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A neurodynamic model of inter-brain coupling in the gamma band

A neurodynamic model of inter-brain coupling in the gamma band The use of EEG to simultaneously record multiple brains (i.e., hyperscanning) during social interactions has led to the discovery of inter-brain coupling (IBC). IBC is defined as the neural synchronization between people and is considered to be a marker of social interaction. IBC has previously been observed across different frequency bands, including theta [4–7 Hz]. Given the prox- imity of this frequency range with behavioral rhythms, models have been able to combine IBC in theta with sensorimotor coordi- nation patterns. Interestingly, empirical EEG-hyperscanning results also report the emergence of IBC in the gamma range [>30 Hz]. Gamma oscillations’ fast and transient nature makes a direct link between gamma-IBC and other (much slower) interpersonal dynamics difficult, leaving gamma-IBC without a plausible model. However, at the intrabrain level, gamma activity is coupled with the dynamics of lower frequencies through cross-frequency coupling (CFC). This paper provides a biophysical explanation, through the simulation of neural data, for the emergence of gamma inter-brain coupling using a Kuramoto model of four oscilla- tors divided into two separate (brain) units. By modulating both the degree of inter-brain coupling in the theta band (i.e., between-units coupling) and CFC (i.e., intraunit theta-gamma coupling), we provide a theoretical explanation of the observed gamma-IBC phenomenon in the EEG-hyperscanning literature. NEW & NOTEWORTHY The last years were marked by an increasing interest in multiple-brain recordings. However, the inter- brain coupling arising across interacting individuals also sparks debates about the underlying biological mechanisms. The inter- brain coupling in the gamma band [>30 Hz] was particularly criticized for lacking a theoretical framework. Here, by using biologi- cally informed neural simulations with the Kuramoto model, we assess the role of intra- and inter-brain neural dynamics in the emergence of inter-brain synchrony in the gamma band. cross-frequency coupling; EEG; hyperscanning; Kuramoto model; synchronization social neuroscience to study interpersonal brain dynamics INTRODUCTION (1–4). Specifically, electroencephalography (EEG) hyperscan- Social interaction is a core feature of human life. However, ning led to the report of a phenomenon called inter-brain the neural mechanisms that support our capacity to interact coupling [IBC, but see also similar terms such as inter-brain with others remain poorly understood due to the fact that synchrony/synchronization (3, 5, 6)], a temporal synchroni- neuroscience has mainly focused on recording single partici- zation of neural signals across brains when participants pants in isolation rather than assessing several interacting interact (7–10). Inter-brain coupling is now widely accepted agents simultaneously. Recently, however, the simultaneous as a marker of social engagement and successful interperso- recording of multiple brains, commonly known as hyper- nal communication, despite the doubt regarding its epiphe- scanning, has become a popular method within the field of nomenal nature not being completely lifted (11, 12). Further- Correspondence: G. Dumas ([email protected]). Submitted 18 May 2022 / Revised 5 September 2022 / Accepted 6 September 2022 www.jn.org 0022-3077/22 Copyright© 2022 The Authors. Licensed under Creative Commons Attribution CC-BY 4.0. 1085 Published by the American Physiological Society. MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND more, the current knowledge on IBC relies on empirical data. As mentioned earlier, our model is composed of four oscil- In this paper, we aimed at simulating EEG hyperscanning lators, two in each brain unit (oscillators A1 and A2 in brain data using a simplistic model to advance our understanding unit A and B1 and B2 in brain unit B, see Fig. 1). The connec- of underlying phenomena captured by inter-brain coupling tivity matrix K is illustrated in Fig. 1B. The inter-brain cou- methods. pling between A1 and B1 in the theta band and the intrabrain IBC has been mostly highlighted using phase synchrony theta-gamma CFC (between A1 and A2 and B1 and B2) were indices such as the phase-locking value (PLV; 13), the phase- programmed to range from 0 to 1, by steps of 0.1. locking index (PLI; 14) and the partial directed coherence Inter-brain Coupling Measure (PDC; 15). This revealed a variety of inter-brain synchroniza- tions across different frequency bands, including in the theta To quantify the coupling between the A2 and B2 gamma (4–7Hz) (10, 16–18) and the alpha/mu (8–13 Hz) ranges (3, 19, oscillators, we used the phase locking value (i.e., PLV) 20). According to the laws