# Complex Pearson Correlation Coefficient for EEG Connectivity Analysis

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Complex Pearson Correlation Coefficient as a Measure of Undirected Connectivity

#### 2.2. Phase Locking Value PLV and Its Relation to CPCC

#### 2.3. Weighted Phase Lag Index wPLI and Its Relation to CPCC

#### 2.4. Connectivity Estimation Based on Phase Difference Histograms

## 3. Results

#### 3.1. Synthetic Signals from the MRC Brain Network Dynamics Unit (University of Oxford)

#### 3.2. Synthetic Signals Generated with the Kuramoto Model

#### 3.3. Real-Life Signals

- The raw brain activity data were imported into MATLAB using the EEGLAB toolbox;
- Electrode positions (also called channel locations) were defined in the software;
- The data were referenced to average;
- The data were filtered with a band pass filter limited to 0.5 and 45 Hz;
- Automatic spectral-based channel suppression (z = 5) was performed using the EEGLAB “pop rejchan” function;
- Artifacts were removed using the ICLabel plugin for EEGLAB (thresholds for removing components were less than or equal to 0.05 for brain activity and greater than or equal to 0.9 for artifacts);
- The data were re-referenced to average;
- Sub-bands of the EEG signal were extracted (delta 0.5–4 Hz, theta 4–8 Hz, alpha 8–13 Hz, low beta 13–18 Hz, high beta 18–30 Hz, gamma 35–45 Hz).

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

absCPCC | absolute value of complex Pearson correlation coefficient |

CPCC | complex Pearson correlation coefficient |

DTI | diffusion tensor imaging |

EC | eyes closed |

EEG | electroencephalography |

EO | eyes open |

HT | Hilbert transform |

imCPCC | imaginary component of complex Pearson correlation coefficient |

MEG | magnetoencephalography |

MRI | magnetic resonance imaging |

PLV | phase locking value |

PLI | phase lag index |

wPLI | weighted phase lag index |

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**Figure 1.**Visualization of averaging used in calculation of the PLV. The PLV is computed from unit vectors representing instantaneous phase differences.

**Figure 2.**Visualization of averaging used in calculation of the PLI. The PLI is computed from unit vectors representing instantaneous phase differences.

**Figure 3.**The relationship between connectivity and phase differences distributions. When volume conduction is not considered (

**a**) higher connectivity reflects in lower variance, while the mean value is irrelevant. In the presence of volume conduction (

**b**), it reflects in higher values for phase differences close to 0 or $\pi $, which, therefore, do not (necessarily) indicate connectivity. Connectivity level is expressed with colors; red is the highest and yellow is the lowest.

**Figure 4.**Connectivity matrices obtained with PLV (

**a**), wPLI (

**b**), absCPCC (

**c**), and imCPCC (

**d**) for signals generated with [26], for 8–13 Hz frequency band.

**Figure 5.**Scatter plots of absCPCC to PLV relationship (

**left**) and imCPCC to wPLI relationship (

**right**). Each dot represents one electrode pair. Dots are colored according to their relative density. The black line represents identity, while the cyan one is the best linear fit. Rows correspond to different frequency bands: (

**a**,

**b**) 0.5–4 Hz; (

**c**,

**d**) 4–8 Hz; (

**e**,

**f**) 8–13 Hz.

**Figure 6.**Phase difference distributions for selected electrode pairs (synthetic signals [26]). Shown are the distributions corresponding to the highest: (

**a**) PLV and absCPCC values, (

**b**) wPLI and imCPCC values, (

**c**) ratio between absCPCC and imCPCC values, (

**d**) ratio between imCPCC and absCPCC values.

**Figure 7.**Connectivity matrices obtained with PLV, wPLI, absCPCC, and imCPCC for signals generated with the Kuramoto model [27].

**Figure 8.**Relationships between absCPCC and PLV (

**a**), and wPLI and imCPCC (

**b**), shown as a scatter plot of values for all signal pairs where signals were generated with the Kuramoto model [27]. The black line represent the identity while the cyan line shows the best linear fit. The colors of the dots represent the relative density of the connectivity values.

