Introduction

Autism spectrum disorder (ASD) is a neurodevelopmental nonfocal brain disorder with the main features of social impairments, interest limitations, and repetitive behaviors (Abrahams & Geschwind, 2008; Li et al., 2021). Diagnosis of ASD usually relies on behavioral observations, clinical interviews, and questionnaires (Eaves et al., 2006; Lord et al., 2000; Tyszka et al., 2014). However, these methods may lead to misdiagnosis (Dekhil et al., 2021), finding objective biomarkers to diagnose ASD becomes crucial (Aggarwal & Angus, 2015; Beljan et al., 2006; Retico et al., 2014).

In elucidating the neurological changes in ASD, it is recognized that ASD is a connectome dysfunction syndrome (Geschwind & Levitt, 2007; Kana et al., 2011), manifested by abnormalities of intrinsic functional connectivity in the brain (Assaf et al., 2010; Elton et al., 2016; Lynch et al., 2013). Functional connectivity is described as a correlation between two brain regions and is widely used in ASD-related studies (Hyatt et al., 2022). Additionally, there is another type of brain connectivity called effective connectivity. Common methods include Granger causality, conditional entropy (Wadhera & Kakkar, 2020), etc., which can assist researchers in understanding the information transmission pathways between different brain regions. It’s worth noting that both effective and functional connectivity only describe the effects or influences between two brain regions. Given the complex relationships within the brain regions, analyzing higher-order interactions across multiple brain regions is essential. Previously, higher-order interactions have been obtained by analyzing the relationships between all brain regions, such as graph-based topological measures (Wadhera, 2023), graph representation learning (Yousefian et al., 2023), motifs (Sporns & Kötter, 2004), mapping cores or mesoscale communities (Fortunato, 2010), and transitivity or clustering coefficients (Watts & Strogatz, 1998). However, the statistical dependencies between multiple brain regions are more likely to provide an objective picture of the relationship between the onset of ASD and brain region interactions. Several different frameworks have been developed in the field of multivariate information theory to capture statistical dependencies other than two-by-two correlations. These frameworks have been used in brain science, such as age-related brain changes (Gatica et al., 2021), and changes induced by isoproterenol anesthesia and severe brain injury (Luppi et al., 2020), which suggest that higher-order dependencies can help us discover diagnostic biomarkers. Partial entropy decomposition (PED) (Ince, 2017) is one of the key frameworks. The PED can be used to capture statistical dependencies across three brain regions (triad) and compute redundancy and synergy information. Redundancy and synergy represent two types of higher-order interactions. Redundancy refers to the duplication of information over brain regions, which means the same information can be obtained from multiple brain regions regardless of their joint state. Synergy refers to information that becomes accessible only when considering the joint states of multiple brain regions, rather than deriving it from individual brain region or simple combinations (Luppi et al., 2022; Varley et al., 2023a, b). In the past, higher-order dependencies were analyzed using the total correlation (Watanabe, 1960), the dual total correlation (Abdallah & Plumbley, 2012), the S-information, and the O-information (Rosas et al., 2019), but these methods either suffer from the double counting of redundant information or compromise higher-order synergistic information (Varley et al., 2023b). Thomas et al. (2011) analyzed and defined new higher-order dependency measures by the PED, which are more suitable for analyzing brain activities. Combining the PED and the surrogate tests allows us to not only analyze higher-order dependencies within triads but also investigate the impact of the brain regions in a triad. This integration forms the basis of the innovation in our work.

Materials and Methods

Resting-state functional magnetic resonance imaging (rs-fMRI) data of ASD and typical controls (TC) were used to calculate redundant and synergistic information in all triads by the PED algorithm. These redundant and synergistic information were utilized to characterize higher-order brain structures and to filter some key triads to distinguish ASD/TC. In addition, a classification model for identifying ASD was constructed using the key triads as features. Figure 1 demonstrates the workflow of this paper.

