In this study, we recruited 48 children diagnosed with ASD, with a mean age of 5.61 years (SD = 1.32), and 48 age-matched TD children, with a mean age of 5.48 years (SD = 1.39). The ASD children were randomly assigned to two groups: (1) the experimental group, consisting of 24 children (mean age = 5.84 years, SD = 1.85), who received 24 sessions of real tDCS; and (2) the control group, also comprising 24 children (mean age = 5.44 years, SD = 2.17), who received 20 sessions of sham tDCS.

The inclusion criteria for children with ASD were: (1) a diagnosis confirmed by a professional child psychiatrist based on the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5) [24]; (2) ages between 4 and 6 years; and (3) written informed consent. Exclusion criteria included: (1) a history of neurological disorders, such as epilepsy, brain injury, or neurosurgery; (2) the use of antipsychotic or anticonvulsant medications during or prior to the experimental period; and (3) previous exposure to tDCS, repetitive transcranial magnetic stimulation (rTMS), or neurofeedback prior to participation. Before the experiment, parents or legal guardians were fully briefed on the study procedures and provided written informed consent. They were also informed that the children could withdraw from the study at any time without any negative consequences. Throughout the study, the children with ASD continued their regular behavioral training. This study adhered to the Declaration of Helsinki and was approved by the Ethics Committee of Ningbo Rehabilitation Hospital (Approval No. 2023006).

This study was conducted in three phases:

Phase 1: Baseline Assessment. In the first phase, detailed EEG recordings were taken from all participating children to establish baseline brain activity patterns. Additionally, comprehensive behavioral assessments were carried out using the ABC to evaluate the children’s behavioral baseline.

Phase 2: Treatment Period. During this phase, children in the experimental group received tDCS treatment with 1 mA anodal stimulation applied to the left DLPFC. Each treatment session lasted 20 minutes and was administered five times a week for four weeks, totaling 20 sessions. Children in the control group received sham stimulation to account for placebo effects. Throughout this phase, the children’s comfort and any potential side effects were carefully monitored to ensure both safety and efficacy.

Phase 3: Post-tDCS Assessment. After the treatment period, EEG recordings and ABC assessments were repeated for all children to evaluate changes in brain activity and behavior following the tDCS intervention.

In this study, to ensure the safety and efficacy of tDCS stimulation, we strictly controlled the impedance between the two saline-soaked sponge electrodes during stimulation, keeping it below 20 k$\mathrm{\Omega }$. This impedance control helped reduce skin irritation and ensured uniform current distribution. For the real tDCS intervention, the anode was placed over the left DLPFC, while the cathode was positioned over the right supraorbital area. During the real tDCS intervention, the current gradually increased to 1 mA over a period of 30 seconds and was then maintained at this intensity for the following 20 minutes, before gradually decreasing to 0 mA over another 30 seconds. This gradual ramp-up and ramp-down of the current was designed to minimize potential discomfort and ensure smooth delivery of stimulation. In contrast, during sham tDCS, although the electrode positions were identical to those used in real tDCS, the current was delivered only through the participant’s scalp during the first 30 seconds of the intervention. This brief current delivery simulated the initial sensation of real tDCS but did not produce any sustained neuromodulatory effects.

EEG data were collected in a soundproof room while participants remained at rest with their eyes open. An eight-channel EEG acquisition system was used to record five minutes of EEG data from electrodes placed at F3, F4, C3, C4, T3, T4, O1, and O2. During the data recording, the impedance of all channels was maintained below 50 k$\mathrm{\Omega }$, and the sampling rate was set to 1000 Hz.

The offline data analysis was performed using MATLAB R2016a (Version 9.0.0.341360, MathWorks, Inc., Natick, MA, USA) and EEGLAB V2022.1 (Swartz Center for Computational Neuroscience, University of California San Diego, La Jolla, CA, USA). To reduce the impact of external noise and experimental conditions on the EEG signals, a bandpass filter with a range of 0.5 to 40 Hz was applied during preprocessing. Independent Component Analysis (ICA) was then used to decompose the EEG signals and separate different independent components. The automatic algorithms in EEGlab were employed to identify and remove artifacts associated with eye movements, muscle activity, and electromyography (EMG) from the independent components. The average number of Independent Components (ICs) removed due to artifacts was 1.6. After artifact-related components were removed, the EEG signal was reconstructed using the remaining independent components, excluding any component with a data length less than half of the original data length. Finally, manual inspection was conducted to detect and eliminate any remaining artifacts to ensure data quality, and all channels were re-referenced to the average reference.

In this study, we aimed to elucidate the global information flow within the complex brain network by examining the transmission of information between various brain regions. To achieve this, we constructed functional brain networks for both children with ASD and TD children across the $\delta$ (1–4 Hz), $\theta$ (4–8 Hz), $\alpha$ (8–13 Hz), and $\beta$ (13–30 Hz) frequency bands. Specifically, our approach focused on analyzing the signal synchrony between paired electrodes within these networks.

