FAPEX: Fractional Amplitude-Phase Expressor for Robust Cross-Subject Seizure Prediction

We sincerely thank the reviewer for their careful reading and constructive feedback. Due to space constraints, we present only a subset of additional results here.

Response to Weakness(es)

(W1): Preprocessing and FPR/h

We clarify that the EEG epoch duration is already specified in Table 1 of the main paper (column “Duration”), where the pretraining window length is 32 s. In response to the reviewer’s suggestion, we have computed and added FPR/h, precision, and specificity metrics for all datasets under the same protocol. Owing to rebuttal length constraints, we present a representative subset here; the complete tables are provided in the revised supplementary materials. Across all tested datasets, FAPEX consistently achieves the lowest FPR/h and highest precision/specificity among all baselines, demonstrating its value for practical deployment.

(W2) Channel Processing

Channel rejection was strictly performed following dataset provider annotations (bad channels flagged in iEEG, and artifact-affected leads in scalp EEG). No manual override was used, preventing introduction of any bias. Typically, 0–37% of channels are rejected per subject (detailed ranges now added in the revision). To standardize input shapes for training, we filled empty slots to 64 channels with the mean value of remaining valid channels; this step is only for hardware optimization and does not increase the information content. When using native low-channel 10–20 EEG without duplication, model performance remains essentially unchanged—confirming that duplication does not inflate results. Importantly, seizure-related regions such as the temporal and frontal lobes, as well as hippocampal electrodes (in the case of iEEG), are consistently preserved due to their critical role in identifying predictive biomarkers. Notably, these key channels have not been reported as damaged across all datasets, as confirmed by their respective providers and maintainers.

(W3) Data Mixing:

1. Clarification of Data Mixing

We clarify that data mixing occurs only during self-supervised pretraining, in line with foundation model practices in NLP and CV. During supervised training and evaluation, no cross-modality mixing is used. Although iEEG and scalp EEG differ in noise characteristics and spectral distributions, our approach consistently outperforms baselines.

2. Further Motivation and Evidence

Mixing samples from multiple sources during pretraining is a widely recognized practice in foundation models across NLP, CV, and emerging EEG models. It alleviates data scarcity and enhances generalization across diverse datasets. Our ablation study confirm that mixed pretraining improves downstream performance compared to single-modality pretraining.

3. HFOs and Noise Concerns

We thank the reviewer for raising this concern. Our study targets clinically relevant high-gamma activity, which is fully preserved at 512 Hz. Ultra-fast ripples (>250 Hz) are rarely recorded reliably at scale and are inaccessible in scalp EEG. Thus, 512 Hz offers a practical balance for consistent processing across modalities. To validate this choice, we conducted an ablation at 1024 Hz. While slight improvements appeared in KAIME and IESS, overall gains were limited—supporting 512 Hz as sufficient for robust performance.

(W4) Generalizability Across Devices and Cohorts

We appreciate the reviewer’s point. Our current study includes strong cross-cohort/device validation—e.g., IESS → BEIRUT (neonatal to adult EEG across hardware) and AGS → BEIRUT (generalized-to-focal transfer)—demonstrating robustness to population and acquisition variability. While FAPEX is currently evaluated on clinical-grade EEG (scalp/iEEG), wearable datasets for preictal forecasting remain limited. Nevertheless, FAPEX’s focus on key brain regions (temporal, frontal, hippocampus) indicates potential adaptability to low-channel or wearable EEG. We see this as a promising future direction.

Response to Questions

(Q0) Imaginary Parts of FrFT

FrNFO preserves the full complex-valued fractional spectrum; both the real and imaginary components are used. The APCE module computes instantaneous amplitude and phase directly from these components. Unlike Mamba and S4, FAPEX does not discard this information.

(Q1) Data Mixing

As detailed in (W3), we have conducted separate analyses for iEEG and scalp EEG, as well as mixed-modality pretraining. The results demonstrate that cross-modality pretraining benefits performance without introducing bias.

