GeoDynamics: A Geometric State‑Space Neural Network for Understanding Brain Dynamics on Riemannian Manifolds
Authors: Tingting Dan
Jiaqi Ding
Guorong Wu
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Neural Network
State Space Model
Brain Dynamics Analysis
Riemannian Manifold
Primary area: neuroscience_and_cognitive_science

Abstract: State‑space models (SSMs) have become a cornerstone for unraveling brain dynamics, capturing how latent neural states evolve over time and give rise to observed signals. By combining deep learning’s flexibility with SSMs’ principled dynamical structure, recent studies have achieved powerful fits to functional neuroimaging data. However, most approaches still view the brain as a set of loosely connected regions or impose oversimplified network priors, falling short of a truly holistic, self‐organized dynamical system perspective. Brain functional connectivity (FC) at each time point naturally forms a symmetric positive definite (SPD) matrix, which lives on a curved Riemannian manifold rather than in Euclidean space. Capturing the trajectories of these SPD matrices is key to understanding how coordinated networks support cognition and behavior. To this end, we introduce *GeoDynamics*, a geometric state space neural network that tracks latent brain state trajectories directly on the high‑dimensional SPD manifold. *GeoDynamics* embeds each connectivity matrix into a manifold‑aware recurrent framework, learning smooth, geometry‑respecting transitions that reveal task‐driven state changes and early markers of Alzheimer’s, Parkinson’s, and autism. Beyond neuroscience, we validate *GeoDynamics* on human action recognition benchmarks (UTKinect, Florence, HDM05), demonstrating its scalability and robustness in modeling complex spatiotemporal dynamics across diverse domains.

GeoDynamics: A Geometric State-Space Neural Network for Understanding Brain Dynamics on Riemannian Manifolds
Tingting Dan
Jiaqi Ding
Guorong Wu*
Departments of Psychiatry and Computer Science
University of North Carolina at Chapel Hill
Chapel Hill, NC 27599
(Tingting_ gww.gggaaannnnggaaaaaa.ae Cmed. unc. edu; iaaiddcss unc unc.edu
Abstract
State-space models (SSMs) have become a cornerstone for unraveling brain dynam- ics, capturing how latent neural states evolve over time and give rise to observed signals. By combining deep learning's 's flexibility with SSMs' principled dynamical structure, recent studies have achieved powerful fits functional neuroimaging data. However, most approaches still view the brain as a set of loosely connected regions or impose oversimplified network priors, falling short of a truly holistic, self-organized dynamical system perspective. Brain functional connectivity (FC) at each time point naturally forms symmetric positive definite (SPD) matrix, which lives on curved Riemannian manifold rather than in Euclidean space. Capturing the trajectories these SPD matrices is key to understanding how co- ordinated networks support cognition and behavior. To this end, introduce GeoDynamics, a geometric state space neural network that tracks latent brain state trajectories directly the high-diig dith-emieniionll SPD manifold. GeoDynamics embeds each connectivity matrix into a manifold-ate aware recurrent framework. learning smooth, geometry-respecting transitions that reveal task-driven state changes and early markers of Alzheimer's ss Parkinson's ss and autism. Beyond neuroscience, we validate GeoDynamics human action recognition benchmarks (UTKinnet, Flo- rence, HDM05), demonstrating its scalability and robustness in modeling complex spatiotemporal dynamics across diverse domains.
Introduction
The human brain is a complex, dynamic system with distinct structural regions specialized for specific functions, which are locally segregated yet interconnected to process diverse information [36] Over recent decades, understanding the brain's functional mechanisms has been a central focus in neuroscience. Functional magnetic resonance imaging (fMRI) widely used non-invasive technique, measures blood oxygen level-dependent (BOLD) signals over time, which are linked to neural activity. While initial research emphasized BOLD signals, the focus has shifted toward functional connectivity (FC), which captures co-activations across brain regions [3). Studies using Pearson's correlation quantify FC have shown that functional brain networks
static but exhibit dynamic topology changes, even in task-free states 4. These dynamic changes have been associated with brain disorders, offering insights into underlying neurobiological processes [8]
Efforts to model brain dynamics have focused on two main approaches: (1) analyzing temporal fluctuations in BOLD signals and (2) capturing topology changes in evolving FC matrices. While
"ooresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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BOLD signals track neural activity, they often struggle with noise and intrinsic fluctuations. Neural mass models, for instance, describe brain dynamics using non-linear equations but often ignore spatial dependencies 633
FC matrices, on the other hand, reveal network-level interactions and, when extended to dynamic FC (dFC), track connectivity evolution over time using methods like sliding windows 42 Recent advances, such as geometric-attention neural network proposed in [15], relate FC topology changes to brain activity. However, sliding window techniques remain sensitive to window size, which can hinder the detection of subtle brain state changes.
