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DevelopmentODE: Structured Neural ODEs for Early Brain Development Dynamics Across a Decade

Kaiqiao Han, Haitao Chen, Bryan Quah, Xiaoda Wang, Janelle Liu, John H Gilmore, Wei Gao, Yizhou Sun

arXiv:2609.33048v1cs.LG

TL;DR

Sparse, irregular longitudinal observations and variation across individuals and developmental ages make long-term brain-development forecasting challenging. DevelopmentODE structures continuous-time dynamics around shared population development; it outperforms competing baselines across prediction horizons, including on all test pairs relative to RNN.

  • Problem

    Long-horizon neurodevelopmental forecasting must learn from sparse, irregular visits while accounting for individual variability and developmental dynamics that change with age.

    Source: §1 · p. 2

  • Method

    DevelopmentODE structures continuous-time dynamics around a shared developmental geometry and adapts the shared vector field through ordered age-dependent modulation.

    Source: §1 · pp. 2–3

  • Results

    On all test pairs, DevelopmentODE reduces MSE by 4.89% and improves PCC by 4.24% relative to RNN, and consistently outperforms baselines across short- and long-horizon predictions.

    Source: §1 · p. 2, §4.2 · p. 7

  • Takeaways & Limitations

    The results suggest that incorporating structured population information is a direction for learning reliable dynamics from sparse longitudinal observations.

    Source: §5 · p. 10

    The empirical evaluation focuses on resting-state functional connectivity during early childhood, and the learned components’ causal or mechanistic correspondence remains for future work.

    Source: D LIMITATIONS · p. 19

Abstract

from arXiv · show

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.

1 INTRODUCTION

Long-horizon developmental forecasting must handle sparse, irregular observations, individual variability, and dynamics that change with age. DevelopmentODE organizes these factors around shared population-level developmental structure and outperforms baselines across short- and long-horizon predictions.

1 INTRODUCTION

  • Long-horizon forecasting is challenging because observations are sparse and irregular, developmental mechanisms change with age, and individual trajectories vary around shared population structure.The introduction frames decade-long prediction as more than extending short-term models farther in time.Source: §1 · p. 2
  • DevelopmentODE organizes subject-specific variation relative to a population-shared developmental canal and adapts shared dynamics through ordered age-dependent deformations.Developmental age determines how deformations are combined, while elapsed time determines the integration horizon.Source: §1 · p. 2
  • DevelopmentODE consistently outperforms direct predictors, sequential models, Neural ODEs, and age-conditioned continuous-time baselines across short- and long-horizon predictions.Evaluation includes all-forward and adjacent-transition settings; controlled ablations support the geometry and age-dependent dynamics.Source: §1 · p. 2
  • The paper formulates multi-year forecasting to distinguish population-level from subject-specific variation and developmental age from elapsed time.Its contributions include evaluation under all-forward and adjacent-transition settings, with ablations supporting the proposed geometry and age-dependent dynamics.Source: §1 · pp. 2–3

2 RELATED WORK

The related work describes childhood brain development as heterogeneous across systems and individuals, with maturation varying in timing and rate. Neural ODEs handle irregular observations but do not by themselves structure population, subject-specific, and age-dependent variation.

2.1 INFANT AND CHILD BRAIN DEVELOPMENT

  • Brain systems mature at different rates and developmental periods, so equal elapsed time at different ages need not correspond to equivalent developmental change.The passage relates this heterochrony to individual variability around shared population patterns, described as developmental canalization.Source: §2.1 · p. 3

2.2 NEURAL ODES FOR DYNAMICAL SYSTEMS

  • Neural ODEs integrate latent dynamics to arbitrary irregularly sampled times, but generic age- or subject-conditioned fields can absorb developmental signals without explicitly structuring them.DevelopmentODE instead uses shared population development as a reference, models subject-specific dynamics relative to it, and modulates shared dynamics with age.Source: §2.2 · p. 3

3 METHOD

DevelopmentODE models sparse longitudinal FC trajectories in a shared geometry, separating population progression from subject-specific motion while allowing dynamics to vary with developmental age. It integrates from a child’s source age to a future target age and trains on forward observation pairs.

