ODEBrain: Continuous-Time EEG Graph for Modeling Dynamic Brain Networks
Authors: Haohui Jia
Zheng Chen
Lingwei Zhu
Rikuto Kotoge
Jathurshan Pradeepkumar
Yasuko Matsubara
Jimeng Sun
Yasushi Sakurai
Takashi Matsubara
Venue: ICLR 2026 Poster (2026)
Keywords: EEG
ODE
Brain network
continuous dynamics
Primary area: applications to neuroscience & cognitive science

Abstract: Modeling neural population dynamics is crucial for foundational neuroscientific research and various clinical applications. Conventional latent variable methods typically model continuous brain dynamics through discretizing time with recurrent architecture, which necessarily results in compounded cumulative prediction errors and failure of capturing instantaneous, nonlinear characteristics of EEGs. We propose ODEBrain, a Neural ODE latent dynamic forecasting framework to overcome these challenges by integrating spatio-temporal-frequency features into spectral graph nodes, followed by a Neural ODE modeling the continuous latent dynamics. Our design ensures that the latent representations can capture stochastic variations of complex brain states at any given time point. Extensive experiments verify that ODEBrain can improve significantly over existing methods in forecasting EEG dynamics with enhanced robustness and generalization capabilities.

Published as a conference paper at ICLR 2026
ODEBRAIN: CONTINUOUS-TIME EEG GRAPH FOR MODELING DYNAMIC BRAIN NETWORKS
Haohui Jia Zheng Chen Lingwei Zhu Rikuto Kotoge*, Jathurshan Pradeepkumar
Yasuko Matsubara Jimeng Sun Yasushi Sakurai*, Takashi Matsubara
Faculty of Information Science and Technology, Hokkaido University, Japan
* SANKEN, The University of Osaka, Japan
& Great Bay University. China
O Department of Computer Science, University of Illinois Urbana-Champaa Champaign, USA "Corresponding author:
chenz8sanken, osaka-u.ac. jp
ABSTRACT
Modeling neural population dynamics is crucial for foundational neuroscientific research and various clinical applications. Conventional latent variable methods typically model continuous brain dynamics through discretizing time with recur- rent architecture, which necessarily results in compounded cumulative prediction errors and failure of capturing instantaneous, nonlinear characteristics of EEGs. We propose ODERRA TN, a Neural ODE latent dynamic forecasting framework overcome these challenges by integrating spectral graph nodes, followed by Neural ODE modeling the continuous latent dynamics. Our design ensures that latent representations can capture stochastic variations of complex brain states any given time point. Extensive experiments verify that ODEBRAIN can improve significantly over existing methods in fore- casting EEG dynamics with enhanced robustness and generalization capabilities. Our code is available at https: //ggt//bgttb//t//// com/
cor---eemorralffreunnne features into
INTRODUCTION
Modeling dynamic activity brain networks or connectivity using electroencephalograns (EEGs) is crucial for biomarker discovery (Rolls et al., 2021; Jones et al., 2022) and supports wide range of clinical applications (Kotoge et al., 2024; Pradeepkumar et al., 2026). Temporal graph networks (TGNs), which integrate temporally sequential models with graph neural networks (GNNs), have recently emerged as a promising approach (Tang et al., 2022; Ho & Armanfard, 2023; Delavari et al., 2024; Li et al., 2024). These methods represent multi-channel EEGs as graphs, where GNNs capture spatial dependencies and sequential models capture fine-grained temporal dynamics, thereby providing mechanistic insights into gradual evolution of brain networks.
However, a critical, yet overlooked problem remains: existing methods transform EEGs into fixed discrete time steps, which conflict with the inherently continuous nature of brain networks. Such discretization imposes rigid windowing assumptions and prevents the models from capturing the unfolding time-course dynamics or irregular transitions in brain networks, as in Figure 1. This paper aims to tackle this issue by developing a novel method that models EEGs in an explicitly continuous manner, while leveraging Neural Ordinary Differential Equations (NODEs) (Chen et al., 2018).
In contrast to sequential models based on recurrent neural networks (RNNs), which discretize time in fixed steps, NODEs parameterize the derivative of the hidden state and integrate it continuously over time (Park et al., 2021). This formulation provides principled method to model the dynamical evolution of neural activity (Hu et al., 2024) and has been studied across domains (Hwang et al., 2021), including brain imaging (Han et al., 2024). In this paper, study novel and critical prob- lem: modeling the dynamics of brain networks with NODEs to learn informative continuous time representations from EEGs. This remains an unexplored and non- trivial task, and we focus on two main challenges:

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Forecasting exploration neuronal population dynamics
Channel
Observed
Forecusring
Larent state
Continuous EBG recordings
Time
Discrete forecusting frecument-basel)
Contimuous forecasting (ODE-based)
Figure 1: (Left) Continuous EEG real-time neuronal activity recordings. (Mid) Recurrent- based methods employ discrete modeling. (Right) ODE provides continuous representation for forecast- ing neuronal population dynamics.
(i) Effective spatiotemporal modeling for ODE initialization. NODEs critically depend on the qual- ity of their initial conditions because the ODE solver propagates trajectories starting from this ini- tialization. Accordingly, a poor initialization propagates errors and destabilizes long-term dynamics. However, EEG signals are inherently noisy and stochastic, making learning robust spatiotemporal representations of brain networks particularly challenging. Therefore, designing an initialization that captures meaningful spatiotemporal structures is essential for stable ODE integration and effec- tive downstream learning.
fi) ) Accurate trajectory modeling. Trajectory modeling is essential for NODEs, as their strength lies in learning continuous latent dynamics rather than discrete predictions. Unlike conventional time- series data, which often exhibit stable patterns such as periodicity or long-term trends (Klötergens al., 2025), EEG signals are highly variable, which makes trajectory learning particularly challeng- ing. Thus, a major challenge is to constrain and preserve meaningful trajectories in the latent space so that NODEs can accurately capture the continuous dynamics of EEGs.
In this paper, we introduce a new continuous-time EEG Graph method, ODEBRAIN, based on the NODE, model dynamic brain networks. To address the above challenges, first propose dual-encoder architecture to provide effective initialization for NODEs. One encoder captures de- terministic frequency-domain observations model brain networks, whereas the other integrates raw EEG representations to retain stochastic characteristics. This combination yields robust spa- tiotemporal features for initializing the ODE solver. Second, we introduce a trajectory forecasting decoder that maps latent representations obtained from NODE solutions back to graph structures. Thereafter, a multistep forecast loss function was applied to explicitly predict future brain networks at different time steps. This design enables direct trajectory modeling of dynamic brain networks and improves accuracy. Third, beyond modeling, we are the first to propose the use of the gradient field of NODEs as metric quantify the dynamics of the brain network. Herein, also conduct a case study on seizure data to illustrate the clinical interpretability of this method.
New Problem Formulation. To the best of our knowledge, we are the first to explicitly formulate EEG brain networks as continuous-time dynamical system, where the brain network repre- sented sequence of time-varying graphs whose latent dynamics is governed by NODE. This perspective differs from prior approaches based on recurrent models, which model gradual state transitions in discrete-time framework rather than in a principled continuous-time manner.
Novel Method. We develop the ODEBRAIN framework that integrates three key components. first combines deterministic graph-based features with stochastic EEG representations to produce robust initial state. Then an explicit trajectory forecasting decoder with multi-step forecast- ing loss hat models temporal-spatial dynamics continuously, enabling principled forecasting of evolving brain networks.
Comprehensive Evaluation. We demonstrate strong performance across benchmarks and pro- vide retrospective clinical case studies that highlight interpretability. Our ODEBRAIN outperforms all baselines the TUSZ dataset, achieving 6.0% and 8.1% improvements in F1 and ACC, re- spectively. In the TUAB, ODEBRAIN consistently achieves the best performance, such as 1.2% improved Fl and 2.4% improved AUROC. Moreover, we further evaluate the learned field and its clustering to reveal the dynamic behaviors (varying speed and direction) between the seizure and normal states, and achieve 12.0% improvement for the prediction of brain connectivity.