We first calculated the phase of the EEG for each channel and then determined the phase-locking value (PLV) for each pair of channels, as described in the literature [25]. This approach allowed us to generate an 8 $\times$ 8 adjacency matrix, where higher PLV values indicated stronger phase synchrony between channels. A crucial aspect of constructing the brain networks was defining the nodes and connectivity [26]. In this study, the number of EEG data channels served as the nodes, with each network consisting of 8 nodes. For connectivity, we used functional connectivity based on PLV calculations to quantify the interactions between these nodes.

To build the functional networks, we utilized a 2-second time window with an 80% overlap. This approach facilitated a comprehensive analysis of the dynamic interactions within the brain’s functional architecture across different frequency bands. Specifically, to estimate the instantaneous phases $\phi$ u(t) and $\phi$ v(t) of the EEG signals u(t) and v(t), we established the analytic signal H(t) using the Hilbert Transform (HT), which can be expressed as:

(1)$$\{\begin{array}{c}{H}_u(t)=u(t)+iH{T}_u(t)\hfill \\ H_v(t)=v(t)+iH{T}_v(t)\hfill \end{array}$$

where $H_u\left(t\right)$ and $H_v\left(t\right)$represented the corresponding HT, which can be expressed as:

(2)$$H{T}_u(t)={\displaystyle \frac{1}{\pi }}P\cdot V\cdot {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{u\left(t^{\prime }\right)}{t-t^{\prime }}}𝑑t^{\prime }$$ $$H{T}_v(t)={\displaystyle \frac{1}{\pi }}P\cdot V\cdot {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{v\left(t^{\prime }\right)}{t-t^{\prime }}}𝑑t^{\prime }$$

where P.V. denoted the Cauchy principal value. The phases of the analytic signals $\phi$_u(t) and $\phi$_v(t) was then calculated as follows:

(3)$$\{\begin{array}{c}{\phi }_u(t)=\arctan \frac{H{T}_u(t)}{u(t)}\hfill \\ {\phi }_v(t)=\arctan \frac{H{T}_v(t)}{v(t)}\hfill \end{array}$$

Subsequently, the calculation of the PLV was given by:

(4)$$PLV=\left|\frac{1}{N}\sum _{j=0}^{N-1}{e}^{i\left({\phi }_u(j\mathrm{\Delta }t)-{\phi }_v(j\mathrm{\Delta }t)\right)}\right|$$

where j represented the j-th sampling point, and N was the total number of samples for each signal; t denoted the time point and $\mathrm{\Delta }$t was the sampling period. The PLV reflected the phase synchrony strength between paired nodes.

In this study, we employed fuzzy entropy to quantify the complexity of brain functional networks and to investigate the temporal variability of their connectivity [27]. Specifically, we assumed that each subject had M functional networks, with the connectivity system $W_i$ () defined as follows:

(5)$$\begin{array}{c}\hfill W_i^q=\{w(i),w(i+1),\mathrm{\dots },w(i+q-1)\},\overline{w}(i)i=1,\mathrm{\dots },M-q+1\hfill \end{array}$$

where $W_i^q$ represented the $q$ consecutive $w$ values (i.e., PLV values) of the $i$-th network, which can be derived by excluding the baseline $\overline{w}\left(i\right)=q^{-1}{\sum }_{l=0}^{q-1}w(i+l)$.

Given r, the similarity index between $W_i^q$ and the corresponding $W_l^q$ was denoted as $D_{il}^q$:

(6)$$D_{il}^q=\mu (D_{il}^q,r)$$

where $D_{il}^q$ represented the maximum absolute difference of scalar components between $W_l^q$ and $W_i^q$. For each vector $W_i^q(I=1,2,\mathrm{\dots },M-q-1)$, the average similarity $D_{il}^q$ with all adjacent vectors $W_l^q(l=1,2,\mathrm{\dots },M-q-1,l\ne i)$ was computed as follows:

(7)$${\phi }_i^q(r)={(M-q-1)}^{-1}\sum _{l=1,l\ne i}^{M-q}{D}_{il}^q$$

The $FuzzEn(q,r)$ of the time series $W_i(1\le i\le M)$ was formulated as:

(8)$$FuzzEn(q,r)=\underset{M\to \mathrm{\infty }}{lim}\left[\ln {\phi }^q(r)-\ln {\phi }^{q+1}(r)\right]$$

This can be estimated through statistical data:

(9)$$FuzzEn(q,r,M)=\ln {\phi }^q(r)-\ln {\phi }^{q+1}(r)$$

Where M was the length of the time series to be analyzed, r was the width of the similarity measure boundary, and q was the length of the comparison window. Previous study has suggested that an appropriate value for q ensures a detailed reconstruction of dynamic processes, while values that were too large may result in information loss [28]. Based on prior research [29], we set q to 2. Furthermore, if r was too small, it may introduce noise, while if it was too large, it may lead to information loss. Therefore, we set r to 0.2 times the standard deviation of the time series [27].