(Q2) FPR/h and Latency

We have reported FPR/h, precision, and specificity for all datasets in (W1). Latency is also provided for in-house datasets are provided as follows. These values represent the median latency of FAPEX-Base-SSL. Higher latency reflects more confident and early predictions.

(Q3) Citations

Thank you for your careful reading. We have revised it in the updated version.

(Q4) Channel Counts and Epoch Duration

Thank you for your advice. We have clarified in the manuscript that the "Duration" column in Table 1 indicates the EEG epoch length. Raw channel counts per individual will be presented in the revised supplementary materials due to high variability within datasets.

Summary

We hope these responses address your concerns and strengthen our submission. In light of these, we respectfully ask you to reconsider the score if the new material satisfactorily resolves the earlier reservations.

Thank you for your constructive feedback, which has significantly enhanced the clarity and rigor of our work.

Replies to Weaknesses

(W1) Notation Clarity

We clarify that $Amp$ (amplitude) and $Pha$ (phase) are directly computed from the complex-valued tensor $\hat{X} \in \mathbb{C}^{\cdot}$. Specifically, letting $\hat{X} = \Re(\hat{X}) + i, \Im(\hat{X})$, we have: $Amp = |\hat{X}| = \sqrt{\Re(\hat{X})^2 + \Im(\hat{X})^2},$ $Pha = \angle \hat{X} = \mathrm{atan2}!\left(\Im(\hat{X}), \Re(\hat{X})\right).$ Equivalently, this can be expressed as: $Amp = |\hat{X}|,\quad Pha = \arg(\hat{X}).$ $\hat{X}$ is the same as $\tilde{X}$. $\tilde{X}$ is the output of APCE, therefore, it represents the fused information from phase and amplitude.

(W2) Complexity

We address your concerns as follows.

(a) Eq. (4)

The MLP incurs a computational cost primarily limited to the feedforward computation during training. At inference time, the MLP only needs to be computed once, as it produces a fixed kernel.

Let $N$ = number of temporal samples, $d_{\mathrm{model}}$ = feature channels, $M$ = number of exponential terms, $K$ = highest order of Hermite expansion

1. Exponential‐term GEMM
   $E \in \mathbb{C}^{N\times M},\; W \in \mathbb{C}^{M\times d_{\mathrm{model}}} \;\longrightarrow\; X = E\,W \quad\Rightarrow\quad \mathcal{O}(N \, M \, d_{\mathrm{model}})$

2. Hermite‐term GEMM
   $H \in \mathbb{R}^{N\times (K+1)},\; A \in \mathbb{R}^{(K+1)\times d_{\mathrm{model}}} \;\longrightarrow\; Y = H\,A \quad\Rightarrow\quad \mathcal{O}(N \, K \, d_{\mathrm{model}})$

3. Element‐wise multiplication
   $\Phi = X \circ Y \quad\Rightarrow\quad \mathcal{O}(N \, d_{\mathrm{model}})$

Total Cost is

$\mathcal{O}\bigl(N\,d_{\mathrm{model}}\,(M + K)\bigr)$

Note:

(1) Training time:
These window/kernel computations must be redone every training iteration (since the parameters $w_{i,k}, b_{i,k}, c_{i,k}, a_{n,k}$ are updated).
(2) Test (inference) time:
Because the window/kernel depends only on fixed time positions $t_j$ and learned parameters—not on the dynamic input values—it can be computed once ahead of inference, eliminating per‐sample overhead completely.

(b) Fractional Convolution

The numerical implementation of FrFT algorithm is dominated by FFT operations. A single FrFT of length $N$ involves:

• one FFT and one IFFT of size $\approx 2N$, and
• two elementwise chirp multiplications of size $N$.