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RELATED WORKS
2.1 TEMPORAL GRAPH METHODS FOR MODELING EEG DYNAMICS
GNNs have emerged as a powerful method for effectively capturing spatial dependencies and rela- tional structures in the analysis of brain networks (Li, 2022; Yang & Hong, 2022; Kan et al., 2023). Specifically, EEG-GNN performs a learnable mask to filter the graph structure of EEG for cognitive classification tasks (Demir et al., 2021). ST-GCN formulates the connectivity of spatio-temporal graphs to capture non-stationary changes (Gadgil et al., 2020). (Tang et al., 2022) have introduced the DCRNN approach for graph modeling. setting new standard for SOTA in seizure detection and classification tasks. Following this, GRAPHSAMER (Tang et al., 2023) improved the graph structure and integrated it with the MAMBA framework improve long-term modeling capabili- ties. AMAG (Li et al., 2024) forecasting method has been proposed to effectively capture the causal relationship between past and future neural activities, demonstrating greater efficiency in modeling dynamics. More recently, EvoBrain has investigated the expressive power of TGNs in integrating temporal and graph- based representations to model brain dynamics (Kotoge et al., 2025). However, these studies rely on discrete modeling and may lead to suboptimal representation of continuous dynamics of brain networks.
2.2 DIFFERENTIAL EQUATIONS FOR BRAIN MODELING
Modeling brain function as low-dimensional dynamical systems via differential equations has been long-standing direction in neuroscience (Churchland et al., 2012; Mante et al., 2013; Vyas et al., 2020), and nonlinear EEG analysis for brain activity mining (Pijn et al., 1997; Xue al., 2016; Lehnertz et al., 2003; Lehnertz, 2008; Mercier et al., 2024). Recently, NODEs formulate dynamical systems by parameterizing derivatives with neural networks and have shown impressive achieve- ments in various fields (Fang et al., 2021; Hwang et al., 2021; Park et al., 2021). In the modeling of BCI and epilepsy, controllable formulations and fractional dynamics provide important theoreti- cal foundations for modeling brain dynamics (Gupta et al., 2018b; Tzoumas al., 2018; Lu al., 2021; Martis et al., 2015; Lepeu et al., 2024). In latent-variable dynamics models, the EEG and neu- ronal processes are described fractional dynamics (Gupta al., 2019; 20; Yang et al., 2019; 2025). In neuroscience, (Kim et al., 2021) learn neural activities by modeling the latent evolution of nonlinear single-trial dynamics with Gaussian processes from neural spiking data. (Hu et al., 2024) propose using a smooth 2D Gaussian kernel to represent spikes as latent variables and de- scribe the path dynamics with linear stochastic differentiable equations (SDEs). Another study (Cai et al., 2023) demonstrates robust neuroimaging performance by combining biophysical priors with NODEs, starting from predefined cognitive states. (Chen et al., 2024) have shown the advantage of graph ODE by modeling continuous-time propagation for the EEG emotion task. (Han et al., 2024) further illustrate that integrating spatial structure with NODEs can effectively facilitate the modeling of neuroimaging dynamics, even in the presence of missing data. However, these studies focus on imaging data or neuronal feature engineering, while data-driven modeling of brain networks with fractional dynamics from EEGs remains underexplored.
PRELIMINARY AND PROBLEM FORMULATION
Neural Ordinary Differential Equations. NODEs (Chen et al., 2018) provide a framework for modeling continuous-time dynamics by parameterizing the derivative of a hidden state using neural networks. Intuitively. NODEs solve the trajectory of the hidden state continuously at any arbitrary time т. rather than restricting updates to fixed discrete steps At in RNNs. Specifically, the hidden dynamics are computed via an adaptive numerical ODE solver:
r++
(t + 1) ODEsolver(zo.f)) zo +
fo x1) dt,
(1)
where fo is a continuous, differentiable function parameterized by a neural network. This formula- tion yields unique continuous trajectory z(t) over an interval [to, to + 7].
Intuition in Modeling EEG Dynamics. Conventional sequential models, such as RNNs, have been standard tool for modeling EEG. However, they implicitly assume that time can be discretized into
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fixed steps and that state transitions, such the onset of seizure, must oceur exactly these steps (Kotoge et al., 2025). While this assumption simplifies the computation, it poorly matches the reality of EEG, where brain activity evolves continuously and transitions occur at arbitrary points in time. In contrast, NODEs address this limitation by modeling EEG dynamics through a continuous function So whose integration yields smooth latent trajectories. Within this framework, discrete EEG signals recorded at sampling intervals are treated as observations sampled from an underlying continuous process f (t) dt. This perspective allows NODEs to capture both gradual oscillatory rhythms and abrupt transitions in neural activity, thereby providing more faithful representation of the EEG brain dynamics.
However, applying the NODE to EEG is non non-trivial, and we recognize two questions needing to be answered:
1. Robust initialization 2o against transients and stochasticity in EEGs. NODE requires a well- cablibrated starting condition 2g effectively forecast future behavior. This because EEGs are highly stochastic, or even chaotic to an extent. Their key features are transient and may appear without any preindicator (Chen al., 2022). Without proper initialization 2g as a guide, the integration of the model So over time alone cannot accurately forecast future states.
2. Meaningful objectives of fo , 21) capture the underlying EEG dynamics. Standard NODE training typically relies on regression-like objectives aimed at forecasting future states. key challenge lies in identifying which representations best capture the underlying neural dynamics, so that (t, zt) is guided toward modeling the true evolution of brain networks rather than only surface-level predictions. For example, in seizure analysis, the model must also learn to discern not only seizure but also any leading states that herald an impending seizure (Li et al., 2021).
Problem Statement (Modeling Dynamic Brain Networks). Given the observed EEG up to time f, denoted as Xgt. the objective is to model the dynamics of the brain network and forecast their future evolution. The predicted dynamics act as representations of brain states, enabling the distinction between conditions such as seizure and non-seizure. Following prior work (Tang et al., 2022; Chen et al., 2025), we represent the brain as a graph and aim develop an EEG-based NODE (I)) to predict sequence of time-varying graphs:
GI+1:+K (X,+1, X (22 fe(g..)).
Over the next K steps graph, where GI. denotes the observed brain networks up to time and Get1++ represents the predicted dynamic brain networks. These graphs characterize dynamic brain networks, but this problem poses two key challenges: (i) obtaining robust initialization Zo that can resist the transient and stochastic nature of EEGs; and (ii) defining an objective for Sa that faithfully captures the underlying neural dynamics.
4 METHODOLOGY
Figure shows the system overview of CDEBRAIN. Specifically, graph representations obtained from each EEG segment (Section A.2), entering stage 1: attaining reverse initial state encoding 29 and temporal encoding =* (Section 4.... Stage consists of Neural ODE that takes as input 2 (Section 4.2). Finally, forecasting loss between ODE output and ground truth is computed.
4.1 STAGE REVERSE INITIAL STATE ENCODING
Spectral Node Encoding. Previous discrete forecasting studies have shown that the capacity to estimate future neural dynamics depends on past activity in (Li et al., 2024). We also define this forecasting paradigm within our CDEBRAIN. Intuitively, both the latent initial state 2o and the field f. i.c., dz(t) are described by encoding the past observation Gi to govern the latent continuous evolution. The works of (Rubanova et al., 2019; Chen et al., 2018) suggest that the construction of an effective latent initial state requires an autoregressive model capable of jointly extracting both the initial condition and the latent evolution. Accordingly. introduce graph state descriptor : Rd + R that represents latent graph states d E R using the autoregressive and graph network modules.
di
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Stage 1 Reverse initial state encoding
Inpul graphs Pest
Z++ +
f t),
True States
Multi-steps forecasting loss
f
T
T
ST
27--3
Stage 2 Forward temporal-spatial ODE Solving
Figure 2: Continuous neural dynamics modeling via ODEBRAIN with graph forecasting. In stage I, multi-channel EEG signals are encoded into spectral graph snapshots and fused with raw features to construct noise-robust initial states for ODE integration, predicting future spectral graphs. In stage 2, ODEBRAIN propagates latent states through time, generating dynamic field f that captures continuous trajectories. Finally, future graph node embeddings are obtained by 2y and compared with the ground-truth graph nodes.