Regarding the temporal variability of functional networks, we utilized preprocessed EEG signals to calculate the PLV with an 80% overlap and a 2-second time window. This approach generated M networks with a resolution of 200 milliseconds. To construct dynamic functional networks, we employed fuzzy entropy as a measure. Given that fuzzy entropy calculations can be easily influenced by false connectivity within the network, we implemented a thresholding strategy to identify and remove the false connectivity. A threshold that was too high may retain problematic connections, while one that was too low may lead to the loss of important connectivity [30]. Drawing from previous studies, we defined the top 20% of network edges with the highest fuzzy entropy as indicative of flexible network organization. This approach has been shown to help accurately classify different groups and predict cognitive performance [31]. Based on these findings, we applied a 20% threshold to eliminate false connectivity in our analysis.

To quantify the potential for functional segregation and integration in the brain and to reflect both global and local features of brain networks [32], four network characteristics were calculated: CC, CPL, GE, and local efficiency (LE) [21].

The clustering coefficient served as an indicator of the degree of interconnectivity among nodes in the brain network, reflecting the local clustering of the network. A higher clustering coefficient indicated greater efficiency in information transfer.

(10)$$CC=\frac{1}{n}\sum _{i\in \theta }\frac{{\sum }_{j,h\in \theta }\left(w_{ij}^{plv}{w}_{ih}^{plv}{w}_{jh}^{plv}\right)}{{\sum }_{jϵ\theta }{w}_{ij}^{plv}\left({\sum }_{j\in \theta }{w}_{ij}^{plv}-1\right)}$$

The characteristic path length (CPL) was the average length of the shortest paths between all pairs of nodes in the network, reflecting the efficiency of global information processing in the brain network. A smaller value indicated higher efficiency in information transfer within the network.

(11)$$CPL=\frac{1}{n}{\sum }_{iϵ\theta }\frac{{\sum }_{jϵ\theta ,j\ne i}{d}_{ij}}{n-1}$$

GE was a characteristic that reflected the speed of information transfer in the brain network and served as an important metric for assessing the network’s information transmission capacity. It was the inverse of the harmonic mean of the shortest path lengths between all pairs of nodes. A higher global efficiency indicated better information transmission performance within the network.

(12)$$GE=\frac{1}{n}{\sum }_{i\in \theta }\frac{{\sum }_{j\in \theta ,j\ne i}{\left(d_{ij}\right)}^{-1}}{n-1}$$

LE reflected the capacity for local information transfer within the network and the closeness of neighboring nodes. It was defined as the harmonic mean of the shortest path lengths between all pairs of nodes within the same local sub-network.

(13)$$\mathrm{LE}=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{i}\in \theta }\frac{{\sum }_{\mathrm{j},\mathrm{h}ϵ\theta ,\mathrm{j}\ne \mathrm{i}}{\left({\mathrm{w}}_{\mathrm{ij}}^{\mathrm{plv}}{\mathrm{w}}_{\mathrm{ih}}^{\mathrm{plv}}{\left[{\mathrm{d}}_{\mathrm{jh}}\left({\theta }_{\mathrm{i}}\right)\right]}^{-1}\right)}^{\frac{1}{3}}}{{\sum }_{\mathrm{j}ϵ\theta }{\mathrm{w}}_{\mathrm{ij}}^{\mathrm{plv}}\left({\sum }_{\mathrm{j}\in \theta }{\mathrm{w}}_{\mathrm{ij}}^{\mathrm{plv}}-1\right)}$$

Here, $\theta$ represented the set of all network nodes, n was the number of nodes, $w_{ij}^{plv}$ denoted the weighted connectivity strength between node i and node j, and $d_{ij}$denoted the shortest path length between node i and node j.

In this study, we used the ABC [33] as an assessment tool to monitor behavioral changes before and after the tDCS intervention. The ABC was completed by the parents or guardians of the children and consisted of 57 items designed to comprehensively assess the behavior of children with autism across multiple domains. These items were categorized into five subscales: sensory processing, social relationships, use of body and objects, language and communication skills, and social and adaptive skills.