of coordination dynamics, behav- which provides a frequency-specific phase synchrony ioral rhythms of participants during an interaction can both measure between two signals across time (13)and is influence and be reciprocally influenced by the behavior of widely used in both intra- and inter-brain EEG studies (3, the partner, resulting in a convergence of the dyad’sbehav- 24, 45, 46). We applied a Hilbert transform to extract the ioral rhythms toward a common frequency (21, 22). Given instantaneous phase of the signals from oscillators A2 and the proximity of theta and alpha/mu frequencies with the B2 (see Fig. 1A)and computed the c-PLV via the following rhythms of behavioral sensorimotor coordination, IBC in equation: these ranges can be modeled according to the same coordi- iðh ðtÞh ðtÞÞ nation dynamics principles and reciprocal exchanges of in- A2 B2 PLV ¼ e ; A2;B2 formation across members of an interaction, leading to the t¼1 brain-behavior coordination dynamics framework (3, 21–24). where T is the number of sampled time points and h (t)and A2 However, inter-brain synchronizations in higher frequen- h (t) are the instantaneous phase values of oscillators B and B2 cies such as in the gamma range (>30 Hz) have also been D at time point t. PLV values range from 0 to 1, where 0 reported (3, 25–27). Gamma waves are fast and ultra-fast reflects an absence of phase synchrony and 1 an identical rel- transient oscillations believed to support local computation ative phase between the two signals. (28–31). Hence, the time scale of this frequency band cannot be directly attributed to behavioral coordination rhythms, Signal-to-Noise Ratio leading some to question the validity of observed gamma- By extracting the signal and the noise amplitude of the IBC (12). On the other hand, at the intrabrain level, an simulated time series, we computed the signal-to-noise ratio increase of local gamma amplitude is supported by the phase (SNR) using the following formula: of lower frequencies theta-gamma coupling) through cross- Signal frequency coupling (CFC; 32–34). CFC has been described as 10 SNR ðdBÞ¼ 20  log : a physiological mechanism capable of coordinating neural Noise dynamics across spatial and temporal scales, where the fir- Data Availability ing of local neural populations is controlled by larger whole brain dynamics (35). Based on these characteristics, we pro- The current manuscript only relies on computational simula- pose that previously observed gamma IBC during social tions, no data has been recorded. All codes are available at interactions can be explained by the combination of two https://github.com/ppsp-team/Hyper-Model (archive https://doi. neurophysiological occurrences: 1) inter-brain coupling of org/10.5281/zenodo.7047107). The data folder contains the nu- lower frequency waves according to coordination dynamics merical matrices generated to reproduce Fig. 2. and 2) intrabrain level cross-frequency coupling. RESULTS AND DISCUSSION MATERIAL AND METHODS Schematic Model Dynamical Model of Gamma IBC with Kuramoto To test our hypothesis that IBC in gamma can be Leveraging Python implementation of Kuramoto systems accounted for by the joint effects of theta IBC and theta- (36), we implemented our model in Python 3.7 (37)using gamma CFC, we conceptualized a simple computational libraries such as Numpy (38), and SciPy (39) for the computa- model of coupled oscillators simulating EEG data from two tional analyses, and Matplotlib (40) for the visualization. brains. We opted for a model capable of capturing the ele- The Kuramoto model also holds several assumptions: that all mentary principles of intra- and inter-brain coupling with oscillators are identical, that the oscillators are innately minimal features. As illustrated in Fig. 1A, our model con- coupled, and that the oscillations follow a sinusoidal pattern tains two brains, represented as two separate units (units A (41–44). Finally, the phase h of an oscillator i at time t is and B), that are coupled together through inter-brain cou- described by the following dynamical equation: pling in the theta band (h), while within each unit theta and X gamma (c) are coupled through CFC. dh ðtÞ ¼ x ðtÞþ K sinðh ðtÞ h ðtÞÞ; i ij j i dt j¼1 Kuramoto Simulations and Signal-to-Noise Ratio where K is the coupling matrix with coupling from oscilla- Previous studies used the Kuramoto model for weakly ij tor i to oscillator j and x is the frequency of oscillator i. coupled oscillators (41, 44) to demonstrate the effect of 1086 J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND (±1, i.e., within the gamma range) and without time delays A Output V Output Va ariable: (i.e., we did not include time lags in our model). We simu- Inter-brain Inter-br rain n γ γ-PL -PLV V lated time series with a length of 40 s (by steps of 10 ms). The inter-brain coupling