Specifically, given the observations until now Gi as input, we perform a sequence representation for the node and edge attributes. For node embeddings, the evolution is computed by the gated recurrent unit (GRU) as h"- = GRU (Ni,st) where Vi,g denotes the spectral attribute sequences of node and K, the spectral intensity. Similarly, for edge, the attribute sequences are defined from the adjacency matrices by h = GRu (Atj.st). The resulting node and edge embeddings are integrated into an aggregated graph structure G = (hi hise)) to be learned by a GNN to capture the spatial dependency across epochs: 2 GNN (h", h The forward process of & captures both the epoch variations between frequency bands and explicit channel correlations.
Temporal Embedding with Stochasticity. Accurate modeling of the temporal evolution of EEG signals is crucial because neural dynamics inherently exhibits nonuniform temporal fluctuations and asynchronous activations across channels. Although the graph descriptor & effectively captures the evolution of node and edge attributes, the short-time Fourier transform (STFT) segments EEG signals by constant windows, which inevitably disrupts the continuous temporal correlation between the raw EEG observations.
Morcover, fully deterministic latent representations lack the flexibility necessary to effectively repre- sent transient motions of EEG analyzed in Sec- tion 3. Conversely, introducing controlled random- ness into temporal embeddings serves natural regularization strategy, effectively increasing the ro- bustness and preventing premature convergence suboptimal. Here, we apply the temporal descriptor V T b RRC Re,c & m quantify the random- ness of the raw EEG epochs across N channels into z E R Given EEG segments X from N channels within a sliding window length L, we define stochas- The
(3)
=(1)
f0
fo
fo
z(1 Ad)
(())
x(1)
((1)
(b)
(9(*)) + 1) o h(*,)
Atlaplive nvural maniic
W
x(=') ø
Stuchaatic regularization
tie temporal embedding as x = V (XTxxx controlled stochasticity further acts form of la- tent space regularization, enhancing generalization and robustness noise in EEG data collection.
Proposed temporal-spatial ODE-RK4 function
Figure 3: The structure of the temporal- spatial ODE. (a) RK-4 step numerical solver. (b) Procedure of temporal-spatial fg.
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4.2 STAGE 2: FORWARD TEMPORAL-SPATIE ODE SOLVING
Depending on the above encoding process, we define the initial state [=*, with & o V + Rm+e, which summarizes stochastic temporal variability and deterministic spectral connectivity, respectively. Given the initial state Zy E II Iet+, the general approach models the ODE vector field following the classical neural network solution fe with residual connection as:
(3) where fo R + R m-te represents a vector field capture complicated dynamics and its continuous evolution is governed by fo with the learnable @ across the entire epoch sequences. However, this introduces the challenge of optimizing the deep network-based fo over highly variable EEG states, making large solver errors.
(z (t) O)t,
20 [==, "=, E 1.
Considering the deep architecture-based multi-step numerical solver design (Lu et al., 2018; Oh et al., 2024) and the logic gating interaction of brain dynamics (Goldental et al., 2014), we design a temporal-spatial ODE solution to incorporate the initial state 2g for additive and gate operations as shown in Figure 3. In addition, we further introduce on the stochastic temporal state z", to adjust the vector field fo. accounting for the complexity and dynamic nature of the brain as system. As shown in Figure 3(b), the fo used in the proposed ODE function is defined as follows:
adaptive decay component conditioned
fo (z)) (g(zo) + +1 o ((z)) A(x") zo: 20- 1=" =0]
(4)
where @ represents the alement-wise multiplication. Initially, the vector field is computed by the general residual block h(z)) and updated by gated vector field with a sigmoid function as:
g(zo) oW, + by) E (0, 1)m+
which provides adaptive modulation of the dynamics. Finally, to regularize trajectories under noisy EEG inputs, we add an adaptive decay conditioned on the temporal stochastic state z":
X(=*) = Softpl us (W (2) a tanh (W + b²) + b²) >
The latent trajectory zt) at an arbitrary time can be solved by:
2
K
+
fo
dt
The state solutions are calculated by solving with efficient numerical solvers in Figure 3(a), such as Runge-Kutta (RK) (Schober et al., 2019). The latent state at the next timestamp is updated as follows:
z(t At) x(1) Al (ki + 2k + 2kg ka).
4.3
GRAPH EMBEDDING FORECASTING
Depending on Equation 7, the latent dynamic function and neural forecasting are presented as fol- lows:
[z Z+4x) k = ODESolver (So, [=",, , 1,, + K]), G = 9(2(4)) Vi (1,2....,K') K}.
(10)
where continuous latent trajectories {=(t)}{{ are projected back to the future EEG node attributes with V the set of all possible unique nodes in G.+1E4K via a predictive module Ω gm+tc + Rd, explicitly capturing spatial correlations across the EEG channels over future K time steps. Here, X,>t [X. X,,++] integrate all future node attributes.
Unlike the previous works, which focus on forecasting the temporal neural population dynam- ics. Our learning objective is to predict the graph structure rather than the simple temporal dynamics, since neuron firing generally activates in the asynchronous channels simultancously
Lg Eg ( G--1-K We first train the model in an unsupervised manner using dynamic graph forecasting loss capture continuous neural dynamics via ODE solvers. Then pool the la- tent continuous trajectory 2(t) extracted from the ODE solver with entire timesteps for downstream fine-tuning, such as classification.

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Table Main results on TUSZ (12s seizure detection) and TUAB. Bold and underline indicate best and second- bestt results. *: The performance depends on the discrete multi-steps forecasting. i: The performance depends on the continuous multi-steps forecasting. t: The performance depends on the continuous single-step forecasting.
TUSZ
TUAB
Method
Acc
F1
AUROC
Acc
F1
AUROC
CNN-LSTM
0.735 + 0.003
0.347 = 0,012
0.757 + 0.003
0.741 0.002
0.736 + 0.007
0.813 + 0.003
BIOT
0,702 = 0.003
0,291 = 0.006
0.772 = 0.006
0.717 = 0.002
0.713 + 0.001
0.788 + 0.002
EvolveGCN
0.769 + 0.002
0.385 = 0.005
0.791 +: 0.004
0.708
0.003
0.707 + 0,002
0.777 + 0.003
DCRNN
0.816 à: 0.002
0.416 :: 0.009
0.825 1: 0.002
0.768
0.004
0.769 4: 0.002
0.848 1 0.002
Intent-ODE
0.827 = 0.004
0.470 = 0.005
0.849 = 0.004
0.749 0.003
0.745 + 0.002
0.829 + 0.004
latent-ODE (RK4)
0.821 = 0.003
0.465 = 0.001
0.815 = 0.001
0.716
0.739 + 0.002
0.823 + 0.003
-DE-RNN
0.802 0.002
0.455 1: 0.007
0.855 1: 0.003
0.761 0.003
0.744 0.004
0.838 1 0.005
noural SDE
0.857 0.002
0.467 + 0.003
0.851 0.002
0.768 0.003
0.751 + 0.003
0.834 +: 0.002
GDEs
0.849 + 0.003
0.475 = 0.005
0.841 + 0.003
0.757 0.003
0.737 + 0.006
0.823 + 0.004
ODEBRAIN'
0.869 1 0.003
0.488 1: 0.015
0.875 1. 0.005
0.771 4. 0.005
0.770 4. 0.005
0.849 1 0.003
ODFRRATN?
0.004
0.496 0.017
0.881
0.006
0.778 0.003
0.774 0.005
+ 0.005
EXPERIMENTS
In this section, we conduct experiments to answer the following research questions:
-RQ1. Does ODEBRAIN strengthen seizure detection capability through continuous forecasting?
-RQ2. How does the initial state Zg affect the development of the latent neural trajectory?
RRQ3. Does the objective of Ω facilitate dynamic optimization?