The widespread success of recurrent neural networks (RNNs, Fig. I (a)) [59, including long short- term memory (LSTM) [31] and gated recurrent units (GRU) [1]. in sequential modeling tasks such as natural language processing (NLP), has inspired numerous efforts to apply these architectures for Recently, state space models (SSMs) (as shown in Fig | (b. black solid box)) (2 26 have emerged as powerful tool for capturing a system's 's behavior using hidden variables, or "states", marked as 8t (i.e., s(()), which effectively model temporal dependencies in sequential data with well-established theoretical properties. These models have gained significant attraction in fields like computer vision (CV) and natural language processing (NLP) due to their ability to represent complex temporal patterns. A more inclusive literature survey can be found in the Appendix A.I
characterizing brain dynamics 47 48
(a)
X2
X
Downstream
Relevant work of SSM on brain func- tional studies. While SSMs have achieved success in CV and NLP ap- plications, their use in brain functional studies has primarily focused on event- related fMRI data [22 (35, limiting their applicability to resting-state IMRI (rs- (MRI). To address this, [64] combined an auto-encoder for learning BOLD sig- nal relationships with a hidden Markov model (HMM) for state transitions, but the separate training of these compo- nents reduced efficiency and neglected brain network spatial structures. Sim- ilarly, (66) proposed a linear SSM us- ing variational Bayesian methods to in- fer effective connectivities from EEG and [MRI data. However, this approach struggles to capture the complex dynam- ies between evolving functional connec- tivity and underlying cognitive or behav- ioral outcomes.
lasks
MLP
S0
$2
S's
Sx
X1
X2
X3
X
A
50
S
D
Downstrcam
V/3
tasks
Figure I: The architecture of RNNs (a) typically relies on multi-layer perceptron (MLP) to project the system state space into the output space, where various downstream tasks are then performed. These models operate entirely within Euclidean space. In contrast, vanilla SSMs (b, black solid box) incorporate two ordinary differential equations (ODEs), the state equation (upper) and observation equa- tion (lower), which can directly perform downstream tasks through the inferred observed output, also within Euclidean space, focusing primarily temporal information. Our geometric deep model of SSM (b, purple dashed box) ex- tends this approach by capturing both temporal and spatial information, operating on a manifold space.
Our work. The dynamic nature of complex system cannot be understood by thinking of the system as comprised of independent elements. Rather, approach is needed to utilize knowledge about the complex interactions within a system to understand the behavior of the system overall. In light of this, modeling the fluctuation of functional connectivities on the Riemannian manifold provides holistic view of understanding how brain function emerges in cognition and behavior. In this paper, we integrate the power of geometric deep learning on Riemannian manifold and the mathematical insight of SSM to uncover the interplay between evolving brain states and observed neural activities. First, our method structural in that we propose to learn intrinsic FC feature representations on the Riemannian manifold of symmetric positive-definie (SPD) matrices [17]. which allows us to take the whole-braid wiring patterns into account by considering each FC matrix as manifold instance. Second, our method is behavioral in that we leverage the SSM to model temporal dynamics. As shown in Fig. (b), SSMs operate through two core ODEs, the state equation and the observation equation, which describe the relationship between the input ztt) (short for 22) of the dynamic system and the system output y(t) (short for 31) at given time f, mediated by a latent state s(I) (short for st). Taken together, our contributions have threefold. (1) We present novel geometric deep model by integrating state space model and manifold learning. By incorporating Riemannian geometry, our deep model provides an in-depth insight into system dynamics and state transitions, enhancing the model's ability to capture both temporal and spatial complexities in a data-driven manner. (2) We replace

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M and use the the Euclidean algebra of conventional SSMs with Riemannian geometric algebra (accompanied by theoretical analysis) to effectively capture the spatio-tempore information, which allows us to better handle irregular data structures and harness the geometric properties of SPD matrices. (3) We have significantly improved the computational efficiency compared to manifold-based deep models by using modern machine techniques such as geometric deep model (Sec. 3.1 and SPDD-preeerving attention mechanism (Sec. 3.2).
We apply our proposed method (GeoDynamics) to two types of system dynamics: brain dynamics and action recognition 28 While brain dynamics is our primary focus, action recognition serves as a validation task to assess the method's generalization performance across different domains. In the application of understanding brain dynamics, upon which we refer to as GeoDynamics, we have evaluated model performance on the large-scale human brain connectome (HBC) databases, including one Human Connectome Project 79 and four disease-related resting-state fMRI data: (1) Alzheimer's s Disease Neuroimaging Initiative (ADNI) [52), (2) Open Access Series of Imaging Studies (OASIS) [46) (3) Parkinson's Progression Markers Initiative (PPMI) [50], and (4) the Autism Brain Imaging Data Exchange (ABIDE). For action recognition,
use three classic human action
recognition (HAR) datasets including the Florence 3D Actions dataset [6]], the HDM05 database [53] and the UTKinect-Action33 (UTK) dataset [75]. Our GeoDynamics has achieved significant results across both brain dynamics and action recognition tasks, demonstrating its effectiveness and practicality. These upplications both neuroscience and computer vision highlight the scalability and robustness GeoDynamics in understanding complex ppatio-tempora dynamics across diverse
systems.
2 Preliminary
2.1 Classical State Space Model
The continuous-time state space model is defined as
s'(1) - A st) + Bx(() and
+ D
(I)
where s(t) E R. is the system state, x(t) E & is the control input, and y(t) M the output (we follow the single-ing inggee-npp/siigl-ouuuuuu convention unless otherwise stated). The matrices A NNNN Nx x CER MxN CERMAN and D (often D=0) encode transition and observation, as sketched in Fig. [b). While Eq. ① assumes Euclidean variables, many neuroscientific signals, notably functional connectivity matrices, are naturally manifold-valued (SPD), motivating a geometry-aware extension.
