ERG-Graph: Graph Signal Processing of the Electroretinogram for Classification of Neurodevelopmental Disorders
Abstract
1. Introduction
- (1)
- To develop and optimize the ERG-Graph construction methodology, including the adaptation of amplitude quantization and graph generation to the ERG waveform, the systematic evaluation of quantization resolution, and the comparison of multiple graph construction strategies (quantization-based, visibility graph, recurrence network, k-nearest neighbor, -ball, and ordinal partition networks).
- (2)
- To extract, evaluate, and compare graph-theoretic features across centrality, topology, clustering, and spectral domains for their ability to differentiate ASD, ADHD, and control groups across multiple classification scenarios (two-group, three-group, and four-group), sex strata, flash strengths, and eye laterality conditions.
- (3)
- To demonstrate that ERG-Graph features provide complementary and superior classification performance compared to traditional time-domain and time–frequency (DWT, VFCDM) features, both independently and in fusion, thereby advancing objective biomarkers for neurodevelopmental disorder screening.
2. Materials and Methods
2.1. Participants and Electrophysiology
2.2. ERG-Graph Construction
2.3. Alternative Graph Construction Methods
2.4. Graph-Theoretic Feature Extraction
- (1)
- Total Load Centrality (TLC). Load centrality quantifies the fraction of shortest paths passing through each node. For a node v, the load centrality is the proportion of all shortest paths between pairs of other nodes that pass through v [59,67]. TLC sums this measure over all nodes:where is the total number of shortest paths from node s to node t, and is the number of those paths passing through v. Higher TLC indicates that more amplitude states serve as obligatory transitions, reflecting waveform complexity.
- (2)
- Total Harmonic Centrality (THC). Harmonic centrality handles disconnected components gracefully by using inverse distances [68]. THC aggregates the harmonic closeness across all nodes:where is the shortest-path distance between nodes v and u. THC reflects global reachability within the graph: signals with diverse amplitude transitions produce graphs where nodes are mutually accessible through short paths.
- (3)
- Number of Cliques (GNC). A clique is a maximally complete subgraph in which every pair of nodes is connected [69]. The number of maximal cliques is:
- (4)
- (5)
- Graph Radius. The radius is the minimum eccentricity over all nodes, where eccentricity = max() for all u [60]:
- (6)
- Average Clustering Coefficient (CC). The local clustering coefficient measures the tendency of a node’s neighbors to be interconnected [70]. The global average is:where is the degree of node v and are edges between neighbors of v. High CC indicates that amplitude levels visited in sequence tend to form closed triangles, reflecting localized oscillatory dwelling.
- (7)
- (8)
- Algebraic Connectivity (λ2). The second-smallest eigenvalue of the graph Laplacian L, known as the Fiedler value, measures the robustness of graph connectivity [45,71,72]:where is the first (constant) eigenvector. A higher indicates that the graph is tightly connected with few structural bottlenecks separating amplitude regions. In the ERG context, ASD graphs exhibit elevated because the signal remains within a compact amplitude range, producing redundant transitions between nearby levels. Conversely, ADHD graphs show low because the signal traverses a broader amplitude range, creating a more fragmented trajectory through amplitude space.
- (9)
- Graph Density (ρ). The ratio of observed edges to the maximum possible (Equation (5)). Dense graphs indicate that many different amplitude transitions occur, while sparse graphs indicate a constrained trajectory [60].
2.5. Classification Scenarios
2.6. Machine Learning Pipeline and Statistical Analysis
3. Results
3.1. Quantization Resolution Optimization
3.2. Graph Structure Analysis
3.3. Two-Group Classification Performance
3.4. Three-Group Classification
3.5. Four-Group Classification
3.6. Feature Importance
3.7. Graph Construction Variant Comparison
| Study | Features | Classification Scenario | Best BA | n | Notes |
|---|---|---|---|---|---|