between A1 and B1 in the h band and Intra-brain θ-γ Intra-brain θ-γ CFC C the intrabrain theta-gamma CFC (between A1 and A2 and B1 CF CFC C and B2) were programmed to range from 0 to 1, by steps of 0.1. Simulations were run 10,000 times to obtain stable results. We applied a Gaussian noise (μ =0, r = 0.6), resulting in a signal-to-noise ratio of 6.575 dB, comparable with SNR found in the EEG literature (49). Inter-brain Coupling in Inter-brain Co oupling in θ θ Inter-brain c Connectivity Bra Brain in Un Unit it A A Brain Unit B Brain Unit B To estimate inter-brain connectivity between the simu- lated time series of oscillator A2 and B2, we computed the phase locking value (see MATERIAL AND METHODS). The c-PLV matrix containing the inter-brain connectivity val- ues between the oscillators A2 and B2 is illustrated by the heatmap in Fig. 2.The first observation is that constant high PLV values in gamma occur for low CFC values (i.e., between 0 and 0.2). This is shown by consistent high PLV scores along the x-axis on the heatmap in Fig. 2.This ob- C servation seems paradoxical, as intuitively one would expect an increase in PLV values with increasing theta IBC (i.e., a gradient from low to high PLV values along the x-axis). However, these PLV values can be labeled as spu- rious coupling, given that the constant high PLV values aresimply a result of thelack of modulation in CFC strength (0–0.2). Whenintrabraingamma is not modu- lated by theta within either of the units, the PLV measure confuses the similarity of the oscillators in each brain for inter-brain synchrony (45). Our result is a good illustra- tion of the fact that absolute PLV values alone are not meaningful for empirical hyperscanning data, but that a careful choice of contrasting conditions (e.g., synchrony vs. nonsynchrony) is necessary to interpret IBC values correctly (50). Figure 1. Overview of the model. A: schematic representation of our Our second crucial finding is that the increase of h-c two-brain model, capable of capturing the elementary principles of CFC (above 0.3 on the y-axis of the heatmap in Fig. 2)to- intra- (A1-A2 and B1-B2) and inter-brain (A1-B1) coupling. B: the connec- gether with an increase of inter-brain coupling in the h tivity matrix K, where 1 means the presence and 0 the absence of a band is associated with higher PLV scores (see top-right coupling between the oscillators. C: example of time and time- frequency series of the simulated neural data. CFC, cross-frequency corner values of the heatmap in Fig. 2). In addition, we coupling; PLV, phase-locking value. subtracted the values (i.e., DPLV) with the highest degree of IBC (i.e., inter-brain coupling in the theta band = 1) from the values with the lowest degree of coupling (i.e., intrabrain anatomical and functional connectivity on IBC inter-brain coupling in the theta band = 0) and performed (23), as well as interpersonal behavioral synchronization aone-sample z test (n = 10,000) on DPLV values against 0 strategies and how they rely on the relationship between for each values of CFC, confirming incremental effect of intra- and interunit coupling (47). Generally, the Kuramoto CFC on c-PLV (see bar plots in Fig. 2): Z =0 = 0.164, CFC model describes a system of coupled oscillators where the P =0.565; Z =0.1 =2.050, P < 0.001; Z =0.2 = CFC CFC individual oscillators are attracted and entrained to the aver- 13.657, P < 0.0001; Z =0.3 = 30.943, P < 0.0001; Z = CFC CFC age rate (in our case, this refers to phase convergence rather 0.4 = 48.661, P < 0.0001; Z =0.5 =57.844, P < 0.0001; CFC than frequency convergence (23, 42). Even though Kuramoto Z =0.6 =64.354, P < 0.0001; Z =0.7 =75.685, P < CFC CFC models do not account for the nonstationarity nature of neu- 0.0001; Z =0.8 =77.022, P < 0.0001; Z =0.9 =76.077, P < CFC CFC ral data, they can still uncover essential concepts of neuro- 0.0001; Z =1 =72.245, P < 0.0001. These results highlight CFC oscillatory dynamics and explain synchronous coupling in the impactof the jointincrease of h inter-brain coupling and complex systems (48). Here, we implemented our model h-c cross-frequency coupling on c-PLV. Future empirical using Kuramoto oscillators, following the connectivity ma- research in both humans (using M/EEG) and animal models trix K (Fig. 1B). The mean frequency of the oscillators A1 and (51–53) should account for these two phenomena by target h B1 was set at 6 Hz (±1, i.e., within the theta range), and the and c dynamics in social contexts, both at the intra- and inter- mean frequency of the oscillators A2 and B2 was set at 40 Hz brain level. Causal relationship between c IBC and h-c CFC J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org 1087 MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND Figure 2. Effect of inter-brain coupling in h and h-c cross-frequency on inter-brain coupling in