Journal Article

FZJ-2026-05276

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Durrmeyer deep fractional graph networks: A variable-order fractional framework for adaptive memory depth in spatiotemporal brain dynamics

Kadak, U. (Corresponding author)FZJ*

2026
Elsevier Science Amsterdam [u.a.]

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Please use a persistent id in citations: doi:10.1016/j.chaos.2026.118897  doi:10.34734/FZJ-2026-05276

Abstract: Modeling spatiotemporal dynamics in networked systems with heterogeneous memory effects remains a fundamental challenge in both machine learning and neuroscience. Classical Neural Ordinary Differential Equations assume memoryless Markovian dynamics, while fixed-order Neural Fractional Differential Equations enforce a single global memory timescale across all system components—neither framework can capture the spatially varying temporal integration windows observed in complex brain networks during naturalistic cognition. We introduce Durrmeyer deep variable-order fractional graph networks (DDFGN), a principled framework that extends Neural FDEs by incorporating three critical innovations: (i) symmetrized Durrmeyer-type deep neural operators with provable uniform convergence guarantees, (ii) variable-order fractional derivatives where memory depth αv(t)∈(0,1) adapts across both spatial locations and time as a learned function of local graph-weighted dynamics, and (iii) explicit graph coupling that captures state-dependent asymmetric information flow through continuous-time message passing. We establish rigorous theoretical foundations including local existence and uniqueness of solutions, Lipschitz continuity of the graph-weighted operator, and pointwise/uniform convergence results. Numerical experiments confirm systematic operator convergence with empirical rate ≈n−2.1 and demonstrate that variable-order adaptation enables functional role separation in multi-node networks, where fast-reset drivers, slow-memory integrators, and intermediate relays emerge from optimized dynamics. Application to naturalistic fMRI data (Forrest Gump dataset, N=19 participants) validates DDGNN’s neurobiological relevance across multiple organizational scales. Parameters are estimated via a two-stage optimization procedure with held-out evaluation, achieving 92.0%±2.9% reduction in one-step-ahead prediction error (t(18)=136.1, p=1.4×10−28) and significantly outperforming a fixed-order fractional baseline in which αv(t)≡ᾱ (t(18)=14.1, p=3.7×10−11; Wilcoxon p<10−4). A phase-randomized null model confirms that the optimization exploits genuine temporal structure rather than spectral artifacts (p=0.001). At the regional level, single-subject analysis using optimized parameters reveals a sensory-to-association memory hierarchy: the Visual network operates at higher fractional order (ᾱ=0.456, shorter memory) relative to the Default Mode network (ᾱ=0.437, longer memory), consistent with the known gradient of cortical temporal receptive windows. This ordering is independently validated by negative correlations between optimized αv and BOLD signal properties computed without reference to the operator output (lag-1 autocorrelation r=−0.030; spectral exponent r=−0.243). Group-level analysis reveals that cross-subject averaging attenuates the inter-network α range, consistent with the known between-subject variability of fMRI-derived functional hierarchies; a sensory–association contrast nonetheless approaches significance (t(18)=−1.75, p=0.097, Cohen d=−0.40). Inter-network synchrony analysis further reveals that Default and Control networks maintain strongly coupled memory trajectories (Pearson r=0.85). Critically, graph neural network analysis establishes that topological centrality and memory depth constitute orthogonal functional dimensions: among eight identified hub nodes (degree centrality ≥0.625), mean αv ranges from 0.459 to 0.588 – a 28% difference corresponding to approximately 2.3-fold variation in effective memory horizon – demonstrating that network hubs are functionally heterogeneous in their temporal processing strategies. Classical fixed-order fractional models would enforce a single compromise value (αglobal≈0.45) that accurately represents no individual region, thereby masking this functionally meaningful diversity. DDFGN thus provides a complete graph-theoretic framework for learning spatially heterogeneous, temporally adaptive memory architectures in continuous-time networked systems, with implications for understanding how distributed neural circuits coordinate region-specific integration timescales during naturalistic, ecologically valid cognition.

Note: Highlights•Durrmeyer Deep Graph Neural Networks (DDGNN) enable adaptive memory modeling.•Provable uniform convergence and local existence for variable-order graph dynamics.•Numerical experiments confirm an empirical operator convergence rate of n−2.5.•fMRI analysis reveals a sensory-to-association hierarchy of memory timescales.•Topological centrality and memory depth are orthogonal in brain network hubs.

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Contributing Institute(s):

1. Gehirn & Verhalten (INM-7)

Research Program(s):

1. 5251 - Multilevel Brain Organization and Variability (POF4-525) (POF4-525)

Appears in the scientific report 2026

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Database coverage:
; ; ; Clarivate Analytics Master Journal List ; Current Contents - Physical, Chemical and Earth Sciences ; Ebsco Academic Search ; Essential Science Indicators ; IF >= 5 ; JCR ; Nationallizenz ; SCOPUS ; Science Citation Index Expanded ; Web of Science Core Collection

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