2.2 Riemannian Geometry on the SPD Manifold
Let M -SPD(N) denote the manifold of N N SPD matrices. We write X,Y Y Stein metric for computational efficiency:
X + Y
d(x,Y) Y)
log det
log det(
2
This distance avoids repeated eigendeccomposiiions while preserving key geometric structure.
Group action and isometric "translation". Let = GL(N) be the general linear group acting
on M via y.X : gXg" Under the Stein metric, such actions are isometric, restricting to the orthogonal subgroup ((N) C G preserves distances, d(g.X,, y-Y d(X, Y) for all X, Y e M (proof in Appendix A.2 We call this an isometric translation on M and denote by
(x(9) == =
g ONN.
(3)
Weighted Fréchet mean (wFM). Given (X C. CM and nonnegative weights [w, J with
Σn W. 1, the wFM
((X)).(u))) arg min FEM n=1
Σ
wn (X F).
(4)
Assuming } lies in a suitable geodesic ball, existencc/uniqueness holds (see Ref. (]).

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SPD convolution on M.
For matrix-valle valued features, the SPD convolution at layer & is
x
Hu, X (k--1) stu
(5)
with kernel H E TR To preserve SPD structure, we parameterize
H = = +el,
c 0+
(6)
so the learned parameter is Z while H is guaranteed SPD, consequently X remains SPD (proof in Appendix p.30.
Method
Geometric formulation of state space dynamics. We extend the classical state space formulation to the Riemannian manifold of SPD matrices. In this setting, the system impul ( system state S()), and system oulput (l) no longer vectors in RR but elements of the SPD manifold M SPD(N). equipped with the Stein metric. This design enables the system dynamics to evolve in geometrically consistent manner that respects the intrinsic structure of functional connectivity representations. Formally, at time t, we denote: X() E M, S(t) E M, Y() E M. The temporal evolution of the system is driven by a of learnable operators defined on M, replacing the Euclidean linear operators of conventional SSMs.
3.1 State Update and Observation on the SPD Manifold
Let r denote the length of the temporal receptive field (i.e., sliding window). We model the evolution of g(k) and (*) discrete time step k {1, K} through two components: (1) Temporal aggregation of past states and inputs using the weighted Fréchet mean (wFM) to ensure intrinsic averaging along (2) Group action translation on the manifold to represent state transitions and input effects in an isometry preserving manner.
geodesics rather than Euclidean linear combinations.
isometry-preserving
The state update and observation equations defined
s(k) = (F(()) 1 (11 F(XX)) 1/ (B y
(B1))=x=7))
(C F(X
where F denotes the wFM operator on M (Eq. 4h, and T denotes the translation on the manifold induced by orthogonal group actions (Eq. 3). The learnable parameters {Ay. Bj, Cj2 Dý} control how the history of states and inputs contributes to the current state.
Weighted Fréchet mean aggregation. Given set of SPD matrices { and nonnegative weights (w,} with E3 Wy 1, the weighted Fréchet mean defined as:
((Zz),(w))) arg min Σ wy (Z,, PEM j=
(8)
where d(.,-) is the Stein geodesic distance on M. Unlike Euclidean averaging, this operation produces an intrinsic barycenter on the manifold, ensuring that the aggregated representation remains SPD.
Group action translation. To propagate states forward in time, we introduce a translation operator T:
T(U.V) -V)
where ((V) is an orthogonal transformation inferred from V. Since g(V) O(N), this operation is an isometry under the Stein metric and guarantees that the output remains SPD. The translation acts multiplicative update the manifold, analogous to additive transitions in Euclidean SSM.

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Discretized dynamics and control. We parameterize the temporal dynamics by discretizing the continuous-time operator using a matrix exponential scheme. Specifically, for each step A:
Ā == exp(AA),
== (exp(AAA - A)AB.
(10)
This formulation ensures stable temporal integration while preserving geometric structure, allowing smooth evolution of the system's hidden states on M. The role of A is to propagate the previous hidden state, while B injects control-dependen information from the input. Their influence is integrated within the wFM in Eq. (7) to produce a manifold-consisennt transition.
Task readout via logarithmic mapping. The output y(k) is SPD matrix, which cannot be directly passed to standard classifier. We therefore map it to the tangent space at the identity using the logarithmic map:
(k) log ()og(((())) log(A))*,
(11)
where { = oAΦ T is the cigendecomposition. This yields a symmetrie matrix in Euclidean space that preserves local manifold geometry.
For classification task and brain state decoding, we employ a softmax layer:
exp(w, vec(i(
P(Q q ((K)
(12)
Σg expl W. vee(((())
where vec(-) denotes vectorization of the symmetric matrix and Q is the number of task classes or clinical outcomes. Model training the cross-entropy loss:
L
Oig log P(Q = 01 ())
(13)
where Oig is the ground truth label. E denotes the number of subjects/samples.
Global convolutional reformulation. Although the above formulation describes the step-wise state evolution, it can be equivalently expressed as convolution operation over time (25]. Expanding the recurrence gives:
-CB+ + D. CAB, ..... CA.B...... CA'B...,CA'B.....
y = K.
