Understanding how individual brain development unfolds over childhood requires modeling developmental trajectories from sparse longitudinal observations. Long-term neurodevelopmental forecasting is challenging because each child is typically observed at only a few irregularly spaced visits, while developmental dynamics vary across individuals and age. Generic continuous-time models accommodate irregular timing but often absorb these factors into a single flexible transition function, providing little structure for how population progression, individual variability, and developmental age shape the dynamics. We propose DevelopmentODE, a structured continuous-time framework that organizes population- and subject-specific variation within a shared developmental geometry while allowing the governing dynamics to evolve with age. The model builds this geometry around a developmental canal representing the population trajectory, whose local direction provides a reference for organizing subject-specific variation. Subject deviation velocities are constrained relative to this direction, while a shared nonlinear deviation field captures individual developmental motion without disrupting population-level progression. DevelopmentODE further models developmental non-stationarity through ordered age-dependent deformations of the shared vector field, progressively adapting a common dynamical structure as age changes, while elapsed time determines the integration horizon. This formulation uses population-level developmental structure to guide learning from sparse individual trajectories while allowing dynamics to evolve smoothly with age. We evaluate DevelopmentODE on longitudinal fMRI by predicting future functional connectivity of the same child from earlier observations. DevelopmentODE consistently outperforms competing baselines across short- and long-horizon predictions.
Figures & tables
Figure 1: Overview of DevelopmentODE. Sparse and irregular longitudinal FC observations are mapped into a shared developmental geometry, where DevelopmentODE models continuous trajectories and predicts future brain states at arbitrary target ages. Latent dynamics are organized around a population developmental canal μη(t) , with subject-specific evolution modeled as deviation motion in the complementary geometry (bottom left). Smoothly ordered age-dependent modulation adapts the shared deviation field across development (bottom right).
Model
Short-term
Long-term
All
MSE ↓
MAE ↓
R2↑
PCC ↑
MSE ↓
MAE ↓
R2↑
PCC ↑
MSE ↓
MAE ↓
R2↑
PCC ↑
RNN
3.07
1.23
34.51
59.36
3.10
1.23
31.65
57.19
3.09
1.23
32.80
58.05
GRU
3.09 -0.4%
1.23 0.0%
34.29 -0.6%
59.18 -0.3%
3.10 -0.1%
1.23 +0.1%
31.64 -0.1%
57.18 0.0%
3.10 -0.2%
1.23 +0.1%
32.69 -0.3%
57.97 -0.1%
LSTM
3.07 0.0%
1.22 +0.3%
34.59 +0.2%
59.40 +0.1%
3.09 +0.4%
1.22 +0.6%
31.96 +1.0%
57.43 +0.4%
3.08 +0.3%
1.22 +0.5%
33.01 +0.7%
58.22 +0.3%
Transformer
3.04 +1.3%
1.22 +0.6%
35.32 +2.3%
60.02 +1.1%
3.02 +2.5%
1.21 +1.4%
33.42 +5.6%
58.47 +2.3%
3.03 +2.0%
1.21 +1.1%
34.17 +4.2%
59.09 +1.8%
Direct MLP
3.03 +1.5%
1.22 +0.2%
35.58 +3.1%
60.09 +1.2%
3.07 +1.1%
1.23 -0.1%
32.41 +2.4%
57.66 +0.8%
3.05 +1.2%
1.23 0.0%
33.68 +2.7%
58.63 +1.0%
Table 1: Evaluation results on short-term, long-term, and all prediction pairs. Metrics include MSE ( ↓ ), MAE ( ↓ ), R2 ( ↑ ), and PCC ( ↑ ). The values following each metric denote the relative improvement (%) over the RNN baseline, where positive values indicate better performance according to the direction of each metric. The best results are in bold , and the second-best are underlined . MSE, R2 , and PCC are multiplied by 100, and MAE by 10, for readability.
Variant
MSE ↓
PCC ↑
Ours
2.939
60.51
w/o Canal
2.975 (+1.22%)
59.96 (-0.91%)
w/o Time
2.997 (+1.97%)
59.58 (-1.54%)
w/o Stage
2.997 (+1.97%)
59.61 (-1.49%)
Table 2: Ablation study on all test pairs. Relative changes from the full model are shown in parentheses.
Figure 2: Forecasting error under different amounts of supervision.
Figure 3: Temporal analysis of longitudinal FC forecasting. (a) Prediction performance as a function of the source–target temporal interval. (b) Performance across source-time and target-time pairs. (c) Performance across different target time.
Figure 4: Robustness of DevelopmentODE to numerical solver, latent representation dimension, and developmental stage-boundary choices.