3 METHOD

  • The method is a structured continuous-time framework that organizes population and subject-specific dynamics in shared developmental geometry while allowing governing dynamics to vary with age.The geometry is organized around a developmental canal representing population-level progression.Source: §3 · p. 3, Figure 1 · p. 4

3.1 PROBLEM FORMULATION

  • DevelopmentODE predicts the same subject’s future FC from an earlier observation, with absolute developmental age shaping dynamics separately from elapsed integration time.Equal-duration intervals may induce different transformations when they occur at different developmental ages.Source: §3.1 · p. 4, §3.4 · p. 6

3.2 STRUCTURED DEVELOPMENTAL DYNAMICS AND GEOMETRY

  • A differentiable population canal and projected deviation field separate shared developmental progression from subject-specific motion, preventing the individual field from freely re-explaining population development.The deviation input excludes its component along the canal’s local direction, and the generated velocity is projected the same way.Source: §3.2 · pp. 4–5

3.3 ORDERED STAGE-DEPENDENT DEVELOPMENTAL FIELD

  • Ordered, smoothly overlapping age stages deform a shared deviation field, allowing dynamics to vary with age without fragmenting sparse supervision across independent stage-specific fields.Stage centers are fixed from training ages, while the shared field pools transitions across developmental ages.Source: §3.3 · p. 5

3.4 DEVELOPMENTODE

  • The full dynamics combine the canal’s population progression, a shared deviation field, and stage-dependent deformations of individual motion.The canal pools cross-subject observations, while the deviation field pools longitudinal transitions.Source: §3.4 · pp. 5–6

3.5 LEARNING OBJECTIVE

  • Training forms forward source–target pairs and samples subjects uniformly before sampling a valid pair, reducing overrepresentation of subjects with more observations.A normalized progress variable allows trajectories with different prediction intervals to be integrated jointly while retaining absolute-age dependence.Source: §3.5 · p. 6

4 EXPERIMENTS

The experiments evaluate multi-year within-child functional connectivity forecasting on longitudinal fMRI, comparing DevelopmentODE with direct, sequential, and continuous-time baselines. DevelopmentODE leads across settings, with ablations and further analyses examining its structure, supervision, temporal patterns, sensitivity, and external-cohort performance.

4.1 EXPERIMENT SETUP

  • The main evaluation uses EBDS longitudinal resting-state fMRI and compares direct predictors, discrete sequence models, and continuous-time models using MSE, MAE, R2, and PCC.The dataset retains 484 subjects with at least two visits; evaluation separates short-term and long-term forecasting.Source: §4.1 · pp. 6–7

4.2 MAIN RESULTS

  • On all test pairs, DevelopmentODE reduces MSE by 4.89% and improves PCC by 4.24% relative to RNN, with the advantage larger for long-term than short-term transitions.It achieves the best results across metrics and evaluation settings.Source: Table 1 · p. 7, §4.2 · p. 7

4.3 ABLATION STUDY

  • Removing the canal, temporal conditioning, or ordered stage modulation degrades performance, with temporal conditioning and stage modulation producing the largest reported losses.The respective changes are +1.22% MSE/−0.91% PCC, +1.97%/−1.54%, and +1.97%/−1.49%.Source: Table 2 · p. 8, §4.3 · p. 8

4.4 SPARSITY ANALYSIS

  • DevelopmentODE is more sensitive to reductions in population supervision than longitudinal supervision, yet remains stable and outperforms representative baselines under reduced longitudinal supervision.The analysis varies population and longitudinal supervision separately and jointly across multiple availability levels.Source: §4.4 · p. 8, Figure 2 · p. 8

4.5 PREDICTION HORIZON ANALYSIS

  • Forecasting difficulty generally increases with interval length, but source–target age pairs show that transitions of similar duration can differ in difficulty.The transition map retains both endpoint ages rather than grouping pairs solely by temporal gap.Source: §4.5 · p. 8, Figure 3 · p. 9

4.6 SENSITIVITY ANALYSIS

  • Performance is generally stable across tested ODE solvers and latent dimensions, while developmental stage-boundary placement has a more noticeable effect.The sensitivity analyses vary each factor independently while holding other model and optimization settings fixed.Source: §4.6 · p. 9, Figure 4 · p. 9

4.7 INDEPENDENT COHORT EVALUATION

  • On the independently collected BCP cohort, DevelopmentODE achieves lower prediction error and higher structural agreement than competing methods.Methods are independently trained and evaluated on BCP using the same experimental protocol.Source: Table 3 · p. 9, §4.7 · p. 9

5 CONCLUSION

The conclusion presents DevelopmentODE as a structured continuous-time approach for sparse, irregular longitudinal data. Across internal and external evaluations, it reports improvements over discrete-time, generic Neural ODE, and age-conditioned alternatives.

