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Permutation entropy (PE) has been widely exploited to measure the complexity of the electroencephalogram (EEG), especially when complexity is linked to diagnostic information embedded in the EEG. Recently, the authors proposed a spatial-temporal analysis of the EEG recordings of absence epilepsy patients based on PE. The goal here is to improve the ability of PE in discriminating interictal states from ictal states in absence seizure EEG. For this purpose, a parametrical definition of permutation entropy is introduced here in the field of epileptic EEG analysis: the permutation Rényi entropy (PEr). PEr has been extensively tested against PE by tuning the involved parameters (order, delay time and alpha). The achieved results demonstrate that PEr outperforms PE, as there is a statistically-significant, wider gap between the PEr levels during the interictal states and PEr levels observed in the ictal states compared to PE. PEr also outperformed PE as the input to a classifier aimed at discriminating interictal from ictal states.

Electroencephalography (EEG) is an essential tool for the diagnosis of epilepsy and many other neurological diseases. It generally consists of a non-invasive recording of brain electrical activity by means of electrodes integrated into a cap, which is worn by the patient and connected to the acquisition system through a certain number of wires. Once the EEG has been recorded (which can last for minutes to days), a careful review of the entire recording is needed, in order to detect the presence of critical events, such as epileptic seizures or interictal spikes.

In this paper, we will focus in particular on childhood absence epilepsy (CAE), which is a common idiopathic generalized epilepsy syndrome [

As regards other applications of PE to the EEG in the medical field, evaluating the effects of anesthetic drugs is another key topic. Silva

In recent years, the authors proposed a spatial-temporal analysis of epileptic EEG recordings based on measures of complexity of the EEG [

The present work aims to improve the ability of PE in discriminating interictal states (seizure-free EEG segments) from ictal states (seizure EEG segments) in CAE. For this purpose, a parametrical definition of permutation entropy, the permutation Rényi entropy (Per), which was introduced in [

The paper is organized as follows: Section 2 will introduce the methodology, providing details of the EEG recording, Per definition and theoretical comparison

The flowchart of the procedure (

In order to carry out a parameter optimization, 23 EEG recordings of patients diagnosed with CAE were acquired and processed. The children’s mean age was 7.44 years, with a standard deviation of 1.67 years, and 16 of them were female. The EEG were acquired according to the international 10/20 system [

The

Going into the details of PE calculation, let us denote a given EEG time series as _{t}

The values of _{t}_{t}_{t}

Therefore, each vector _{t}_{1}, _{2}, …, _{m}

In this work, a new definition of permutation entropy, according to Rényi’s theory, will be exploited [

It is named permutation Rényi entropy (PEr). A new parameter,

Rényi’s entropy is linked to the distribution of the time series through

We tested the computational time over a simulated random time series (20000 samples); computing _{i}_{i}_{i}^{α}

In particular, we aim to test the ability of PEr, against PE, in discriminating interictal EEG from ictal EEG in absence epilepsy patients. For this purpose, we will compare the average PEr of the interictal EEG segments and the average PEr of the ictal EEG ones, against the same indices estimated with PE. First of all, we will carry out a theoretical comparison between PE and PEr.

In order to test the ability of PEr against PE in detecting dynamical changes in a time series, we first applied it to the transient logistic map. We generated a transient time series

Our aim was to find out what the optimal

Several trials were carried out under different parameter settings:

Order

Lag

Once the EEG recording is converted into PE and PEr matrices (see Section 2.2), it is partitioned into interictal and ictal sub-matrices. The two sub-matrices are then averaged with respect to the rows (channels) and the columns (PE sample vectors), so that an average interictal PE value (

the difference

the ratio

The optimization process was carried out for every patient of the dataset described in Section 2.1. In order to clarify how the optimal parameter configuration was selected, we will now show the details of the procedure for one patient (Patient 4) and then summarize the results obtained for the remaining patients in Section 3.2, showing the optimal parameters selected patient by patient.

According to the procedure described in Section 2.2.2, PE and PEr are estimated in every possible parameter configuration. Each

First of all, we can observe that PEr (red circles) provided better results, because its (_{i}_{i}

As regards Patient 4, the overall optimal configuration was

In order to visually evaluate the effects of

In PEr estimation,

In order to give an overall view of the PEr profiles of the entire EEG recording, estimated for each channel,

The same optimization analysis was extended to the remaining 22 patients. Therefore, for every patient, 350 simulations were carried out (50 for PE estimation and 300 for PEr estimation, as described in Section 2.2.2), for a total of 8050 simulations.