To investigate the differences in brain functional networks and their characteristics between TD children and children with ASD, we first performed independent sample t-tests for preliminary analysis, followed by False Discovery Rate (FDR) correction to adjust for multiple comparisons. Additionally, we conducted a 2 $\times$ 2 repeated measure analysis of variance (ANOVA) to compare the experimental and control groups. After the ANOVA, we applied FDR correction to the p-values and performed post hoc analyses to further explore significant interaction effects. Before conducting these statistical tests, we assessed the normality of the data using the Kolmogorov-Smirnov (K-S) test. For data that did not meet the assumption of normal distribution, we performed log transformation to achieve normality. For data that met normal distribution criteria, we used independent sample t-tests or 2 $\times$ 2 repeated measure ANOVA for further analysis.

We first constructed the brain networks using the PLV algorithm and examined the differences in functional connectivity between TD children and children with ASD across four different frequency bands. As shown in Fig. 1, there were significant differences in functional connectivity between the two groups, with ASD children exhibiting weaker connectivity in the $\alpha$ band, while alternating patterns of enhanced and diminished connectivity were observed in the other three bands (p 0.05).

Fig. 1.

Differences in functional connectivity across four frequency bands between TD and ASD children. A positive value (red line) indicated that functional connectivity was higher in TD children compared to ASD children, while a negative value (blue line) indicated that functional connectivity was lower in TD children compared to ASD children. TD, typically developing; ASD, autism spectrum disorder.

Subsequently, we further investigated the differences in the temporal variability of brain network functional connectivity. As shown in Fig. 2, both groups demonstrated significant differences in temporal variability, with ASD children showing higher temporal variability across all four frequency bands (p 0.05).

Fig. 2.

Differences in the temporal variability of functional connectivity between TD and ASD children across four frequency bands. A positive value (red line) indicated that temporal variability was higher in TD children compared to ASD children, while a negative value (blue line) indicated that temporal variability was lower in TD children compared to ASD children.

In exploring the network characteristics (see Fig. 3), we found that ASD children had lower CC and LE in the $\alpha$ band, while displaying higher CC and LE in the $\beta$ band. No significant differences in network characteristics were observed between the two groups in the $\delta$ and $\theta$ bands. Additionally, the $\alpha$ band exhibited lower GE and longer CPL, whereas the $\beta$ band showed higher GE and shorter CPL. No significant differences were observed in the other frequency bands.

Fig. 3.

Differences in network properties across four frequency bands between TD and ASD children. The comparison included four network characteristics: CC, LE, GE, and CPL in each frequency band. “*” indicated a p-value less than 0.05.

This study aimed to explore the impact of tDCS on brain network functionality in children with ASD. We compared the differences in functional connectivity between the experimental and control groups before and after the tDCS intervention across four different frequency bands. As presented in Table 1, significant interaction effects were observed between the two groups. Further post hoc analyses indicated that, as shown in Fig. 4, the experimental group exhibited enhanced functional connectivity in all four frequency bands following tDCS, while the control group did not show significant differences.

Fig. 4.

Differences in functional connectivity across four frequency bands before and after tDCS in the experimental group. Positive values indicated that the functional connectivity of children in the experimental group after tDCS was higher than before tDCS, while negative values indicated that the functional connectivity after tDCS was lower than before tDCS. tDCS, transcranial direct current stimulation.

We also investigated the temporal variability of brain network functional connectivity before and after the tDCS intervention. As shown in Table 2, the results indicated that the experimental group showed reduced temporal variability in functional connectivity in the $\alpha$ and $\beta$ bands after receiving tDCS, while the control group exhibited no significant differences, as shown in Fig. 5.

Fig. 5.

Differences in the temporal variability of functional connectivity across four frequency bands in the experimental group before and after tDCS intervention. Positive values indicated that the temporal variability of functional connectivity in children from the experimental group after tDCS was higher than before tDCS, while negative values indicated that the temporal variability of functional connectivity after tDCS was lower than before tDCS.

Additionally, we conducted a comparative analysis of network characteristics before and after tDCS intervention. Based on the data in Table 3 and Fig. 6, we found that ASD children who received 20 sessions of real tDCS treatment exhibited higher CC and LE in the $\alpha$ band, as well as shorter CPL and higher GE in the $\alpha$ and $\delta$ bands. However, no significant differences in network characteristics were observed in the $\theta$ and $\beta$ bands.

Fig. 6.

Differences in network properties across four frequency bands before and after tDCS for the experimental children. “*” indicated a p-value less than 0.05.

To assess the impact of tDCS intervention on symptom improvement in children with ASD, we conducted repeated measures analysis of variance on ABC scores for both the experimental and control groups before and after tDCS treatment, with the results presented in Table 4. Post hoc analyses were then performed on the scores that showed significant interaction effects, as shown in Table 5. The results revealed that after 20 sessions of real tDCS treatment, the total ABC scores of children in the experimental group showed a significant decrease (p 0.05), indicating symptom improvements. In contrast, the control group children who received sham tDCS treatment did not show significant differences in their ABC scores. These findings provided preliminary evidence for the effectiveness of tDCS intervention in improving symptoms in children with ASD.