# Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Signal Analysis

#### 2.1.1. Data Quantification

#### 2.1.2. Permutation Entropy

#### 2.1.3. Lempel–Ziv Complexity

- Reproduction: it consists of extending a sequence ${S}_{1:T}$ to a sequence ${Q}_{1:N}$ via recursive copy–paste operations, which leads to ${S}_{1:T+N}={S}_{1:T}{Q}_{1:N}$, i.e., where the first letter ${Q}_{1}$ is in ${S}_{1:T}$, that is to say ${Q}_{1}={S}_{i}$, the second one is the following one in the extended sequence of size $T+1$, i.e., ${Q}_{2}={S}_{i+1}$, etc.: ${Q}_{1:N}$ is a subsequence of ${S}_{1:T+N-1}$. In a sense, all of the “information” of the extended sequence ${S}_{1:T+N}$ is in ${S}_{1:T}$.
- Production: the extended sequence ${S}_{1:T+N}$ is now such that ${S}_{1:T+N-1}$ can be reproduced by ${S}_{1:T}$, but the last symbol of the extension can either follow the recursive copy–paste operation (thus is a reproduction) or can be “new”. Note that a reproduction is a production, not the other way round. Let us denote a production by ${S}_{1:T}\Rightarrow {S}_{1:N+T}$.

#### 2.1.4. Complexity vs. Entropy Map

- Step 1: Record and reprocess the signal—using band pass filter and notch filter at 60 Hz (Figure 2A).
- Step 2: Discretize the raw signal using a permutation vector approach (Figure 2B).
- Step 3: Calculate the Lempel–Ziv complexity for the sequence taken from step 2 (Figure 2C).
- Step 4: Take the sequence from step 2 and, using a histogram, estimate the probability distribution (PD) of permutation vectors. Then, calculate the Shannon entropy related with this PD.
- Step 5: Finally, with the two measures, we have the coordinates $({h}_{PE},{C}_{LZ})$ of the map corresponding to the recorded signal.

#### 2.2. Patient Recording

#### Technical Procedures and Clinical Description

#### 2.3. Clinical Description of the Cases

#### 2.3.1. Case I

#### 2.3.2. Case II

## 3. Results

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Example of the permutation vector discretization. (

**A**) for the parameter $d=3$, we have d! = 6 possible patterns; (

**B**) to each data point $t=1,\dots ,n-(d-1)$ is assigned the corresponding pattern, depending on the relative values of the neighbors; (

**C**) the quantified sequence to the original signal.

**Figure 2.**The steps to analyze a recording signal through the Complexity vs. Entropy map. (

**A**) signal recording and preprocessing; (

**B**) quantify the signal using the permutation vector approach; (

**C**) calculate the permutation Lempel–Ziv complexity and the permutation entropy for the quantify sequence.

**Figure 3.**EEG recording for the four states of treatment (Case I). (

**A**) ${T}_{0}$ corresponds to the first monitoring performed, and it can be observed that before the time 5:52 and after the time 5:55, the discharges occur in both hemispheres and had a duration greater than 10 s. (

**B**,

**C**) In the studies ${T}_{1}$ and ${T}_{2}$, bilateral short-term discharges of 2–3 s are observed. There is little clinical differentiation between studies ${T}_{1}$ and ${T}_{2}$. (

**D**) In the study ${T}_{3}$, a trace with a normal voltage with a predominance of bilateral slow rhythms in the frequency domain is observed. All studies correspond to intercritical records.

**Figure 4.**EEG traces belonging to patient diagnosed with narcolepsy (Case II). (

**A**,

**B**) ${T}_{0}$ corresponds to sleep paths, and in both can see the slow waves between 3 and 5 Hz. In addition, K and spindle complexes, characteristic of phase 2 of non-REM sleep, are observed. (

**C**,

**D**) The ${T}_{1}$ and ${T}_{2}$ traces correspond to normal records in the waking eyes closed condition. ${T}_{2}$ trace corresponds to the second hour of the monitoring. A trace of low voltage with alpha and theta rhythms is observed in both records.

**Figure 5.**PLZC vs. HPE map analysis belonging to a patient diagnosed with Generalized Idiopathic Epilepsy, in four different stages of treatment (square and triangles), compared to the mean value of a control group and their respective errors (circle and ellipse). Each subplot shows the analysis for a particular EEG channel (despite starting in the inner plot). The parameters used were $d=4$ and $\tau =1$.

**Figure 6.**PLZC vs. HPE map analysis belonging to a patient diagnosed with generalized narcolepsy, in three different stages of treatment (square and triangles), compared to the mean value of a control group and their respective errors (circle and ellipse). Each subplot shows the analysis for a particular EEG channel (despite starting in the inner plot). The parameters used were $d=4$ and $\tau =1$.

Subject | Time Record (MIN) | Segment Time Analyzed (MIN) | Gender | Age |
---|---|---|---|---|

1 | 60 | 44 | M | 24 |

2 | 60 | 49 | M | 20 |

3 | 60 | 44 | M | 31 |

4 | 60 | 44 | M | 48 |

5 | 60 | 44 | M | 25 |

6 | 60 | 48 | M | 38 |

7 | 30 | 30 | F | 27 |

8 | 30 | 30 | F | 28 |

9 | 30 | 30 | M | 19 |

10 | 60 | 30 | F | 32 |

11 | 30 | 30 | F | 44 |

12 | 30 | 25 | F | 40 |

13 | 30 | 25 | F | 38 |

14 | 60 | 48 | M | 48 |

15 | 60 | 39 | M | 34 |

16 | 60 | 50 | M | 50 |

17 | 60 | 40 | M | 23 |

18 | 30 | 30 | M | 23 |

19 | 60 | 41 | M | 31 |

20 | 60 | 45 | F | 27 |

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**MDPI and ACS Style**

Diaz, J.M.; Mateos, D.M.; Boyallian, C.
Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs. *Entropy* **2017**, *19*, 540.
https://doi.org/10.3390/e19100540

**AMA Style**

Diaz JM, Mateos DM, Boyallian C.
Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs. *Entropy*. 2017; 19(10):540.
https://doi.org/10.3390/e19100540

**Chicago/Turabian Style**

Diaz, Juan M., Diego M. Mateos, and Carina Boyallian.
2017. "Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs" *Entropy* 19, no. 10: 540.
https://doi.org/10.3390/e19100540