For every EEG recording, both PE and PEr time series were estimated, according to the personalized parameter configuration, and stored as two matrices with _{PE}_{PE}

In order to compare the ability of PEr and PE in discriminating interictal from ictal, the boxplots of either

PEr showed a higher

Since PEr was shown to outperform PE, we tested its general ability in discriminating interictal EEG from ictal EEG. The boxplots of the

On average, the PEr extracted from interictal EEG was higher than the PEr extracted from ictal EEG. This result will be statistically tested in the

Finally, in order to investigate what was the effect of the EEG channel on the discrimination ability of PEr for ictal/interictal segments, for every patient, we computed an average PEr profile for every cerebral area of interest (frontal, temporal, parietal, central, occipital). We averaged the PEr profiles of the EEG channels belonging to each specific area and then we plotted the boxplot of the

Looking at

In order to assess whether this increased gap might result in an improved classification of the cerebral state (interictal or ictal), we decided to test PE and PEr as possible input to a classifier. In recent years, the authors proposed a spatial-temporal analysis of epileptic EEG recordings based on measures of complexity of the EEG [

As here, the goal is to assess whether PEr overcomes PE in differentiating interictal and ictal states, we decided to associate the particular ictal PE topography with the ictal state. In this way, we will find out how the spatial distribution of the entropy extracted from the EEG matches the brain state. As this can be interpreted as a problem of clustering preliminary to supervised classification, we decided to classify through learning vector quantization (LVQ) [

In our experiment, once the PE and PEr profiles were estimated from the EEG (a

Seventy-five percent of ictal entropy vectors and of interictal entropy vectors were used for training, and the remaining 25% of both was used for testing. The LVQ network was trained with PE vectors and with PEr vectors separately. The LVQ network was trained with three hidden neurons, because, looking at the topographical distribution of PE values [

Permutation entropy can measure the complexity of the electroencephalogram, which is a key feature when the aim is to diagnose neurological pathologies that affect the complexity of the EEG. This is the case with patients affected by absence seizures. The goal of the present research was to improve the ability of PE in discriminating interictal (seizure free) states from ictal (seizure) states in absence epilepsy EEG. For this purpose, a parametrical definition of permutation entropy and the permutation Rényi entropy is here introduced in the field of epileptic EEG analysis and tested against PE, tuning the different parameters involved (order, delay time and alpha). The results show that PEr performs better than the standard PE, as it provides a wider gap between the average interictal and ictal entropy levels. After the optimization of the parameter setting, PEr was tested over 23 EEG recordings in order to assess its ability in discriminating interictal from ictal states. The ability of PEr to discriminate between interictal and ictal states was statistically significant, and it outperformed PE, as endorsed by the statistical hypothesis tests and by the classification of the cerebral states based on PEr. In the future, PEr could replace PE in the techniques mentioned in the Introduction to assess if such methodologies for the analysis of the EEG may benefit from PEr’s superior sensitivity.

This work was co-funded by the Italian Ministry of Health, Project Code GR-2011-02351397.

The result _{PEr} > D_{PE}_{PEr}_{PE}_{PEr}_{PE}_{PEr}_{PE}_{t}

This result was tested statistically considering the two _{PEr}_{PE}_{PEr}_{PE}

This result was tested statistically considering the two avgPEr vectors as the two populations to be compared:

Nadia Mammone designed the research, implemented the algorithm and ran the simulations. Francesco C. Morabito carried out an intensive research on the state-of-the-art in this topic and provided suggestions about algorithm design and results presentation. Jonas Duun-Henriksen and Troels Wesenberg Kjaer recorded the EEG data, set up the database and helped with setting the goals of the research. Every author contributed in co-writing the manuscript. All authors have read and approved the final manuscript.

Jonas Duun-Henriksen is employed by HypoSafe A/S, a company that develops EEG recording equipment.

The flowchart of the procedure. (1) The

The transient logistic map data

R

Effect of alpha on the behavior of PEr. The Figure shows the PE and PEr profiles of an EEG trace (from Channel Fp2), estimated in the same parameter configuration (

PEr profiles estimated for all of the channels, computed with

Boxplot of R and D vectors. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.

Boxplot of

Investigation on the effect of the EEG channel on the discrimination ability of PEr for ictal/interictal segments. Given a patient, PEr profiles are averaged over every cerebral area of interest (frontal, temporal, parietal, central, occipital), then the elements of

Comparison of the accuracy in the cerebral state classification (interictal or ictal) provided by PE + learning vector quantization (LVQ) and by PEr + LVQ.

Optimal PE and PEr parameter settings for each patient.

Pt | PE | PEr | |||
---|---|---|---|---|---|

| |||||

m | L | m | L | ||

1 | 4 | 3 | 5 | 4 | 6 |

2 | 4 | 2 | 5 | 3 | 3 |

3 | 4 | 10 | 4 | 7 | 7 |

4 | 4 | 8 | 4 | 7 | 7 |

5 | 3 | 10 | 3 | 5 | 7 |

6 | 4 | 3 | 5 | 5 | 7 |

7 | 4 | 2 | 5 | 3 | 7 |

8 | 4 | 10 | 5 | 4 | 7 |

9 | 4 | 3 | 5 | 6 | 7 |

10 | 4 | 3 | 5 | 5 | 6 |

11 | 4 | 3 | 4 | 3 | 5 |

12 | 4 | 3 | 5 | 5 | 5 |

13 | 4 | 8 | 4 | 7 | 7 |

14 | 4 | 3 | 5 | 4 | 7 |

15 | 4 | 2 | 4 | 3 | 4 |

16 | 4 | 3 | 5 | 4 | 6 |

17 | 5 | 3 | 4 | 6 | 7 |

18 | 4 | 2 | 4 | 2 | 2 |

19 | 4 | 3 | 5 | 4 | 6 |

20 | 4 | 3 | 4 | 3 | 4 |

21 | 4 | 3 | 5 | 5 | 6 |

22 | 4 | 3 | 5 | 3 | 4 |

23 | 4 | 3 | 4 | 7 | 7 |