(14)
To ensure SPD structure, we constrain each kernel K!t to be SPD by parameterizing
(15)
where >> 0 is stability constant. The nonlinearity is introduced via the exponential map exp(-) on the Riemannian algebra, which guarantees that the output remains SPD after each convolutional block [15]. Frobenius normalization is applied to control eigenvalue growth and maintain numerical stability during training.
3.2 SPD-Preserving Attention
To enhance the model's ability to identify informative disease-related spatio-temporal patterns, we introduce SPDD-preserving attention (SPA) module embedded within the manifold convolutional backbone. Unlike conventional attention mechanisms that are defined in Euclidean space, SPA operates directly on the SPD manifold and designed to preserve the positive-defintt structure of the signal throughout the attention process. Given the convolutional response X K. M, we define attention weights as
5(-) = exp ([X * XC])
X := diag(p, X),
(16)
max(expp[X XC]))
where diag(p, X) pads the borders of the SPD matrix with zeros size p - and introduces a small positive diagonal component to ensure strict positive definiteness (see Appendix A.5). The exponential operation exp(-) plays a role analogous to the sigmoid function, smoothly mapping the weighted response into the interval [0. while retaining manifold consistency. The resulting d(-)
5

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acts as a soft mask that udaptively modulates local connectivity patterns. The wteention-weighted representation then computed
= K.),
(17)
where @ denotes elementwise multiplication. Because 8()) is strietly positive and bounded, and the base convolution is SPD-constrained, X remains SPD after modulation. This preserves both the local geometry (SPD structure at each time point) and global temporal consistency.
Disease-relevant pattern localization. Many neurological and psychiatric disorders, including neurodegenerative diseases such as Alzheimer's, 's, are characterized by localized but propagating
disruptions in functional connectivity. These disruptions often manifest as: (1) spatially specific abnormal hubs or subnetworks (e.g., default mode, limbic regions), (2) temporal shifts in network activation or coupling, (3) altered covariance structure in FC trajectories reflecting pathological propagation. conventional SSM or GNN models, such disease-related signals may be smoothed out by global averaging fixed-weight message passing. In contrast, the SPA mechanism learns spatially and temporally adaptive weighting functions directly on the SPD manifold, enabling: Enhanced sensitivity to local network deviations: the multiplicative modulation amplifies abnormal signal components (c.g., hyper- or cypo-conncctivity in key subnetworks) while attenuating background fluctuations. Preservation of intrinsic geometry: since both the attention and the convolution are SPD- preserving, pathological patterns expressed as topological changes (e.g., altered covariance structure) remain valid on the manifold, avoiding distortions induced by Euclidean projections. Temporal adaptivity: 8()) is computed at each time step, enabling the network to detect evolving abnormal FC trajectories associated with disease progression. Importantly, the learned attention weights 5(-) anatomically and temporally interpretable, i.e high attention scores highlight regions or subnetworks where deviations from healthy connectivity trajectories are most pronounced. This makes SPA not only an effective inductive bias for improving predictive performance, but also tool for identifying candidate discase biomarkers.
3.3 Summary of the GeoDynamics Framework
The proposed GeoDynamics integrates: (I) Geometric state evolution: Input, system states, and outputs are represented as SPD matrices evolving on M, ensuring geometric consistency. (2) Manifold-aware operators: Temporal aggregation via weighted Fréchet means and state transitions via orthogonal group actions guarantee that system trajectories remain on the manifold. (3) Stable discretization: Matrix exponential schemes provide stable integration and controllable temporal fonolutionaloormaaation and scalable implementation. (5) Geometric attention: SPA enhances interpretability and sensitivity to task-relevant regions.
dynamics. (4) Efficient convolutional formulation: A global kernel expansion enables parallel training
This formulation provides a principled and computationally efficient way to model non- non-Euclidean temporal dynamics, making it particularly well suited for functional connectivity sequences and other manifold-valus time series.
4 Experiments
4.1
Experimental Setup
Dataset. We apply our method to two types of datasets including human brain connectome (HBC) and human action recognition (HAR), more detailed data information is shown in Tf Appendix A.6 For HBC dataset. We select one dataset of healthy young adults and four disease-related human brain datasets for evaluation: the HCP Working Memory (HCP-WM) [79]. ADNI [52), OASIS [46), PPMI [50], and ABIDE 20]. We selected a total of 1,081 subjects from the HCP-W CCP-WMM dataset. The working memory task included eight task events. Brain activity was parcellated into 360 regions based on the multi-modal parcellation from (4.. For the OASIS (924 subjects) and ABIDE (1,025 subjects) datasets, which are binary-class, datasets, one class represents a disease group and the other represents healthy controls. the ADNI dataset, subjects are categorized based on clinical outcomes into four distinct cognitive status groups. The PPMI dataset also consists of four classes. We employ Automated Anatomical Labeling (AAL) atlas 67 (116 brain regions) on ADNI, PPMI, ABIDE datascts, while Destrieux atlas [18] (160 brain regions) are used in OASIS to verify the scalability

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of the models. For HAR dataset. We validate the performance of the proposed GeoDynanics on three uidely-used HAR benchmarks: the Florence 3D Actions dataset [6]]. the HDM05 database [53) and the UTTKinet-AAtiionDD (UTK) dataset (75] The Florence 3D Actions dataset consists of 9 activities performed by 10 subjects, with each activity repeated 2 to times, resulting 215 samples. The actions are captured by the motion of 15 skeletal joints. For the HDM05 dataset, we follow the protocol from [71]. focusing on 14 action classes. This dataset contains 686 samples, each represented by 31 skeletal joints. Lastly, the classes. Each action was performed twice by 10 subjects, yielding a total of 199 samples.