5 CONCLUSION

  • Across internal and external evaluations, DevelopmentODE improves over discrete-time, generic Neural ODE, and age-conditioned alternatives.The framework combines shared population-level developmental structure with subject-specific variation.Source: §5 · p. 10

AI USE STATEMENT

Generative AI tools assisted with language polishing, manuscript presentation and organization, and minor formatting. The authors reviewed the assisted text and retained responsibility for the final content.

AI USE STATEMENT

  • Generative AI assisted with language polishing, presentation and organization, and minor formatting, but not experimental data or results.The authors reviewed and revised AI-assisted text and take responsibility for the final content.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

The study analyzes previously collected EBDS and BCP data, without new recruitment, data collection, or intervention. It is methodological research and makes no clinical decisions or subject-level diagnostic conclusions.

ETHICS STATEMENT

  • The study is a secondary analysis of existing EBDS and BCP datasets, with no new recruitment, data collection, or intervention.The work focuses on methodological development and does not make clinical decisions or subject-level diagnostic conclusions.Source: ETHICS STATEMENT · p. 10

A DATASET DETAILS

The study uses a decade-spanning longitudinal fMRI cohort and evaluates within-child forecasting across irregular visit intervals, with an independent BCP cohort for auxiliary validation.

A.1 LONGITUDINAL PREDICTION COHORT

  • EBDS contributes 1,476 rsfMRI scans from 633 children across seven developmental ages, from 3 weeks to 10 years.Usable scan counts by age are 372, 273, 220, 125, 175, 155, and 156.Source: §A.1 · p. 12
  • Scanners and acquisition conditions varied across ages; preprocessing and feature harmonization addressed scanner- and acquisition-related effects.Scans with fewer than three minutes of usable data after motion scrubbing were excluded.Source: §A.1 · p. 12

A.2 LONGITUDINAL PREDICTION PROTOCOL

  • Forecast pairs use one child’s earlier observation as the source and that same child’s later observation as the target, with subject-level data splits.The protocol includes short- and long-horizon intervals because visit spacing varies substantially.Source: §A.2 · p. 13

A.3 BCP AUXILIARY COHORT

  • The auxiliary BCP cohort contains 121 resting-state fMRI observations from children aged 0–5 years.The passage also specifies its processing pipeline and checks for matrix validity and consistent ROI ordering.Source: §A.3 · p. 13

B IMPLEMENTATION DETAILS

The implementation fits fold-specific representations and a population developmental canal, then models subject deviations and age-dependent dynamics in continuous time. Analyses describe how these fitted components vary across age and transitions.

B IMPLEMENTATION DETAILS

  • Each fold fits a 32-dimensional PCA state space, age normalization, canal, and checkpoint using training subjects only; all-forward uses every forward pair, while adjacent uses consecutive visits.Subjects are split before source–target pairs are constructed.Source: B IMPLEMENTATION DETAILS · p. 13
  • The population canal is initialized from fold-training age-group means, refined for 500 AdamW steps, and jointly optimized with subject-specific dynamics during longitudinal training.The primary objective is target FC reconstruction, with a centering objective weighted by 0.05.Source: B IMPLEMENTATION DETAILS · pp. 13–14
  • Prediction integrates canal change, a shared nonlinear deviation field, and age-dependent components from source age to target age, using three smoothly ordered stages centered at normalized ages 0.2 and 0.6.The population canal’s age derivative is obtained by automatic differentiation.Source: B IMPLEMENTATION DETAILS · pp. 13–14
  • Training samples subjects before selecting a valid transition, uses validation FC-space MSE for checkpoint selection, and applies fixed-step RK4 by default; baselines share the fold-specific data and optimization protocol.Transitions are mapped to a common unit integration interval while the vector field is scaled by the true normalized age difference; short-term spans are at most 0.2 normalized age units.Source: B IMPLEMENTATION DETAILS · p. 14

C.1 DEVELOPMENTAL GEOMETRY ACROSS AGE

  • The deviation dynamics combine a shared nonlinear field, age-dependent contraction, and low-rank stage-specific directional deformations around the population canal.Ordered stage weights combine learned positive contraction coefficients; the components distinguish population progression, individual variation, and age-dependent changes.Source: §C.1 · pp. 14–15
  • The fitted canal velocity is highest at 3 weeks (284.007) and falls to 33.623 at 10 years, while mean projected deviation rises to 14.072 at 10 years.The projected deviation averages 11.361 at 3 weeks, peaks at 13.882 at 2 years, and then reaches 14.072 at 10 years; these are properties of the fitted representation, not direct biological estimates.Source: Table 4 · p. 15, §C.1 · p. 15