Thus, the total time complexity is:

For fractional convolution of two signals of length $N$, the computation proceeds as:

1. Forward FrFT of the input: $\mathcal{O}(N \log N)$,
2. Forward FrFT of the kernel: $\mathcal{O}(N \log N)$,
3. Pointwise multiplication: $\mathcal{O}(N)$,
4. Inverse FrFT: $\mathcal{O}(N \log N)$.

This gives an overall complexity of:

This approach yields performance comparable to FFT-based conventional convolution, differing only by constant factors, while offering greater flexibility in the fractional domain.

Response to Questions

(Q1) Ictal Windows

We sincerely thank the reviewer for this insightful question.

To directly address the reviewer’s concern, we conducted additional experiments in which ictal windows were used in the training data. Consistent with the reviewer’s intuition, the inclusion of ictal segments led to an improvement in overall predictive performance; however, this boost did not reach statistical significance across the datasets (Wilcoxon rank-sum test: $p > 0.05$) and is therefore marginal.

This minor gain further supports the robustness of our main conclusions and confirms that restricting training and evaluation to preictal windows does not compromise model effectiveness. Indeed, adhering to preictal-only windows aligns with established seizure prediction protocols, where the clinical objective is to deliver timely warnings before seizure onset, not during or after.

For completeness, we clarify that all our core experiments presented in the current manuscript systematically exclude ictal windows from all supervised training, fine-tuning, and evaluation, in line with community standards.

In short, our findings affirm that while incorporating ictal activity yields benefits, albeit marginal. Strict preictal-only modeling remains the field’s standard and is sufficient for robust, clinically meaningful seizure prediction. Once again, we thank the reviewer for prompting this additional analysis and hope this clarifies our methodological choices.

Summary

We hope these additions could address your concerns and strengthen our submission. In light of these, we respectfully ask you to reconsider the score if the new material satisfactorily resolves the earlier reservations.

We sincerely thank the reviewer for their careful reading and constructive feedback.

Response to Weaknesses and Questions

(W1 & Q1): Novelties

We thank the reviewer for requesting clarification. FAPEX introduces a principled and generalizable framework for seizure prediction, with the core novelty being the Fractional Neural Frame Operator (FrNFO)—a learnable fractional-order transform that adaptively captures both low- and high-frequency EEG dynamics. Unlike fixed STFT or wavelets, FrNFO provides data-driven time–frequency representation, enabling better detection of diverse preictal signatures. FrNFO overcomes the low-frequency bias of CNNs via fractional decomposition with refined instantaneous phase and amplitude representation. Learnable fractional orders allow flexible trade-offs between temporal and spectral detail. Theoretical analysis (Appendix A) shows FrNFO is stable under signal deformation and robust across noises. The APCE and SCA modules further enhance cross-channel fusion and multi-band learning, and extensive experiments—across modalities, cohorts, and devices—demonstrate strong generalizability.

We address the reviewer's concerns as follows:

(a) Clarification on APCE & PAC

(1) We acknowledge that our introduction, in emphasizing the clinical relevance of phase–amplitude interactions, may have given the impression that our APCE module is designed to explicitly estimate or model physiological PAC mechanism. This is not true. When we talk about phase-amplitude interaction, we refer to simultaneous utilization or feature fusion of instantaneous phase and amplitude representations in neural networks, which is under-explored in deep learning-based seizure prediction. We clarify that our intention is not to model or estimate PAC in a neuroscientific sense. Rather, PAC is cited to motivate the use of both amplitude and phase. The APCE block is a neural fusion mechanism inspired by PAC, designed to extract joint, discriminative features for forecasting—not to compute validated PAC metrics.

(2) Explicitly modeling PAC incurs a time complexity of at least $\mathcal{O}(C^2 N\log N + C N\log N) = \mathcal{O}(C^2 N\log N),$ where $C$ is the number of EEG channels and $N$ is the sequence length. In contrast, our proposed approach achieves a time complexity of only $\mathcal{O}(N\log N)$, which is orders of magnitude more efficient. This distinction is crucial, as our design goal is to develop a neural network inspired by (but not strictly replicating) biological mechanisms for seizure prediction.