More detailed experiment settings can be found in the Appendix A.
5.1 EXPERIMENTAL SETUP
Tasks. In this study, we evaluate our ODEBRAIN for modeling neuronal population dynamics with seizure detection and abnormal EEG classification. We select these tasks because they reflect rapid, highly oon-stationary transitions and clinically abnormal neural patterns. We provide detailed information about the datasets in Section A.3.
Baseline methods. We select two studies of dynamic neural population: EvolveGICN and DCRNN (Li et al., 2017; Pareja et al., 2020) with a reconstruction objective. We also com- pare it with the benchmark Transformer BIOT (Yang al., 2023), which captures temporal- spatial information for EEG tasks, and a stan- dard CNN-LSTM Aahmedt-Arisizzaba et al., 2020). For continuous modeling methods, select the normal NODE (Chen et al.,
Table I: Results (AUROC1, F1f) on TUSZ (12s and 60s seizure detection) against discrete and continuous baselines, with options on the gate and stochastic regularization. (-: w/o, +Random: gate with random coefficients for stochastic regulariza- tion.) Bold best.
Model Method
T(s)
AUROC
F1
Discrete & Continuous
BIOT
12 0.772110.006 0.29411.0006
60 0.642+:0.009
0.256+000003
2018), an enhanced ODE-RNN for irregular time modeling with autoregressive vector field (Rubanova et al., 2019), bochassic-regularizze neural SDE (Liu al., 2019) and gradg-aaphhiie GDEs (Poli et al., 2019).
DCRNN
12 0.816+000022 0.416=#0.009
60
0.802+000003
0.413=#0.008
latent-ODE
12
0.791 0.385=
0.745+:0.33 0.331=0.033 60
ODEBRAIN
12
0.881 0.496=10.017
0.828+0.003 60
0.420
Metrics. To answer RQ1, we evaluate the model using the Area Under the Receiver Op- crating Characteristic Curve (AUROC) and the Fl score. The AUROC measures the ability of the models across varying thresholds, while the Fl score highlights the balance between pre-
ODEBRAIN IN
Gate
12 0.86710.004 0.48810.000
60 0.82110.034 0.424110.003
Stochastic
12 0.84810.011 0.462:10.013
60 0.817#00029 0.414+0.047
+
+Random
12 0.860+0.017 0.474=0.033
cision and recall at its optimal threshold for classification. For RQ2, we measure the struc- tural similarity of the predicted graph using the Global Jaccard Index (GJI) GJI Eprea) EErue nE Peedd Perre Urrred
60 0.819+0.026 0.418+0.017
(Castrillo et al., 2018). For RQ3, We
compute the cosine similarity of predicted node embeddings.
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(a) Seizure
(b) Normal & Pre seizure
(c) Seizure
Fast
Field
(d) ODE-defieed dynamic field
Slow
Field direction
X
oenter
Figure 4: Visualization results between the multichannel EEG signal (upper and lower) and its latent dynamic field fe (middle) obtained by ODEBRAIN. Local minima appearing in (a) and (c) indicate rapid changes, corresponding to seizure states.
5.2 RESULTS
5.2.1 MAIN RESULT
RQ1 concerns the continuous forecasting capability EEG. Table 2 summarizes seizure detec- tion accuracy across models on the TUSZ and TUAB datasets for a duration of 12 seconds. Our ODEBRAIN consistently outperforms all baselines based the AUROC and F1 score, demon- strating the superiorty continuous forecasting. Notably, our single-step forecasting achieves an AUROC of 0.881 0.006 and Fl score of 0.496 + 0.017, surpassing latent-ODE. Our multi-step forecasting attains a recall of 0.563 + 0.015, balancing overall detection capability and positive- instance coverage. These results indicate that ODEBRA IN is more effective in capturing the transient dynamics of EEGs than the fixed-time interval or reconstruction baselines.
Field direction
Fast
Field speed
Slow
Field center
ST-ODE solver (Ours) F-ODE solver
Seizure
Non-seizure
Seizure
Non-seizure
(c) F-ODE solver Figure 5: Visualizing learned dynamic fields between our spatial-temporrl(SSTOOO solver and the
(a) Dynamic field clustering
(b) ST-ODE solver (Ours)
frequency (F)-ODE solver.
To further illustrate this point, we visualize the dynamic field fo the latent space in Figure This dynamic field characterizes the difference between seizure and normal states. This is most apparent the centers the seizure Figure 4(a) and 4(c) while absent from the normal & pre-seizure states in Figure 4(b). These centers depict area where gradients point and eventually the flows converge. This aligns well with the corresponding EEGs that show wild-type oscillations with high present
frequency components. In contrast, for the normal & pre-seizure data, such centers in the field, showing that the dynamics is driven mainly by low-frequency oscillations. It is worth noting that such visualization is only available continuous dynamics modeling of our method.
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answer RQ1 follows: through continuous forecasting, ODEBRAIN out- performs existing baselines in terms of seizure detection by accurately depicting neural popula- tion dynamics. The learned field fo clearly delineate the boundary between seizure and nor- mal states via its vector field representation reconstruction-bait hased baselines, ODEBRAIN provides arbitrary temporal resolution and it is sen- sitive to transient neural changes. We have verified that it helps capture the transition process of different brain states.
In summary,
neuronal activity. Unlike discrete-time interval and
5.2.2 DYNAMIC GRAPH FORECASTING EVALUATION
RQ2 concerns the initial state Zg- Figure 5 depicts the learned latent dynamic fields in- duced by different initialization strategies. The proposed temporal-spatial initialization (ST- ODE solver) yields markedly more structured latent field than the frequency- initializa- tion (F-ODE solver). In particular, the ST-ODE field exhibits clearer clustering behavior and more distinguishable dynamic behaviors between seizure and non-seizure states. ODEBRAIN can preserve the transient dynamics and guide the trajectory toward meaningful dynamical region. We answer RQ2, given that our 20. ODEBRAIN can generate more remarkable vec- tor fields that respect the dynamics of EEG and maintain continuous evolutionary properties. Table 1 describes the performance of seizure de- tection under 12s and 60s, comparing discrete and continuous baselines with ODEBRAIN. ODEBRAIN achieves the best or tied-best results at both horizons, indicating that the adaptive vector field effectively strengthens stability. The ablations further validate our design by removing the gating mechanism lead- ing to performance drop from 0.881 to 0.867, high- lighting the adaptive vector field can achieve stable trajectory evolution. Removing stochastic regular- ization also degrades F1 from 0.496 0.462, prov- ing that stochastic regularization mitigates the in- stability caused by noise. In contrast, using gate with random coefficients for stochastic regulariza- tion still underperforms the full model, implying that learnable regularization is more effective.
Out metti staps (tost) Our single-sier step (Cest.
0.63
0.60
Outs win remoorall (Cost)
03
AMAG (DRc.)
Structural Similnty
Cont.: continuous forecasting. forecasting.
(a) Predicted graph structural similarity scores result
Front
Right
Top
Groundtruth
RQ3 concerns consistency in the graphs. Figure 6 shows the effectiveness of our objective f that helps predict dynamic graph structures. It is visible that ODEBRAIN achieves higher similarity scores (0.53 + 0.63) than the discrete predictor, indicating that ODEBRAIN more accurately captures the true graph structure with the help of Q. The similarity matrices reveal that ours aligns more closely terms of lo- cal correlation distribution, in which the discrete pre- dictor exhibits notable discrepancies in certain block structures. Now we answer RQ3: the explicit graph embedding target improves forecasting accu- racy. This is achieved by guiding the vector field fo to learn continuous trajectories that align well with neural activity, leading to more reliable prediction.
Ours (Continuous modeling)
Discrete modeling
(b) Predicted graph structures
Figure 6: Results on (a) graph similarity and (b) functional connections.
5.2.3 ABLATION STUDY
We perform ablation study on the following factors of ODEBRAIN: initialization 2o, loss objective Ω and forecasting horizon, the results summarized in Figure 7.
