ABFR-KAN: Neural FC Analysis

• ABFR-KAN is an advanced brain function representation system that replaces atlas-based parcellation with data-driven patch sampling and Kolmogorov–Arnold Network nonlinearities.
• It uses randomized anchor selection and iterative patch sampling to generate individualized, multi-scale functional connectivity matrices while mitigating atlas bias.
• The transformer-based architecture with KAN feed-forwards significantly improves diagnostic accuracy in neurological disorders, as evidenced by robust cross-site evaluations.

ABFR-KAN (Advanced Brain Function Representation with Kolmogorov–Arnold Networks) is a transformer-based neural architecture for functional connectivity (FC) analysis of brain fMRI, optimized for classification tasks such as diagnosis of neurological disorders. It replaces conventional atlas-based parcellation and multi-layer perceptrons (MLPs) with data-driven patch sampling and Kolmogorov–Arnold Network (KAN) nonlinearities. Extensive empirical evaluations on multi-site datasets demonstrate improved subject specificity, reduction of atlas selection bias, and state-of-the-art diagnostic accuracy in autism spectrum disorder (ASD) classification (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

1. Theoretical Foundations: Kolmogorov–Arnold Networks

The KAN replaces standard MLP blocks in deep networks by leveraging the Kolmogorov–Arnold representation theorem: any continuous function $f:[0,1]^n \to \mathbb R$ decomposes into superpositions and sums of continuous univariate functions. In particular,

$$f(x_1, \ldots, x_n) = \sum_{q=0}^{2n} \Phi_q\left( \sum_{p=1}^n \Psi_{p,q}(x_p) \right),$$

with $\Psi_{p,q}$, $\Phi_q$ univariate and continuous. KANs implement this via learnable basis expansions (e.g., B-spline, radial, wavelet, or Chebyshev bases) for edge-specific univariate nonlinearities: $$y_i = \sum_{j=1}^d \phi_{ij}(x_j), \qquad \phi_{ij}(x) = \sum_{\ell=1}^m w_{ij\ell} B_\ell(x),$$ where $B_\ell(x)$ are basis functions and $w_{ij\ell}$ learnable weights. Efficient KAN implementations (e.g., FastKAN, Wav-KAN, ChebyKAN) reduce computational costs by factorizing basis function evaluations and pooling features via global matrix multiplication.

This approach provides strictly greater expressivity than fixed-activation MLPs for the same architectural width and depth, supporting more flexible approximations of complex FC mappings derived from neuroimaging data (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

2. Advanced Brain Function Representation Module

Conventional FC analysis relies on atlas-based spatial segmentation, introducing selection bias and impairing subject specificity. ABFR-KAN circumvents this via two data-driven procedures:

1. Randomized Anchor Selection: Anchor patches are sampled uniformly in the brain volume with enforced minimal gray matter (GM) content, yielding a subject-specific, anatomically-conforming anchor set. Given 3D fMRI $V \in \mathbb{R}^{T \times X \times Y \times Z}$ and GM mask $M$, patches of side $s$ are accepted only if $\sum_{\text{patch}} M \geq \tau$ (e.g., $\tau=100$).
2. Iterative Patch Sampling: Subject patches are drawn with uniform centers and multiple spatial scales. To ensure GM inclusion, only patches containing at least one GM voxel are retained. This sampling is repeated for $R$ iterations and scales.

For each scale and subject, the mean BOLD time course per patch and anchor is computed, followed by Pearson correlation estimation of patch-anchor FC: $$F_{ij}^{(r)} = \mathrm{Corr}(v_i^{(r)}, a_j), \quad i=1, \ldots,N, \; j=1,\ldots,H,\; r=1, \ldots,R.$$ Final FC matrices are aggregated by averaging over $r$ and concatenating positional embeddings, producing robust, multi-scale, individualized FC measurements (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

3. Transformer-Based Classification Architecture

The FC-position fused matrix and normalized patch coordinates are embedded as token vectors and input to a Vision Transformer (ViT)-style encoder. Key steps include:

• Input Embedding and Top-K Pooling: Features are projected and scored, with the top percentile (e.g., 80%) of most informative patches retained to maintain computational efficiency.
• Transformer Encoder with KAN Feed-Forwards: Standard feed-forward (FFN) MLP layers in each encoder block are replaced with KAN blocks (e.g., FastKAN, Wav-KAN), augmenting the network's ability to learn complex nonlinear FC structures.
• Global Pooling and Classification Head: Output tokens are mean-pooled and input to a classification head, which can be a further KAN or hybrid MLP/KAN module, yielding logits for diagnostic classification via softmax and cross-entropy loss (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