Model
MSE ↓
PCC ↑
DevelopmentODE
4.0467
53.6233
Time-MLP
4.6256
48.6872
Direct MLP
4.6728
48.3735
Age-NODE
5.0071
44.4017
Generic NODE
5.0102
44.3411
Table 3: Comparison on the external BCP cohort. MSE and PCC are multiplied by 100 for readability.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Age
N
∥μ˙∥2
λ(t)
E∥r~∥2
3 weeks
228
284.007
27.782
11.361
1 year
233
109.156
25.219
13.384
2 years
199
101.897
21.251
13.882
4 years
116
73.443
14.464
12.672
6 years
156
52.219
11.598
12.197
8 years
152
37.022
10.192
13.739
Appendix
Table 4: Developmental geometry across age on held-out subjects. Values are averaged over five subject-disjoint folds.
Transition
N
Population
Deviation
3 weeks → 1 year
159
0.530
0.470
3 weeks → 2 years
109
0.489
0.511
3 weeks → 4 years
71
0.022
0.978
3 weeks → 6 years
65
0.033
0.967
3 weeks → 8 years
90
0.040
0.960
3 weeks → 10 years
76
0.106
0.894
Appendix
Table 5: Population-aligned and complementary components of observed longitudinal changes on held-out subjects.
Transition
N
λˉ
log∥r~a∥2∥r~b∥2
3 weeks → 1 year
159
26.693
0.181
1 year → 2 years
130
23.344
0.038
2 years → 4 years
54
17.215
-0.073
4 years → 6 years
53
12.813
-0.056
6 years → 8 years
82
10.715
0.106
8 years → 10 years
87
9.942
0.004
Appendix
Table 6: Learned age-dependent modulation and observed deviation persistence for adjacent developmental transitions.
Model
Short-term
Long-term
All
MSE
↓
MAE
↓
R2
↑
PCC
↑
MSE
↓
MAE
↓
R2
↑
PCC
↑
MSE
↓
MAE
↓
R2
↑
PCC
↑
RNN
3.06
–
1.22
–
34.78
–
59.61
–
3.17
–
1.24
–
29.96
–
56.00
–
3.13
–
1.23
–
31.88
–
57.44
–
GRU
3.06
-0.09
1.22
0.13
34.75
-0.09
59.59
-0.02
3.20
-0.64
1.24
-0.21
29.51
-1.48
55.71
-0.53
3.14
-0.42
1.23
-0.07
31.60
-0.87
57.26
-0.32
LSTM
3.06
0.01
1.22
0.24
34.81
0.10
59.63
0.03
3.18
-0.21
1.24
0.09
29.81
-0.49
55.93
-0.13
3.13
-0.12
1.23
0.15
31.81
-0.22
57.41
-0.06
Transformer
3.04
0.79
1.22
0.24
35.28
1.46
60.01
0.68
3.06
3.69
1.21
2.02
32.63
8.94
57.86
3.31
3.05
2.56
1.22
1.32
33.69
5.68
58.72
2.23
Direct MLP
3.01
1.66
1.22
0.35
35.92
3.28
60.36
1.26
3.13
1.53
1.24
0.22
31.07
3.73
56.70
1.24
3.08
1.58
1.23
0.27
33.01
3.54
58.16
1.25
Appendix
Table 7: Evaluation results when models are trained only on adjacent within-subject transitions and evaluated on all forward test pairs. Metrics include MSE ( ↓ ), MAE ( ↓ ), R2 ( ↑ ), and PCC ( ↑ ). The change rate (%) is computed relative to the RNN baseline. The best results are in bold , and the second-best are underlined .
Model
Short-term
Long-term
All
MSE
SD
MAE
SD
R2
SD
PCC
SD
MSE
SD
MAE
SD
R2
SD
PCC
SD
MSE
SD
MAE
SD
R2
SD
PCC
SD
RNN
3.07
0.064
1.23
0.011
34.51
0.488
59.36
0.354
3.10
0.040
1.23
0.008
31.65
0.804
57.19
0.585
3.09
0.039
1.23
0.007
32.80
0.562
58.05
0.395
GRU
3.09
0.064
1.23
0.010
34.29
0.510
59.18
0.379
3.10
0.037
1.23
0.004
31.64
0.723
57.18
0.531
3.10
0.036
1.23
0.004
32.69
0.483
57.97
0.342
LSTM
3.07
0.067
1.22
0.010
34.59
0.450
59.40
0.346
3.09
0.035
1.22
0.005
31.96
0.881
57.43
0.628
3.08
0.037
1.22
0.005
33.01
0.547
58.22
0.381
Transformer
3.04
0.070
1.22
0.011
35.32
0.530
60.02
0.387
3.02
0.036
1.21
0.007
33.42
0.854
58.47
0.601
3.03
0.040
1.21
0.007
34.17
0.607
59.09
0.408
Direct MLP
3.03
0.062
1.22
0.008
35.58
0.421
60.09
0.317
3.07
0.035
1.23
0.005
32.41
0.806
57.66
0.583
3.05
0.038
1.23
0.004
33.68
0.545
58.63
0.401
Appendix
Table 8: Evaluation results on short-term, long-term, and all prediction pairs. Metrics include MSE ( ↓ ), MAE ( ↓ ), R2 ( ↑ ), and PCC ( ↑ ). The SD columns report the standard deviation. The best results are in bold , and the second-best are underlined .