Fig. 1
figure 1

A workflow for higher-order dependency analysis based on the PED, used to discover diagnostic biomarkers in ASD

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Subjects

The brain blood oxygen level-dependent (BOLD) signaling dataset for all subjects was obtained from NYU Langone Medical Center in Autism Brain Imaging Data Exchange I (ABIDE I, information on scanning details, diagnostic protocols, and ethics statements is available from: https://fcon_1000.projects.nitrc.org/indi/abide/abide_I.html) (Di Martino et al., 2014), which has been anonymized and doesn’t contain protected health information according to HIPAA guidelines and 1000 Functional Connectome Project/INDI protocol. The dataset was screened by three anthropologists who examined incomplete brain coverage, high peak motion, the presence of ghosting, and scanner artifacts (Abraham et al., 2017). We selected the subjects referred to http://preprocessed-connectomes-project.org/abide/quality_assessment.html. The deleted subjects include 50953, 50971, 50975, 50980, 50998, 51071, 51108, 51115, 51119, 51120, 51121, and 51125. There are 172 subjects, consisting of 74 ASDs and 98 TCs. All subjects’ brain regions were defined by the Craddock 200 (CC200) atlas (Craddock et al., 2012), which divides the brain into 200 different regions. The selected preprocessing pipeline is the Configurable Pipeline for the Analysis of Connectomes (C-PAC) (Craddock et al., 2013). C-PAC is an open-source software pipeline for connectome analysis that performs preprocessing including slice timing correction, and intensity normalization, and image realignment to correct for motion and other preprocessing. Reference (Abraham et al., 2017) provided a detailed description of this data preprocessing. We discretized the BOLD data by binarizing the z-scored time series. The BOLD signals can be expressed as: \(\varvec{X}={({X}_{1},\dots ,{X}_{i},\dots ,{X}_{j},\dots ,{X}_{m},\dots ,{X}_{200})}^{{\prime }}\), and the subscript indicates the serial number of the brain region.

Partial Entropy Decomposition and Higher-order Dependency Measures

The Shannon entropy of multidimensional random variable is expressed as:

$$H\left(\varvec{X}\right)=-\sum\nolimits_{x\in \chi }P\left(x\right)\,{\text{log}}_{2}\,P\left(x\right)$$
(1)

where \(x\) indicates a particular configuration of \(\varvec{X}\), and \(\chi\) is the support set of \(\varvec{X}\). \(H\left(\varvec{X}\right)\) measures the information knowable about variable \(\varvec{X}\). However, this is only a crude measure, ignoring the complex structure in \(\varvec{X}\). Through PED, the extent to which uncertainty about the overall state can be resolved by acquiring information about the states of its constituent parts can be determined.

PED is developed from the mathematical framework of PID (Williams & Beer, 2010) applied to multivariate entropy. For utilizing the PED on rs-fMRI data, it is necessary to select an approach for defining redundant entropy. In this study, we used the partial entropy function (Makkeh et al., 2021) \({\varkappa }_{sx}\):

$${\varkappa }_{sx}^{\varvec{X}}\left(\alpha \right)={\text{log}}_{2}\frac{1}{P\left({a}_{1}\cup \cdots \cup {a}_{k}\right)}$$
(2)

For the set α, there are k elements, which are subsets of the variable \(\varvec{X}=\left\{{X}_{1},{X}_{2},\dots ,{X}_{200}\right\}\) and may overlap. If \(\varvec{X}=\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\) and \(\alpha =\left(\right\{i,j\left\}\right\{i,m\left\}\right\{j,m\left\}\right)\), then \({\varkappa }_{sx}^{\varvec{X}}\left(\alpha \right)\) is interpreted as the information that can be learned about the state of the whole \(\varvec{X}\) by observing \({X}_{i}\, \& \,{X}_{j}\) or \({X}_{i}\, \& \,{X}_{m}\) or \({X}_{j}\, \& \,{X}_{m}\).

Here we introduce the PED in the binary system \(\varvec{X}=\left\{{X}_{1},{X}_{2}\right\}\):

$$H\left({X}_{1}\right)={\varkappa }_{sx}^{12}\left(\left\{1\right\}\left\{2\right\}\right)+{\varkappa }_{sx}^{12}\left(\left\{1\right\}\right)$$
(3)
$$H\left({X}_{2}\right)={\varkappa }_{sx}^{12}\left(\left\{1\right\}\left\{2\right\}\right)+{\varkappa }_{sx}^{12}\left(\left\{2\right\}\right)$$
(4)
$$H\left(\varvec{X}\right)={\varkappa }_{sx}^{12}\left(\left\{1\right\}\left\{2\right\}\right)+{\varkappa }_{sx}^{12}\left(\left\{1\right\}\right)+{\varkappa }_{sx}^{12}\left(\left\{2\right\}\right)+{\varkappa }_{sx}^{12}\left(\left\{\text{1,2}\right\}\right)$$
(5)