Initial state. Temporal-spatinn initial state option yields the best performance, achieving the highest AUROC (0.877) and surpassing spatial-only (0.862) and mix up (0.851). It mitigates sensitivity to initial conditions and delivers the largest gains at the longest horizon (11s). Loss objective. Our structural multi-step forecasting consistently outperforms reconstruction-only and raw-signal

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F1 Score
Rocal
AUROC
Spocial only Mixed
Temporal-Sppe
Infial State Methoe Ablation
Loss function Ablation
Forccesting Length Ablation
0,922
1922
0.862
0.797
0.83
0,00
0.502
0 565
0474
0.480
0,502
D776
2.00
0.185
04
(8)
(c)
Figure 7: Summary of ablation study. (a) State initialization. We compare spatial-only, mixed, and temporal-spatial initialization and summarized results in F1, Recall and AUROC. Temporal-Spatial achieves the best F1 (0.502) with competitive recall. (b) Loss function. Replacing our struc- tural forecasting loss with reconstruction-only or raw-signal forecasting degrades performance on AUROC. (c) Forecast horizon. AUROC decreases as the horizon grows (Is + 3s + 11s), and Tem- Poral-Spatial remains the best across all horizons over others.
Table 3: Computational cost with wall-clock
Table Ablation study on Top-r-3 and different regularizer options. Bold denotes the best.
time (s) and NFEs.
Model
Param.
Wall
NFEs
Type
Discretc
CNN-LSTM 5976K 0.586.10.004
Model
Regularizer
AUROC
Recall
BIOT 3174K
0.508:10.003
ODEBRAIN laten-ODE
Shrinkage
0.833#0.032
0.567+0.021
DCRNN
281K 0.418:10.006
Graphical lasso
0.846=0..225 0.557=0.022
Norm
0.849=00.004 0.575=0.005
latent -DEE 386K 0.421=0.002
102
Continnous
Shrinkage
0.872=#0.023 0.606=0.0035
ODE-RNN 675K
0.601
189
Graphical lasso 0.872:10.017 0.613:10.33
neural SDE 346K
0.482
153
ODEBRAIN 459K 0.516.1.0022 164
Norm
0.88110.006 0.605-10.003
forecasting across F//RecalI/AUROC, indicating that geometry-aware regularization improves dy- namical modeling. We attribute the gains to ODEBRAIN that couples the ppectral-spation structure with EEG dynamics and enables more stable integration and stronger generalization.
Table 3 shows single-batch inference cost for discrete vs. continuous baselines, including parame- ters, wall-clock time, and NFEs (only for solver-based models). Discrete methods have fixed-depth computation, so latency mainly follows model size/sequence length. NFEs are shown only for the ODE solver-based models. ODEBRAIN IN contains 459k parameters with 164 NFEs, and 0.516s per batch, which falls in the same latency band as discrete models with fixed-depth computation. These results indicate that ODEBRAIN does not introduce prohibitive cost in practice, and the reduced NFEs suggest a more stable integration than other complicated continuous baselines.
Table 4 evaluates sensitivity to top-T=3 sparsity and regularization options. Adding regularization improves Recall, confirming that norm correlation graphs are noisy and susceptible to volume con- duction, while regularized connectivity is more reliable. The performance is stable across regular- ization options. Concretely, an ODE solver can achieve better performance with sparser, regularized graphs. Graphical lasso or Norm with sparsity yields the best both AUROC and Recall. For ODEBRAIN, Norm with 3 sparsity achieves the best AUROC (0.881 1), and Graphical lasso gets the highest Recall (0.613), demonstrating robust dependence on graph-constructtoo choices.
CONCLUSION
In this work, we introduced ODEBRAIN, a novel continuous-time dynamic modeling framework for modeling EEGs, designed explicitly to overcome critical limitations associated with discrete-time recurrent approaches. By adopting a neural bDE- approach with adaptive vector field strategy, our model effectively captures the continuous neural dynamics and spatial interactions in EEG data. Although ODEBRAIN models latent dynamics continuous time, the inputs and supervision are still based on epoched segments, which limits long-term continuous modeling, The generalization to other neurological disorders or cognitive tasks remains to be explored.
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ACKNOWLEDGMENT
This study was partly supported by JST PRESTO (JPMUPR24TB), CREST (JPMJCR24QS, JP- MJCR23M3, JPMICR20C6), ASPIRE (IPMIAP2329), and Moonshot R&D (JPMIMS2033-14), ISPS KAKENHI Grant-in-Aid for Scientific Research Number JP24K20778, JST START JP- MJST2553, JST Program JPMJKP25Y6, JST COI-NEXT (JPMIPF2009, JPMJPF2115), as well as the Future Social Value Co-Creation Project (Osaka University).
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A EXPERIMENTAL SETTINGS
A.1 DISCUSSION: KEY INSIGHTS OF ODEBRAIN
Conceptually, the main gain of our work comes from explicitly modeling continuous dynamics over graph structures. By capturing the dynamic evolution of EEG signals, the model can effectively handle substantial noise, randomness, and fluctuations. Our comparison with the baseline without continuous dynamics (i.e., using only a temporal GNN backbone) clearly supports this observa- tion. Methodologically, our improvements arise from two key aspects: (i) obtaining a high-quality initialization Zo. and (ii) formulating a vector field fo that captures informative and stable dynam- ics. First, reverse initial encoding provides bigh-quality continuous representation that enables the model unfold temporal information embedded in EEGs. This is achieved through a dual-encoder architecture that integrates spectral graph features with stochastic temporal signals. Second, the tem- poral-spatial ODE solver fo incorporates initialization into additive and gating operations, enabling adaptive emphasis on informative EEG connectivity patterns that encode richer dynamics (Figure 4). Furthermore, the stochastic regularizer mitigates the classical ccrumuccuuulttionn problem of ODEs by modeling stochasticity in the EEG time domain, thereby improving long-term stability, We also include new ablation table (Table 1) validate the contribution of each component and support the above points.
A.2 DYNAMIC SPECTRAL GRAPH STRUCTURE
Raw EEG signals consist of complicated neural activities that overlap in multiple frequency bands, each potentially encoding different functional neural dynamics. Directly analyzing EEG signals in the time domain often misses subtle state transitions that occur uniquely within specific frequency bands (Yang & Hong. 2022; Chen et al., 2023). Hence, it is beneficial to represent the intensity variations of frequency bands and waveforms by decomposing raw EEG signals into frequency components. To effectively provide detailed insights for subtle state transitions, we perform the short-time Fourier transform (STFT) to each EEG epoch, preserving their non-negative log-spectral. Consequently, the multi-channel EEG recordings processed as:
X,
xtt wit m]e- mjwt
(11)
and sequence of EEG epochs with their spectral representation is formulated as X E NxxxT
We then apply a graph representation by measuring the similarity between the spectral representation X across the EEG channels. Specifically, we define an adjacency matrix (i.jj) )) at each epoch as sim(X,,r. j,)) and compute the normalized correlation between nodes and
follows: je( Ug. where the structure of the graph and its associated edge weight matrix Ai,j are inferred from X, for each t-th epoch. We only preserve the highest top-7 correlations to construct the evident graphs without redundancy. To avoid redundant connections and clearly represent dominant spatial struc- tures, retain only the top-7 strongest connections at each epoch for sparse and meaningful graph representations. Thus, we obtain a temporal sequence of EEG spectral graphs (G, = (V,, A.) )
Temporal Graph Representation. Taking EEG X consisting of N channels and T time points, we represent X as graph, denoted G = AV X), where V (uI,..., eN) represents the set of nodes. Each node corresponds an EEG channel. The adjacency matrix A E NxxNXI XT encodes the connectivity between these nodes over time, with each element a.,iæ indicating the strength of connectivity between nodes U; and vy at the time point t. Here, we redefine as a sequence of EEG segments, termed epochs, obtained using moving window approach. The embedding of node v; at the 1-th epoch is represented as ht,, E R.m. Specifically, we perform the short-time Fourier transform (STFT) on each EEG epoch, referring to (Tang et al., 2022). Then we measure the similarity among the spectral representation of the EEG channels to initial the At j) for each epoch
A.3 DATASETS AND EVALUATION PROTOCOLS
Datasets. We use Temple University Hospital EEG Seizure (TUSZ) and the TUH Abnormal EEG Corpus (TUAB) (Shah et al., 2018), the largest publicly available EEG seizure database. TUSZ contains 5,612 EEG recordings with 3,050 annotated seizures. Each recording consists of 19 EEG
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channels following the 10-20 system, ensuring clinical relevance. A key strength of TUSZ lies in its diversity, as the dataset includes data collected over different time periods, using various equipment, and covering wide age range of subjects. To provide normal controls, we sample studies from the normal subset of TUAB. Unless stated otherwise, recordings are processed with the same pipeline across corpora (canonical 10-20 montage with 19 channels and unified resampling), ensuring con- sistent preprocessing for cross-dataset evaluation.