Metric
RNN
GRU
LSTM
Transformer
Direct MLP
Time-cond. MLP
Generic NODE
Age-cond. NODE
MoE NODE
Stage NODE
DevelopmentODE
MSE ↓
4.7677
4.8221
4.8894
4.7032
4.6728
4.6256
5.0102
5.0071
5.0075
5.0134
4.0467
PCC ↑
46.3742
46.1490
46.0146
47.0882
48.3735
48.6872
44.3411
44.4017
44.3976
44.5096
53.6233
Appendix
Table 9: Comparison on the external BCP cohort. MSE and PCC are multiplied by 100 for readability. The best results are in bold , and the second-best are underlined .
Pairs
Model
α
MSE ↓
R2↑
MAE ↓
PCC ↑
All
Fixed comp.
0
2.9403±0.0326
36.128±0.509
1.2009±0.0046
60.495±0.383
Learnable aligned
0.153±0.088
2.9390±0.0321
36.158±0.527
1.2005±0.0052
60.515±0.396
Adjacent
Fixed comp.
0
2.9451±0.0262
36.534±0.321
1.2033±0.0041
60.773±0.256
Learnable aligned
0.198±0.025
2.9428±0.0266
36.586±0.307
1.2032±0.0042
60.808±0.242
Appendix
Table 10: Sensitivity to a learnable population-aligned component. Results are mean ± SD over five subject-disjoint folds. MSE, R2 , and PCC are ×100 ; MAE is ×10 .
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 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.
Haohui Jia, Zheng Chen, Lingwei Zhu +6
Information Science and Techinology, Hokkaido University, Japan · SANKEN, The University of Osaka, Japan · Great Bay University, China +1
Accurately predicting the temporal evolution of clinical biomarkers is crucial for the early diagnosis and management of neurodegenerative diseases such as Alzheimer's disease. However, this relies on longitudinal data to capture biomarker changes over time, which is often sparse and irregular due to the high cost, labor-intensive nature, and patient burden. To address these challenges, we propose ENC-ODE, an Event-level Neurodegenerative modeling in Continuous time with neural Ordinary Differential Equations. ENC-ODE predicts future biomarker evolution by modeling clinical events through diagnosis-conditioned continuous dynamics. A target-conditioned attention mechanism weights and aggregates event-level predictions for the target time and modality without history compression. Extensive experiments on Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset demonstrate that ENC-ODE outperforms representative sequence models while offering a scalable and neuroscientifically grounded solution for clinical support. The code is available at https://github.com/JardinDelSol/enc-ode.
Yujee Song, Seunghun Baek, Guorong Wu +1
Pohang University of Science and Technology, Pohang, South Korea · University of North Carolina at Chapel Hill, Chapel Hill, USA
Brain activity unfolds as coordinated patterns across regions over time, and modeling these dynamics is a central goal in computational neuroscience. Because activity is distributed over connected brain regions, it is naturally represented on a graph whose nodes are brain regions or recording electrodes and whose edges capture relationships between these nodes, with message-passing modeling regional communication. However, standard message-passing networks place every node in a single shared feature space and typically aggregate messages from neighbors by scalar summation, which does not align with neurobiological evidence of brain communication. In order to model the heterogeneous nature of the brain, we introduce BrainDyn, a graph ordinary differential equation (ODE) network built on a cellular sheaf with dynamic restriction maps, recomputed from the forecasting context window rather than fixed once, mediating region-to-region communication through learned linear transformations that drive the time evolution of neural activity. We evaluate BrainDyn on resting-state fMRI, EEG, and simulated activity from the NEST simulator. Across these modalities, we test multi-region forecasting and autoregressive generation of brain transients, and, using NEST, prediction of altered transients following controlled perturbations. BrainDyn outperforms competing models across modalities, while analysis of the learned features shows that they are expressive, interpretable, and biologically aligned.
Siddharth Viswanath, Panayiotis Ketonis, Cassidy All +10
Yale University · Boise State University · University of Wisconsin–Madison