where \({\varkappa }_{sx}^{12}\left(\left\{1\right\}\left\{2\right\}\right)\) denotes the uncertainty about the state of \(\varvec{X}\) acquired while learning either \({X}_{1}\) alone or \({X}_{2}\) alone; \({\varkappa }_{sx}^{12}\left(\left\{1\right\}\right)\) denotes the uncertainty about the state of \(\varvec{X}\) that can be learned only from \({X}_{1}\); similarly, \({\varkappa }_{sx}^{12}\left(\left\{2\right\}\right)\) denotes the uncertainty about the state of \(\varvec{X}\) that can be learned only from \({X}_{2}\); and \({\varkappa }_{sx}^{12}\left(\left\{\text{1,2}\right\}\right)\) denotes the uncertainty about the state of \(\varvec{X}\) that must be learned from both \({X}_{1}\) and \({X}_{2}\). The joint and marginal entropies are known. We can calculate \({\varkappa }_{sx}^{12}\left(\left\{1\right\}\left\{2\right\}\right)\) through the partial entropy function, then the remaining unknown variables are solved.

To better study higher-order dependencies, Thomas et al. proposed new higher-order dependency measures (Varley et al., 2023b). Higher-order redundancy can be thought of as information that is duplicated between three or more brain regions. For the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\), higher-order redundancy is defined as:

$$S_R(X_{i,}X_{j,}X_m)={\varkappa }_{sx}^{ijm}(\{i\}\{j\}\{m\})$$
(6)

Similarly, higher-order synergy can be thought of as information duplicated with the joint state of two or more brain regions. For the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\), higher-order synergy is defined as:

$$\begin{aligned}{S}_{S}({X}_{i},{X}_{j},{X}_{m})=&\;{\varkappa }_{sx}^{ijm}{(\textit{{i}{j, m}})}+{\varkappa}_{sx}^{ijm}{(\textit{{j}{i, m}})}+{\varkappa }_{sx}^{ijm}{(\textit{{m}{i, j}})}\\& +{\varkappa }_{sx}^{ijm}{(\textit{{i, j}{i, m}{j, m}})}+{\varkappa}_{sx}^{ijm}{(\textit{{i, j}{i, m}})}\\& +{\varkappa }_{sx}^{ijm}{(\textit{{j, m}{i, m}})}+{\varkappa }_{sx}^{ijm}{(\textit{{i, j}{j, m}})}\end{aligned}$$
(7)

The joint entropy and higher-order dependency measures were computed for \(\left(\begin{array}{c}200\\ 3\end{array}\right)=1313400\) triads.

PED-based Higher-order Brain Structures

For 1313400 triads, the distribution of redundancy and synergy in brains was analyzed to determine the relation of the interactions between brain regions with ASD. To ensure that the statistical dependencies can reflect non-trivial interactions, we conducted significance tests on a null distribution containing one million, maximum entropy null models. To be specific, we generated the sets of binary time series with total independence, and computed the joint entropy for one million triads. For the higher-order brain structures, only those triads with a joint entropy smaller than the minimum joint entropy (2.8123) observed in any of the one million maximum-entropy nulls that formed our null distribution were selected. Then, the smallest number of triads necessary for ensuring the inclusion of all brain regions was chosen based on synergy values, sorted from largest to smallest. The redundant triads were selected based on redundancy values, also sorted from largest to smallest. 3010 triads were separately employed for creating a synergistic structure and a redundant structure. Subsequently, the hypergraph modularity maximization algorithm was applied for a higher-order characterization of all brain regions. This algorithm segregates collections of node sets called hyperedges into communities with high intra-community integration and low inter-community integration.

