Subjects consisted of 14 HANS patients (female: 14, mean: 22.79 years, standard deviation: 4.49 years) and 12 young controls (male: 7, female: 5, mean: 27.92 years, standard deviation: 10.94 years). HANS patients had involuntary movements and seizures and were deemed to require electroencephalography (EEG) measurement by neurologists. Young controls visited the hospital at the same time as the HANS patients. They had symptoms such as loss of consciousness and seizures, and a neurologist determined that an EEG measurement was necessary, but they were not diagnosed with HANS. These subjects were chosen as the control because it was assumed that it would be difficult to collect the same number of healthy individuals with no symptoms as HANS patients during the study period. Written informed consents were obtained from all participants. The study protocol, including all EEG data analysis, was approved by Research Ethics Committee, Faculty of Medicine, Teikyo University. EEG measurements were performed at Mizonokuchi Hospital, Teikyo University School of Medicine, through a Nihon Kohden EEG-1224. The device was equipped with 16 Ag/AgCl electrodes (channels) (a Nihon Kohden H503A). Electrodes were attached at Fp1, Fp2, F3, F4, C3, C4, P3, P4, O1, O2, F7, F8, T3, T4, T5, and T6 based on the international 10 - 20 system. In addition to the 16 electrodes, A1 and A2 (reference electrodes) were attached to both earlobes. The sampling frequency was set to 500 Hz and a low-pass filter (=120 Hz), except for the Japanese electrical power noise (50 Hz), was set to removed artifacts. All subjects were asked to lie on their backs in a resting position with eyes closed for at least 5 minutes. All EEG data were checked with EEGLAB (provided by the Swartz Center for Computational Neuroscience). For all subjects, EEG data from the second to third minute of measurement were used as stable EEG data for analysis. All EEG data were reconstructed by Fourier transform and inverse Fourier transform into 5 frequency bands: theta (4 - 8 Hz), lower alpha (8 - 10 Hz), upper alpha (10 - 13 Hz), beta (13 - 30 Hz), and gamma (30 - 45 Hz).

After EEG data processing, Synchronization Likelihood (SL) was applied between all two electrodes in each of the five frequency bands [ 10 , 11 ]. The SL value is a measure of synchronization between the two channels, which avoids bias due to the degrees of freedom of the interacting subsystems and can deal with non-stationary dynamics. The basic principle of SL is to divide each time series into a series of “patterns” (short portions of the time series containing several cycles of the main frequency) and to search for repetitions of these patterns [ 11 ].

Figure 1 shows an outline of calculation of SL at reference time $i$. The calculation consisted of following steps: First, the state vector of channel $k$ at reference time $i$ was defined as follows:

$X_{k,i}=\left(x_{k,i},x_{k,i+L},x_{k,i+2\ast L},\cdots ,x_{k,i+\left(m-1\right)\ast L}\right)$ (1)

where $x_{k,i}$ is the time series of channel $k$. $L$ is the lag and $m$ is the dimension of the embedding vector in state space. $L\ast \left(m-1\right)$ is the length of state vector. $L$, $m$, and $L\ast \left(m-1\right)$ were defined as follows:

$L=fs/\left(3\ast HF\right)$ (2)

$L\ast \left(m-1\right)=fs/LF\leftrightarrow m=3\ast HF/LF+1$ (3)

where $fs$ is the sampling frequency in Hz. In this study, $fs$ was defined as 500 Hz. HF and LF are the highest and lowest frequency of the frequency band of interest.

Second, reference vector $X_{A,i}$ and vectors $X_{A,j}$ were constructed ranging from $i-W_2/2$ to $i-W_2/2$ and from $i+W_1/2$ to $i+W_2/2$ in steps of $1/fs$, respectively. Here, $W_1$ and $W_2$ are defined as follows:

$W_1=2\ast L\ast \left(m-1\right)$ (4)

$n_{rec}=\left[W_2-W_1+1\right]\ast p_{ref}\leftrightarrow W_2=\left(n_{rec}/p_{ref}\right)+W_1-1$ (5)

In this study, $n_{rec}=10$ and $p_{ref}=0.01$.

Third, $S{L}_i$ between channels A and B at time $i$ was calculated as follows:

$S{L}_i=n_{AB}/n_{rec}$ (6)

where $n_{AB}$ is the number of simultaneous repetitions in channels A and B, and is defined as follows:

$n_{AB}={\displaystyle {\sum }_{j=i-W_2/2}^{i-W_1/2}n}+{\displaystyle {\sum }_{j=i+W_1/2}^{i+W_2/2}n}$

$n=\theta \left(r_{A,i}-\left|X_{A,i}-X_{A,j}\right|\right)\theta \left(r_{B,i}-\left|X_{B,i}-X_{B,j}\right|\right)$

$\theta \left(x\right)=\{\begin{array}{l}1\text{if}x\ge 0\\ 0\text{if}x<0\end{array}$ (7)

These three steps were repeated to obtain SL time series. To calculate SL time series, reference time $i$increases with 32 ms increment. The first and last data points of SL time series were deleted (40 data points for other frequency bands). SL value is the average of SL time series after deletion of data points.

