The resting-state fMRI data used in this paper are from the four collection sites, ABIDEII-NYU_1, ABIDEII-NYU_2, ABIDEII-SU_2, and ABIDEII-EMC_1, in the ASD Public Dataset (ABIDE) (Martino and Milham, 2017). The imaging data conditions of the ASD group and the TD control group are as follows: (1) there (Moran et al., 2014) was a strong gender effect in ASD patients, so the subjects were all male; (2) all subjects were dextral; (3) there was an age effect on brain FC, so the age of the subjects was 6–13 years old, including 54 ASD patients [mean (9.3 ± 1.9) years old] and 52 TD controls [mean (9.6 ± 1.9) years old], and there was no significant difference in age between the two groups (p = 0.31).

In this study, the tool DPABI (Yan et al., 2016) (http://rfmri.org/DPABI) is used for preprocessing resting-state fMRI data. The preprocessing of data mainly includes the following steps: (1) Removing the first 10 items from the time series and realigning; (2) Cleaning low frequency scanner drift and applying bandpass filtering (0.01–0.1 Hz); (3) Normalizing spatial data by DARTEL; (4) Selecting anatomical automatic labeling (AAL) (Tzourio-Mazoyer et al., 2002) to define regions of interest (ROIs); and (5) Calculating the FC matrix based on BOLD time series for each pair of brain regions. For more data preprocessing details, please refer to the Supplementary Material.

Let (X, D) be a metric space and a Vietoris-Rips (VR) complex with ε parameters be a simple complex denoted by VR(X, ε), whose vertex is the set X, and {x₀, x₁, ..., x_k} generates a k-complex if and only if d(x_i, x_j) ≤ ε for all 0 ≤ i, j ≤ k.

A correlation matrix of the brain functional network is obtained by using DPABI. Each brain region is a vertex in the brain functional network. We use the Pearson correlation coefficient to define the distance between two vertices P_i and P_j, dX(Pi,Pj)=1-corr(Pi,Pj); therefore, the metric space (X, d_X) is formed from the brain functional network. The correlation matrix is sorted from smallest to largest by the distance between the two vertices: ε₁ < ε₂ < … < ε_M. We can obtain a VR filtration model of the brain network:

JavaPlex software is used to calculate the VR filtration barcode (the barcode-interval set). These intervals represent the lifetime of nontrivial cycles in the growing process of complexes. The left end of the interval represents the birth of a new topological property, and the right end of the interval represents its death. From this, we can describe the change rules of the Betti number of a brain network. The 0-dimensional Betti number β₀ represents the number of components of the brain functional network, and the one-dimensional Betti number β₁ represents the number of one-dimensional holes in the brain functional network, that is, the number of FC neural circuits in the brain network. VR complexes constructed from a point cloud data are described in the following example to help understand persistence homology.

For a set X consisting of 5 points on the plane, we construct VR complexes at different values of ε. Let's draw five circles centered at the points with the radius ε/2. The VR complexes are shown in Figure 1. We explain each subfigure in detail to help understand Figure 1. In Figure 1A, there is no one-dimensional simplex formed at ε = 0.60. There are 5 components, this means β₀ = 5. In Figure 1B, when the distance between the two points is less than ε = 1.37, the two lower right points form 1-dimensional simplex. There are 4 components, this means β₀ = 4. In Figure 1C, the two bottom points are connected, and three bottom points form a new component when ε = 1.62, that is β₀ = 3. In Figure 1D, in this case ε = 1.78, β₀ = 2. In Figure 1E, a loop is formed, i.e., β₁ = 1 and β₀ = 1 when ε = 1.97. Then we calculate the persistence barcode according to persistence homology. The persistence barcodes of the data are shown in Figure 2.

We select AAL to define the ROIs, and a brain network has 90 vertices. Figure 3 presents the results obtained by calculating two original data points of an ASD patient and a TD control in ABIDE II with the above steps. A vertical line is drawn across the x axis, and the vertical coordinate of the intersection point of the vertical line and the 0-dimensional (1-dimensional) Betti curve is the 0-dimensional (1-dimensional) Betti number of the network corresponding to the threshold value. Figure 3 shows the curves of the 0-dimensional and 1-dimensional Betti numbers of VR filtration for the ASD patient and the TD control. It depicts the differences in the topological features of the ASD brain network and the TD brain network at different threshold values.

