# Investigation of Corticomuscular Functional Coupling during Hand Movements Using Vine Copula

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## Abstract

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## 1. Introduction

`→`EMG, uplink EMG

`→`EEG) coupling was observed between EEG and EMG [12]. However, as the functional coupling between EEG and EMG involves nonlinear causality, the GC approach based on the established model cannot effectively describe the characteristics of nonlinear and higher-order coupling between EEG and EMG [24]. In related studies on motion recognition, the GC approach cannot effectively describe nonlinear characteristics and relying on linear characteristics alone may result in the inability to effectively distinguish human actions. Therefore, to overcome the afore-described problems, researchers have introduced a copula-based method for application to corticomuscular coupling analysis [25].

## 2. Materials and Methods

#### 2.1. Vine Copula

#### 2.2. GARCH Model and Marginal Distribution

#### 2.3. Correlation Measure Based on Copula Function

#### 2.4. Modeling Step of Vine Copula

- Calculate the Kendall correlation coefficients between all variables, compare the sum of absolute values, and choose the largest spanning tree as the structure of the first layer tree;
- Select the optimal pair-copula function of the first layer tree structure by using the Akaike information criterion (AIC) and Bayesian information criterion (BIC), and calculate the conditional marginal distribution function;
- According to step (2), calculate the Kendall correlation coefficients between all conditional variables, and set the generating tree with the maximum sum of the absolute values of all $\tau $ correlation coefficients as the structure of the second layer tree;
- Select the second-level tree structure by using the AIC and BIC, and calculate the conditional marginal distribution function. Repeat steps (3) and (4) until only two nodes and one edge are left;
- The decomposition expression of the joint density function of random variables is expressed by the marginal density function and pair-copula function.