(3) We highlight that we have conducted ablation studies (Appendix D), among these experiments we provide architectural variants \model{FAPEX-w/o-Fr-1}: Removes the FrNFO module completely, which means temporal representations are directly sending to Cross-BSSM module without calculating phase and amplitude at all; \model{FAPEX-w/o-Phase}:Removes the phase inputs; They confirm that integrating both amplitude and phase features via APCE is crucial for model performance, even though the module itself does not compute, extract or enforce standard PAC indices. Solely using amplitude information leads to performance drops across all datasets, suggesting that APCE captures additional information beyond amplitude. Both amplitude and phase are necessary.

A new ablation study was conducted to test whether the APCE module’s gain comes from genuine phase–amplitude interaction, rather than simple feature concatenation. We randomly shuffled phase and amplitude vectors independently for each sample, thereby destroying any correspondence. The resulting performance dropped markedly. Only the original APCE (with true cross-encoding) preserved high accuracy. This confirms that meaningful phase–amplitude interaction—not mere concatenation—is critical for the predictive power of our approach.

(4) We appreciate the reviewer's suggestion of a direct comparison with standard PAC estimators. APCE is not a PAC estimator but an amplitude–phase fusion block inspired by PAC concepts. Nevertheless, we implemented PAC-based baselines using MVL-MI and KL-MI computed for frequency bands of delta, theta, alpha, beta coupling with gamma (such combinations are experimentally attested PAC in epileptic brains) across channels as features processed by a two-layer MLP classifier. The results show that FAPEX significantly outperforms these baselines (Wilcoxon rank-sum test: p<0.001):

These suggest that our model achieves better performance than PAC estimator-based seizure prediction. Using standard PAC metrics alone cannot achieve satisfactory classification of preictal vs interictal samples.

(b) Linear Attention

We respectfully clarify that our linear attention in SCA is bidirectional, meaning it fully accounts for interactions among all electrode channels, developed in CV and NLP as a suboptimal yet efficient alternative. It does not lose the ability to capture cross-electrode dependencies, Replacing the linear attention with full one resulted in marginal differences:

(W2): Spatial Analysis

Thank you for the helpful suggestion. Attention weight analysis from the SCA module shows consistent patterns across preictal samples, reliably highlighting electrodes near known epileptogenic zones. In the FMCE dataset, for instance, patients with temporal epilepsy show strong coupling within the temporal lobe and hippocampus, with weak frontal–temporal interactions—matching SOZ annotations. These results confirm the SCA module’s effectiveness in modeling inter-electrode dependencies and will be added to the revised manuscript.

We also highlight that Supplementary Section B2 (Brain Region Analysis) already provides quantitative evidence of spatial relevance. As shown in Figure 3, masking channels from clinically critical regions (e.g., Temporal Lobe, Hippocampus) causes significant performance drops, confirming that FAPEX effectively leverages physiologically meaningful regions rather than relying on arbitrary patterns.

(W3 & Q3): Efficiency Analysis

Thank you for this important question. We address it in two points.

(a) Efficiency

We present a comparison including parameter count, latency (ms) and FLOPs per inference for best-performing models. On average FAPEX has moderate to medium computational efficiency with higher overall performance across datasets.

(b) Wearable Device

We thank the reviewers for raising the question of wearable EEG applicability. FAPEX is currently designed for clinical-grade settings (scalp EEG, iEEG, and animal models) under controlled conditions, as consumer-grade preictal datasets remain scarce and fragmented. As noted in the Supplementary Limitations, this makes rigorous wearable evaluation infeasible at present. Nonetheless, FAPEX demonstrates strong generalization across 12 diverse clinical datasets, with efficient parameter size and latency. Our interpretability analysis further shows that FAPEX relies on clinically relevant brain regions (e.g., temporal, frontal, hippocampus), supporting its potential for future adaptation to low-channel or wearable EEG. We view this as a promising direction and plan to pursue it as suitable benchmarks emerge.

