All patients with nontraumatic cardiac arrest after cardiopulmonary and cerebral resuscitation (CPCR) were screened for study inclusion. The patients were treated by standardized post-resuscitation treatment based on clinical conditions. Based on the post-resuscitation protocol in our institute, the feasibility of target temperature management (TTM) of all sudden cardiac death patients should be evaluated. If the post-resuscitation GCS is less than M5, TTM should be applied, unless the patient has contraindications of TTM such as very unstable hemodynamic status or severe bleeding tendency. Due to facility limitation, these patients received a standard EEG examination in an EEG room. According to the post-resuscitation protocol, standard EEG would be arranged on day 3 and day 7 after resuscitation, including patients who received TTM. If the patient’s consciousness fully recovered after resuscitation, the EEG will be canceled. For patients with clinical suspicions of nonconvulsive seizures or status epilepticus by the neurologist, repeated EEG will be arranged to increase the yield rate. EEG signals with a signal-to-noise ratio (SNR) computed as the power within the lowest frequency and the highest frequency bands of interest (0.15–35 Hz) over the total power of the rest of the signal less than 12 dB are considered having poor EEG quality and were excluded from the study. The neurological outcome was assessed at discharge and 3 months after discharge based on clinical records. Survivors with a cerebral performance category (CPC) score of 1–2 were classified as “good outcome”, and patients with a CPC score of 3–5 or who died during the hospitalization were classified as “poor outcome” (Perkins et al., 2015; Geocadin et al., 2019).

We included 75 comatose patients who were resuscitated following in-hospital or out-of-hospital nontraumatic cardiac arrest between 2013 and 2017 in the intensive care unit (ICU) of National Taiwan University Hospital (NTUH). The mean EEG performing day was day 4 after cardiac arrest. All patients who had good EEG quality were included for further analysis. A total of 12 patients (16%) were defined as good outcome. A total of 32 patients were defined as poor neurological outcome, and 31 patients died in the hospital stay. Most of the patients who underwent the EEG exam were sedation-free. Only 11 patients had sedatives at the time of the EEG exam. Within these 11 patients, two had good neurological outcome, five had poor neurological outcome, and four patients died. Patients with poor neurological outcome and mortality (63 patients, 84%) were defined as poor outcome.

In this section, we established the mathematical relationship between GFS and phase coherence. It was to prove that GFS is mainly determined by the phase difference between signals. In addition, the influence of noise in GFS was explained and demonstrated. Finally, we proposed an anti-noise algorithm to solve the problem concerning noise.

We provided a brief description of band-limited signals and the covariance matrix from complex Fourier transform of multiple channels in the succeeding sections.

We used standard EEG for analysis. The electrodes were attached at 19 standard sites on the scalp (international 10/20 system placements Fp1/2, F3/4, C3/4, P3/4, O1/2, F7/8, T3/4, T5/6, Fz, Cz, and Pz). We separated the EEG frequency bands into delta (0.5-4Hz), theta (4-8Hz), alpha (8-12Hz), and beta (12-40Hz). All EEGs were interpreted by neurologists, and the patterns were defined according to the ACNS EEG terminology (Westhall et al., 2016) as qualitative EEGs. Suppression, burst-suppression, periodic discharges, rhythmic discharges, and evolving seizure activity are highly malignant patterns. Discontinuous background, low-voltage, and unreactive background are malignant patterns. Continuous background indicates good outcome, which is defined as continuous normal voltage (>20 μV) background and preserved reactivity to stimuli.

For quantitative EEG analysis, for each frequency band, we averaged the value in the frequency domain and obtained the data around the middle of the recording in the time domain to avoid the artifact caused by emergency treatment. This value was taken as quantitative EEG. For each frequency band, 19 points of quantitative EEG were acquired.

We computed GFS by transforming an EEG signal with M channels into the frequency domain using Fourier transform (FT) (Eq. 3). Then, the FT of the signal yields a complex form made up of real (Eq. 4) and imaginary parts (Eq. 5) for each frequency. For every particular frequency band, pairs of resulting complex Fourier coefficients from all channels can be mapped as points in a two-dimensional diagram. The resulting pairs of all channels   were then quantified using principal component analysis (PCA) resulting in two eigenvalues per frequency. Using these eigenvalues (Eq. 8), we compute GFS as the ratio of the resulting eigenvalues (Eq. 9). For a more thorough step-by-step explanation, refer to Section 2.2.1.