total of
dataset comprises 10 action
SPD matrices construction. For HBC dataser. Each fMRI scan has been processed into N mean time courses of BOLD signals, each with T time points (where N represents the number of brain parcellations), we employ a sliding window technique to capture functional brain dynamics. Specifically, we construct a N x N correlation matrix at each time point (t = 1=....., T) based on the BOLD signal within the sliding window, centered at time f. This results in a sequence of FC matrices encoding the functional dynamies for each scan, represented as X (X(1) | = 1, ...... T} E R RTNN*N in Fig. (a). For HAR dataset. HAR datasets exhibit variability due to differences in action duration, complexity, the number of action classes, and the technology used for data capture. Therefore, we first apply a preprocessing step following [56] to obtain the SPD matrices. This step involves fixing the root joint at the hip center (red dashed circle in Fig. (b)) and calculating the relative 3D positional differences for all other N - 1 joints. For each timestamp t 1,.... T, obtain a x (N -)-dimensional column vector representing the relative displacements of the joints. Then, compute covariance matrices using the method proposed in [56] to yield the SPD matrices. After that, we apply a sliding window technique to capture the dynamics over time, resulting in a sequence of SPD matrices X' = (t) ,,...,T) E X (1)x)))()3333)3)0000 1)) as illustrated in Fig. (b).
S
(4)
(00 1=
FC Matriees
S(0)
X = (X()) t= L....T)
AIR
Spontancous brain
5(15)
M(S)
Manifold space
sPD Matrico
Learning the system dynamies
The construction of SPD matrices for HBC (a) and HAR (b) datasets. Learning the system dynamics on manifold space illustrated in (c).
Figure 2:
Comparison methods and evaluation metrics. For HAR dataset. There are some popular methods for HAR, such as multi-part bag-of-pose (MBP) 61). Lie group [69, shape analysis on manifold (SAM) 19). elastic function coding (BFC) 2 multi-instance multitask learning (MML) (78. Tensor Representation (TR) [45], LieNeu334) SPGK (72), ST-NBNN (73), GR-GCN (23 and DMT-Net and
F-DMT-Net 80]. We also include Bi-long short-term memory (Bi-LSTM) [5] and part-aware LSTM
(P-LSTM) [62] For HBC dataset. We stratify the comparison methods for HBC into two groups:
spatial and sequential models. Spatial models focus on capturing brain dynamics. Traditional GNNs like GCN 44 and GIN 777 are included for their ability to handle structured data. Subgraph-
41 focus on identifying local patterns, while expressive GNNs like GSN 7 and GNN-AK 8T enhance subgraph encoding for better expressivity. SPDNet [66. a manifold-based model, is chosen for managing high-dimensional data. Plus, an MLP serves as simple, generic baseline. Sequential models target temporal dynamics in BOLD signals. ID-CNN captures temporal patterns, while RNN (59] and L.TM K handle sequential dependencies. MIP- Mixer [65] integrates both temporal and spatial information, and Transformer (TF) (68] captures global dependencies through attention. Mamba (25). vanilla SSM. is included for its ability to model
GNNs like Moment-GNN
system dynamics over time. Two dynamic-FC methods, STAGIN 43), NeuroGraph [60]. Three brain
network analysis methods BrainGNN 49 BNT 40). and ContrastPool [76]. More details are shown in Appendix A.7
Evaluation metrics. For Florence and UTKineett datasets, we adopt the standard deave-one-actor-nn validation protocol as outlined 233 This method generates Q classification accuracy values, which are averaged to produce the final accuracy score. For HDM05 dataset, we follow the experimental setup from [32, conducting 10 random evaluations. each evaluation, half of the samples from each class randomly selected for training, with the remaining half used for testing. In all HBC
7

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experiments, we utilize 10-fold cross cross- validation scheme, reporting accuracy (Acc), precision (Pre), and F1 score to provide thorough evaluation of model performance across various datasets.
4.2 Results on Human Brain Connectivity Datasets
We investigate brain dynamics across both healthy and disease-related cohorts using task-based and resting-state fMRI data.
Task-based IMRI. We first evaluate the task-based working- memory dataset (HCP-WM) using sixteen representative methods. The results in Fig. 3] (green colors) show that sequential models achieve markedly higher accuracy than spatial models (pair-wise t-test, < 10 with performance gains up to 30%. Our proposed GeoDynamics achieves the best overall performance.
Interpretation. From a machine learning perspective, sequential models may better capture the temporal dependencies in BOLD signals, which inherently dynamic during task execution. In contrast, spatial models rely on static functional connectivity patterns that are less sensitive to rapid task-induced variations. Biologically, task-based fMRI paradigms elicit specific neural responses related to cognitive processes such as attention and memory, leading to pronounced fluctuations in brain activity that align naturally with sequential modeling.