gamma. Phase-locking value (PLV) scores between A2 and B2 oscillators reveal that a joint increase of inter-brain coupling in h and h-c cross-frequency coupling account for the observation of inter-brain coupling in the c band. Panels on the right show that subtracting high h-inter-brain coupling and low h-IBC (i.e., DPLV) and Z-testing the values against 0 confirms the pattern observed on the heatmap. CFC, cross-frequency coupling. could also be investigated through perturbation/stimulation IBC. Furthermore, our model confirms the hypothesis that techniques using transcranial electrical stimulation (tES) IBC in gamma can be ascribed to intrabrain theta-gamma techniques (5, 54). cross-frequency and theta inter-brain coupling, by showing higher PLV scores during the joint increase of both parame- Biophysical Explanation for gamma Inter-brain Coupling ters. Hence, our simulations give a biophysical model to the observed gamma-IBC in the EEG-hyperscanning literature Our model provides a biophysical explanation for (3, 25–27). Beyond supporting the empirical observations of gamma inter-brain coupling, an observation that has often gamma-IBC, this model also nuances the claim about their been reported in human EEG hyperscanning studies. epiphenomenalism (12). Indeed, although the causal link is Although IBC in lower frequencies such as theta and alpha not directly carried by gamma frequency exchange of infor- is in line with brain-behavior coordination dynamics (3, mation, gamma rhythms associated with intra and inter- 21–24), doubts regarding the validity and the reliability of brain processes can statistically be synchronized during observed gamma inter-brain coupling (i.e., gamma IBC not social interaction and thus lead to Hebbian learning in net- being a neural correlate of interaction) are not completely works associated with self and other behavior (56), thus lifted (12). Based on evidence from intrabrain, inter-brain, and com- developing mirroring in related brain structure (57, 58). Our putational connectivity studies (3, 23, 47, 55), but also recent results also illustrate the importance of taking into account account of inter-brain correlations in animals including bats both intra- and inter-brain factors in the design, analysis, (53) and mice (51, 52), we hypothesized that gamma IBC and interpretation of hyperscanning experiments. could be explained and modeled according to two distinct Altogether, through computational modeling, our approach neurophysiological processes, namely inter-brain coupling and results advance our mechanistic understanding of IBC, which is crucial to reaching a coherent theoretical framework in theta (10, 16–18) and intrabrain theta-gamma cross-fre- describing causal relations between socio-cognitive factors, quency coupling (33, 34). First, we showed that our 4-oscilla- tor Kuramoto model, divided into two separate units, was behavioral dynamics, and neural mechanisms involved in able to replicate the core characteristics of both CFC and multi-brain neuroscience (59). 1088 J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND 1999. doi:10.1002/(SICI)1097-0193(1999)8:4<194::AID-HBM4>3.0. GRANTS CO;2-C. This study was supported by the Institute for Data Valorization, 14. Tass P, Rosenblum MG, Weule J, Kurths J, Pikovsky A, Volkmann Montreal (IVADO; CF00137433). G.D.’s salary was covered by the J, Schnitzler A, Freund H-J. Detection of n:m phase locking from Fonds de recherche du Quebec (FRQ; 285289). L.A. was funded noisy data: application to magnetoencephalography. Phys Rev Lett 81: 3291–3294, 1998. doi:10.1103/PhysRevLett.81.3291. by the Foundation Mindstrong (2021) of the Jewish General 15. Baccalá LA, Sameshima K. Partial directed coherence: a new con- Hospital in Montrea  l. This study was enabled in part by support cept in neural structure determination. Biol Cybern 84: 463–474, provided by Calcul Quebe  c (www.calculquebec.ca) and Digital 2001. doi:10.1007/PL00007990. Research Alliance of Canada (www.alliancecan.ca). 16. Astolfi L, Toppi J, Borghini G, Vecchiato G, Isabella R, De Vico Fallani F, Cincotti F, Salinari S, Mattia D, He B, Caltagirone C, DISCLOSURES Babiloni F. Study of the functional hyperconnectivity between cou- ples of pilots during flight simulation: an EEG hyperscanning study. No conflicts of interest, financial or otherwise, are declared by Conf Proc IEEE Eng Med Biol Soc 2011: 2338–2341, 2011. doi:10.1109/ the authors. IEMBS.2011.6090654. 17. Kawasaki M, Yamada Y, Ushiku Y, Miyauchi E, Yamaguchi Y. Interbrain synchronization during coordination of speech rhythm in AUTHOR CONTRIBUTIONS human-to-human social interaction. Sci Rep 3: 1692, 2013. 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A neurodynamic model of inter-brain coupling in the gamma band

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0022-3077