| Posada-Quintero et al. [10,11] | VFCDM spectral | ASD vs. Control (binary) | Sens. 0.85 /Spec. 0.78 | 278 | Sensitivity/specificity reported |
| Constable et al. [3] | TD + DWT | ASD/ADHD vs. Control | — | 278 | Wavelet energy bands; BA not reported |
| Manjur et al. [13] | TD + DWT + VFCDM | Three-group (ASD/ADHD/Ctrl) | 0.70 | 278 | Previous three-group benchmark |
| Constable et al. [12] | TD + DWT + VFCDM | ASD vs. Ctrl (males)/ADHD vs. Ctrl (females) | 0.87/0.84 | 278 | Four-group BA = 0.53 |
| This study (ERG-Graph only) | GSP graph features | ASD vs. Ctrl (males)/ADHD vs. Ctrl (females) | 0.91/0.88 | 278 | 9 topological + spectral features |
| This study (ERG-Graph + TD) | GSP + TD fusion | Three-group (ASD/ADHD/Ctrl) | 0.81 | 278 | +11 pp vs. prior benchmark; four-group BA = 0.67 |
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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| Feature | Control | ASD | ADHD | p-Value | δ ASD/Ctrl | δ ADHD/Ctrl | δ ASD/ADHD |
|---|---|---|---|---|---|---|---|
| TLC | 245.3 ± 42.1 | 198.7 ± 38.5 | 278.4 ± 51.2 | <0.001 *** | −0.52(L) | 0.34(M) | −0.61(L) |
| THC | 312.8 ± 55.6 | 289.4 ± 49.3 | 341.2 ± 62.7 | 0.002 ** | −0.22(S) | 0.27(S) | −0.41(M) |
| GNC | 18.4 ± 4.2 | 15.1 ± 3.8 | 21.7 ± 5.1 | <0.001 *** | −0.41(M) | 0.36(M) | −0.58(L) |
| Diam. | 8.2 ± 1.9 | 6.8 ± 1.5 | 9.5 ± 2.3 | <0.001 *** | −0.44(L) | 0.39(M) | −0.56(L) |
| Radius | 4.5 ± 1.1 | 3.9 ± 0.9 | 5.2 ± 1.4 | 0.001 ** | −0.31(M) | 0.28(M) | −0.47(L) |
| CC | 0.42 ± 0.08 | 0.48 ± 0.09 | 0.37 ± 0.07 | <0.001 *** | 0.38(M) | −0.33(M) | 0.59(L) |
| APL | 3.8 ± 0.7 | 3.2 ± 0.6 | 4.3 ± 0.9 | <0.001 *** | −0.47(L) | 0.31(M) | −0.57(L) |
| λ2 | 0.35 ± 0.08 | 0.41 ± 0.09 | 0.29 ± 0.07 | 0.003 ** | 0.35(M) | −0.40(M) | 0.55(L) |
| Density | 0.28 ± 0.05 | 0.32 ± 0.06 | 0.24 ± 0.05 | <0.001 *** | 0.37(M) | −0.42(M) | 0.54(L) |
| Comparison | Sex | Model | Flash/Eye | BA | F1 | Feat. |
|---|---|---|---|---|---|---|
| ASD vs. Ctrl | Male | XGB | 446 Td.s/Right | 0.91 | 0.90 | 7 |
| ASD vs. Ctrl | Female | RF | 446 Td.s/Right | 0.84 | 0.83 | 8 |
| ASD vs. Ctrl | All | XGB | 446 Td.s/R + L | 0.84 | 0.83 | 12 |
| ADHD vs. Ctrl | Male | SVM | 446 Td.s/Right | 0.83 | 0.82 | 6 |
| ADHD vs. Ctrl | Female | RF | 446 Td.s/Right | 0.88 | 0.87 | 7 |
| ADHD vs. Ctrl | All | RF | 446 Td.s/R + L | 0.83 | 0.82 | 11 |
| Feature Set | Model | BA | F1 | Feat. |
|---|---|---|---|---|
| TD only | AdaB | 0.62 | 0.60 | 8 |
| TD + DWT | GradB | 0.67 | 0.65 | 24 |
| TD + VFCDM | XGB | 0.70 | 0.68 | 32 |
| TD + DWT+VFCDM | XGB | 0.70 | 0.69 | 45 |
| ERG-Graph only | XGB | 0.78 | 0.76 | 18 |
| ERG-Graph + TD | XGB | 0.81 | 0.79 | 22 |
| Full fusion | XGB | 0.79 | 0.77 | 38 |
| Feature Set | Model | BA | F1 | #Feat. |
|---|---|---|---|---|
| TD + DWT + VFCDM [12] | XGB | 0.53 | 0.51 | 45 |
| ERG-Graph only | XGB | 0.64 | 0.62 | 18 |
| ERG-Graph + TD | XGB | 0.67 | 0.65 | 22 |
| Full fusion | XGB | 0.65 | 0.63 | 38 |
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Mercado-Diaz, L.R.; Pinzon-Arenas, J.O.; Constable, P.A.; Lee, I.O.; Loh, L.; Thompson, D.A.; Posada-Quintero, H.F. ERG-Graph: Graph Signal Processing of the Electroretinogram for Classification of Neurodevelopmental Disorders. Bioengineering 2026, 13, 446. https://doi.org/10.3390/bioengineering13040446
Mercado-Diaz LR, Pinzon-Arenas JO, Constable PA, Lee IO, Loh L, Thompson DA, Posada-Quintero HF. ERG-Graph: Graph Signal Processing of the Electroretinogram for Classification of Neurodevelopmental Disorders. Bioengineering. 2026; 13(4):446. https://doi.org/10.3390/bioengineering13040446
Chicago/Turabian StyleMercado-Diaz, Luis Roberto, Javier O. Pinzon-Arenas, Paul A. Constable, Irene O. Lee, Lynne Loh, Dorothy A. Thompson, and Hugo F. Posada-Quintero. 2026. "ERG-Graph: Graph Signal Processing of the Electroretinogram for Classification of Neurodevelopmental Disorders" Bioengineering 13, no. 4: 446. https://doi.org/10.3390/bioengineering13040446
APA StyleMercado-Diaz, L. R., Pinzon-Arenas, J. O., Constable, P. A., Lee, I. O., Loh, L., Thompson, D. A., & Posada-Quintero, H. F. (2026). ERG-Graph: Graph Signal Processing of the Electroretinogram for Classification of Neurodevelopmental Disorders. Bioengineering, 13(4), 446. https://doi.org/10.3390/bioengineering13040446

