A total of 44 children with ASD (mean age = 5.52 years, SD = 1.78) and 44 age-matched typically developing (TD) children (mean age = 5.81 years, SD = 1.73) were recruited. No significant differences were observed between the two groups in terms of age (t (86) = 0.15, p = 0.873, Cohen’s d = 0.03) or sex distribution ($\chi$² (1) = 0.08, p = 0.768, Cramer’s V = 0.03) (see Table 1).

The inclusion criteria for children with ASD were as follows: (a) a diagnosis of ASD made by a qualified child psychiatrist according to the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5); (b) aged between 4 and 6 years; and (c) written informed consent obtained from their legal guardians.

The exclusion criteria were as follows: (a) a history of neurological disorders, such as epilepsy, brain injury, or having undergone neurosurgery; and (b) use of antipsychotic or anticonvulsant medications during or prior to the study period.

In this study, the ASD children had limited language abilities and reduced self-care skills, and could only understand and follow simple instructions; therefore, the ASD participants in this study were all classified as low-functioning individuals.

The inclusion criteria for TD children were as follows: (a) aged between 4 and 6 years with normal intelligence, and (b) written informed consent provided by their guardians. Some participating TD children underwent intellectual assessments, with Intelligence Quotient (IQ) scores ranging from 90 to 105, which fell within the normal range.

All resting-state EEG data were collected in a quiet laboratory with a comfortable ambient temperature. Children were accompanied by their parents and seated in a comfortable chair, while the experimenters provided simple and clear instructions to help them remain relaxed with eyes open. During data acquisition, the children’s behavioral state was monitored in real time to ensure data quality. An eight-channel EEG recording system (model JL-EEG-8, Jielian Medical, Jiangxi, China) was used, with electrodes placed at F3, F4, T3, C3, C4, T4, O1, and O2. All electrode impedances were maintained below 20 k$\mathrm{\Omega }$, with Cz used as the reference electrode. The data were sampled at 1000 Hz, and the recording duration for each child was approximately 5 min.

Offline preprocessing was performed using MATLAB R2020a software (https://www.mathworks.com) in conjunction with the EEGLAB v2022.0 toolbox (https://eeglab.org). The raw continuous EEG data were first visually inspected to identify channels with severe noise or instability, which were later restored through interpolation at the end of preprocessing. Considering that artifacts may be introduced by electromagnetic interference, power-line noise, participant movements, electrocardiographic, and ocular activities, the preprocessing pipeline included the following steps: A zero-phase narrowband notch filter centered at 50 Hz was first applied to suppress power-line interference, followed by a zero-phase FIR band-pass filter of 0.5–45 Hz to retain the primary physiological frequency bands. Subsequently, an integrated approach combining ensemble empirical mode decomposition and independent component analysis (EEMD-ICA) was applied to remove artifacts such as eye movements, blinks, and electromyographic activity [25], with a mean of 1.6 independent components removed per dataset. Residual artifact segments were further excluded through visual inspection to ensure data quality. All EEG signals were then re-referenced to the average reference. The final duration of EEG data retained for each child was three min, with no significant differences between groups in terms of preprocessed data length. The data were subsequently segmented into 2-s epochs for further analysis.

We quantified the temporal complexity of EEG signals using the LZC algorithm [26]. LZC is a widely used nonlinear dynamic analysis metric that evaluates the structural diversity and informational richness of signals in the temporal domain. It reflects the brain’s information processing capacity and functional flexibility, which are often reduced in neurodevelopmental disorders such as ASD [27, 28].

Preprocessed EEG signals were segmented into non-overlapping 4-s epochs, consistent with the approach used in previous EEG complexity studies. Each epoch was binarized using the median threshold method, converting the continuous time series into a binary (0-1) sequence suitable for LZC calculation [29]. LZC was computed for each channel and epoch, and the results were subsequently averaged across all epochs to obtain the overall LZC metric for each participant.

Traditional EEG spectral analysis typically uses standardized, fixed-frequency-band boundaries (e.g., the alpha band is uniformly defined as 8–13 Hz). However, substantial evidence has indicated that the peak frequency of neural oscillations, particularly the alpha rhythm, varies considerably across individuals, with this variability being especially pronounced in children with ASD [8, 11]. Using fixed frequency bands may misalign individual oscillatory activity, leading to inaccurate power spectral estimates. To address this, the present study used the gedBounds method [30], which is based on Generalized Eigen Decomposition (GED), to automatically extract individualized frequency-band boundaries from multi-channel EEG signals. By maximizing the separability between the signal and reference covariance matrices, this method identifies frequency features from spatially coherent oscillatory activities. The processing pipeline is as follows:

We introduced the Genuine Symbolic Nonlinear Granger Causality (GSNGC) framework [32] to assess functional connectivity in children with ASD. This method integrates symbolic dynamics, nonlinear causality analysis, and surrogate testing to eliminate spurious connections and enhance the noise robustness of EEG signals.