#### 2.5. Granger Causality

#### 2.6. Computation of Network Characteristics

## 3. Experimental Procedure

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, Z.; Huang, Z.; He, W.; Su, C.-Y. Adaptive impedance control for an upper limb robotic exoskeleton using biological signals. IEEE Trans. Ind. Electron.
**2016**, 64, 1664–1674. [Google Scholar] [CrossRef] - Namazi, H.; Ala, T.S. Decoding of simple and compound limb motor imagery movements by fractal analysis of Electroencephalogram (EEG) signal. Fractals
**2019**, 27, 1950041. [Google Scholar] [CrossRef] - Qi, J.; Jiang, G.; Li, G.; Sun, Y.; Tao, B. Surface EMG hand gesture recognition system based on PCA and GRNN. Neural Comput. Appl.
**2020**, 32, 6343–6351. [Google Scholar] [CrossRef] - van Wijk, B.C.; Beek, P.J.; Daffertshofer, A. Neural synchrony within the motor system: What have we learned so far? Front. Hum. Neurosci.
**2012**, 6, 252. [Google Scholar] [CrossRef] [PubMed] [Green Version] - van Vliet, M.; Liljeström, M.; Aro, S.; Salmelin, R.; Kujala, J. Analysis of functional connectivity and oscillatory power using DICS: From raw MEG data to group-level statistics in Python. Front. Neurosci.
**2018**, 12, 586. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lebedev, M.A.; Nicolelis, M.A. Nicolelis, Brain-machine interfaces: From basic science to neuroprostheses and neurorehabilitation. Physiol. Rev.
**2017**, 97, 767–837. [Google Scholar] [PubMed] - Zhang, J.; Wang, B.; Zhang, C.; Xiao, Y.; Wang, M.Y. An EEG/EMG/EOG-based multimodal human-machine interface to real-time control of a soft robot hand. Front. Neurorobot.
**2019**, 13, 7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yang, Y.; Solis-Escalante, T.; Yao, J.; Van Der Helm, F.C.T.; Dewald, J.P.A.; Schouten, A.C.; Van Der Helm, F.C.T. Nonlinear Connectivity in the Human Stretch Reflex Assessed by Cross-Frequency Phase Coupling. Int. J. Neural Syst.
**2016**, 26, 1650043. [Google Scholar] [CrossRef] - Larsen, L.H.; Zibrandtsen, I.C.; Wienecke, T.; Kjaer, T.W.; Christensen, M.S.; Nielsen, J.B.; Langberg, H. Corticomuscular coherence in the acute and subacute phase after stroke. Clin. Neurophysiol.
**2017**, 128, 2217–2226. [Google Scholar] [CrossRef] [Green Version] - Dal Maso, F.; Longcamp, M.; Cremoux, S.; Amarantini, D. Effect of training status on beta-range corticomuscular coherence in agonist vs. antagonist muscles during isometric knee contractions. Exp. Brain Res.
**2017**, 235, 3023–3031. [Google Scholar] [CrossRef] - Babiloni, C.; Vecchio, F.; Bares, M.; Brazdil, M.; Nestrasil, I.; Eusebi, F.; Rossini, P.M.; Rektor, I. Functional coupling between anterior prefrontal cortex (BA10) and hand muscle contraction during intentional and imitative motor acts. NeuroImage
**2008**, 39, 1314–1323. [Google Scholar] [CrossRef] [PubMed] - Witham, C.L.; Riddle, C.N.; Baker, M.R.; Baker, S.N. Contributions of descending and ascending pathways to corticomuscular coherence in humans. J. Physiol.
**2011**, 589, 3789–3800. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lóopez-Larraz, E.; Birbaumer, N.; Ramos-Murguialday, A. Ramos-Murguialday, A hybrid EEG-EMG BMI improves the detection of movement intention in cortical stroke patients with complete hand paralysis. In Proceedings of the 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Honolulu, HI, USA, 18–21 July 2018; IEEE: Piscataway, NJ, USA; pp. 2000–2003. [Google Scholar]
- Antelis, J.M.; Montesano, L.; Ramos-Murguialday, A.; Birbaumer, N.; Minguez, J. Decoding upper limb movement attempt from EEG measurements of the contralesional motor cortex in chronic stroke patients. IEEE Trans. Biomed. Eng.
**2016**, 64, 99–111. [Google Scholar] [CrossRef] [PubMed] - Edwards, L.; King, E.M.; Buetefisch, C.M.; Borich, M.R. Putting the “Sensory” Into Sensorimotor Control: The Role of Sensorimotor Integration in Goal-Directed Hand Movements After Stroke. Front. Integr. Neurosci.
**2019**, 13, 16. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chowdhury, A.; Raza, H.; Dutta, A.; Prasad, G. EEG-EMG based hybrid brain computer interface for triggering hand exoskeleton for neuro-rehabilitation. In Proceedings of the Advances in Robotics, New Delhi, India, 28 June–2 July 2017; pp. 1–6. [Google Scholar]
- Youssofzadeh, V.; Zanotto, D.; Wong-Lin, K.; Agrawal, S.K.; Prasad, G. Directed functional connectivity in fronto-centroparietal circuit correlates with motor adaptation in gait training. IEEE Trans. Neural Syst. Rehabil. Eng.
