# Identification of a Group’s Physiological Synchronization with Earth’s Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Procedures

#### 2.1. Participants

#### 2.2. Ethics Statement

#### 2.3. Computational Estimation of the Synchronization of a Group’s HRV Time Series with Earth’s Magnetic Field Data

#### 2.3.1. Magnetic Field Data

#### 2.4. Computation of the Power of Local Magnetic Field

- (1)
- Compute the spectrogram $S\left(\theta ,\omega \right)$ (as described previously).
- (2)
- Crop the spectrogram $S=\mathrm{min}\left\{S;{S}_{crop}\right\}$ in order to eliminate intermittent chaotic outbreaks in the measured data due to manmade noise, lightening, etc.
- (3)
- Apply the Gaussian median filter of dimensions $3\times 3$ to $S$ for the reduction of noise.
- (4)
- Compute the signal power as $P={\displaystyle \sum}_{\omega ={\omega}_{min}}^{{\omega}_{max}}\left(\frac{1}{\mathsf{\Delta}\theta}{\displaystyle \sum}_{t={t}_{0}}^{{t}_{1}}S\left(t,\omega \right)\right)$.

#### 2.4.1. Example: Computation of the Local Magnetic Field Power

#### 2.5. Algorithm for the Computation of Geometrical Synchronization between Two Time Series

#### 2.5.1. Computation of the Area of an Attractor in the State Space

- (1)
- Compute the center of the mass of the points comprising the attractor. Move the origin of the state space to the center of the mass.
- (2)
- Divide the state space of the attractor into the slices with equal central angles of a circle centered on the origin. The number of slices depends on the number of points in the observation window of the time series.
- (3)
- Set the radius of each slice to the maximal distance between a point belonging to that slice and the origin.
- (4)
- Compute the area of the attractor ${S}_{\tau}$ as the sum of areas of all slices.

#### 2.5.2. Example 1: Identification of the Optimal Time Lag

#### 2.5.3. Construction of the Algorithm for the Estimation of the Geometrical Synchronization between Two Time Series

- (1)
- Divide signals $X$ and $Y$ into $T$ observation windows of size $m$ ($m$ should be large enough to enable the reconstruction of a meaningful attractor in the state space):$$\left({X}_{1},\dots ,{X}_{m}\right),\left({X}_{m+1},\dots ,{X}_{2m}\right),\dots ,\left({X}_{n-m+1},{X}_{n}\right);\phantom{\rule{0ex}{0ex}}\left({Y}_{1},\dots ,{Y}_{m}\right),\left({Y}_{m+1},\dots ,{Y}_{2m}\right),\dots ,\left({Y}_{n-m+1},{Y}_{n}\right).$$
- (2)
- Compute optimal time lags for each observation window for both time series using Algorithm B. Such computations result in two vectors of optimal time lags: ${\tau}_{*j}^{\left(X\right)},{\tau}_{*j}^{\left(Y\right)}\left(j=\overline{1,T}\right)$. This information reduction algorithm allows the identification of similarities between attractors reconstructed from different time series from the geometrical point of view. The variation of optimal time lags reconstructed for a pair of time series is used for the quantification of the generalized geometrical synchronization between those time series.
- (3)
- Calculate the vector of absolute differences between obtained optimal time lags for each observation window: ${\tau}_{*j}^{\left(X,Y\right)}=\left|{\tau}_{*j}^{\left(X\right)}-{\tau}_{*j}^{\left(Y\right)}\right|\text{}\left(j=\overline{1,T}\right)$. The differences between the optimal time lags are used as the metric of geometrical similarity between the analyzed time series.
- (4)
- In order to identify the slow dynamics reflecting averaged changes in absolute differences between optimal time lags for each data signal, divide the vector of absolute differences into $F=\frac{T}{h}$ segments: $\left[{\tau}_{*\left(h\xb7\left(i-1\right)+1\right)}^{\left(X,Y\right)},\dots ,{\tau}_{*\left(h\xb7i\right)}^{\left(X,Y\right)}\right]\left(i=\overline{1,F}\right)$. The number of points $h$ in each segment should be large enough to produce a meaningful averaging.
- (5)
- Calculate the mean absolute difference ${\overline{\tau}}_{i}^{\left(X,Y\right)}=\frac{1}{h}{\displaystyle \sum}_{j=1}^{h}{\tau}_{*\left(h\xb7\left(i-1\right)+j\right)}^{\left(X,Y\right)}\left(i=\overline{1,F}\right)$ between optimal time lags for each segment. The obtained vector of mean absolute differences ${A}^{\left(X,Y\right)}=[{\overline{\tau}}_{1}^{\left(X,Y\right)}{\overline{\tau}}_{2}^{\left(X,Y\right)}\dots {\overline{\tau}}_{F}^{\left(X,Y\right)}]$ is defined as a measure representing the geometrical synchronization between data signals $X,Y$.