The frontal lobes participate in high levels of cognition and are extremely important for the development of social behavior. The orbitofrontal cortex, located on the ventral side of the frontal lobe, plays an important role in social emotional cognition and self-regulating behavior in social communication. This provides a basis for effective social communication. Some scholars (Mori et al., 2013; Jonker et al., 2015) believe that neurological functional deficits in the orbitofrontal cortex can cause ASD in patients with social function disorders. The posterior cingulate gyrus is an important part of emotional circuits (Miao et al., 2011), which participate in the processes of emotion and self-evaluation. Some scholars (Lai et al., 2010; Lee et al., 2013; Sophia et al., 2013) have found that the functions of the posterior cingulate gyrus are reduced in ASD patients. Loveland et al. (2008) found that the functional activity of brain regions (including the orbitofrontal cortex and the posterior cingulate gyrus) related to the judgment of facial expressions in emotional tasks was decreased in ASD patients. The opercular part of the inferior frontal gyrus is connected to the amygdala through the insula lobe so that when observing emotional expressions, an unconscious imitation of facial expression can promote the subjective emotional experience as well as the understanding of others' emotional intentions, which is closely related to emotional face processing (Fusar-Poli et al., 2009). Compared with the TD control group, the ASD group had abnormally fewer FC neural circuits in some involved brain regions, including the right orbital part of the superior frontal gyrus, bilateral orbital part of the middle frontal gyrus, left opercular part of the inferior frontal gyrus, and bilateral posterior cingulate gyrus. This indicates that there are FC deficits in these brain regions, which may be the cause of the cognitive dysfunction experienced by ASD patients and their inability to effectively implement social interaction behaviors. The supplementary motor area is related to the voluntary movement control of the trunk muscles. Huang et al. (2015) found that the FC deficit in the premotor cortex in ASD patients may be one of the mechanisms of habituate behaviors. The number of FC neural circuits involved in the left supplementary motor area was abnormally decreased in the ASD group. This finding indicates that the dysfunction of planning and executing movement in ASD patients may result in abnormal cognitive integration.

The temporal lobes process auditory information and participate in emotion and memory functions. The hippocampus is mainly responsible for the storage, conversion, and orientation of long-term memory, which is related to social episodic memory and self-awareness of memory (Yonelinas et al., 2001; Greenberg et al., 2005). In 1937, Papez found that the “parahippocampal gyrus—hippocampus—papillary body—anterior thalamic nucleus—cingulate gyrus—parahippocampal gyrus” formed the hippocampal circuit, which is related to emotion, learning, memory, and other higher functions. Rudie et al. (2012) showed that the hippocampus and the parahippocampal gyrus in ASD patients are strongly connected, and the habituate behaviors of ASD patients are related to FC of the parahippocampal gyrus. Deficts in the parahippocampal gyrus may cause the dysfunction in emotional and cognitive behaviors. Compared with the TD control group, the ASD group had abnormally fewer FC neural circuits in the left hippocampus and the right parahippocampal gyrus. This indicates that memory dysfunction in ASD patients may be a potential cause of social dysfunction. The Heschl gyrus is located on the superior surface of the temporal lobe. The researcher in Long et al. (2018) found that the Heschl gyrus is responsible for processing the complex spectrum of acoustic information, such as pitch, intensity, and timbre. This finding indicates that speech recognition disorder in ASD patients may result in cognitive dysfunctions.

The parietal lobes participate in memory processing, which is related to sensory and linguistic functions (Roth and Courtney, 2007; Johnson et al., 2011). The angular gyrus is the visual language center, which facilitates the understanding of the meaning of symbols and words. The left angular gyrus connects the visual, auditory, and somatosensory cortices, and various forms of information generate integrated language activities through the function of the angular gyrus (Joseph, 1993; Hofmann et al., 2008). Compared with that in the TD control group, the number of FC neural circuits in the left angular gyrus was abnormally decreased in the ASD group. This may lead to the cognitive dysfunction in ASD patients and the loss of the connection between what they view and what they hear.

The caudate nucleus (Sesack and Grace, 2010) is a key brain region in the reward network and plays an important role in behavior monitoring. Many scholars (Portmann et al., 2014; Staal, 2015; Qiu et al., 2016) believe that the caudate nucleus is related to the habituated behavior of ASD patients. Compared with that in the TD control group, the number of FC neural circuits in the right caudate nucleus was abnormally decreased in the ASD group. This indicates that the dysfunction of the caudate nucleus is associated with repetitive and stereotypical behaviors in ASD patients.

In addition, this study shows that most brain regions are unilateral rather than bilateral abnormalities, which indicates that there is brain laterality for FC in ASD patients.

In conclusion, the present study analyzes the FC neural circuits of ASD patients aged 6–13 years in the resting state and finds that there are significant differences in the topological properties of the FC neural circuits of ASD patients compared with those of the TD controls. These abilities of information transmission and the functional integration of brain networks are reduced. The number of FC neural circuits with different length ranges and the number of FC neural circuits in some involved brain regions are different, particularly those associated with social cognition, executive functioning, and memory. These abnormalities indicate that impairments in social cognition and social communication abilities in ASD patients are related to abnormal changes in FC neural circuits. The shortcomings of this study are the small sample size and cross-sectional data. In view of these shortcomings, the next step is to increase the sample size for research and analysis on individual longitudinal data. In addition, networks of different scales influence brain networks differently. Therefore, topographic maps of different scales should be combined for network construction to obtain more abundant local and global information. Therefore, more reliable biological markers for the diagnosis of ASD patients can be found, and more effective neuroimaging evidence for treatment will be provided.