**2016**, 24, 1265–1275. [Google Scholar] [CrossRef] [PubMed] - Conway, B.A.; Halliday, D.M.; Shahani, U.; Maas, P.; Weir, A.I.; Rosenberg, J.R.; Farmer, S.F. Common frequency components identified from correlations between magnetic recordings of cortical activity and the electromyogram in man. J. Physiol.
**1995**, 483, P35. [Google Scholar] - de Vries, I.E.J.; Daffertshofer, A.; Stegeman, D.F.; Boonstra, T.W. Functional connectivity in the neuromuscular system underlying bimanual coordination. J. Neurophysiol.
**2016**, 116, 2576–2585. [Google Scholar] [CrossRef] [Green Version] - Hu, G.; Yang, W.; Chen, X.; Qi, W.; Li, X.; Du, Y.; Xie, P. Estimation of time-varying coherence amongst synergistic muscles during wrist movements. Front. Neurosci.
**2018**, 12, 537. [Google Scholar] [CrossRef] - Fallani, F.D.V.; Pichiorri, F.; Morone, G.; Molinari, M.; Babiloni, F.; Cincotti, F.; Mattia, D. Multiscale topological properties of functional brain networks during motor imagery after stroke. Neuroimage
**2013**, 83, 438–449. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Sheng, Y.; Zeng, J.; Liu, H. Corticomuscular coherence for upper arm flexor and extensor muscles during isometric exercise and cyclically isokinetic movement. Front. Neurosci.
**2019**, 13, 522. [Google Scholar] [CrossRef] - Sameshima, K.; Baccala, L.A. Methods in Brain Connectivity Inference Through Multivariate Time Series Analysis; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Marinazzo, D.; Liao, W.; Chen, H.; Stramaglia, S. Nonlinear connectivity by Granger causality. NeuroImage
**2011**, 58, 330–338. [Google Scholar] [CrossRef] [PubMed] - Gao, X.; Shen, W.; Ting, C.-M.; Cramer, S.C.; Srinivasan, R.; Ombao, H. Estimating Brain Connectivity Using Copula Gaussian Graphical Models. In Proceedings of the International Symposium on Biomedical Imaging, Venice, Italy, 8–11 April 2019; pp. 108–112. [Google Scholar]
- Sklar, M. Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris
**1959**, 8, 229–231. [Google Scholar] - Dauwels, J.; Yu, H.; Wang, X.; Vialatte, F.; Latchoumane, C.-F.V.; Jeong, J.; Cichocki, A. Inferring brain networks through graphical models with hidden variables. In Machine Learning and Interpretation in Neuroimaging; Springer: Berlin/Heidelberg, Germany, 2012; pp. 194–201. [Google Scholar]
- Aas, K.; Czado, C.; Frigessi, A.; Bakken, H. Pair-copula constructions of multiple dependence. Insur. Math. Econ.
**2009**, 44, 182–198. [Google Scholar] [CrossRef] [Green Version] - Bedford, T.; Daneshkhah, A.; Wilson, K.J. Approximate uncertainty modeling in risk analysis with vine copulas. Risk Anal.
**2016**, 36, 792–815. [Google Scholar] [CrossRef] [Green Version] - Schepsmeier, U. A goodness-of-fit test for regular vine copula models. Econom. Rev.
**2019**, 38, 25–46. [Google Scholar] [CrossRef] [Green Version] - Krithikaivasan, B.; Zeng, Y.; Deka, K.; Medhi, D. ARCH-based traffic forecasting and dynamic bandwidth provisioning for periodically measured nonstationary traffic. IEEE/ACM Trans. Netw.
**2007**, 15, 683–696. [Google Scholar] [CrossRef] - Kim, S. Forecasting internet traffic by using seasonal GARCH models. J. Commun. Netw.
**2011**, 13, 621–624. [Google Scholar] [CrossRef] - Cormen, T.H.; Leiserson, C.E.; Rivest, R.L. Introduction to Algorithms (3); MIT Press: Cambridge, MA, USA, 2009. [Google Scholar]
- Xi, X. Construction and analysis of cortical–muscular functional network based on EEG-EMG coherence using wavelet coherence. Neurocomputing
**2021**, 438, 248–258. [Google Scholar] [CrossRef] - Muthuraman, M.; Galka, A.; Deuschl, G.; Heute, U.; Raethjen, J. Dynamical correlation of non-stationary signals in time domain—A comparative study. Biomed. Signal Process. Control
**2010**, 5, 205–213. [Google Scholar] [CrossRef] - Zhao, J.; Zhou, W.; Liu, K.; Cai, D. Application of SVM and Wavelet Analysis in EEG Classif ication. J. Biomed. Eng.
**2011**, 28, 277–279. [Google Scholar] - Clemens, B.; Puskás, S.; Besenyei, M.; Spisák, T.; Opposits, G.; Hollódy, K.; Fogarasi, A.; Fekete, I.; Emri, M. Neurophysiology of juvenile myoclonic epilepsy: EEG-based network and graph analysis of the interictal and immediate preictal states. Epilepsy Res.
**2013**, 106, 357–369. [Google Scholar] [CrossRef] [PubMed] - Xi, X. Emotion-movement relationship: A study using functional brain network and cortico-muscular coupling. J. Neurosci. Methods
**2021**, 362, 109320. [Google Scholar] [CrossRef] [PubMed] - Bezruchko, B.P.; Ponomarenko, V.I.; Prokhorov, M.D.; Smirnov, D.A.; Tass, P.A. Modeling nonlinear oscillatory systems and diagnostics of coupling between them using chaotic time series analysis: Applications in neurophysiology. Phys. Uspekhi
**2008**, 51, 304–310. [Google Scholar] [CrossRef] - Granger, C.W.J. Testing For Causality: A Personal Viewpoint. J. Econ. Dyn. Control
**1980**, 2, 329–352. [Google Scholar] [CrossRef]