Metries. To answer RQI, we evaluate the model using the Area Under the Receiver Operating Characteristic Curve (AUROC) and the Fl score. AUROC measures the ability of models across varying thresholds, while the Fl score highlights the balance between precision and recall at its optimal threshold for classification. For RQ2, we measure the predicted graph structural similarity using the Global Jaccard Index (GJI) (Castrillo et al., 2018):
|Etrue n Epred
GJI Epred) /Eerue Epreal
(12)
Model training, All models are optimized using the Adam optimizer (Kingma, 2014) with an initial learning rate of 1 x 100$ in the PyTorch and PyTorch Geometric libraries on NVIDIA A6000 GPU and AMD EPYC 7302 CPU. We adopt the adaptive Runge-Kutta NODE integration solver (RK45)
with relative tolerance set to x 10 for training.
A.4 HYPERPARAMETERS
All experiments are conducted on the TUSZ and TUAB dataset using CUDA devices and a fixed random seed of 123. EEG signals are preprocessed via the Fourier transform, segmented into 12- second sequences with a I-second step size, and represented as dynamic graphs comprising 19 nodes (EEG channels). Graph sparsification is achieved with Top-r 3 neighbors. Both dynamic and individual graphs use dual random-walk filters, whereas the combined graph employs a Laplacian filter. The default backbone GRU-GCN for reverse initial state encoding. consisting of 2-layer GRU with 64 hidden units per layer. We also apply a CNN encoder with 3 hidden layers to extract the stochastic feature =" to obtain the final initial value 2g- The convolution adopts 2 x kernel size with batch normalization and max pooling The input and output feature dimensions are both tasks.
100, with the number of classes set to for detection/
We train models using initial learning rate of 3e-4, weight decay 5e-4, dropout rate 0.0. batch sizes of 128 (training) and 256 (validation/test), and a maximum of 100 epochs. Gradient clipping with maximum norm of 5.0 and early stopping with patience of 5 epochs are applied. The model checkpoints are selected by maximizing AUROC in the validation set (weighted averaging). When the metric is loss, instead minimize it; all other metrics (c.g.. F1, ACC) are maximized. Data augmentation is enabled by default, while curriculum learning is disabled unless otherwise stated.
B ADDITIONAL RESULTS
Fig. 8 shows the visualization of the dynamic field of the latent space. reveals distinct neural tivity patterns: during synchronous low-frequency oscillations, dynamic field appears steady state, while high-frequency bursts trigger localized positive gradients, driving system activation. Asyn- chronous cross-channel interactions manifest as vortex-like flows, reflecting dynamic balance. No- tably, continuous dynamic evolution offers finer temporal resolution at an arbitrate time. ODEBRAIN enables early detection of neural transitions, better than discrete- time methods.
Fig. 9() depicts the predicted connectivity patterns and edge densities of ODEBRAIN closer to the true connectivity than the discrete predictor-based AMAG, leading to a significant topology con- sistency. These structural features are crucial for modeling consistent brain dynamics, as small topological offsets lead to correct brain activity for downstream tasks. The stochastic components of the raw EEG signal can be regard as an implicit regularity term, which helps to enhance the gen- eralization ability of continuous trajectory inference and maintains consistency with the structure. The latent variable trajectories generated by ODEBRAIN IN not only maintain continuous evolutionary properties, but also enhance the predictive ability of spatial consistency.
Fig. 9 shows the effectiveness of predicting the dynamic graph structure depending on our meaning- ful forecasting objective Ω. Fig. 9(b) shows that ODEBRATN can achieve higher similarity than the
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acticiy
Dyramic
Choundl
PCA2
PCA1
Figure 8: Visualization results between the multichannel EEG signal (upper) and its latent dynamic
field fo (lower) in our cenporal-spatiin neural ODE.
Table 5:
Ablation of pooling options over ODE-trajectory on TUSZ (12s seizure detection) and TUAB. Bold indicates best result.
TUSZ
TUAB
Method
Acc
F1
AUROC
Acc
F1
AUROC
Max pooling
0.877 +: 0.004
0.496 + 0.017
0.881
0.006
0.778 +: 0.003
0.774 + 0.005
0.857
0.005
Mean pooling
0.842 + 0.002
0.385
0.005
0.827
0.003
0.748 + 0.002
0.635
0.002
0.827 0.004
Sum pooling
0.851 0.002
0.466 0.005
0.867 0.004
0.753 0.003
0.755 0.002
0.831 0.004
discrete predictor, indicating that the continuous prediction model more accurately captures the true graph structure. The similarity matrices reveal that ours aligns more closely in terms of local cor- relation distribution, in which the discrete predictor exhibits notable discrepancies in certain block structures. The explicit graph embedding target improves the forecasting accuracy, while effectively guides the vector field fo to learn continuous trajectories aligned with neural activity, leading to more reliable prediction.
Table 5 concerns the sensitivity with Top-7 options (r=3,7) and different graph regularizers, eval- uated under both latent-ODE and CDEBRAIN. Overall, regularized graph construction consistently improves both metrics for the two frameworks, indicating that raw correlation graphs can be vul- nerable to noise and volume conduction, while statistical regularization yields more reliable func- tional connectivity. Specifically, for latent-ODE, graphical lasso and norm regularization with 7=3 achieve the strongest AUROC/Recall, suggesting that a sparser, regularized partial-correlatin struc- ture is preferable for continuous dynamics modeling. For ODEBRA IN norm with 7=3 gives the best
17

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Published as a conference paper at ICLR 2026
Groundtruth
Ours (Continnous modeling)
Discrete modeling
(a) Comparison among groundtruth, graph output of continuous predictor, graph output of discrete predictor.
Simlriit
18
Our continuous prediction
Oues w new signal loss
Discrete prediction
(b) Comparison of correlation scores between graph output of our continuous predictor, and graph output of discrete predictor.
Figure 9: Comparison the predicted graph output between our continuous predictor and discrete predictor.
AUROC (0.881), whereas graphical lasso with r=3 achieves the highest Recall (0.613); the perfor- mance gap is small between and regularizers, demonstrating robust behavior to graph-construction choices.
Table shows the effects of GNN backbones on TUSZ under 12s and 60s forecasting hori- zons. We find that the GNN choice has non- trivial impact on continuous seizure forecast- ing. GRU-GCN yields the best overall perfor- mance, reaching 0.881 AUROC / 0.496 Fl at 12s and 0.828 AUROC 0.430 Fl at 60s. This indicates that recurrent gating over graph mes- sages better captures fast and non-stationary dynamics, especially for short-term prediction. DCRNN performs competitively but consis- tently below GRU-GCN (0.823/0.433 12s; 0.818/0.417 at 60s), suggesting diffusion-based spatiotemporal propagation effective but less expressive without explicit gating. In contrast, EvolveGCN degrades substantially, particularly for horizon forecasting (0.729 AUROC 0.378 F1 60s), implying that merely evolving GCN pa-
Table 6: Ablation GNN options on TUSZ (12s and 60s seizure detection) (AUROC1, FI1) Bold best.
Method
T(Sec.)