PED-based Analysis of Triads

Two-sample t-tests (two-tailed) were used to evaluate differences in higher-order dependency measures, with the significance level set at 0.001. Then, we proposed a method to explore whether the brain region is extraordinary or ordinary (i.e., the impact of individual brain region in triads). Specifically, the time series of each brain region in a triad was replaced 100 times with surrogate data which is the shuffled time series. The higher-order dependency measures of the new triad were recalculated. In the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\), each brain region had a corresponding surrogate set. Then the one-tailed Z-test could test whether the original higher-order dependency measures were significantly different from the shuffled ones. Three p-values (\({p}_{i}^{ijm}\), \({p}_{j}^{ijm}\), \({p}_{m}^{ijm}\)) were obtained in the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\), revealing a pattern composed of the states of individual brain region. If the p-value \(<0.05\), this brain region is deemed to be in an extraordinary state; otherwise, it is considered in an ordinary state. We examined not only the cases with p-values less than 0.05 but also those with p-values exceeding 0.95 (P-values less than 0.05 indicate that the original higher-order dependencies are larger than the samples in the surrogate set, and p-values greater than 0.95 indicate that the original higher-order dependencies are smaller than the samples in the surrogate set.). We do not consider the p-values greater than 0.95, as they are rare. We separately calculated the proportions \((ASD\text{: } {R}_{i}^{ijm},{R}_{j}^{ijm},{R}_{m}^{ijm};TC\text{: } {r}_{i}^{ijm},{r}_{j}^{ijm},{r}_{m}^{ijm})\) of the extraordinary state in brain regions \(i\), \(j\), and \(m\). In the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\), if the relative sizes of \({R}_{i}^{ijm}\), \({R}_{j}^{ijm},\) and \({R}_{m}^{ijm}\) are inconsistent with those of \({r}_{i}^{ijm}\), \({r}_{j}^{ijm}\), and \({r}_{m}^{ijm}\), the triad has the potential to become a key triad.

Patterns Analysis Based on the Triads

The patterns of the triads were considered. If \({p}_{i}^{ijm}<0.05\) and \({p}_{j}^{ijm}<0.05\) and \({p}_{m}^{ijm}<0.05\) in the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\) of a subject, the pattern is (1,1,1); if \({p}_{i}^{ijm}\ge 0.05\) and \({p}_{j}^{ijm}<0.05\) and \({p}_{m}^{ijm}<0.05\) in the triad \(\left\{{X}_{i},{X}_{j},{X}_{m}\right\}\) of a subject, the pattern is (0,1,1). The total number of patterns is 8 (Fig. 2). Similar criteria are applied to derive other patterns.

Fig. 2
figure 2

The pattern diagram of the triads. The 0 and 1 indicate two states of the brain regions respectively, corresponding to the color of each node in the triad. 0 (green) indicates ordinary state, and 1 (yellow) indicates extraordinary state. The three vertices \(i,\, j,\) and \(m\) of the triangle indicate three brain regions in the triad

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PED-based ASD Recognition

After obtaining the differences in higher-order dependency measures, we constructed a classification model. The triads’ crucial information was extracted as a feature vector for each subject. Each element of the feature vector represented the triad of redundant information or synergistic information. The feature vectors were input into a support vector machine (SVM) classifier for training. The optimal parameters were determined by using a grid search method (see Supplementary Information). To minimize the potential influence of the training set/test set division on the classification results, we implemented five distinct methods for splitting the data into training and testing sets. Additionally, we performed ten-fold cross-validation to validate the evaluation robustness. Classification quality was measured as Accuracy (ACC), Sensitivity (SEN), Specificity (SPE) (Artiles et al., 2023), F1-Score, and Area Under Curve (AUC). The mathematical definitions of these metrics are as follows:

$$ACC=\frac{TP\,+\,TN}{TP\,+\,TN\,+\,FP\,+\,FN}$$
(8)
$$SEN=\frac{TP}{TP\,+\,FN}$$
(9)
$$SPE=\frac{TN}{TN\,+\,FP}$$
(10)
$${F}_{1}=\frac{2TP}{2TP\,+\,FP\,+\,FN}$$
(11)

where, \(TP\): the number of positive classes predict to be positive; \(FN\): the number of positive classes predict to be negative; \(FP\): the number of negative classes predict to be positive; \(TN\): the number of negative classes predict to be negative.

$$AUC= \frac{\sum I\left({P}_{Postive},{P}_{Negative}\right)}{M*N}$$
(12)
$$I\left({P}_{Postive},{P}_{Negative}\right)=\left\{\begin{array}{c}1,\quad{ P}_{Postive}>{P}_{Negative}\\ 0.5,\quad{ P}_{Postive}={P}_{Negative}\\ 0,\quad{P}_{Postive}<{P}_{Negative}\end{array}\right.$$
(13)

where \(M\) is the number of positive samples (ASD) and \(N\) is the number of negative samples (TC). \({P}_{Postive}\) is the positive sample prediction score and \({P}_{Negative}\) is the negative sample prediction score.

