delta x_{\varvec{i}} (0)}} = 2^{{\lambda_{i} t}} ,\quad \forall t \to \infty ,$$

(14)

$$\lambda_{\text{i}} = \mathop {\lim }\limits_{t \to \infty } \frac{1}{t}\log_{2} \frac{{\delta x_{i} (t)}}{{\delta x_{\varvec{i}} (0)}},$$

(15)

Two general methods used for the calculation of the LE from time series are the geometrical and Jacobian approaches. In this paper, the first method was used. The first is based on following the time-evolution of nearby points in the phase space. This algorithm estimates the LLE only (Nasrolahzadeh et al. 2015).

3. BIS: BIS is the Fourier transform of the third order correlation of the time series and is defined as:

$$B\left( {f_{1} ,f_{2} } \right) = E\left[ {X\left( {f_{1} } \right)X\left( {f_{2} } \right)X^{*} \left( {f_{1} + f_{2} } \right)} \right],$$

(16)

where X is the Fourier transform of the signal x, X^* is the complex conjugate of X and E[] is an average over an ensemble of realizations of a random signal (Nasrolahzadeh et al. 2016).

Equation (16) shows the bispectrum is a function of two frequency variables and complex-valued.

Bispectrum can be estimated through various methods (Chua et al. 2009). In this paper, direct (FFT-based) (Nasrolahzadeh et al. 2018) is used to estimate Bispectrum.

The extracted bispectral based features in this study are:

• 1. Mean of bispectral magnitude:

  $${\text{M}}_{\text{avg}} = \frac{1}{\text{L}}\mathop \sum \limits_{\varOmega } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|,$$

  (17)

  where L is the number of points within the region $\varOmega .$

• 2. Max of bispectral magnitude within the region.

  $${\text{Max}} = {}_{\varOmega }^{ \hbox{max} } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|,$$

  (18)

• 3. Min of bispectral magnitude within the region.

  $${\text{Min}} = {}_{\varOmega }^{ \hbox{min} } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|,$$

  (19)

• 4. The sum of the logarithmic amplitudes of the bispectrum (Mookiah et al. 2012):

  $${\text{H}} = \mathop \sum \limits_{\varOmega } {\text{Log}}\left( {\left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|} \right),$$

  (20)

• 5. Bispectral phase entropy (Ph) (Mookiah et al. 2012):

  $${\text{Min}} = {}_{\varOmega }^{ \hbox{min} } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|,$$

  (21)

  where

  $$\text{p}\left( {\uppsi_{\text{n}} } \right) = \frac{1}{\text{L}}\mathop \sum \limits_{\varOmega } I\left( {\upphi\left( {\text{B}\left( {\text{f}_{1} ,\text{f}_{2} } \right)} \right) \in\uppsi_{\text{n}} } \right),$$

  (22)
  $$\uppsi_{\text{n}} = \left\{ {\upphi |-\uppi + \frac{{2\uppi{\text{n}}}}{\text{N}} \le\upphi < - \pi + \frac{{2\uppi\left( {{\text{n}} + 1} \right)}}{\text{N}}} \right\},$$

  (23)

  n = 0, 1, …, N − 1where $\phi$ is the phase angle of the bispectrum, and l(.) is a function which obtains a value of 1 when $\phi$ is within the range bin $\uppsi_{\text{n}}$ depicted by in Eq. (23).

• 6. Bispectrum entropies (Mookiah et al. 2012):

  $${\text{P}}_{1} = - \mathop \sum \limits_{{\rm k}} {\text{p}}_{{\rm k}} {\text{logp}}_{{\rm k}} ,$$

  (24)

  where

  $${\text{p}}_{\text{k}} = \frac{{\left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|}}{{\mathop \sum \nolimits_{\varOmega } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|}},$$

  (25)
  $${\text{P}}_{2} = - \mathop \sum \limits_{{\rm i}} {\text{q}}_{{\rm i}} {\text{logq}}_{{\rm i}} ,$$

  (26)

  where

  $${\text{q}}_{\text{i}} = \frac{{\left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|^{2} }}{{\mathop \sum \nolimits_{\varOmega } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|^{2} }},$$

  (27)
  $${\text{P}}_{3} = - \mathop \sum \limits_{{\rm n}} {\text{r}}_{{\rm n}} {\text{logr}}_{{\rm n}} ,$$

  (28)

  where

  $${\text{r}}_{\text{n}} = \frac{{\left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|^{3} }}{{\mathop \sum \nolimits_{\varOmega } \left| {{\text{B}}\left( {{\text{f}}_{1} ,{\text{f}}_{2} } \right)} \right|^{3} }},$$

  (29)