(W4 & Q4): Formatting

We sincerely thank the reviewer for the careful reading of our manuscript and for pointing out the formatting issues. We have thoroughly proofread and corrected the entire paper and appendices.

Summary

We hope these additions could address your concerns and strengthen our submission. In light of these, we respectfully ask you to reconsider the score if the new material satisfactorily resolves the earlier reservations.

Thank you for your constructive feedback, which has significantly enhanced the clarity and rigor of our work. Due to space limitations in the rebuttal, we present only a subset of additional results below; the complete results will be included in the camera-ready version.

Response to Weaknesses

(W1) Complexity

We appreciate the reviewer’s interest in the computational complexity of FrFT-based operations. We reconfirm that the code will be released upon acceptance.

We clarify that numerical implementation of the FrFT algorithm is fundamentally driven by FFT operations: Each FrFT (length $N$) consists of a single FFT, a single IFFT (each of length $\approx 2N$), and two $N$-length elementwise chirp multiplications, resulting in an overall complexity of $\mathcal{O}(N \log N)$. Fractional convolution between two $N$-length signals consists of two FrFTs, one pointwise multiplication,and one inverse FrFT, so the total complexity remains $\mathcal{O}(N \log N)$—comparable to standard FFT-basedconvolution but with enhanced flexibility in the fractional domain.

For transparency and reproducibility, we have provided pseudocode for our FrFT implementation (which could be further optimized for differentiable computation on GPU), and the full source code will be made available upon publication.

Algorithm: Fast Fractional Fourier Transform (PyTorch-style pseudocode)

Input:

• x: input tensor of shape [..., T, F] (T: time, F: feature/channel)
• alphas: fractional orders, shape broadcastable to [..., F]

Output:

• y: output tensor, same shape as x, complex-valued

1. (Optional) Move time and feature axes to last for ease of broadcasting.
2. Ensure x is cast to complex-valued tensor.
3. Compute parameters:
   • phi = alphas * (π / 2)
   • cot_phi = cos(phi) / sin(phi)
   • csc_phi = 1 / sin(phi)
   • t = arange(T) - (T - 1) / 2
4. Pre-chirp:
   • For each feature/channel:
     x_pre = x * exp(-i * π * cot_phi * t^2)
5. Chirp convolution (FFT-based):
   • Zero-pad to length 2T.
   • Create chirp filter:
     g[n] = exp(i * π * csc_phi * n^2) for n = -T,...,T-1
   • X_fft = FFT(x_pre, n=2T)
   • G_fft = FFT(g, n=2T)
   • conv = IFFT(X_fft * G_fft, n=2T)
   • Crop middle T samples:
     conv_cropped = conv[..., T-1:T-1+T, :]
6. Post-chirp and scaling:
   • y = scale * exp(-i * π * cot_phi * t^2) * conv_cropped
   • Where scale = exp(-i * (1 - alphas) * π / 4) / sqrt(abs(sin(phi)))
7. Handle integer orders:
   • If alpha ≈ 0: y = x
   • If alpha ≈ 1: y = FFT(x)
   • If alpha ≈ -1: y = IFFT(x)
   • If alpha ≈ 2: y = reverse(x, axis=time)
8. Restore original tensor layout if axes were permuted.

Return: y

Response to Questions

(Q1) False Positives Metrics

We thank the reviewer for emphasizing the importance of false positive–oriented metrics. In response, we now report precision, specificity, and false positives per hour (FPR/h) for FAPEX across all datasets. As suggested, we have computed these metrics under the same evaluation protocol for all datasets. Due to space limitations, we present representative results for selected datasets in the table below, with the complete results available in the revised supplementary materials.

These results confirm that FAPEX maintains strong performance not only in balanced accuracy and AUPRC, but also in minimizing false alarms—supporting its clinical utility.

Summary

We hope these additions could address your concerns and strengthen our submission.