The preprocessed EEG signals were used to generate surrogate time series through the Iterative Amplitude Adjusted Fourier Transform (IAAFT) algorithm [33]. For each channel, 20 surrogate sequences were generated, preserving the amplitude distribution and power spectrum of the original data while randomizing the phase information.

Both the original and surrogate signals were transformed into symbolic sequences [34]. For a time series $x=\{x_t\},t=1,2,\mathrm{\dots },n$, the state space was reconstructed using an embedding dimension of m = 4 and a time delay $\tau$ (optimized based on the dataset; see Parameter Selection) as follows:

(4)$$X_t=\left[\begin{array}{c}\hfill x_t,x_{t-\tau },x_{t-2\tau },\mathrm{\dots },x_{t+(m-1)\tau }\hfill \end{array}\right]$$

The selection of $\tau$ was based on both empirical ranges and experimental validation: first, candidate $\tau$ values were set within the reasonable range of 1–3 according to the literature; then, with the embedding dimension fixed at m = 4, data segment length L = 2000, and 80% overlap, the effect of different $\tau$ values on the GSNGC network’s ability to capture group-level statistical differences was systematically tested. Finally, $\tau$ = 2 was chosen as the optimal parameter, as it most clearly and stably distinguished functional networks between the ASD and TD groups.

Ordinal patterns were generated by sorting the elements within the reconstructed vectors in ascending order, converting the continuous EEG signals into symbolic sequences (e.g., $x⟶x_{\mathrm{sym}}$).

The directed causal strength between pairs of symbolic sequences $(x_{\text{sym}},y_{\text{sym}})$ was quantified using kernel-based nonlinear Granger causality [35].

(5)$${\text{Nonlinear GC}}_{x\to y}=\ln \left(\frac{{\stackrel{`}{o}}_y}{{\stackrel{`}{o}}_{y\mid x}}\right)$$

where ${\stackrel{`}{o}}_y$ and ${\stackrel{`}{o}}_{\mathrm{y|x}}$ represent the prediction errors of $x_{\text{sym}}$ without and with the historical information of $x_{\text{m}}$, respectively. A Gaussian kernel ($\delta$ = 4) was used to map the symbolic sequences into a reproducing kernel Hilbert space (RKHS).

To eliminate false positive connections, a Wilcoxon signed-rank test (p 0.001) was applied to compare the original connectivity (Nonlinear $G{C}_{\text{original}}$) with the null distribution generated from surrogate data (Nonlinear $G{C}_{\text{surr}}$) [36]. Effective connections were required to meet the following criteria.

(6)

Based on the functional-connectivity matrices extracted using the GSNGC method, brain networks were constructed, and topological features were further calculated to investigate the functional patterns of brain networks in ASD. For each participant, the GSNGC connectivity matrix was treated as a weighted functional-connectivity graph, with nodes corresponding to EEG channels and edge weights representing significant functional-connectivity strengths. Subsequently, the Brain Connectivity Toolbox [37] was used to extract the following graph-theoretical metrics to quantify the topological characteristics of the brain networks:

Clustering coefficient (CC): represents the density of connections surrounding each node.

(7)$$CC=\frac{1}{N}\sum _{i=1}^N\frac{2\times e_i}{k_i\times (k_i-1)}$$

Characteristic path length (CPL): reflects the shortest paths between any two nodes within the network.

(8)$$CPL=\frac{1}{N(N-1)}\sum _{i\ne j}{d}_{ij}$$

Global Efficiency (GE): Measures the information integration capability of the entire network.

(9)$$GE=\frac{1}{N(N-1)}\sum _{i\ne j}\frac{1}{d_{ij}}$$

Local Efficiency (LE): Measures the efficiency of information integration and transmission within a node’s neighborhood, reflecting the fault tolerance of the local network.

(10)$$LE=\frac{1}{N}\sum _{i=1}^N\left(\frac{1}{k_i(k_i-1)}\sum _{j,k\in N_i}\frac{1}{d_{jk}^{(i)}}\right)$$

Here, $e_i$ is the actual number of edges between the neighbors of node i, $k_i$ is the degree of node i (the number of its neighbors). N is the total number of nodes in the network. $d_{\text{ij}}$ is the shortest path length from node i to node j. dd is the shortest path length between nodes i and j after removing node i.