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1522-1598
DOI
10.1152/jn.00224.2022
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Abstract

The use of EEG to simultaneously record multiple brains (i.e., hyperscanning) during social interactions has led to the discovery of inter-brain coupling (IBC). IBC is defined as the neural synchronization between people and is considered to be a marker of social interaction. IBC has previously been observed across different frequency bands, including theta [4–7 Hz]. Given the prox- imity of this frequency range with behavioral rhythms, models have been able to combine IBC in theta with sensorimotor coordi- nation patterns. Interestingly, empirical EEG-hyperscanning results also report the emergence of IBC in the gamma range [>30 Hz]. Gamma oscillations’ fast and transient nature makes a direct link between gamma-IBC and other (much slower) interpersonal dynamics difficult, leaving gamma-IBC without a plausible model. However, at the intrabrain level, gamma activity is coupled with the dynamics of lower frequencies through cross-frequency coupling (CFC). This paper provides a biophysical explanation, through the simulation of neural data, for the emergence of gamma inter-brain coupling using a Kuramoto model of four oscilla- tors divided into two separate (brain) units. By modulating both the degree of inter-brain coupling in the theta band (i.e., between-units coupling) and CFC (i.e., intraunit theta-gamma coupling), we provide a theoretical explanation of the observed gamma-IBC phenomenon in the EEG-hyperscanning literature. NEW & NOTEWORTHY The last years were marked by an increasing interest in multiple-brain recordings. However, the inter- brain coupling arising across interacting individuals also sparks debates about the underlying biological mechanisms. The inter- brain coupling in the gamma band [>30 Hz] was particularly criticized for lacking a theoretical framework. Here, by using biologi- cally informed neural simulations with the Kuramoto model, we assess the role of intra- and inter-brain neural dynamics in the emergence of inter-brain synchrony in the gamma band. cross-frequency coupling; EEG; hyperscanning; Kuramoto model; synchronization social neuroscience to study interpersonal brain dynamics INTRODUCTION (1–4). Specifically, electroencephalography (EEG) hyperscan- Social interaction is a core feature of human life. However, ning led to the report of a phenomenon called inter-brain the neural mechanisms that support our capacity to interact coupling [IBC, but see also similar terms such as inter-brain with others remain poorly understood due to the fact that synchrony/synchronization (3, 5, 6)], a temporal synchroni- neuroscience has mainly focused on recording single partici- zation of neural signals across brains when participants pants in isolation rather than assessing several interacting interact (7–10). Inter-brain coupling is now widely accepted agents simultaneously. Recently, however, the simultaneous as a marker of social engagement and successful interperso- recording of multiple brains, commonly known as hyper- nal communication, despite the doubt regarding its epiphe- scanning, has become a popular method within the field of nomenal nature not being completely lifted (11, 12). Further- Correspondence: G. Dumas ([email protected]). Submitted 18 May 2022 / Revised 5 September 2022 / Accepted 6 September 2022 www.jn.org 0022-3077/22 Copyright© 2022 The Authors. Licensed under Creative Commons Attribution CC-BY 4.0. 1085 Published by the American Physiological Society. MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND more, the current knowledge on IBC relies on empirical data. As mentioned earlier, our model is composed of four oscil- In this paper, we aimed at simulating EEG hyperscanning lators, two in each brain unit (oscillators A1 and A2 in brain data using a simplistic model to advance our understanding unit A and B1 and B2 in brain unit B, see Fig. 1). The connec- of underlying phenomena captured by inter-brain coupling tivity matrix K is illustrated in Fig. 1B. The inter-brain cou- methods. pling between A1 and B1 in the theta band and the intrabrain IBC has been mostly highlighted using phase synchrony theta-gamma CFC (between A1 and A2 and B1 and B2) were indices such as the phase-locking value (PLV; 13), the phase- programmed to range from 0 to 1, by steps of 0.1. locking index (PLI; 14) and the partial directed coherence Inter-brain Coupling Measure (PDC; 15). This revealed a variety of inter-brain synchroniza- tions across different frequency bands, including in the theta To quantify the coupling between the A2 and B2 gamma (4–7Hz) (10, 16–18) and the alpha/mu (8–13 Hz) ranges (3, 19, oscillators, we used the phase locking value (i.e., PLV) 20). According to the laws of coordination dynamics, behav- which