The data were expressed as mean ± standard deviation for continuous variables and percentage for categorical variables. The baseline characteristics between good vs poor outcome were compared by using t-test for continuous variables and chi-square test for categorical variables. Logistic regression analysis was performed to find the predictors for outcome. The baseline characteristics including age, gender, out-of-hospital cardiac arrest (OHCA), shockable rhythm, cardiopulmonary resuscitation (CPR) duration, use of target temperature management (TTM), use of extracorporeal membrane oxygenation (EMCO), Acute Physiology and Chronic Health Evaluation II (APACHE II) score, and comorbidities such as coronary artery disease (CAD), diabetes mellitus (DM), hypertension (HTN), chronic kidney disease (CKD), and atrial fibrillation (AF) were entered as covariates. Receiver operating characteristic (ROC) analysis was used to determine the optimal prediction model and evaluate the ability of GFS to discriminate good from poor outcome with the area under curve (AUC) as an indicator. Sample size analysis of the area under ROC curve using Medcalc Version 20.027, with type I error 0.05, type II error 0.2, area under ROC curve 0.8, and ratio of sample size in negative/positive groups 5, estimated the sample size was 54, in which nine were defined as good outcome and 45 were defined as poor outcome.

In the previous sections, we discussed the covariance matrix (Eq. 7) and approximated the signal model with narrow phase variation (Eq. 16) to prove that the synchronized phase has a single eigenvalue and high GFS score, that is, we could assume that phase coherence is the dominant factor to GFS.

As seen in the previous result, we have already discovered that the GFS is sensitive to phase distribution, and GFS somehow has a relationship with the phase coherence. Based on the process in Section 3.1.1, we also adapted a 20-channel signal model. In this section, we altered the strength of modulation function φ i ( t ) (Eq. 1) with the constant amplitude. We assumed φ i ( t ) as a random variable with normal distribution and used 10 different standard distributions ( σ : 0 : 2 π ), that is, we were going to observe the score of GFS and global phase coherence during the phase manipulation.

The signals become chaotic when standard distribution of φ i ( t ) increase, so it could also be regarded as a desynchronization indicator. The result in Figure 3A revealed that both GFS and global phase coherence decrease when the strength of standard distribution of φ i ( t ) (desynchronization indicator) increases. Therefore, GFS has a similar function with the phase coherence–based method. Nonetheless, the curve of GFS decays faster than the curve of global phase coherence to the strength of desynchronization. On the other hand, GFS was proven as a much sensitive measurement tool than global phase coherence.

Moreover, in addition to the comparison in sensitivity of both methods to signal synchronization, we also performed a comparison of the sensitivity to the length of the signal. In clinical scenarios, EEG is vulnerable to different kinds of noise, which may badly affect the resulting GFS scores. Based on our proposed robustness algorithm, we could divide signals into several segments and reject the noise events, that is, the longer signals help us remove the noise events and acquire a genuine GFS score.

In addition, we not only considered the temporal noise event but also made GFS robust to the poor signal quality condition. We added different levels of noise into signals to study the robustness between different signal length and the background noise. (Figure 3B).

In Figure 3B, we tested 50-s and 10-s signals with the phase coherence–based method and GFS under different SNR scenarios. Though both methods decreased as the noise level increased, changes in signal length do not affect the global phase coherence score result, while longer signals showed significant improvement in the GFS score.

The demographics and clinical parameters of all SCA survivors are shown in Table 1 and Figure 4. The mean age in the good-outcome group was younger than that in the poor-outcome group (51.6 ± 15.7 vs 68.1 ± 12.9, p < 0.001). The patients in both the groups were male-dominant. The patients with initial shockable rhythm were more in the good-outcome group (75 vs 15%, p < 0.001). The patients in the good-outcome group had a higher percentage of receiving TTM than that in the poor-outcome group (83.3 vs 44.4%, p = 0.014). The APACHE II score was higher in the poor-outcome group (25.4 ± 6.2 vs 19.3 ± 5, p = 0.0019). There was no statistical difference in the percentage of OHCA, duration of CPR, and rate of receiving ECMO therapy between the two groups. There was also no statistical difference in rates of comorbidities including CAD, DM, HTN, CKD, and AF between the two groups. Multivariate logistic regression analysis showed that only initial shockable rhythm was an independent factor for predicting good outcome (OR 2.15, 95% CI 1.6–46.1, p = 0.012).