Results

Inter-group Differences in Redundancy and Synergy Distributions

For each subject, the synergistic and redundant structures were created utilizing the hypergraph modularity maximization algorithm. We correlated the synergistic and redundant structures with the canonical seven Yeo systems (Thomas et al., 2011) (The canonical seven Yeo systems consist of seven networks. Due to the existence of some brain regions within the CC200 that do not match with the canonical seven Yeo systems, we classified these brain regions into an eighth subnetwork S8.). The left side illustrates the top one hundred pairs of brain regions with the significant differences, while the right side specifically highlights the association between the five pairs of brain regions with the greatest differences and the canonical seven Yeo systems, as shown in Fig. 3. Among these, brain regions 45, 111, and 161 exhibited significant differences in redundancy, while brain regions 60 and 170 displayed significant differences in synergy. In redundant communities, brain region pairs 18–45 were affiliated with subnetwork S6, 118–199 with subnetwork S3, 161–184 with subnetwork S7, while 96–111 and 119–161 belonged to distinct functional subnetworks. Only brain region pair 118–199 displayed a higher proportion of being classified into the same community in ASD compared to TC. Turning to synergy, brain region pairs 20–84, 21–28, 60–123, 61–170, and 174–191 were linked to diverse functional networks. In ASD, brain region pairs 20–84 and 61–170 showed a higher proportion of being classified into the same community, while the remaining pairs of brain regions showed a lower proportion compared to TC. Notably, each pair of brain regions (18–45, 118–199, 161–184, 20–84, 60–123, and 61–170) corresponds to distinct brain hemispheres, and these distinctions are evident in Fig. 1 within the context of structural differences in ASD.

Fig. 3
figure 3

The differences in redundant and synergistic structures between ASD and TC. The left figure illustrates the top one hundred brain region pairs, each of which displayed a difference in the proportion of being classified into the same community in ASD compared to TC. The brain regions enclosed within the red box align with those in the right figure. In the right figure, red dots represent a higher proportion of being classified into the same community in ASD compared to TC, while blue dots indicate a lower proportion of being classified into the same community in ASD compared to TC

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Inter-group Differences in Higher-order Dependency Measures

After the analysis of triads, the redundant key triads with larger \(\left|{R}_{i}^{ijm}- {r}_{i}^{ijm}\right|+\left|{R}_{j}^{ijm}-{r}_{j}^{ijm}\right|+\left|{R}_{m}^{ijm}-{r}_{m}^{ijm}\right|\) were discovered (Fig. 4). The synergistic key triads were discovered by the two-sample t-tests. ASD displayed an increased redundant interaction in triad 25-183-190 (P = 1.44e-05) and an increased synergistic interaction in triads 5-34-78 (P = 7.14e-04), 8-9-190 (P = 9.27e-04), 21-173-197 (P = 3.78e-04), and 28-65-96 (P = 9.36e-04). In contrast, ASD demonstrated a reduced redundant interaction in triads 18-31-42 (P = 1.26e-04), 18-108-150 (P = 5.98e-07), 20-62-102 (P = 1.55e-05), 24-135-199 (P = 5.25e-06), 37-148-176 (P = 2.51e-05), 39-68-111 (P = 2.75e-07), and 148-176-197 (P = 6.28e-06), as well as a reduced synergistic interaction in triads 26-55-126 (P = 7.20e-04), 33-60-178 (P = 4.63e-04), 44-69-141 (P = 2.62e-04), and 78-155-178 (P = 6.35e-05). Figure 4 shows the relative sizes of \({R}_{i}^{ijm}\), \({R}_{j}^{ijm}\), and \({R}_{m}^{ijm}\) and the relative sizes of \({r}_{i}^{ijm}\), \({r}_{j}^{ijm}\), and \({r}_{m}^{ijm}\) within the key triads.

Fig. 4
figure 4

Extraordinary state of brain regions in the key triads. In ASD, the largest among \({R}_{i}^{ijm}\), \({R}_{j}^{ijm}\), and \({R}_{m}^{ijm}\) is marked in red, the second largest in gray, and the smallest in blue. Similarly, the same color-coding scheme is applied in TC

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Inter-group Differences in the Patterns of the Key Triads

We noted variations in the occurrence of the key triads’ patterns between ASD and TC, as depicted in Fig. 5. In redundancy, the most common occurrences were represented by only two patterns: (0, 0, 0) and (1, 1, 1), denoting all extraordinary and all ordinary. In ASD, the most common pattern for 18-108-150, 18-31-42, and 20-62-102 is (0, 0, 0), while the most common pattern for 25-183-190 is (1, 1, 1). In TC, the most common pattern for 18-108-150, 18-31-42, and 20-62-102 is (1, 1, 1), while the most common pattern for 25-183-190 is (0, 0, 0). In synergy, we found that there was only one pattern (0, 0, 0), i.e., all ordinary.