provides a frequency-specific phase synchrony ioral rhythms of participants during an interaction can both measure between two signals across time (13)and is influence and be reciprocally influenced by the behavior of widely used in both intra- and inter-brain EEG studies (3, the partner, resulting in a convergence of the dyad’sbehav- 24, 45, 46). We applied a Hilbert transform to extract the ioral rhythms toward a common frequency (21, 22). Given instantaneous phase of the signals from oscillators A2 and the proximity of theta and alpha/mu frequencies with the B2 (see Fig. 1A)and computed the c-PLV via the following rhythms of behavioral sensorimotor coordination, IBC in equation: these ranges can be modeled according to the same coordi- iðh ðtÞh ðtÞÞ nation dynamics principles and reciprocal exchanges of in- A2 B2 PLV ¼ e ; A2;B2 formation across members of an interaction, leading to the t¼1 brain-behavior coordination dynamics framework (3, 21–24). where T is the number of sampled time points and h (t)and A2 However, inter-brain synchronizations in higher frequen- h (t) are the instantaneous phase values of oscillators B and B2 cies such as in the gamma range (>30 Hz) have also been D at time point t. PLV values range from 0 to 1, where 0 reported (3, 25–27). Gamma waves are fast and ultra-fast reflects an absence of phase synchrony and 1 an identical rel- transient oscillations believed to support local computation ative phase between the two signals. (28–31). Hence, the time scale of this frequency band cannot be directly attributed to behavioral coordination rhythms, Signal-to-Noise Ratio leading some to question the validity of observed gamma- By extracting the signal and the noise amplitude of the IBC (12). On the other hand, at the intrabrain level, an simulated time series, we computed the signal-to-noise ratio increase of local gamma amplitude is supported by the phase (SNR) using the following formula: of lower frequencies theta-gamma coupling) through cross- Signal frequency coupling (CFC; 32–34). CFC has been described as 10 SNR ðdBÞ¼ 20  log : a physiological mechanism capable of coordinating neural Noise dynamics across spatial and temporal scales, where the fir- Data Availability ing of local neural populations is controlled by larger whole brain dynamics (35). Based on these characteristics, we pro- The current manuscript only relies on computational simula- pose that previously observed gamma IBC during social tions, no data has been recorded. All codes are available at interactions can be explained by the combination of two https://github.com/ppsp-team/Hyper-Model (archive https://doi. neurophysiological occurrences: 1) inter-brain coupling of org/10.5281/zenodo.7047107). The data folder contains the nu- lower frequency waves according to coordination dynamics merical matrices generated to reproduce Fig. 2. and 2) intrabrain level cross-frequency coupling. RESULTS AND DISCUSSION MATERIAL AND METHODS Schematic Model Dynamical Model of Gamma IBC with Kuramoto To test our hypothesis that IBC in gamma can be Leveraging Python implementation of Kuramoto systems accounted for by the joint effects of theta IBC and theta- (36), we implemented our model in Python 3.7 (37)using gamma CFC, we conceptualized a simple computational libraries such as Numpy (38), and SciPy (39) for the computa- model of coupled oscillators simulating EEG data from two tional analyses, and Matplotlib (40) for the visualization. brains. We opted for a model capable of capturing the ele- The Kuramoto model also holds several assumptions: that all mentary principles of intra- and inter-brain coupling with oscillators are identical, that the oscillators are innately minimal features. As illustrated in Fig. 1A, our model con- coupled, and that the oscillations follow a sinusoidal pattern tains two brains, represented as two separate units (units A (41–44). Finally, the phase h of an oscillator i at time t is and B), that are coupled together through inter-brain cou- described by the following dynamical equation: pling in the theta band (h), while within each unit theta and X gamma (c) are coupled through CFC. dh ðtÞ ¼ x ðtÞþ K sinðh ðtÞ h ðtÞÞ; i ij j i dt j¼1 Kuramoto Simulations and Signal-to-Noise Ratio where K is the coupling matrix with coupling from oscilla- Previous studies used the Kuramoto model for weakly ij tor i to oscillator j and x is the frequency of oscillator i. coupled oscillators (41, 44) to demonstrate the effect of 1086 J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND (±1, i.e., within the gamma range) and without time delays A Output V Output Va ariable: (i.e., we did not include time lags in our model). We simu- Inter-brain Inter-br rain n γ γ-PL -PLV V lated time series with a length of 40 s (by steps of 10 ms). The inter-brain coupling between A1 and B1 in the h band and Intra-brain