In one previous study by Grmec and Gasparovic (2001), among the scoring systems including Mainz Emergency Evaluation System (MEES), Glasgow Coma Scale (GCS), and APACHE II, GCS performed the best in the prediction of mortality for nontraumatic coma patients. Schefold et al. (2009) also considered GCS as a powerful tool in predicting the neurological outcome of cardiac arrest patients treated with therapeutic hypothermia. APACHE II (Acute Physiology and Chronic Health Evaluation II) is a severity-of-disease classification system that includes a GCS score as part of evaluation. The APACHE II, although it encompasses the GCS score as part of the evaluation, was only a modest indicator of illness severity and predictor of mortality/neurologic morbidity (Donnino et al., 2013). In this study, the APACHE II score was higher in the poor-outcome group in the univariate analysis but became insignificant after the adjustment of other covariates in regression analysis.

In this study, the mean age in the good-outcome group was significantly less. The patients in the good-outcome group had more initial shockable rhythm, higher percentage of receiving TTM, and lower APACHE II score. The previous studies showed that TTM increased the rate of a favorable neurologic outcome and reduced mortality (Lascarrou et al., 2019). However, the APACHE II score of patients with poor outcomes in our study was higher than that of patients with good outcomes. Therefore, more patients in the poor-outcome group might have contraindications of TTM, and the multivariate logistic regression analysis showed that only initial shockable rhythm was an independent factor for predicting good outcome, rather than TTM or not or APACHE II score.

The prediction of clinical outcomes will help optimize the treatment and benefit the patients. In the clinical setting, neurocritical illnesses are diseases requiring expensive treatment associated with poor outcomes. Survival with poor neurological outcomes is very resource-demanding. The treatment plans may be altered by the socioeconomic condition of the patient or his family. The family and clinician may avoid futile medical care if poor outcomes can be identified early. On the other hand, if we can recognize patients who are potential independent survivors, aggressive treatment is indicated, and the patients will be beneficial to the treatment.

Neurologists interpret EEG according to the ACNS EEG terminology (Westhall et al., 2016) as qualitative EEGs. In this study, highly malignant and malignant patterns both showed high sensitivity but low specificity in predicting poor outcome. Continuous background indicates good outcome, which is defined as continuous normal voltage (>20 μV) background and preserved reactivity to stimuli. In this study, benign patterns showed high specificity but low sensitivity. There is interrater variability in qualitative EEG interpretation. Westhall et al. (2015) published a multinational study on the interrater reliability of EEG interpretation using the 2013 revised ACNS terminology among comatose patients following cardiac arrest. There was a high interrater agreement (κ 0.71) for highly malignant patterns and moderate agreement (κ 0.42) for malignant patterns. Identifying an unreactive EEG was fair (κ 0.26). Therefore, traditional qualitative evaluation of EEG should be interpreted with caution. Using quantitative methods may lower the interrater variation. Duez et al. (2018) published a quantitative method to evaluate EEG reactivity. In the study, agreement among experts on overall EEG reactivity varied from 53 to 83% (κ 0.05–0.64) and reached 100% (κ1) between two quantitative EEG reactivity calculators.

Neurophysiological monitoring tests, such as EEG and somatosensory evoked responses (SSEPs), may play a role in the prediction of neurological outcome prediction (Rossetti et al., 2016). EEG is a signal with a complex structure, and several EEG features have been proven to be correlated with the degree of neuronal injury (Westhall, 2017). However, EEG signal evaluation requires specific expertise and can suffer from interrater variability (Westhall et al., 2015). Computational processing of the EEG or quantitative EEG (qEEG) has been utilized for neurological outcome prediction after cardiac arrest. The parameters derived from qEEG such as bispectral index (BIS), amplitude-integrated EEG, burst-suppression ratio, or entropy, have been demonstrated to be valuable in clinical practice (Leary et al., 2010; Rundgren et al., 2010; Seder et al., 2010; Noirhomme et al., 2014; Selig et al., 2014; Stammet et al., 2014). Among the parameters, some studies showed that BIS had better discriminative power in predicting the outcome (Leary et al., 2010; Seder et al., 2010; Selig et al., 2014; Stammet et al., 2014), while other computer-assisted analyses had high specificity but relatively low sensitivity for outcome prediction.