Fig. 5
figure 5

The patterns of the key triads. Yellow indicates pattern (1, 1, 1), green indicates pattern (0, 0, 0). The shade of the color indicates the size relationship in ASD and TC, with lighter colors indicating smaller and darker colors indicating larger. The light blue brain region belongs to two triads and its color cannot be defined

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ASD-TC Classification Model

We used the higher-order dependency measures of the key triads as features to construct a classification model. Table 1 illustrates the average results of classification tests utilizing different features, including the average accuracy of multiple test sets (Test-ACC) and the average accuracy of multiple validation sets (Validation-ACC). Utilizing the redundant key triads’ redundant information (Redundant information) as features yielded an accuracy of 85%, while using the synergistic key triads’ synergistic information (Synergistic information) resulted in an accuracy of 80%. Utilizing the redundant key triads’ redundant information and the synergistic key triads’ synergistic information (All information) achieved an accuracy of 83%. Upon investigating different training/test set divisions, employing the redundant key triads’ redundant information and the synergistic key triads’ synergistic information as features led to an accuracy of 97%, 94% for the redundant key triads’ redundant information, and 89% for the synergistic key triads’ synergistic information (see Supplementary Information).

Table 1 SVM classification results based on the key triads
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Discussion

The PED provides a new approach to quantify higher-order dependencies in the brain. By comparing the differences in higher-order dependency measures, we are able to gain a deeper understanding of how specific changes in the ASD brains relate to the aspects of functional organization and cognitive behavior. Additionally, given PED’s limitation in investigating changes at the individual brain region level, we proposed utilizing the surrogate tests to study the impact of brain regions within a triad on both redundancy and synergistic information. We found some key triads associated with ASD. Following SVM validation, these key triads may serve as diagnostic biomarkers for ASD. To better utilize previous reports to validate our findings, CC200 was matched with the anatomical automatic labeling (AAL) atlas. All subsequent discussions are framed in terms of AAL, represented as AAL brain region name (CC200 brain region serial number).

Characterizing the higher-order brain structures can reveal differences that were hard to spot. Significantly, ASD exhibited some structural abnormalities across brain hemispheres. Among the redundant information, brain region pair right thalamus (18)-left thalamus (45) belonged to subnetwork S6, while right supplementary motor area (161)-left insula (184) belonged to subnetwork S7. The link between right thalamus (18) and the left thalamus (45) and the link between right supplementary motor area (161) and left insula (184) are more loose in terms of redundancy in ASD, indicating a potential association with a decline in sensory processing capabilities in this population (Baran et al., 2023; Ingalhalikar et al., 2021; Karabanov et al., 2023; Wagner et al., 2023). Brain region pair right cerebellum superior 6 (118)-left parahippocampal gyrus (199) showed a higher proportion of being classified into the same community in ASD compared to TC, indicating a more similar pattern of information transmission or states within these brain regions. Brain region pairs left postcentral gyrus (96)-left paracentral lobule (111) and right supplementary motor area (161)-left insula (184) also showed a loose relationship in ASD. This implied a marked decrease in communication within the two pairs of brain regions in ASD. In synergistic information, each of the five pairs of brain regions belonged to distinct functional subnetworks, elucidating the functional synergy impairment observed in ASD brains. The link between left precuneus (174) and left superior frontal gyrus dorsolateral (191) in terms of synergy in ASD are more loose, and these brain regions are thought to be associated with ASD (Cheng et al., 2023; Fang et al., 2020; Guo et al., 2023).