**Figure 1.**Setup for measuring motor tasks. (

**a**) Experimental scene, (

**b**) hand open, (

**c**) hand closed, (

**d**) wrist flexion, (

**e**) wrist extension, (

**f**) Delsys sEMG sensor diagram.

**Figure 2.**Experimental paradigms. HO: hand open, HC: hand close, WF: wrist flexion, WE: wrist extension.

**Figure 3.**Electrode placement for (

**a**) EEG, (

**b**) EMG. FDS: flexor digitorum superficialis; BR: brachioradialis; BB: biceps brachii; ED: extensor digitorum; FCU: flexor carpi ulnaris; ECU: extensor carpi ulnaris. As shown in Figure 3a, EEG data were recorded by a digital EEG apparatus (g.MOBllab + MP—2015) at the following eight positions of the 10–20 systems: Cz, C3, C4, Cp1, Cp2, FC3, FC4, and Fz (Fpz was selected as the grounding electrode). Considering that all the subjects were right-handed, some of the electrodes were only placed in the left brain region to build a much leaner function network.

**Figure 4.**Heatmaps of the Kendall rank correlation coefficient of the EEG and EMG signals corresponding to (

**a**) hand open, (

**b**) hand closed, (

**c**) wrist flexion, and (

**d**) wrist extension.

**Figure 5.**Kendall rank correlations of (

**a**) C3-FDS, (

**b**) C3-ED, (

**c**) C3-ECU, and (

**d**) C3-FCU for the four different activities. (* means p < 0.05, *** means p < 0.001, **** means p < 0.0001). HO: hand open, HC: hand closed, WF: wrist flexion and WE: wrist extension.

**Figure 6.**Vine copula first-layer tree structure corresponding to different activities: (

**a**) hand open, (

**b**) hand closed, (

**c**) wrist flexion, (

**d**) wrist extension. It can be observed from Figure 4 that the Kendall rank correlation coefficient between EEG and EMG is low for all the four activities. Nevertheless, the EEG signals are highly correlated. We also note that the correlation coefficient between the signals of the same type (EMG–EMG and EEG–EEG) is always higher than that between two different types of signals (EEG–EMG). Generally, the correlation between the Fz and other signals is strong, and the same is true for FC3 and FC4. For the HO and HC movements, a similarly strong correlation exists between the EEG signals. For WF and WE, the correlation between the EEG signals is similar. Meanwhile, the Kendall rank correlation of the FDS and FCU is higher than that of the other EMG channels for WF. However, in the case of WE, the correlation between BR and ECU is higher compared with that for other EMG channels.

**Figure 7.**Determination of threshold. K: average node degree; SD: standard deviation of the average node degree.

**Figure 8.**Corticomuscular network for different motor activities: (

**a**) hand open, (

**b**) hand closed, (

**c**) wrist flexion, (

**d**) wrist extension.

**Figure 9.**Clustering coefficient (CC) and characteristic path length (L) of the network for the different motor activities considered: (

**a**) CC obtained with GC, (

**b**) L obtained with GC, (

**c**) CC obtained with vine copula, (

**d**) L obtained with vine copula. (* means p < 0.05, ** means p < 0.01,*** means p < 0.001).

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**MDPI and ACS Style**

Ye, F.; Ding, J.; Chen, K.; Xi, X.
Investigation of Corticomuscular Functional Coupling during Hand Movements Using Vine Copula. *Brain Sci.* **2022**, *12*, 754.
https://doi.org/10.3390/brainsci12060754

**AMA Style**

Ye F, Ding J, Chen K, Xi X.
Investigation of Corticomuscular Functional Coupling during Hand Movements Using Vine Copula. *Brain Sciences*. 2022; 12(6):754.
https://doi.org/10.3390/brainsci12060754

**Chicago/Turabian Style**

Ye, Fei, JinSuo Ding, Kai Chen, and Xugang Xi.
2022. "Investigation of Corticomuscular Functional Coupling during Hand Movements Using Vine Copula" *Brain Sciences* 12, no. 6: 754.
https://doi.org/10.3390/brainsci12060754