C.2 POPULATION AND INDIVIDUAL COMPONENTS OF LONGITUDINAL DEVELOPMENT

  • Observed displacement aligned with the population direction is 0.530 for 3 weeks→1 year and 0.489 for 3 weeks→2 years, while many other transitions are mostly complementary-space.The aligned and complementary fractions sum to one, and their relative contributions vary across transitions.Source: Table 5 · p. 16, §C.2 · p. 16
  • Learned age-dependent modulation decreases smoothly, but observed deviation magnitude changes non-monotonically, decreasing most clearly over 2→4 and 4→6 years and increasing over 3 weeks→1 year and 6→8 years.The authors interpret λ(t) as a dynamical modulation, not a direct proxy for observed deviation change.Source: Table 6 · p. 16, §C.2 · p. 16

C.3 TRAINING ON ADJACENT LONGITUDINAL TRANSITIONS

The adjacent-only training protocol tests whether models trained on consecutive observations remain effective on all forward test intervals; the section also reports standard deviations for the main evaluation.

C.3 TRAINING ON ADJACENT LONGITUDINAL TRANSITIONS

  • The adjacent-only training experiment evaluates both short- and longer-range forward transitions, despite training on consecutive within-subject transitions.The comparison includes all forward test pairs.Source: §C.3 · pp. 16–17
  • DevelopmentODE achieves the best performance across metrics and horizons when trained on adjacent transitions and evaluated on all forward pairs, reducing MSE by 6.09% versus RNN overall and 7.36% on long-term transitions.The short-term MSE reduction versus RNN is 4.13%; the model also outperforms the listed Neural ODE alternatives.Source: Table 7 · p. 17, §C.3 · p. 17

C.4 MAIN TABLE WITH STANDARD DEVIATION

  • Standard deviations are reported for each metric in Table 8 for the main evaluation.Table 8 covers short-term, long-term, and all prediction pairs.Source: §C.4 · p. 17, Table 8 · p. 17

C.5 FULL INDEPENDENT VALIDATION RESULTS

DevelopmentODE is evaluated on an independent BCP cohort and in a relaxation of its population-aligned constraint. The external comparison favors DevelopmentODE on MSE and PCC, while the relaxation yields only small aggregate performance changes.

C.5 FULL INDEPENDENT VALIDATION RESULTS

  • On the independent BCP cohort, DevelopmentODE achieves the lowest MSE and highest PCC among the evaluated methods.The cohort comparison is presented as complementary evidence under an additional cohort setting.Source: §C.5 · p. 17

C.6 SENSITIVITY TO A LEARNABLE POPULATION-ALIGNED COMPONENT

  • The sensitivity variant replaces the fixed complementary-space operation with a learned coefficient that can retain population-aligned state information and velocity.The coefficient is initialized at zero; the compared variants otherwise use identical splits, representations, training settings, checkpoints, and evaluation pairs.Source: §C.6 · p. 18
  • The learned α is 0.153 ± 0.088 for all-forward pairs and 0.198 ± 0.025 for adjacent pairs, while mean MSE decreases by approximately 0.043% and 0.078%, respectively.Changes in other metrics are similarly small; the authors retain the simpler complementary-space formulation with essentially the same predictive performance.Source: §C.6 · p. 18, Table 10 · p. 18

D LIMITATIONS

The discussion identifies limits in evaluation scope and model flexibility, and says the learned components' causal or mechanistic correspondence remains for future work.

D LIMITATIONS

  • Evaluation centers on resting-state FC during early childhood; the primary cohort covers the first decade, while BCP covers a shorter interval.The BCP experiment provides complementary evidence of cross-cohort robustness within early development.Source: D LIMITATIONS · p. 19
  • To accommodate limited longitudinal supervision, the model uses fold-specific PCA and a small number of ordered developmental stages with fixed transition centers.The authors note that jointly learning the representation and stage organization would require greater flexibility with relatively limited supervision.Source: D LIMITATIONS · p. 19
  • Learned components show systematic developmental patterns, but their causal or mechanistic correspondence to neurodevelopmental processes remains unestablished.The authors identify establishing that correspondence as a direction for future work.Source: D LIMITATIONS · p. 19

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