The extraction of key triads is based on PED’s proficiency in uncovering higher-order dependencies, complemented by surrogate tests to address PED’s limitations. The key triads (25-183-190, 37-148-176, 148-176-197, 8-9-190, 26-55-126) involving the cerebellum exhibited changes in either redundant or synergistic information. Functional, structural, and cytoarchitectural differences of the cerebellum have been reported several times in studies of ASD (Frosch et al., 2022). Thalamus, right middle temporal gyrus, left paracentral lobule, left precuneus, and left temporal pole of the middle temporal gyrus belong to the default mode network (Ingalhalikar et al., 2021; Joo et al., 2020; Liloia et al., 2023; Raichle et al., 2001). The abnormalities in these brain regions can lead to changes in the functionality of the default mode network. The key triads (37-148-176, 148-176-197), which include left thalamus all incorporate the involvement of the cerebellum, potentially contributing to impairments in sensory processing and attention in ASD (Baran et al., 2023; Wagner et al., 2023). The redundant interactions within the key triads that involve either left thalamus or right thalamus (18-108-150, 39-68-111) were reduced, correlating with a decline in sensory processing abilities (Wagner et al., 2023). The interactions within the key triads (39-68-111, 44-69-141) involving right middle temporal gyrus experienced a decrease. This brain region is potentially linked to higher cognitive function (Raichle et al., 2001). Left paracentral lobule (21-173-197, 39-68-111) has also been reported to have insufficient local functional connectivity (Liloia et al., 2023). The key triad (25-183-190) involving right middle frontal gyrus is the only one where the redundant interaction has increased. In previous studies, this brain region has been implicated in face recognition and alterations of stress response in ASD (Chen et al., 2023; Patel et al., 2016). The redundant interactions within the key triads involving left middle occipital gyrus (18-108-150), left precuneus (148-176-197), and left insula (20-62-102), as well as the synergistic interactions within the key triads involving right lenticular nucleus (putamen) (33-60-178, 78-155-178) have all weakened. Left middle occipital gyrus is involved in the processing of visual information and communication with the cerebral cortex and participates in the perception of facial emotions (Teng et al., 2018). Left precuneus plays an important role in the continuous gathering of information about the world around us and within us (Raichle et al., 2001). Left insula is crucial for integrating different functional systems involved in processing emotion, sensory-motor processing, and general cognition (Chang et al., 2013; Pua et al., 2021). Structural or histological abnormalities in right lenticular nucleus (putamen) may underlie ASD pathology (Sato et al., 2014). The synergistic interactions within the key triads involving left postcentral gyrus (8-9-190, 28-65-96) and right postcentral gyrus (28-65-96) have increased. Left postcentral gyrus is responsible for receiving and processing sensory inputs, and increased tactile sensitivity is a recognized feature of ASD (James et al., 2022). Right postcentral gyrus is implicated in multiple large brain subnetworks, and its internal synergy may be related to the functional synergy between subnetworks, a phenomenon associated with ASD (Fatemi et al., 2018). Left temporal pole of the middle temporal gyrus (5-34-78, 78-155-178) is associated with situational memory, self-reference, and information processing (Ingalhalikar et al., 2021; Joo et al., 2020). Notably, one study found that functional connectivity between left temporal pole of the middle temporal gyrus and left superior temporal gyrus was positively correlated with adaptive skills and language development quotient in children with ASD (Jiang et al., 2021).

In recent years, despite the increasing interest in ASD research, accurately recognizing ASD remains a challenging dilemma. Zhao et al. achieved 83% classification accuracy on the NYU dataset by combining conventional functional connectivity networks, low-order dynamic functional connectivity networks, and high-order dynamic functional connectivity networks (Zhao et al., 2020). Cong et al. used Liang information flow computed DMN causal connectivity and network properties as features to construct a classification model with the classification accuracy of 78.12% on the NYU dataset (Cong et al., 2023). We employed higher-order dependency measures for ASD and TC classification, attaining an average accuracy of 85% (using triads’ redundant information as features) and the highest accuracy of 97% (using triads’ redundant information and synergistic information as features). Our classification results performed well when compared to SVM-based functional connectivity classification studies on the NYU dataset. To some extent, this finding highlights the crucial role of both redundant information and synergistic information in the classification of ASD and TC.

Conclusion

In this study, the proposition to combine the PED with the surrogate tests effectively addresses the limitation posed by the PED in studying individual brain region. Our findings suggested that the changes occurred in the interactions or transmission of information pathways in the triads in ASD, which were either increased or decreased. Additionally, these differences existing at the triad level were indeed the result of changes in individual brain region, especially in the extraordinary state of individual brain region. After comparison with previous studies, our results were verified. The classification accuracy provided strong evidence, underscoring the potential significance of the key triads as diagnostic biomarkers for recognizing ASD.

Information Sharing Statement

The data and code used to support the findings of this study are available from the corresponding author upon reasonable request.