AUROC
F1
ODE
Temporal spatta
EvolveGCN
12
0.791 0.401
60
0.729:10.002 0.378=0.003
DCRNN
12
0.823+10.003 0.433:10.005
60
0.818++0.004 0.417=000007
GRU-GCN
12
0.881+0.006 0.496=000117
60
0.828+0.003 0.430==0.021
rameters is insufficient under noisy epoch-wist correlation graphs. In general, these results demon- strate that continuous latent dynamics ODEBRAIN IN benefit the most from temporally gated graph modeling, and the superiority is consistent across horizons.
Table 7 illustrates the robustness of CDEBRAIN when 30% of EEG segments are randomly masked, comparing it with latent-ODE. When 30% segments are randomly masked, ODEBRAIN exhibits smaller AUROC drops from 0.881 to 0.845, and F1 from 0.496 to 0.464; exceeding the AUROC and F1 of latent-ODS by 0.124 and 0.067, respectively. This demonstrates that ODEBRAIN maintains stable vector fields and detection performance under incomplete observations by leveraging adaptive gating operations within the vector field and stochastic regularization to suppress irregular time step
18

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good trade-off between predictive performance and robustness of Published as a conference paper at ICLR 2026
Table Ablation on TUSZ dataset for 12s seizure detection with different top-7 options. Bold and underline indicate best and second-best results.
Top-7
AUROC
Recall
F1
2
0.867 0.003
0.575-0.003
0.48441.000
3
0.881+00006
0.605=0.003
0.496+0.011
0.870=
0.602=0.004
0.488=0.0113
9
0.808:10.004
0.589=00004
0.487=10.011
0.806
0.571=00002
0.400
13
0.805
0.562-10.04
0.474
jumps. The results indicate that ODEBRAIN achieves robustness to trajectory uncertainty under the effects of missing values, enhancing the capacity of ODE solvers.
Table shows the effects the sparsity level of the correlation graph, controlled by the Top- r neighbors per node. Overall, AUROC re- mains stable performance across from to
Table 7: Ablation of missing value (MV) on TUSZ (12s seizure detection) with AUROC1, FI1, and predicted missing graph structural simi- larity (Sim.)f (Bold best).
13 (0.865-0.881), indicating that ODEBRAIN not overly sensitive Top-r options.
3 achieves the best AUROC (0.881) and F1 (0.496), while both too sparse (r 2) and too
Method
Sim.
AUROC
MV
latent-ODE 0.53 0.791 0.401
dense graphs (r 2 9) lead to slight degrada- tion. When the values of 7 is small, the graph becomes too sparse making the edge GRU for- ward stage affect the quality of the graph de- scriptor. As increases, edges become much denser and correlation- based connectivity con- tains propagated noise, which makes the edge GRU forward more over-smoothing and injects noise structure into the initial state žn- The denser Top-r reduces the robustness of the vector field fo- Therefore, adopt ODE dynamics.
ODEBRAIN 0.63 0.881110.006 0.496:10.017
30%
latent-ODE 0.41 0.721+00004 0.377+10.003
ODEBRAIN 0.55 0.845+-0.002 0.464-10.007
19


Reviews:
[{"review_id": "PlaT3LyTay", "reviewer": "Reviewer_itCT", "summary": "This paper presents ODEBRAIN, a novel continuous-time EEG graph framework built upon Neural Ordinary Differential Equations (NODEs) to model dynamic brain networks. To tackle key challenges in learning temporal brain dynamics, the study introduces three main contributions.\n\nFirst, a dual-encoder architecture is proposed to effectively initialize NODEs: one encoder extracts deterministic frequency-domain features to represent brain connectivity, while the other processes raw EEG signals to preserve stochastic variability. Their integration provides robust spatiotemporal representations for initializing the ODE solver.\n\nSecond, a trajectory forecasting decoder is designed to reconstruct graph structures from NODE latent trajectories. By incorporating a multi-step forecasting loss, the model explicitly predicts the evolution of brain networks over time, enabling accurate and continuous trajectory modeling.\n\nThird, the paper introduces a novel gradient field–based metric derived from NODEs to quantify the dynamics of EEG brain networks. A case study on seizure data demonstrates the clinical interpretability and practical value of this approach.", "questions": "- What is GRU (Gated Recurrent Unit)?\n- In the formula of $z^g$ in line 260, is it should be $z^g_i$?\n- Can you explain more $z^s$? Why is it called stochastic temporal embedding?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper writing is generally good although some parts can be further improved.\n- The idea of using ODE solver in forecasting and predicting graph structure are interesting.\n- The experimental results are good.", "weaknesses": "- Some parts are not clear.\n- No discussion of the computational cost.\n- No discussion of the architecture of $f_\\theta$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2V910sIyss", "reviewer": "Reviewer_GEpk", "summary": "This paper introduces ODEBRAIN, a framework for modeling dynamic brain networks from multi-channel EEG. Unlike existing work modeling EEG dynamics in discrete-time manner, the method aims to learn the continuous-time representation of brain state evolution by leveraging Neural Ordinary Differential Equation (NODE). To address the challenges of applying NODEs to EEG, the authors propose a dual-encoder architecture to obtain robust initialization for ODE solver, combining spectral graph features with stochastic temporal signals. Then a novel objective function is proposed to capture underlying EEG dynamics. Experiments on two seizure detection datasets demonstrate model effectiveness.", "questions": "- Can we always assume that the data supports that f_theta is continuous and differentiable function? Some nonlinearity of EEG may not support this assumption.\n- Several Technical details of the proposed method are not clearly stated / explained. How do the the temporal descriptor \\Psi and the objective \\Omega are defined? How the pooling of latent continuous trajectory z_t for downstream task is conducted?\n- The paragraph “RQ3 concerns consistency in the graphs…” seems confusing. From Fig.4 I cannot interpret the similarity scores or similarity matrices from discrete or continuous predictor, and other discussion. It seems the figure misaligned with the content.\n- Hyperparameter selection is not discussed. Why the paper uses top 3 correlation neighbors for each node when constructing the graph? How about the effectiveness of different GNN architectures?\n- The complexity analysis for the model is missing. How is the training/inference time of ODEBRAIN compared to other baselines.\n- What is the motivation for objective loss to predict graph structure rather than EEG signal forecasting? The proposed loss shows effectiveness for the prediction task, how it would perform on the typical EEG signal forecasting?\n- The discussion of prior work on modeling EEG signals, capturing their nonlinearity for a variety of brain activity mining [1][3][4][8][14], modeling and control for analyzing EEG signals in the context of brain machine interfaces and epileptic seizures [5][6][12][13][15], modeling latent variables in EEG [7][11] and other brain activity data [9][10], as well as related papers using graph ODE for EEG data [1] are all missing. This needs improvement in both discussing existing methods and comparing against these methods.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses an important problem that of modeling brain network in continuous time (with NODE).  \n- The novel contributions mainly come from the proposed dual-encoder architecture and objective loss. \n- The learned dynamic field demonstrate potential for qualitative analyses of brain networks.\n- The proposed method shows some improvement, especially on TUSZ.", "weaknesses": "- Several Technical details of the proposed method are not clearly stated / explained. How do the the temporal descriptor \\Psi and the objective \\Omega are defined? How the pooling of latent continuous trajectory z_t for downstream task is conducted?\n- The paragraph “RQ3 concerns consistency in the graphs…” seems confusing. From Fig.4 I cannot interpret the similarity scores or similarity matrices from discrete or continuous predictor, and other discussion. It seems the figure misaligned with the content.\n- Hyperparameter selection is not discussed. Why the paper uses top 3 correlation neighbors for each node when constructing the graph? How about the effectiveness of different GNN architectures?\n- The complexity analysis for the model is missing. How is the training/inference time of ODEBRAIN compared to other baselines.\n- What is the motivation for objective loss to predict graph structure rather than EEG signal forecasting? The proposed loss shows effectiveness for the prediction task, how it would perform on the typical EEG signal forecasting?