The pseudocode for the GSNGC part is shown below (see Algorithm 2):

All statistical analyses were performed using MATLAB R2020b software. Prior to analysis, the Kolmogorov–Smirnov test was used to assess data normality. For non-normally distributed data, a logarithmic transformation was applied; if normality was still not met, non-parametric tests (such as the Mann–Whitney U test or Wilcoxon signed-rank test) were used. Baseline differences between the ASD and TD groups were assessed using independent samples t-tests (or non-parametric tests). All results were reported after false discovery rate (FDR) correction to control for multiple comparisons. For significant group differences, Cohen’s d was also calculated as a measure of effect size, providing a quantitative assessment of the practical significance of between-group differences.

The present study compared the temporal complexity LZC of resting-state EEG signals between the ASD and TD groups using independent-samples t-tests. Significant t-test results were further corrected for multiple comparisons using FDR. The results are shown in Fig. 1. The global LZC values showed that the ASD group was significantly lower than the TD group (t = –4.42, p 0.001, Cohen’s d = 0.94). In the band-specific analysis (see Fig. 2), no significant difference was observed in the delta band between the two groups (t = –1.12, p = 0.267, Cohen’s d = 0.94). In the theta band, the ASD group exhibited significantly lower LZC than did the TD group in the frontal-central regions (F4, C3, C4; t = –2.35, p = 0.021, Cohen’s d = 0.51). In the alpha band, LZC was significantly reduced across the whole brain in the ASD group (t = –4.07, p = 0.002, Cohen’s d = 0.87). In the beta band, the frontal regions (F3, F4) also showed significant reductions in LZC (t = –3.79, p = 0.005, Cohen’s d = 0.81), whereas in the parietal–occipital regions, the ASD group exhibited significantly lower LZC (t = 2.89, p = 0.031, Cohen’s d = 0.62).

Fig. 1.

Comparison of global lempel-ziv complexity (LZC) between the autism spectrum disorder (ASD) and typically developing (TD) groups. The box plot shows the distribution of LZC values, with the ASD group significantly lower than the TD group. * indicates a significant difference after false discovery rate (FDR) correction (p 0.05).

Fig. 2.

Group comparison of lempel-ziv complexity (LZC) between the autism spectrum disorder (ASD) and typically developing (TD) groups across different frequency bands. The figure shows the differences in LZC values between groups in the main brain regions for the Delta, Theta, Alpha, and Beta bands. t-values surviving false discovery rate (FDR) correction at p 0.05 are displayed.

The GED method was used to determine the individualized frequency bands based on the peak spectral curves (see Table 2). The results showed that children with ASD had a significantly lower lower-frequency limit in the theta band, a wider band range, and less stable boundaries; this was significantly different from children with TD (see Fig. 3). Although the re-test reliability of the band boundaries was not directly assessed in this study, it has been shown that eigenvalue spectral decomposition-based band delineation has good stability across time points [38] (Intraclass Correlation Coefficient, ICC $>$0.8), supporting its reproducibility basis as a tool for individualized spectral analysis.

Fig. 3.

Boxplots of group differences in the lower and upper bounds of the Theta band between autism spectrum disorder (ASD) and typically developing (TD) groups. L-fre, Lower frequency limit; U-fre, Upper frequency limit; * indicates a significant difference after false discovery rate (FDR) correction (p 0.05).

After applying FDR correction to the significant results in the low-frequency bands (delta and theta), TD children exhibited significantly higher individualized power than did children with ASD. These differences were primarily distributed from the central to occipital regions, particularly around C3, C4, and O1. ASD children exhibited an overall reduction in power in the alpha band, whereas TD children showed higher alpha power in the occipital region. In contrast, in the high-frequency beta band, ASD children demonstrated significantly higher power than did TD children in the prefrontal and some central regions (see Fig. 4).

Fig. 4.

Comparison of power spectral topographies between autism spectrum disorder (ASD) and typically developing (TD) children across the Delta, Theta, Alpha, and Beta frequency bands. t-values after false discovery rate (FDR) correction at p 0.05 are displayed.

Significant functional-connectivity differences emerged between the ASD and TD groups across multiple frequency bands (Fig. 5). Within the delta band, the ASD group exhibited significantly enhanced functional connectivity, primarily concentrated from the frontal to central regions (e.g., F3, F4 to C3, C4) and from central to occipital areas (e.g., O1, O2). In the theta band, the ASD group showed stronger connectivity from the prefrontal to parieto-occipital regions (e.g., F3–O1, F3–O2), whereas the TD group displayed stronger connectivity only in a few posterior pathways (e.g., O1–O2). In the alpha band, the TD group demonstrated stronger functional connectivity across multiple central and occipital pathways (e.g., C3–T3, O1–O2), whereas the ASD group generally showed weakened connectivity. In the beta band, the ASD group again exhibited stronger connectivity, covering the frontal, central, and occipital regions (e.g., F3–C4, C4–T4, O1–O2). These results suggested differences in resting-state functional connectivity patterns across multiple frequency bands and brain regions between ASD and TD children.