HEP
ADNI
0ASIS
ABIDE $6.26
80.60
Acuneey
89.33
38.16
Prectsion
76/07
0.05
B5.47
68.63
87.34
Fi-xcore
10.CCM
1000
Mioeer
Mambe
C114
0.5H
OM
SPOWet
STAGIN NeureGreph GeoDynatto
Figure 3: Evaluation performance for different methods across HBC datasets. The best performance highlighted in bold, while the second-best is underlined.
Neurodegenerative diseases. Next, focus early diagnosis of neurodogenerative diseases (ND), including Alzheimer's 's Disease (AD) (yellow and purple colors) and Parkinson's 's Disease (PD) (pink colors), using resting-state fMRI. We assess the classification between cognitively normal (CN) and neurodegenerative (ND) groups. Spatial models show slightly better average performance than sequential models 0.37), although the difference is statistically significant. Notably, GeoDynamics substantially outperforms all baselines, achieving significant improvement over the second-best method (p < 0.05).
Interpretation. Unlike task-based fMRI, resting-state fMRI measures spontaneous neural activity, reflecting intrinsic connectivity rather than stimulus-driven dynamics. This may explain why spatial models perform competitively in ND classification tasks. From a biological standpoint, neurodegener- ation primarily manifests gradual network disconnection rather than transient dysfunction [55] 12). Functional disruption often precedes measurable cognitive decline by many years [70]. The results in Fig. thus highlight the clinical promise of deep learning models for detecting early ND-related alterations in large-scale brain networks.
Neuropsychiatric disorders. We further analyze Autism Spectrum Disorder (ASD) using the ABIDE dataset. As shown Fig. (blue color), spatial models slightly outperform sequential ones, with SPDNet consistently ranking second only to our GeoDynamics. Both SPDNet and GeoDynamics share two key design principles: (1) manifold-based representation learning that preserves the intrinsic geometry of functional connectivity matrices (Fig. 2 (c)). and (2) ppatio-tempora modeling that
8

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captures the evolution of connectivity patterns over time. The consistent advantage of these two approaches suggests that robust spatio-temporal representation, grounded in Riemannian geometry, is crucial for reliable diagnosis of neuropsychiatric conditions.
HCP-WM
Central Default mode
Somatosensory
Semtorimotor
Temporal lobe
Frontal lobe
Visual
Cerebellum
Figure 4: Critical connections from SPDD-preservigg attention map on HBC datasets.
Interpretation. Neuropsychiatric disorders such as Autism and Schizophrenia are characterized by atypical neural connectivity and abnormal variability in BOLD dynamics [58] 51 38 These alterations affect both spatial topology and temporal synchronization, suggesting that effective diagnostic models must jointly capture both aspects. In contrast, neurodegenerative discases primarily disrupt large-scale connectivity due to neuronal loss, making static spatial features more informative. Integrating such disease-speciff pathophysiological insights is therefore essential for model design and interpretation.
Interpretable attention patterns. Finally, we visualize the attention weights 5(-) from the SPA module (Sec. 3.2) to identify critical brain regions and connections across datasets. The top-20 mapped onto the brain (Fig. 4 In HCP-WM, the dominant connections
strongest connections reside within the default mode network (DMN, blue) and central executive network (orange), both essential for working- memory tasks. In AD (OASIS and ADNI), the most affected regions include the DMN and somatosensory cortex (green), suggesting that AD may influence sensory processing and bodily awareness in addition to memory loss. In PD, key connections appear in the sensorimotor (red), frontal lobe (purple), DMN, and cerebellum (black), indicating that PD involves both motor and cognitive-emotiona dysfunctions. In Autism, prominent connections emerge in the temporal (brown) and visual cortices (yellow), consistent with known deficits in language, perception, and social interaction.
Interpretation. Despite disease heterogeneity, certain shared patterns emerge across neurodegener- ative conditions, particularly AD and PD. The proposed attention mechanism not only highlights disease-specific alterations but also uncovers common pathways underlying different disorders. Such interpretability could provide valuable insights into shared pathophysiological mechanisms and guide sypothesis-driver neuroscience research.
4.3
Ablation Studies
Sliding window size (PPMI dataset): We examined the effect of varying the sliding window size on model performance. As shown in Table [ GeoDynamics demonstrates relative robustness to window size, achieving optimal performance with moderate values (lypically 35-45). This stability likely arises from the SSM module's ability capture dynamic temporal characteristics, reducing sensitivity to specific window lengths.
Table 1: Ablation studies in terms of sliding window size.
Window size
15 71.35 = 10.26
25
35 71.69
45
55
Acc
70.83 # 15.74
10.10
72,01 # 8.51
71.01 # 14.23
Pre
76.07 + 7.33
74.72
10.00
73.54 # 6.50
71.73
7.56
71.00
7.89
Fl
70.60 a: 9.73
71.71
7.29
70.72 + 3.90
68.56
6.74
67.67
7.81
Multi-class classification (ADNI dataset): We extended the binary classification task to multiple
clinical stages. As reported in Table 2 GeoDynamics consistently achieves higher accuracy, precision,

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and Fl-score across all classes compared with both spatial- and sequence based baselines. This demonstrates robust generalization more challenging multi-class settings.
sruuence-based
Table 2: Multi-class classification results on the ADNI dataset.