#### 2.5.4. Computational Validation of the Geometrical Synchronization Algorithm

#### 2.6. Clusterization of Multivariate Time Series Based and Their Synchronization with a Master Time Series

- (1)
- Compute the vector of mean absolute differences ${A}^{({X}^{\left(k\right)},M)}=[{\overline{\tau}}_{1}^{({X}^{\left(k\right)},M)}\dots {\overline{\tau}}_{F}^{({X}^{\left(k\right)},M)}]$, describing the relationship between ${X}^{\left(k\right)}$ and $M$ as described in Algorithm C, for each ${X}^{\left(k\right)},k=\overline{1,K}$.
- (2)
- Calculate the Euclidean distance (the measure used to estimate the geometrical similarity of two data vectors) which represents the similarity between all $K$ data signals, using the following formula:$${\u2551{A}^{({X}^{\left(i\right)},M)}-{A}^{({X}^{\left(j\right)},M)}\u2551}_{2}=\sqrt{{\left({\overline{\tau}}_{1}^{\left({X}^{\left(i\right)},M\right)}-{\overline{\tau}}_{1}^{\left({X}^{\left(j\right)},M\right)}\right)}^{2}+\dots +{\left({\overline{\tau}}_{F}^{\left({X}^{\left(i\right)},M\right)}-{\overline{\tau}}_{F}^{\left({X}^{\left(j\right)},M\right)}\right)}^{2}},\text{}i,j=\overline{1,K}.$$The above equation yields the symmetric matrix of Euclidean distances.
- (3)
- Construct a dendrogram plot (UPGMA) [37] using the obtained matrix. The main goal of the dendrogram is to identify the clusters of similar time series, i.e., the clustering process involves grouping the analyzed time series based on the similarity of the slower rhythm dynamics of their synchronization with master time series $M$.

## 3. Results

#### 3.1. The Application of the New Analysis Technique on HRV and Magnetic Field Data

#### 3.1.1. Obtaining the Power of Local Magnetic Field during the Experiment

#### 3.1.2. Identification of Clusters in the Groups Based on the Similarity/Synchronization between Participants’ HRV and Magnetic Field Activity

- (1)
- One of the steps of Algorithm C is splitting the participants’ HRV and local magnetic field power time series into segments. The standard length of analysis for HRV is five minutes [38]. Thus, inter-beat (RR) interval and magnetometer data was split into five-minute segments for analysis. Note that since HRV data consists of time intervals between each pair of heartbeats, the number of samples in the data vectors corresponding to each five-minute segment varies due to changes in the participants heart rate and other factors that influence HRV, such as stress and emotional states [39]. Since the power of the local magnetic field was computed for one-second time intervals, the resulting five-minute segments consisted of the same number of elements (300 data points). However, the difference in the size of the segments of HRV and the power of the local magnetic field time series did not impact the overall result of the study, since all of the segments represented the same concurrent five-minute time intervals.
- (2)
- We selected the number of slices in Algorithm B to be 60 because it was empirically observed that a higher number would result in some empty slices.
- (3)
- The maximal value of $\tau $ in Algorithm B was set to 50. Higher values of $\tau $ would generate too short trajectory matrices, because the five-minute segments consisted of approximately 300 elements.
- (4)
- The value of the parameter $h$ in Algorithm C, used for identification of slow dynamics of the synchronization between the two time series, was set to 48. This corresponded to a four-hour averaging of the difference of the optimal time lags. It was observed that this value of $h$ produced the most meaningful averaging.
- (5)
- As noted in Section 3.2 the magnetometer data contained one minute-long periods of missing data at the end of each hour. Since these periods in the time series did not contain any information, it was necessary to remove those periods in such a way that would not disrupt the timing between the HRV and magnetic field time series. The solution we implemented was to remove the missing data segments from both the five-minute magnetometer data and from the five-minute RR interval series. Since the cropped series obtained after this procedure fully defined the five-minute series, they were used in the data reduction step.

#### 3.2. The Relation between Synchnorization Results and the Psychological Interactions between Participants

#### 3.2.1. Psychological Survey Data

#### 3.2.2. Comparison of Survey Data and the HRV/Magnetic Field Synchronization Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

EEG | Electroencephalogram |

HRV | Heart rate variability |

IBI | Inter-beat-interval |

Pc | pulsations continuous |

Pi | pulsations irregular |

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**Figure 1.**An example of the local magnetic field intensity data (measured in Lithuania during the time period between 2015/02/26 01:00:01 and 2015/02/26 03:00:01).

**Figure 2.**An example of the spectrogram for the magnetic field data presented in Figure 1. Frequency resolution is $\frac{1}{4096}$, $\mathsf{\Delta}\theta =4\text{h},\mathsf{\Delta}\omega =52\mathrm{Hz},\omega \in \left[0;52\right]\mathrm{Hz}$.

**Figure 3.**Time series $X=\left({X}_{1},\dots ,{X}_{n}\right)$ representing a numerical solution to Equation (4) with initial conditions $x\left(0\right)=0;{x}^{\prime}\left(0\right)=0.8$.