**Figure 9.**Phase difference distributions for selected synthetic signal pairs generated using the Kuramoto model [27]. Shown are the distributions corresponding to the highest: (

**a**) PLV and absCPCC values, (

**b**) wPLI and imCPCC values, (

**c**) ratio between absCPCC and imCPCC values, (

**d**) ratio between imCPCC and absCPCC values.

**Figure 10.**Connectivity matrices for the the alpha band (8–13 Hz) of the real-life signals [28] for eyes closed (EC) and eyes open (EO) states, computed with PLV, wPLI, absCPCC (

**g**), and imCPCC (

**h**).

**Figure 11.**Scatter plots of the absCPCC to PLV relationship (

**left**) and the imCPCC to wPLI relationship (

**right**), for all electrode pairs and for 10 test subjects (EC state). The black line represents the identity, while the cyan line shows the best linear fit. The colors of the dots represent the relative density of the connectivity values. Each row is shown for a different frequency band: (

**a**,

**b**) 0.5–4 Hz; (

**c**,

**d**) 4–8 Hz; (

**e**,

**f**) 8–13 Hz.

**Figure 12.**Scatter plots of the absCPCC to PLV relationship (

**left**) and the imCPCC to wPLI relationship (

**right**), for all electrode pairs and for 10 test subjects (EO state). The black line represents the identity, while the cyan line shows the best linear fit. The colors of the dots represent the relative density of the connectivity values. Each row is shown for a different frequency band: (

**a**,

**b**) 0.5–4 Hz; (

**c**,

**d**) 4–8 Hz; (

**e**,

**f**) 8–13 Hz.

**Figure 13.**Phase difference distributions for selected electrode pairs (real-life signals [28]). For each distribution, all four measures are calculated. The figure shows the electrode pair with the highest: (

**a**) PLV and absCPCC values, (

**b**) wPLI, and imCPCC values, (

**c**) ratio between absCPCC and imCPCC values, (

**d**) ratio between imCPCC and absCPCC values.

**Table 1.**Correlation values between compared connectivity measures (real-life signal). Here, ${r}_{abs}$ and ${r}_{im}$ denote $r(absCPCC,PLV)$ and $r(imCPCC,wPLI)$ respectively.

Frequency | State-EC | State-EO | ||
---|---|---|---|---|

(Hz) | ${\mathit{r}}_{\mathit{abs}}$ | ${\mathit{r}}_{\mathit{im}}$ | ${\mathit{r}}_{\mathit{abs}}$ | ${\mathit{r}}_{\mathit{im}}$ |

0.5–4 | 0.93 | 0.86 | 0.94 | 0.89 |

4–8 | 0.98 | 0.91 | 0.97 | 0.94 |

8–13 | 0.98 | 0.86 | 0.97 | 0.91 |

13–18 | 0.99 | 0.94 | 0.99 | 0.96 |

18–30 | 0.98 | 0.96 | 0.99 | 0.96 |

35–45 | 0.96 | 0.95 | 0.99 | 0.95 |

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**MDPI and ACS Style**

Šverko, Z.; Vrankić, M.; Vlahinić, S.; Rogelj, P. Complex Pearson Correlation Coefficient for EEG Connectivity Analysis. *Sensors* **2022**, *22*, 1477.
https://doi.org/10.3390/s22041477

**AMA Style**

Šverko Z, Vrankić M, Vlahinić S, Rogelj P. Complex Pearson Correlation Coefficient for EEG Connectivity Analysis. *Sensors*. 2022; 22(4):1477.
https://doi.org/10.3390/s22041477

**Chicago/Turabian Style**

Šverko, Zoran, Miroslav Vrankić, Saša Vlahinić, and Peter Rogelj. 2022. "Complex Pearson Correlation Coefficient for EEG Connectivity Analysis" *Sensors* 22, no. 4: 1477.
https://doi.org/10.3390/s22041477