For each functional connectivity, SL values consisting of data from 26 subjects was compared to the same number of 0. In this study, no functional connectivity was defined that SL value was 0. T-tests were performed with the null hypothesis “there is no functional connectivity between EEG signal pairs.” The null hypothesis has been rejected for all frequency bands.

BFCN consisted of nodes and edges [ 3 ]. In this study, nodes correspond to electrodes where EEGs were recorded. Functional connectivity was represented by SL value. SL values of ${}_{16}{}^{}C{}_2=120$ pairs of nodes were calculated because EEGs were recorded at 16 electrodes. In addition, 120 SL values $S{L}_{ij}\left(i<j;i,j=1,2,\cdots ,16\right)$ were standardized and exponentially transformed ( $\exp \left(Z\left(S{L}_{ij}\right)\right)$).

To verify identifiability between HANS patients and young controls, accuracies of classification were evaluated by machine learning. Leave-one-out cross validation (LOOCV) was applied in this study since the number of subjects is relatively small. We used the gradient boosting for discrimination between HANS patients and young control. This algorithm is one of the ensemble learning methods in machine learning, a method of combining weak learners to build a strong learner. $S{L}_{ij}$ and $\exp \left(Z\left(S{L}_{ij}\right)\right)$ were used as features for each frequency band. Finally, the accuracies were calculated. Accuracies were obtained according to,

$\text{Accuracy}=\left(\text{TP}+\text{TN}\right)/\left(\text{TP}+\text{TN}+\text{FP}+\text{FN}\right)$ (8)

where TP, TN, FP, and FN mean the number of true positives, true negatives, false positives, and false negatives, respectively.

To define high accuracy of classification, random prediction of 26 subjects was executed. 100,000 random predictions were run and a distribution chart of the accuracy was created. We defined the top 2% of accuracies as high accuracy. The top 2% of accuracies was 19/26 = 73.1% or more ( Figure 2 ).

In this study, BFCN was constructed. BFCN is composed of nodes and edges. Nodes refer to 16 electrodes. Edges between pairs of nodes were quantified by $S{L}_{ij}$ or $\exp \left(Z\left(S{L}_{ij}\right)\right)$. Clustering coefficient ( $C$), characteristic path length ( $L$) and small-worldness ( $SW$) were calculated as network indices. These were calculated to each subject and each frequency band.

$C$ represents the local connectivity of the network and has been interpreted as a measure of resilience to random error [ 18 ]:

$C=\left(1/N\right)\ast {\displaystyle {\sum }_{i=1}^N\left({\displaystyle {\sum }_{k\ne i}{\displaystyle {\sum }_{l\ne i,l\ne k}{w}_{ik}{w}_{il}{w}_{kl}}}/{\displaystyle {\sum }_{k\ne i}{\displaystyle {\sum }_{l\ne i,l\ne k}{w}_{ik}{w}_{il}}}\right)}$ (9)

where $w_{ik}$ represents the weights of edges between node $i$ and $k$. $N$ is the number of nodes. $C$ was calculated based on whether there was edge between two nodes connected to node $i$.

$L$ represents the global connectivity of the network and indicates how well integrated a graph is, and how easy it is to transport information or other entities in the network [ 21 ]:

$L=1/\left(N\ast \left(N-1\right)\right){\displaystyle {\sum }_{i,j\in N,i\ne j}{d}_{i,j}}$ (10)

where $d_{ij}$ is the shortest path length between $i$ and $j$. $N$ is the number of nodes. $L$ is the average shortest path length any two nodes. In this study, $L$ was calculated by the average.path.length function (in the igraph package) in R.

$SW$ represents the efficacy of network information transmission [ 22 ]:

$SW=\left(C/C_{rand}\right)/\left(L/L_{rand}\right)$ (11)

where $C_{rand}$ and $L_{rand}$ denote $C$ and $L$ averaged over an ensemble of 50 random networks that were derived from the original networks by randomly reshuffling the edge weights ( $S{L}_{ij}$ or $\exp \left(Z\left(S{L}_{ij}\right)\right)$) [ 21 ].

To compare $C$, $L$, and $SW$ between HANS patients and young controls, we performed two-sided t-tests. HANS patients and young controls were compared for 3 indices and for all 5 frequency bands.

Classification of $S{L}_{ij}$ or $\exp \left(Z\left(S{L}_{ij}\right)\right)$ using gradient boosting algorithm, accuracies of LOOCV were calculated for each frequency band. The results were reported in Table 1 . This table shows high accuracies using $\exp \left(Z\left(S{L}_{ij}\right)\right)$ for theta and lower alpha bands. Accuracies were 73.1% and 76.9%, respectively ( Table 1 ).

Clustering coefficients ( $C$), characteristic path length (L) and small-worldness (SW) of BFCN by $S{L}_{ij}$ or $\exp \left(Z\left(S{L}_{ij}\right)\right)$ were calculated in young controls and HANS patients ( Table 2 , Table 3 ). Significant differences were verified using p-values (p = 0.05). The result showed significant differences at $SW$ using $\exp \left(Z\left(S{L}_{ij}\right)\right)$ for gamma band (p = 0.024) ( Figure 3 ).