θ-γ Intra-brain θ-γ CFC C the intrabrain theta-gamma CFC (between A1 and A2 and B1 CF CFC C and B2) were programmed to range from 0 to 1, by steps of 0.1. Simulations were run 10,000 times to obtain stable results. We applied a Gaussian noise (μ =0, r = 0.6), resulting in a signal-to-noise ratio of 6.575 dB, comparable with SNR found in the EEG literature (49). Inter-brain Coupling in Inter-brain Co oupling in θ θ Inter-brain c Connectivity Bra Brain in Un Unit it A A Brain Unit B Brain Unit B To estimate inter-brain connectivity between the simu- lated time series of oscillator A2 and B2, we computed the phase locking value (see MATERIAL AND METHODS). The c-PLV matrix containing the inter-brain connectivity val- ues between the oscillators A2 and B2 is illustrated by the heatmap in Fig. 2.The first observation is that constant high PLV values in gamma occur for low CFC values (i.e., between 0 and 0.2). This is shown by consistent high PLV scores along the x-axis on the heatmap in Fig. 2.This ob- C servation seems paradoxical, as intuitively one would expect an increase in PLV values with increasing theta IBC (i.e., a gradient from low to high PLV values along the x-axis). However, these PLV values can be labeled as spu- rious coupling, given that the constant high PLV values aresimply a result of thelack of modulation in CFC strength (0–0.2). Whenintrabraingamma is not modu- lated by theta within either of the units, the PLV measure confuses the similarity of the oscillators in each brain for inter-brain synchrony (45). Our result is a good illustra- tion of the fact that absolute PLV values alone are not meaningful for empirical hyperscanning data, but that a careful choice of contrasting conditions (e.g., synchrony vs. nonsynchrony) is necessary to interpret IBC values correctly (50). Figure 1. Overview of the model. A: schematic representation of our Our second crucial finding is that the increase of h-c two-brain model, capable of capturing the elementary principles of CFC (above 0.3 on the y-axis of the heatmap in Fig. 2)to- intra- (A1-A2 and B1-B2) and inter-brain (A1-B1) coupling. B: the connec- gether with an increase of inter-brain coupling in the h tivity matrix K, where 1 means the presence and 0 the absence of a band is associated with higher PLV scores (see top-right coupling between the oscillators. C: example of time and time- frequency series of the simulated neural data. CFC, cross-frequency corner values of the heatmap in Fig. 2). In addition, we coupling; PLV, phase-locking value. subtracted the values (i.e., DPLV) with the highest degree of IBC (i.e., inter-brain coupling in the theta band = 1) from the values with the lowest degree of coupling (i.e., intrabrain anatomical and functional connectivity on IBC inter-brain coupling in the theta band = 0) and performed (23), as well as interpersonal behavioral synchronization aone-sample z test (n = 10,000) on DPLV values against 0 strategies and how they rely on the relationship between for each values of CFC, confirming incremental effect of intra- and interunit coupling (47). Generally, the Kuramoto CFC on c-PLV (see bar plots in Fig. 2): Z =0 = 0.164, CFC model describes a system of coupled oscillators where the P =0.565; Z =0.1 =2.050, P < 0.001; Z =0.2 = CFC CFC individual oscillators are attracted and entrained to the aver- 13.657, P < 0.0001; Z =0.3 = 30.943, P < 0.0001; Z = CFC CFC age rate (in our case, this refers to phase convergence rather 0.4 = 48.661, P < 0.0001; Z =0.5 =57.844, P < 0.0001; CFC than frequency convergence (23, 42). Even though Kuramoto Z =0.6 =64.354, P < 0.0001; Z =0.7 =75.685, P < CFC CFC models do not account for the nonstationarity nature of neu- 0.0001; Z =0.8 =77.022, P < 0.0001; Z =0.9 =76.077, P < CFC CFC ral data, they can still uncover essential concepts of neuro- 0.0001; Z =1 =72.245, P < 0.0001. These results highlight CFC oscillatory dynamics and explain synchronous coupling in the impactof the jointincrease of h inter-brain coupling and complex systems (48). Here, we implemented our model h-c cross-frequency coupling on c-PLV. Future empirical using Kuramoto oscillators, following the connectivity ma- research in both humans (using M/EEG) and animal models trix K (Fig. 1B). The mean frequency of the oscillators A1 and (51–53) should account for these two phenomena by target h B1 was set at 6 Hz (±1, i.e., within the theta range), and the and c dynamics in social contexts, both at the intra- and inter- mean frequency of the oscillators A2 and B2 was set at 40 Hz brain level. Causal relationship between c IBC and h-c CFC J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org 1087 MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND Figure 2. Effect of inter-brain coupling in h and h-c cross-frequency on inter-brain coupling in gamma. Phase-locking value (PLV) scores between