In this study, we demonstrated that patients with a good outcome had higher EEG power in all four frequency bands. The higher EEG power represents higher variability of neural activities and might be associated with better brain function after cardiac arrest. Our result also showed that among the frequency bands, the power of the alpha band was the best in predicting the outcome. This result was correlated with visual analysis findings (Westhall, 2017) and was compatible with previous studies (Wiley et al., 2018; Kota et al., 2020).

Alpha waves are neural oscillations in the frequency range of 8–12 Hz, likely originating from the synchronous and coherent electrical activity of thalamic pacemaker cells in humans (Hughes and Crunelli, 2005). Alpha waves appear in the EEG during the resting but wakeful state, while beta waves are seen in more alert status, and delta/theta waves are seen in the sleep state. In this study, the alpha band showed the best predictive value among all four frequency bands, which is compatible with the clinical neurological status of these patients: comatose but going to wake.

Understanding the mathematical perspective of a system describes how a system works. It uncovers the underlying essential aspects. Based on the original concept by Koenig et al. (2001), GFS focuses on the local distribution between phase and amplitude variation. They explained that channels are synchronized if they have the same phase, counter phase, and a much elongated distribution on the sine–cosine plane, which could indicate a higher degree of synchronization. The data simulations presented in this study have proven the initial claim that both parameters, indeed, play a part in the calculation of GFS. The distribution of amplitude may determine the tolerance of phase variation with the GFS method. Wide amplitude variation signals could get a higher GFS score with wider phase variation than the narrow amplitude variation signals. However, regardless of the amplitude variation, the phase distribution is still the dominant factor to decide the GFS score. Figure 1 shows that the narrower the distribution for phase, the higher the value for GFS, while for amplitude, an increase in amplitude does not correlate to higher GFS if the distribution of phase is wide. This claim is true to both traditional GFS and GFS without the zero-mean process. Hence, although both amplitude and phase contribute to obtaining GFS, interference in phase variation drastically changes the possibility of generating a good GFS score.

In addition, we want to address the critical limitation of the GFS method. When the different channels in a signal share uniform amplitude, even though they have a perfect phase-locking effect, the resulting GFS score would decrease. At the same time, GFS will not be able to detect a perfect phase-locking signal with uniform phase and signals with random phase and amplitude. Fortunately, in clinical applications, we usually record multichannel EEG signals, and it is rare for the amplitude of the different electrodes to have equal magnitude.

Furthermore, in Figure 2, we showed different scenarios of simulated data and provided the sine–cosine plot for GFS. Moreover, the values for GFS and phase coherence were also included. Comparing all scenarios, GFS and phase coherence were at their highest at constant phase even though the amplitude varies for every channel. The random phase with constant amplitude resulted in a much lower value both in GFS and phase coherence compared to data simulation with constant phase due to more random phases related to wider distribution. In addition, we showed that the two measures declined as the interference gets more chaotic, but GFS showed more sensitivity than the phase coherence measure in terms of increased interference. Moreover, longer signal length has better robustness to attenuate the background noise affect in the GFS method, resulting in a better GFS score since we could also reject the spike-like artifact in longer length, though this scenario does not apply with global phase coherence.

GFS is a novel method to analyze the functional synchronization of brain activity recorded by EEG, which was introduced by Koenig et al. (2001). GFS can estimate the connectivity of brain electric activities among different brain regions and is effective in the diagnosis of some neurologic or psychiatric diseases such as Alzheimer’s disease (Ma et al., 2014) or obsessive–compulsive disorder (Ozcoban et al., 2018). To our knowledge, our study is the first one to utilize GFS as an EEG analysis method for neurological outcome prediction in cardiac arrest survivors. In addition, we believe that we are the first to present and explain the true dynamics between GFS and phase coherence from a mathematical point of view. Delving deeper into the method gives new perspective and an in-depth understanding on how it works. Based on the mathematical review presented in the Materials and Methods section of this article, we provided solid proof that GFS is mainly based on phase coherence than its amplitude. Furthermore, based on the results, among the four frequency bands, the alpha band showed the best sensitivity and specificity to predict neurological outcome. By combining EEG power and GFS, the predictive value is even better. This implies that the connectivity of brain electric activities is an important marker of brain function after cardiac arrest.