**1. Introduction**

Different levels of cognitive demand can accommodate the complexity and variability of the everyday tasks and the environments, and can result in different cognitive loads [1–3]. Continuous high cognitive load will not only lead to inefficient work but also accidents that might lead to life-threatening consequences. In addition, it also has negative effects on physical and mental health, such as insomnia, decreased immunity, susceptibility to infection, and migraines [4–8]. As a practical necessity, the evaluation of cognitive load or mental load has become a hot topic of research. Therefore, it is of practical significance to design and build a system capable of detecting cognitive load. The use of such a system will not only make it possible to assess the impact of different tasks on the cognitive load, but more importantly, a timely and accurate estimate of cognitive load will help to determine the optimum level of mental load, in order to prevent accidents and make workers more compatible with the work environment. Conventionally, the measurement of cognitive load can be divided into subjective and objective measures [9]. Subjective measures are collected via interviews or questionnaires. They are usually unreliable due to the subjective opinions of the participants [10–12]. In contrast, objective measures that are mainly based on task performances or derived from physiological recordings are less intrusive to the task and independent of the participants' opinion. With the development of technology, neurophysiological activities from brain, heart, and eye movement can be recorded and analyzed to reflect the mental state objectively in a noninvasive way [13]. Previous studies have confirmed that signals such as near-infrared spectroscopy (NIRS), functional magnetic

**Citation:** Chen, X.; Xu, G.; Zhang, S.; Zhang, X.; Teng, Z. Building Networks with a New Cross-Bubble Transition Entropy for Quantitative Assessment of Mental Arithmetic Electroencephalogram. *Appl. Sci.* **2022**, *12*, 11165. https://doi.org/ 10.3390/app122111165

Academic Editors: Alexander N. Pisarchik and Alexander E. Hramov

Received: 20 September 2022 Accepted: 2 November 2022 Published: 3 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

resonance imaging (fMRI), electrocorticography (ECoG), or electroencephalography (EEG) are closely correlated with brain status and can provide a useful way to assess cognitive load [14–18]. Among these physiological signals, EEG has been widely concerned by researchers because of its high time resolution, noninvasiveness, convenience, security, cheapness, and portability [19,20].

In general, the EEG signal is nonstationary and nonlinear. Linear analysis techniques in the time–frequency domain can be used to detect rhythmic oscillations, but the contained nonlinear information cannot be effectively extracted [21]. Therefore, many scholars have attempted to extract various nonlinear parameters from EEG signals and combine them with the machine learning technique in order to effectively capture the subtle information related to the physiological states. Nilima Salankar et al. used the empirical mode decomposition (EMD) and the variational mode decomposition (VMD) to decompose the EEG signals, respectively, and then used the second-order difference plots for feature mining of the decomposed intrinsic modes. The results showed that alcoholic (A) and nonalcoholic (NA) subjects could be accurately classified when using short-duration EEG recordings [22]. Mohammad Shahbakhti et al. proposed extracting Katz and Higuchi's fractal dimensions, dispersion entropy, and bubble entropy from the sub-band of a single-channel frontal EEG recording to construct the nonlinear feature set and then differentiate between the arousal and the sleep stage I [23]. Jose Kunnel Paul et al. used seven nonlinear parameters, including the sample entropy (SampEn), fractal dimension (FD), higher-order spectrum (HOS), maximum Lyapunov index (LLE), Kolmogorov complexity (KC), Hurst index (HE), and the band power of the EEG signal in sleep stage 2 and 3 as the features to classify between patients with fibromyalgia and healthy controls. The accuracy, sensitivity, and specificity of the classification results were 96.15%, 96.88%, and 95.65%, respectively [24]. The nonlinear parameters in the above-mentioned methods were taken from individual EEG channels and involve no information on the interaction between different channels. However, previous research has shown that the brain should be treated as a complex network system based on the many features it shares with networks of other biological and physical systems [25]. Complex network analysis is a powerful technique based on the graph theory that typically uses a small number of valid and reliable measures to capture the features of the brain network [26]. There is a growing interest in the cognitive load assessment through the construction of complex networks, and various methods have been proposed to convert time series into networks [27–30]. Complex networks constructed using different network construction algorithms may have distinct, significantly different properties [31]. A variety of methods have been proposed so far to define the concept of connectivity between nonlinearly coupling components and investigate the characteristics of the topological properties of networks. Among different methods, for example, the mutual information (MI) (including its time-delayed version) [32,33], transfer entropy (TE) [34], inner composition alignment (IOTA) [35] and cross-sample entropy (CSE) [36], the TE is widely used in particular as a nonparametric measure that does not rely on any assumption of some model and can capture the directional and dynamic interaction between the different components of a time series [37,38]. However, in practice, an unavoidable pitfall of TE is that robust estimation of the interactions requires long-term data recordings. In order to meet the need for interaction estimation using finite data samples, Shi et al. proposed the CPE by fusing inner composition alignment (IOTA) and permutation entropy, and validated it in financial time series analysis [39], noting that CPE was simple, stable, and efficient.

In the original CPE method, only the probability distribution of the symbols after coarse graining of the affected time series is considered during the calculation of entropy, ignoring the transition relationship between the symbols in the temporal domain. For example, given the symbolized set A = [2 2 4 3 5 1 2] and B = [1 2 2 5 3 2 4 ], the probability distributions of the elements in set A and set B are the same and, therefore, the original CPE method would obtain the same entropy value. In addition, like other nonlinear measures, the CPE method involves the manual selection of parameters to ensure the effectiveness of

the results. In order to address these issues, a new method to construct the complex network based on the cross-transition network was proposed in this study to assess cognitive load. The novelty of the method lies in incorporating the advantages of transition network and bubble entropy [40] into the CPE to estimate the coupling strength of two time series from a cross-network perspective. The node-wise out-link transition entropy of two time series cross-transition networks was proposed as the edge weights between two time series to construct the complex network, and the network parameters were extracted as a quantitative measurement of the cognitive load. Referring to the symbolization process of the bubble entropy, the number of swaps required to sort the phase space reconstruction vectors of the affected time series in the ascending order was used instead of the number of intersections calculated by the OITA method in the original CPE. In order to verify the effectiveness of the proposed method, the unidirectional coupled Honen model with different coupling strengths was used, and the results were compared with those obtained using the original CPE. The proposed method and the original CPE method were further compared by constructing the complex network on the realistic EEG recordings from the mental arithmetic task. The significance of the selected network indicator and the capability of the proposed method to differentiate different levels of brain cognitive load were verified using the nonparametric permutation test.

The contributions of this paper are as follows.

1. Based on the cross-transition network, a novel method is proposed that reflects the information interaction between two time series in more detail.

2. The symbolization process with reference to the bubble entropy minimizes the effect of parameter setting on the analysis results.

3. The topological characteristics of complex networks constructed using the nodewise out-link transition entropy of cross-transition networks as the edge weights have the potential to provide useful indicators for physiological complex networks.

This paper is organized as follows. In Section 2, the implementation process of the proposed method in this study is described in detail. In Section 3, the CPE and the proposed method are used to analyze the unidirectional coupled Honen model with its parameters varied, respectively, and their performance is compared. Next, a realistic EEG dataset recorded during the mental arithmetic task is analyzed by constructing the complex networks using the two methods, respectively, in order to further demonstrate the effectiveness of the proposed method. The discussion and conclusions are given in Sections 4 and 5.

## **2. Materials and Methods**

In this section, the CPE method is briefly introduced, and then the detailed implementation process of the proposed method is described.