Fig. 5.

Differences in brain functional connectivity between autism spectrum disorder (ASD) and typically developing (TD) groups across different frequency bands. Each subplot shows significant connectivity differences extracted by generalized symbolic nonlinear Granger causality (GSNGC) in the Delta, Theta, Alpha, and Beta bands. Red lines indicate connections stronger in the ASD group; blue lines indicate connections stronger in the TD group (false discovery rate, FDR correction p 0.05). L, left; R, right.

Network analysis further revealed distinct topological alterations in the ASD group, primarily within the theta and beta bands (Fig. 6). In the theta band, the ASD group showed a significantly higher clustering coefficient (t = 2.25, p = 0.025), global efficiency (t = 2.67, p = 0.008), and local efficiency (t = 2.42, p = 0.016) than did the TD group, although the characteristic path length (t = –2.05, p = 0.041) in the ASD group was slightly shorter than that of the TD group. In the high-frequency beta band, multiple network metrics including clustering coefficient (t = 2.51, p = 0.013), global efficiency (t = 3.942, p 0.001), and local efficiency (t = 2.41, p = 0.017) were significantly higher in the ASD group than in the TD group. No consistent significant differences were observed in the delta and alpha bands.

Fig. 6.

Comparison of brain-network graph-theory metrics between autism spectrum disorder (ASD) and typically developing (TD) groups across different frequency bands (Delta, Theta, Alpha, Beta). The figure shows the distribution of common graph metrics for each band: (a) clustering coefficient, (b) characteristic path length, (c) global efficiency, and (d) local efficiency, * indicates a significant difference after false discovery rate (FDR) correction (p 0.05).

To evaluate the diagnostic efficacy of multi-dimensional EEG features in ASD classification, we constructed an SVM classification model based on integrated temporal (4 $\times$ 8), frequency (4 $\times$ 8), and spatial (8 $\times$ 7 $\times$ 4 + 8 $\times$ 4 $\times$ 4) features. A radial basis function (RBF) was used as the kernel for the SVM, with a penalty parameter C = 1. Although other hyperparameters were not optimized, the model still achieved relatively good classification performance.

A subject-level 5-repetition 10-fold cross-validation was employed, in which all epoch-level features from each participant were aggregated for analysis. This approach ensured complete independence between the training and testing sets, effectively preventing data leakage. The model achieved an average accuracy of 89.21% (95% CI: 85.1%–92.8%), a sensitivity of 87.54% (95% CI: 82.3%–91.9%), a specificity of 89.65% (95% CI: 85.0%–93.2%), an F1-score of 89.72% (95% CI: 85.5%–93.2%), and an area under the curve (AUC) of 0.91 (95% CI: 0.90%–0.97%), demonstrating robust classification performance. Further comparison of classification performance across different feature dimensions showed that using LZC features alone yielded an accuracy of 64.62% ($\pm$4.2%), individualized frequency-band power alone achieved 76.11% ($\pm$5.1%), and functional connectivity alone reached 81.67% ($\pm$4.8%) (see Table 3). After integrating all three feature types, accuracy increased significantly, indicating that multi-dimensional feature-fusion enhanced the discriminative capability of the model. For comparison, a baseline model using only age and sex achieved an accuracy of 52.6% (95% CI: 41.2%–63.1%) and an AUC of 0.51. All EEG feature–based models significantly outperformed the baseline model, indicating that EEG features played a critical role in distinguishing children with ASD from TD children.

To verify the robustness of our findings, a sensitivity analysis was conducted. All male participants were selected from the original sample (ASD: n = 37; TD: n = 38), and the SVM modeling and evaluation were performed using the same multi-dimensional EEG features and classification procedures as in the full-sample analysis. The results showed that, even within the male-only sample, the integrated model (T + F + S) still achieved an accuracy of 87.9% ($\pm$4.3%), comparable to the performance observed in the full sample (89.2%).

Despite providing valuable preliminary findings, this study had several limitations. First, due to the limited number of EEG channels, the spatial resolution was relatively low, making it difficult to resolve neural activity in deep brain regions precisely. Future studies should combine high-density EEG or fMRI to provide complementary validation. Second, the sample in this study was restricted to children aged 4–6 years, which limited the generalizability of the findings. Subsequent research should include a broader age range to characterize the developmental trajectory of neural dynamics in ASD. Third, this study did not systematically quantify the relationship between EEG features and core clinical symptoms of ASD. Future work could integrate behavioral scales (e.g., Social Responsiveness Scale, SRS, Autism Behavior Checklist, ABC) to perform correlation and modeling analyses. Finally, before translation into clinical tools, the findings must be validated in larger, multi-center independent cohorts. Future studies should recruit more representative samples to examine the generalizability of multi-dimensional EEG features.