# Criticality as a Determinant of Integrated Information Φ in Human Brain Networks

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## Abstract

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## 1. Introduction

## 2. Methods

_{jk}denotes the anatomical connections between oscillator j and k, yielding 1 if a connection exists and 0 otherwise. τ

_{jk}is the time delay between node j and k. θ

_{j}(t) is the phase of oscillator j at time t. ω

_{j}is the intrinsic frequency of oscillator j. The Kuramoto model is the canonical model for coupled oscillators as a first-order approximation of more complex couple oscillatory systems. In general, at a sufficient coupling strength, a system of near-identical coupled limit-cycle oscillators can be approximated by this general phase model [29]. In other words, simulation results with the Kuramoto model hold for other complex network models when simplified with first-order approximation or mean-field approximation. Considering that EEG reflects superficial activity, the EEG network is a good case for the application of the Kuramoto model.

## 3. Results Model Study: Correlation Between Criticality and $\overline{\Phi}$

## 4. A Network Mechanism of the Maximal $\overline{\Phi}$ in a Critical State

## 5. EEG Study: Correlation Between Criticality, $\overline{\Phi}$, and Human Consciousness

## 6. Discussion

## 7. Conclusions

## Supplementary Materials

**Table S1**. Statistical tests of Pair Correlation Function(PCF) and integrated information($\overline{\mathsf{\Phi}}$) among supercritical, critical, and subcritical states in model,

**Table S2**. Statistical tests of Pair Correlation Function(PCF) and integrated information($\overline{\mathsf{\Phi}}$) among baseline, induction, anesthesia, and recovery states in experiment. The baseline, induction, anesthesia, and recovery states are the average of 10-min with 0%, 0.4%, 0.8%, 0% drug concentrations, respectively,

**Figure S1**. In order to test the robustness for the relationship between integrated information and criticality (measured by pair correlation function), we tested another integrated information measure, Φ_SI, recently introduced with its toolbox [50]. We compared the two integrated information measures, (A) $\overline{\mathsf{\Phi}}$ in Figure 2 and (B) Φ_SI based on Queyrannes’s submodular optimization algorithm [48,50]. For Φ_SI, we iterated the simulation 10 times. The error bar indicates the standard deviation at each coupling strength. Both the integrated information measures based on different algorithms demonstrate the maximal Φ values near the critical state. Notably, we did not remove trivial partitions (e.g., partitioning the 78 nodes into 1 node and 77 nodes) in the calculation of Φ_SI, which frequenctly observed with the model data. However, for the prescise calculation of Φ_SI, it should be handled carefully in the future application.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic flow diagram for studying the relationship between criticality (defined as the pair correlation function, PCF), integrated information (Φ), and human consciousness. (

**A**) Model study: A simple coupled oscillator model (Kuramoto model) was implemented on an anatomically informed human brain network structure constructed from diffusion tensor imaging (DTI). The Φ was calculated while modulating the level of criticality with a control parameter (coupling strength). The level of criticality was defined using PCF, a susceptibility measure, of the simulated brain activity. (

**B**) Empirical study: 64-channel electroencephalography (EEG) data derived from seven healthy volunteers were recorded while gradually increasing sevoflurane concentration from 0.4% to 0.6% to 0.8%, then decreasing it from 0.8% to 0.6% to 0.4%. The changes of PCF and Φ were compared with the response rate to verbal commands.

**Figure 2.**Criticality, $\overline{\mathsf{\Phi}}$, and structured functional connectivity in the brain network. (

**A**) When modulating the coupling strength (a control parameter), the maximum $\overline{\mathsf{\Phi}}$ coincides with the maximum criticality, as measured by the pair correlation function (PCF), while the order parameter (dotted line) increases in a monotonic way. (

**B**) Only in the critical state (blue region in Figure 2A), a salient structured functional connectivity, which resembles the structural brain network, emerges in the brain network. Color bar indicates the node degree.

**Figure 3.**Correlations between the functional and structural brain networks near to and far from a critical state. (

**A**) The Spearman correlation coefficient between the 78 phase lag index (PLI) values in the functional brain network and the 78 node degrees in the structural brain network is maximal in the critical state (black line, blue shaded region). (

**B**) The scatter plots (the 78 PLI values versus the 78 node degrees) for the supercritical (green), critical (blue), and subcritical (red) states. The large correlation in the critical state implies a large constraint of the structural network on the functional network. (

**C**) The Spearman correlation coefficients between the instantaneous phases of alpha oscillations and the node degrees of the 78 nodes in the structural brain network. The large temporal variation indicates a large repertoire of functional connectivity, which is a characteristic of a critical state.

**Figure 4.**Criticality, integrated information, and level of human consciousness during general anesthesia. (

**A**) The PCF and $\overline{\mathsf{\Phi}}$ of 64-channel EEG correlate with the response rate (grey area), which was modulated with increasing anesthetic concentrations. (

**B**) The Spearman correlation coefficients between the PLI networks of the EEG and the simulated EEG based on the anatomical brain network and critical state. The conscious states (baseline, induction, and recovery) show larger correlations, which imply a stronger constraint of the structural brain network on the EEG in conscious states. As a result, the balance between the large constraint of the structural brain network and the large repertoire (i.e., large PCF) of the functional brain network may be the network basis of the large $\overline{\mathsf{\Phi}}$ of EEG in the conscious brains.

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Kim, H.; Lee, U.
Criticality as a Determinant of Integrated Information Φ in Human Brain Networks. *Entropy* **2019**, *21*, 981.
https://doi.org/10.3390/e21100981

**AMA Style**

Kim H, Lee U.
Criticality as a Determinant of Integrated Information Φ in Human Brain Networks. *Entropy*. 2019; 21(10):981.
https://doi.org/10.3390/e21100981

**Chicago/Turabian Style**

Kim, Hyoungkyu, and UnCheol Lee.
2019. "Criticality as a Determinant of Integrated Information Φ in Human Brain Networks" *Entropy* 21, no. 10: 981.
https://doi.org/10.3390/e21100981