**Figure 4.**Examples of attractors for different time lag values. In (

**a**) $\tau =4$; (

**b**) $\tau =12$; (

**c**) $\tau =23$; (

**d**) $\tau =31$; (

**e**) $\tau =43$; (

**f**) $\tau =50$.

**Figure 5.**Shifting the origin to the center of mass of the attractor for $\tau =4$: (

**a**) the original attractor; (

**b**) the origin shifted to the center of the mass of the attractor.

**Figure 6.**Sliced diagrams of the attractors shown in Figure 4.

**Figure 8.**Time series $X$ (

**a**) and $Y$ (

**b**) and the difference $X-Y$ (

**c**), obtained from numerical integration of the coupled nonlinear pendulum model (Equation (5)). Dotted lines separate time intervals with different values for the coupling parameters.

**Figure 9.**The sets of optimal time lags (

**a**) ${\tau}_{*j}^{\left(X\right)}$ and (

**b**) ${\tau}_{*j}^{\left(Y\right)}$ for the time series depicted in Figure 8. Circles in (

**c**) denote the absolute differences ${\tau}_{*j}^{\left(X,Y\right)}$ between optimal time lags for $X$ and $Y$. The solid red line in (

**c**) corresponds to the averaged absolute differences ${A}^{\left(X,Y\right)}=\left[{\overline{\tau}}_{1}^{\left(X,Y\right)}{\overline{\tau}}_{2}^{\left(X,Y\right)}\dots {\overline{\tau}}_{16}^{\left(X,Y\right)}\right]$.

**Figure 10.**The scheme of the application of Algorithm C on the experimental data. The horizontal axis of the depicted data corresponds to the indices of the time series.

**Figure 11.**Dendrogram plot for the two-day (2015/02/27 18:05:00 through 2015/03/01 18:05:00) data. Numbers on the X axis represent participants (numbered from 1 to 20).

**Figure 12.**The variation of the slow dynamics of the geometrical synchronization constructed from optimal time lags for participant 7 (red line) and participant 20 (blue line) for the time period between 2015/02/27 18:05:00–2015/03/01 18:05:00.

**Figure 13.**The variation of the slow dynamics of the geometrical synchronization constructed from optimal time lags for participant 7 (red line) and participant 15 (blue line) for the time period between 2015/02/27 18:05:00–2015/03/01 18:05:00.

**Figure 14.**Dendrogram plot for the two-week data. Numbers on the X axis represent participants (numbered from 1 to 20).

**Figure 15.**The graph of the evaluated interaction levels between participants. Nodes represent participants (numbered from 1 to 20). A line with an arrow pointing from person $a$ to $b$ ($a,b$), represents that person $a$ feels positive about person $b$. The width of the line is proportional to the overall ($a,b$) interaction value (sum of $a$’s ratings of the interaction with the $b$’s ratings over the 14 days).

**Figure 16.**Participant 15’s change of status during self-evaluation in points (max—10, min—0; Y axis) over the 14 days (X axis). The green, black, red, and blue lines correspond to the self-evaluation of social, general, physical, and emotional states, respectively.

N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | −1 | −1 | 2 | 2 | 1 | −1 | ||||||||||||||

2 | 1 | 2 | ||||||||||||||||||

3 | ||||||||||||||||||||

4 | −1 | 4 | ||||||||||||||||||

5 | ||||||||||||||||||||

6 | ||||||||||||||||||||

7 | 3 | |||||||||||||||||||

8 | ||||||||||||||||||||

9 | 2 | 1 | 6 | |||||||||||||||||

10 | 4 | 1 | 1 | 1 | 4 | 1 | ||||||||||||||

11 | 1 | |||||||||||||||||||

12 | 2 | 1 | 1 | −1 | ||||||||||||||||

13 | 1 | 4 | 2 | 8 | 1 | 3 | ||||||||||||||

14 | 2 | |||||||||||||||||||

15 | 2 | |||||||||||||||||||

16 | 2 | |||||||||||||||||||

17 | ||||||||||||||||||||

18 | ||||||||||||||||||||

19 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | |||||||||

20 | 1 | 1 | 3 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Timofejeva, I.; McCraty, R.; Atkinson, M.; Joffe, R.; Vainoras, A.; Alabdulgader, A.A.; Ragulskis, M.
Identification of a Group’s Physiological Synchronization with Earth’s Magnetic Field. *Int. J. Environ. Res. Public Health* **2017**, *14*, 998.
https://doi.org/10.3390/ijerph14090998

**AMA Style**

Timofejeva I, McCraty R, Atkinson M, Joffe R, Vainoras A, Alabdulgader AA, Ragulskis M.
Identification of a Group’s Physiological Synchronization with Earth’s Magnetic Field. *International Journal of Environmental Research and Public Health*. 2017; 14(9):998.
https://doi.org/10.3390/ijerph14090998

**Chicago/Turabian Style**

Timofejeva, Inga, Rollin McCraty, Mike Atkinson, Roza Joffe, Alfonsas Vainoras, Abdullah A. Alabdulgader, and Minvydas Ragulskis.
2017. "Identification of a Group’s Physiological Synchronization with Earth’s Magnetic Field" *International Journal of Environmental Research and Public Health* 14, no. 9: 998.
https://doi.org/10.3390/ijerph14090998