A2 and B2 oscillators reveal that a joint increase of inter-brain coupling in h and h-c cross-frequency coupling account for the observation of inter-brain coupling in the c band. Panels on the right show that subtracting high h-inter-brain coupling and low h-IBC (i.e., DPLV) and Z-testing the values against 0 confirms the pattern observed on the heatmap. CFC, cross-frequency coupling. could also be investigated through perturbation/stimulation IBC. Furthermore, our model confirms the hypothesis that techniques using transcranial electrical stimulation (tES) IBC in gamma can be ascribed to intrabrain theta-gamma techniques (5, 54). cross-frequency and theta inter-brain coupling, by showing higher PLV scores during the joint increase of both parame- Biophysical Explanation for gamma Inter-brain Coupling ters. Hence, our simulations give a biophysical model to the observed gamma-IBC in the EEG-hyperscanning literature Our model provides a biophysical explanation for (3, 25–27). Beyond supporting the empirical observations of gamma inter-brain coupling, an observation that has often gamma-IBC, this model also nuances the claim about their been reported in human EEG hyperscanning studies. epiphenomenalism (12). Indeed, although the causal link is Although IBC in lower frequencies such as theta and alpha not directly carried by gamma frequency exchange of infor- is in line with brain-behavior coordination dynamics (3, mation, gamma rhythms associated with intra and inter- 21–24), doubts regarding the validity and the reliability of brain processes can statistically be synchronized during observed gamma inter-brain coupling (i.e., gamma IBC not social interaction and thus lead to Hebbian learning in net- being a neural correlate of interaction) are not completely works associated with self and other behavior (56), thus lifted (12). Based on evidence from intrabrain, inter-brain, and com- developing mirroring in related brain structure (57, 58). Our putational connectivity studies (3, 23, 47, 55), but also recent results also illustrate the importance of taking into account account of inter-brain correlations in animals including bats both intra- and inter-brain factors in the design, analysis, (53) and mice (51, 52), we hypothesized that gamma IBC and interpretation of hyperscanning experiments. could be explained and modeled according to two distinct Altogether, through computational modeling, our approach neurophysiological processes, namely inter-brain coupling and results advance our mechanistic understanding of IBC, which is crucial to reaching a coherent theoretical framework in theta (10, 16–18) and intrabrain theta-gamma cross-fre- describing causal relations between socio-cognitive factors, quency coupling (33, 34). First, we showed that our 4-oscilla- tor Kuramoto model, divided into two separate units, was behavioral dynamics, and neural mechanisms involved in able to replicate the core characteristics of both CFC and multi-brain neuroscience (59). 1088 J Neurophysiol doi:10.1152/jn.00224.2022 www.jn.org MODEL OF INTER-BRAIN COUPLING IN THE GAMMA BAND 1999. doi:10.1002/(SICI)1097-0193(1999)8:4<194::AID-HBM4>3.0. GRANTS CO;2-C. This study was supported by the Institute for Data Valorization, 14. Tass P, Rosenblum MG, Weule J, Kurths J, Pikovsky A, Volkmann Montreal (IVADO; CF00137433). G.D.’s salary was covered by the J, Schnitzler A, Freund H-J. Detection of n:m phase locking from Fonds de recherche du Quebec (FRQ; 285289). L.A. was funded noisy data: application to magnetoencephalography. Phys Rev Lett 81: 3291–3294, 1998. doi:10.1103/PhysRevLett.81.3291. by the Foundation Mindstrong (2021) of the Jewish General 15. Baccalá LA, Sameshima K. Partial directed coherence: a new con- Hospital in Montrea  l. This study was enabled in part by support cept in neural structure determination. Biol Cybern 84: 463–474, provided by Calcul Quebe  c (www.calculquebec.ca) and Digital 2001. doi:10.1007/PL00007990. Research Alliance of Canada (www.alliancecan.ca). 16. Astolfi L, Toppi J, Borghini G, Vecchiato G, Isabella R, De Vico Fallani F, Cincotti F, Salinari S, Mattia D, He B, Caltagirone C, DISCLOSURES Babiloni F. Study of the functional hyperconnectivity between cou- ples of pilots during flight simulation: an EEG hyperscanning study. No conflicts of interest, financial or otherwise, are declared by Conf Proc IEEE Eng Med Biol Soc 2011: 2338–2341, 2011. doi:10.1109/ the authors. IEMBS.2011.6090654. 17. Kawasaki M, Yamada Y, Ushiku Y, Miyauchi E, Yamaguchi Y. Interbrain synchronization during coordination of speech rhythm in AUTHOR CONTRIBUTIONS human-to-human social interaction. Sci Rep 3: 1692, 2013. 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Journal of Neurophysiology – The American Physiological Society

Published: Nov 1, 2022

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