(%)
GCN
GIN
GSN
MGNN
GNN-AK
SPDNet
MLP
Acc
50.0016.51
5..60:..200
52.80:15.31
48.8015.31
52.40:1..56
52.40:1.220
46.40:7.42
Pre
14.08:+14222
13.55
13.23=10.33
10.58=#0.28
46.07=99.88
37.01+8.89
40.52:112005
F1
38.4919.73
41.76.17.65
48.21
38.7116.11
43.20.16.55
31.63.1,766
43.83_9.113
(%)
IDN
RNN
LSTM
Mixer
TF
Mamba
GeoDynamics
Ace
46.00+55:4
15.00+6.25
46.00=4744
48.40+4.18
52.00=66.93
47.20+66144
56.00=3.36
Pre
36.40.19.72
40.95.11..9
25.89
18.06
19..3
13.22
60.36.7.677
F1
39.21#7.71
39.24=7.14
31.8749.93
39.40:5.47
44.03=11.32
37.1943.53
50.83=5.73
Model complexity: We systematically varied the total number of trainable parameters by adjusting hidden-state dimensionality in Mamba. In the final column in T we compare GeoDynamics against the best-tuned Mamba configurations the same parameter budget. This demonstrates that our geometry-awart state-space formulation yields measurable gains in predictive performance, even when network capacity is held constant.
Table 3: Comparison between various Mamba configurations and the proposed GeoDynamics model on HCP-WM dataset (brain regions N = 360). For a fair comparison, the hidden dimension and network depth of Mamba are adjusted to match the parameter scale of GeoDynamics (highlighted by underline).
Mamba
GeoDynamics
Hidden dim
2048
1024
1024
1024
512
N
Network layer
5
Para (M)
132
33.71
27.05
14.07
13.93
14.60
Accuracy
97.22:1:063
97.06:0062
96.7620.86
95.92:20.50
96.17-0.11
98.29:10.26
Precision
97.27:10.62
97.09=0.60
96.80=0.84
95.93=0049
96.20=0.10
98.18:10.34
F1-score
97.22:10.63
97,06:0.62
96.7610.86
95.92=0.50
96. 18:111
98.16:10.35
group
4.4 Model Validation
GooDynaniss
MBP
Lie group
41 BrOup
To evaluate the generality of our framework, applied it to the HAR dataset using standard joint-coordinate bench- marks (Fig- 5). Despite the differences from neuroimaging data, GeoDynamics remains competitive, owing to its uni- fied treatment of spatio-temporal dynamics and manifold geometry. By embedding 3D joint positions on a Rieman- nian manifold and employing global convolution kernel, the model captures coordinated movements across distant joints while effectively suppressing noise. This demon- strates the broad upplicability of our geometric state-space approach beyond neuroimaging. Furthermore, our method captures higher-order correlations among 3D joint coordi- nates over time and models spatio-temporal co-occurrences through the tailored convolution kernel, enhancing robustness to noisy joints and improving overall action recognition accuracy.
LieNer
UTKinect HDMDS
0.4
P-LSTM SPONet
F-MM-Nee
S400
5904
DMT-Net
EFC
MML
ST. ST-NBINN
BW.STM
Figure 5: Results on HAR dataset.
More experimental details are provided in Appendix A.7 A.7
5 Conclusion
This work presents geometric deep model of SSM, GeoDynamics, for understanding behav- ior/cognition through deciphering brain dynamics. line with theoretical analysis, our method integrates the principles of geometric deep learning and efficient feature representation learning on non-Euclidean data, specifically designed for learning on sequential data with inherent topological connections. We have achieved promising experimental results on human connectome data as well as human action recognition, indicating great applicability real-worle data for neuroscience and computer vision.
10

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Acknowledgement
This work was supported by the National Institutes of Health (AG091653, AG068399, AG084375) and the Foundation of Hope. The views and conclusions contained in this document are those of the authors and should not interpreted as representing the official policies, either expressed or implied, of the NIH.
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A Appendix
A.1 Literature Survey
RNN and its variants on manifold to neuroimaging application. Recurrent neural networks (RNNs) have been reformulated as ordinary differential equations (ODEs) with continuous-timg system states, as highlighted by LTCNet [30]. These models serve as effective algorithms for modeling time series data and are widely utilized across medical, industrial. and business domains. For instance, [ has demonstrated its potential for brain state recognition and [29] achieves continuous modeling of dynamic brain signals using ODEs. Furthermore, the survey proposed by [54] provides a comprehensive overview of ODE applications in the field of medical imaging, showcasing their practicality and impact in this domain. Following this, several manifold-bas based RNN models have emerged. For instance. [T]] introduced a statistical recurrent model defined on the manifold of symmetric positive definite (SPD) matrices and evaluated its diagnostic potential for neuroimaging applications. This approach underscores the effectiveness of utilizing manifold- based techniques to enhance the performance of RNNs in complex medical contexts. The RNN model formulated on Riemannian manifolds is robustly supported by mathematical theory, as it utilizes covariance information to dynamically model time-series data [37]. This capability allows to capture richer and more subtle representations within a higher-dimeensioal latent space. Such an approach is particularly effective in modeling complex data structures, such as capturing the functional dynamics 33). where the relationships among data points are inherently geometric. By operating within the manifold framework, these models adeptly accommodate the intricacies of underlying data distributions, thereby enhancing both interpretability and predictive performance.