\n- The discussion of prior work on modeling EEG signals, capturing their nonlinearity for a variety of brain activity mining [1][3][4][8][14], modeling and control for analyzing EEG signals in the context of brain machine interfaces and epileptic seizures [5][6][12][13][15], modeling latent variables in EEG [7][11] and other brain activity data [9][10], as well as related papers using graph ODE for EEG data [1] are all missing. This needs improvement in both discussing existing methods and comparing against these methods.\n[1] JPM. Pijn et al., \"Nonlinear dynamics of epileptic seizures on basis of intracranial EEG recordings.\" Brain topography 9, no. 4 (1997): 249-270.\n[2] Y. Chen et al., \"EEG emotion recognition based on ordinary differential equation graph convolutional networks and dynamic time wrapping.\" Applied Soft Computing 152 (2024): 111181.\n[3] Y. Xue et al., \"Minimum number of sensors to ensure observability of physiological systems: A case study.\" In 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1181-1188. IEEE, 2016.\n[4] K. Lehnertz et al. \"Seizure prediction by nonlinear EEG analysis.\" IEEE Engineering in Medicine and Biology Magazine 22, no. 1 (2003): 57-63.\n[5] G. Gupta et al., \"Re-thinking EEG-based non-invasive brain interfaces: Modeling and analysis.\" In 2018 ACM/IEEE 9th International Conference on Cyber-Physical Systems (ICCPS), pp. 275-286. IEEE, 2018.\n[6] V. Tzoumas et al., \"Selecting sensors in biological fractional-order systems.\" IEEE Transactions on Control of Network Systems 5, no. 2 (2018): 709-721.\n[7] G. Gupta et al., \"Learning latent fractional dynamics with unknown unknowns.\" In 2019 American Control Conference (ACC), pp. 217-222. IEEE, 2019.\n[8] K. Lehnertz, \"Epilepsy and nonlinear dynamics.\" Journal of biological physics 34, no. 3 (2008): 253-266.\n[9] R. Yang et al., \"Data-driven perception of neuron point process with unknown unknowns.\" In Proceedings of the 10th ACM/IEEE International Conference on Cyber-Physical Systems, pp. 259-269. 2019\n[10] R. Yang et al., \"Spiking dynamics of individual neurons reflect changes in the structure and function of neuronal networks.\" Nature Communications 16, no. 1 (2025): 6994\n[11] G. Gupta et al., \"Dealing with unknown unknowns: Identification and selection of minimal sensing for fractional dynamics with unknown inputs.\" In 2018 Annual American Control Conference (ACC), pp. 2814-2820. IEEE, 2018.\n[12] X. Lu, \"Detection and classification of epileptic EEG signals by the methods of nonlinear dynamics.\" Chaos, Solitons & Fractals 151 (2021): 111032.\n[13] R. Martis et al. \"Epileptic EEG classification using nonlinear parameters on different frequency bands.\" Journal of Mechanics in Medicine and Biology 15, no. 03 (2015): 1550040.\n[14] M. Mercier et al. \"The value of linear and non-linear quantitative EEG analysis in paediatric epilepsy surgery: a machine learning approach.\" Scientific reports 14, no. 1 (2024): 10887.\n[15] G. Lepeu et al. \"The critical dynamics of hippocampal seizures.\" Nature communications 15, no. 1 (2024): 6945.\nClaiming that the novelty of this approach is that it captures the connectivity of spatio-temporal graphs to capture nonstationary changes is incorrect as can be seen from existing works that do just that.\n- The paper states that the proposed approach “combines deterministic graph-based features with stochastic EEG representations to produce a robust initial state” but it is unclear how the stochastic EEG is accounted for.\n- Can we always assume that the data supports that f_theta is continuous and differentiable function? Some nonlinearity of EEG may not support this assumption.\n- The parameters N, d and T are barely mentioned in passing in section 4.2 but how they are selected in the case study and some guidelines to select them are missing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1Djl7V6F1M", "reviewer": "Reviewer_xvZK", "summary": "This well-written paper introduces ODEBRAIN, a continuous-time latent dynamics framework for multi-channel EEG that: (1) builds spectral EEG graphs (where nodes are channels; edges are top-k correlations in STFT space); (2) uses a dual encoder to initialize a Neural ODE (NODE) with a deterministic graph descriptor and a stochastic temporal descriptor; and (3) forecasts future graph node embeddings via a graph-prediction head with multi-step loss. The authors additionally visualize the learned vector field and argue it is clinically interpretable (e.g., “centers” during seizures). They claim the formulation is the first to explicitly cast EEG brain networks as a continuous-time dynamical system governed by a NODE and show gains on two real world data sets over discrete baselines.", "questions": "Please clarify in what sense the approach exceeds a finely sampled discrete model, beyond the solver’s internal substeps; e.g., can ODEBRAIN answer arbitrary-time queries between epochs evaluated against held-out high-rate labels?\n\nEdges use normalized correlation with top-\\taup sparsification. Correlation graphs are sensitive to noise and volume conduction; what happens with different \\tau, similarity metrics, or regularizers (e.g., shrinkage/graphical lasso/lagged connectivity)?\n\nSince the proposed benefit is about continuous-time superiority, why aren’t Latent ODE-style baselines (e.g., NODE on per-channel embeddings without graph, and graph-ODE methods) and irregular-sampling setups considered?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper argues discretized, windowed EEG pipelines miss inherently continuous dynamics, motivating a NODE approach and posing concrete challenges (robust initialization; meaningful trajectory objectives).\n\nThe dual-encoder (graph plus stochastic temporal) initialization and a graph-forecasting head with multi-step loss are well aligned with continuous latent trajectory learning. The forward ODE is standard and the projection aims at future graph prediction, not just signal.\n\nVisualizations of the learned vector field highlight identifiable structures during seizure vs. non-seizure (e.g., attractor-like “centers” only during seizures), which is a promising narrative for clinical insight.\n\nOn TUSZ/TUAB EEG data, ODEBRAIN outperforms CNN-LSTM, BIOT, EvolveGCN, DCRNN, and AMAG in AUROC/F1. Both single-step and multi-step settings are reported and ablations probe initialization choices, loss design, and horizon.\n\nData, preprocessing, solver tolerance, training hyperparameters, and an anonymous code link are provided to aid reproducibility.", "weaknesses": "Although the latent dynamics are continuous via NODE, inputs and supervision remain epoched STFT segments and edges are top-k correlations per epoch. The model forecasts at discrete horizons (1s/3s/11s), and training targets are per-epoch graphs. \n\nSensitivity analyses related to top-\\tau sparsity and normalized correlations are not reported.\n\nThe baselines considered are primarily discrete TGNs/transformers. The related-work cites graph ODEs, but the empirical table omits them.\n\nNODE solvers can be costly/unstable. You specify RK45 and tolerances, but there’s no wall-clock / NFEs / memory vs. discrete baselines, nor sensitivity to tolerances/horizons", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Gh51FEXMvK", "reviewer": "Reviewer_bZrF", "summary": "This paper proposes ODEBRAIN, a neural ODE latent dynamic forecasting framework with integrating spatio-temporal features into spectral graph nodes. The continuous latent dynamics was modeled by a neural ODE. The model shows efficiency in forecasting EEG dynamics.", "questions": "1. In 4.1, how to decide top-tau?\n2. Some typo: line 219, space before ‘Consequently’. Line 21 and line 23, ‘verifies’ to ‘verify’.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper extends graph-based EEG modeling into the continuous domain using Neural ODEs, with separating spatial and temporal encoders, the method combines structured connectivity with temporal uncertainty. The proposed method does show better performance than several baselines.", "weaknesses": "1. The objective function used to train the algorithm is not clearly defined.\n2. The method should include comparisons with other continuous-time baselines, such as BrainODE, which is already cited in the related works. Such comparisons would help demonstrate the specific advantages of introducing the graph-based formulation.\n3. While the proposed method introduces a graph structure, it might be the main distinction from existing ODE approaches, the paper does not clearly explain how and why incorporating the graph improves forecasting of temporal dynamics.\n4. The data is processed by STFT, however, the details of STFT are not included in the paper, and how do different parameters (such as window, frequency bins, log scaling) influence the performance of the model?\n5. The paper does not discuss limitations of the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)