4. Functional Connectivity Estimation and Bias Mitigation

Patch-to-anchor FC is computed using Pearson correlation across time series, with no subsequent normalization. Key metrics:

$$F_{ij} = \frac{\sum_{t=1}^T (v_{i,t} - \bar v_i)(a_{j,t} - \bar a_j)} {\sqrt{\sum_t (v_{i,t} - \bar v_i)^2} \sqrt{\sum_t (a_{j,t} - \bar a_j)^2}}$$

Averaging across multiple scales and anchor sets reduces variance and mitigates spatial sampling noise.

Randomized anchor selection and iterative patch sampling eliminate fixed regional boundaries, thus alleviating atlas bias and capturing individualized spatial-functional brain patterns. This results in anchor patches 45% closer to GM boundaries, significantly improving anatomical fidelity in FC mapping compared to grid-based or atlas-based frameworks (Ward et al., 1 Jan 2026).

5. Experimental Protocols and Ablation Studies

ABFR-KAN is evaluated on the ABIDE I multi-site rs-fMRI dataset, with sites including NYU (171 subjects, ASD and control) and UM (110 balanced subjects). Preprocessing follows DPARSF conventions. Protocols include:

• Single-Site (5-fold cross-validation): e.g., ViT backbone with MLP-KAN head achieves NYU ACC = 0.743, AUC = 0.734, sensitivity = 0.897, specificity = 0.716.
• Cross-Site Generalizability: Training on NYU and testing on UM yields ViT KAN-KAN ACC = 0.676, AUC = 0.687; reverse (UM→NYU) shows best ACC = 0.538, F1 = 0.574 (Ward et al., 4 Apr 2025).
• Backbone Ablations: ViT outperforms CNN (EfficientNetV2), DeiT, and SSM (Vim), with ACC (UM) improving from 0.6818 (CNN) to 0.7182 (ViT) (Ward et al., 1 Jan 2026).
• Anchor/Patch Sampling Ablations: Randomized anchors with iterative patch sampling yield highest ACC (0.7182, UM), outperforming grid/random and grid/iterative approaches.
• KAN Variant Ablations: FastKAN as KAN-KAN configuration is optimal on NYU (ACC = 0.7427), Wav-KAN as MLP-KAN on UM (ACC = 0.7727) (Ward et al., 1 Jan 2026).
• Baseline Comparison: Outperforms SVM, BrainNetCNN, MVS-GCN, GCN, and KD-Transformer in both accuracy and AUC, with statistically significant gains ($p<0.05$ or $p<0.01$ one-tailed paired $t$-test).

6. Performance Summary and Clinical Implications

ABFR-KAN demonstrates superior performance in FC-based brain disorder diagnosis, especially ASD. Representative results:

Model/Setting	NYU ACC	UM ACC	NYU→UM AUC	UM→NYU ACC
RandomFR	0.7079	0.7182	0.6364	—
ABFR-KAN (best config)	0.7427	0.7727	0.7181	0.5707
MVS-GCN (baseline)	0.726	—	—	—
SVM (baseline)	0.649	—	—	—
BrainNetCNN (baseline)	0.696	—	—	—

Key findings include statistically significant improvements in ACC and AUC, robust sensitivity/specificity balance, enhanced cross-site reproducibility, and substantial reductions in FC estimation variance (by 25.7%). The approach supports full automation (sampling and preprocessing), is suitable for deployment, and enables principled extension to other neurological conditions (e.g., MCI, depression) and covariate integration (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

7. Limitations and Prospects

Notable limitations include increased parameter count in fully KANized models (e.g., KAN-KAN variants reach 190k+ parameters for ViT), raising overfitting risks on limited-sample neuroimaging datasets. Interpretation of learned spline or wavelet nonlinearities remains algorithmically feasible but nontrivial. Clinical extensions could involve tailored anchor sampling, integration of demographic/structural MRI data, and hierarchical or multi-class classification settings. Code is open-sourced for reproducibility and further development (Ward et al., 4 Apr 2025, Ward et al., 1 Jan 2026).

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References:

• (Ward et al., 4 Apr 2025) Improving Brain Disorder Diagnosis with Advanced Brain Function Representation and Kolmogorov-Arnold Networks.
• (Ward et al., 1 Jan 2026) ABFR-KAN: Kolmogorov-Arnold Networks for Functional Brain Analysis.