RNNs and their variants, while widely used for sequential modeling tasks, have notable limitations that affect their performance in complex, dynamic systems. One of the key challenges is that RNNs implicitly learn sequential patterns and temporal dependencies, without explicitly modeling the underlying dynamics. This implicit nature makes RNNs harder to interpret, often turning them into "black-box" models where the relationships between input variables and predicted outcomes can be obscured, limiting their utility in scenarios requiring high interpretability. Although advancements like LTCNet [30] have improved the interpretability of RNNs by framing them as an ODE, these models primarily focus on the dynamics of the system states and inputs (as shown in Fig. [ (a)). However, they failed to consider observation equations (but usually use MLP to fit the observations). which describe the relationship between system states and observed data. This formulation reduces their ability to fully model the observable aspects of system, resulting in an incomplete picture of the system's dynamics and limiting their explanatory power.
SSM to neuroimaging application. State Space Models (SSMs) explicitly model temporal dy- namics through latent variables governed by two key ODEs: the state equation, which captures the evolution of the system state over time, and the observation equation, which relates the latent state to observable data. This structured, ODE-based framework allows SSMs to offer a clearer understanding of how systems evolve and provides higher level of interpretability compared to RNNs. This makes SSMs particularly valuable in domains requiring an understanding of underlying system dynamics, such as medical diagnostics and time-series forecasting. In contrast to RNNs and their variants (e.g.. LSTMs, GRUs), which often operate as "black boxes," SSMs like Kalman Filters 39 have well-established theoretical properties. These properties typically include convergence and stability, providing a solid mathematical foundation that is difficult to guarantee with more complex RNN architectures. RNNs, especially deeper ones, suffer from issues like vanishing or exploding gradients, which affect training stability and interoperability. SSMs also naturally incorporate probabilistic structures, allowing them to effectively handle noisy or uncertain data, This is particularly advantageous in low Signal-to-Noiss Ratio (SNR) datasets, such as IMRI (74) and Electroencephalogramn (EEG) data [57]. where the ability to account for noise and uncertainty SSMs have been developed. For instance. [13] explores the modeling of time series observations in sptee forms defined on Stiefel and Grassmann manifolds. This approach utilizes Bayesian methods to estimate state matrices by calculating posterior modes, effectively integrating geometric constraints with probabilistic inference. However, while Bayesian methods excel in handling uncertainty, they often face limitations in scalability, inference speed, and flexibility compared to deep learning models, which offer more efficient and powerful representation capabilities for large-scale data.
is critical. In light of these performance advantages, only few manifold-
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In this context, the introduction of deep geometric SSMs aims to combine the representational power of deep neural networks with the interpretability and structured dynamics inherent traditional SSMs. By incorporating the geometric properties of manifold-based modeling, these models adeptly capture the intrinsic structure of the data, which is crucial for accurately representing complex relationships in nigh-dimensional datasets, such those found in brain imaging. This combination not only enhances interpretability but also allows for a more nuanced understanding of the underlying dynamics, ultimately improving the efficacy of the modeling process.
A.2 Geodesic Distances Isometry
We aim prove that the 'translation' operation defined by orthogonal transformations on the SPD manifold equipped with the Stein metric is an isometry. Specifically, we show:
d()x(g), Tv( d(X,Y), AX,Y E M,ge0. O.
(18)
Given the SPD manifold:
M (XEERN' X X X > 0},
(19)
translation operation for orthogonal matrices g E (where g 9 - 99 I) defined as:
Tx(g) 9XX"
(20)
The Stein metric is defined as:
X + Y
d(x,))= log det
log det seeedee(y (XY).
(21)
2
Orthogonal matrices preserve determinants:
det(gXg") det(X) (X), since det(g) +1.
(22)
We explicitly verify the invariance under orthogonal transformations:
d(Tx (9), Ty (g)) = d(gXgg )
gXgg - -gggg
log det
det ((yX
X Y
1
log det
9
T
log det ()
2
2
X + Y
log del
log del + (el (ogde)(XY) + log del(g) log det T)]-
2
Since det(g) = +1, we have log (g) 0. Thus,
X +
d(Tx(o),Tv(())
log det
log (X
(23)
2
2
This proves translation operation under orthogonal transformations is an isometry on the SPD manifold with the Stein metric.
A.3 SPD Convolution Operation
Proof. Since Il is SPD, can be decomposed follows:
H = ZT
(24)
where Z -= [[21, 22..... =o] is matrix of full rank. The convolutional result of an SPD representation matrix X NxN can then expressed as:
0= x =
= x (ZZ
(25)
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---

X (212)
+ ... + X (2020) )
(26)
X * * 21 + + X 29 2g:
(27)
where the transition from Eq. 26 Eq. 27
uses the property of separable convolution. Suppose
[,a m2...... ..... m.
for = 1, 1....... A. The convolution between X and can be written
as:
X P, X*2 =xx
(28)
where Ps
MM and
m41
722
M&N
0
Mi1
7,2
MAN
Ps.
0
M41
m22
MAN
(29)
0
m41
172,2
m&N
Thus, the following equations hold:
X x # = Ps 2,
(30)
and
0 == 7 = P + +....
XP
(31)
Since the rank of Pz, equals M -N the matrix P, XPP, also SPD. Therefore, for any E MM where q + 0, we have:
00 Σ q P. PPP
(32)
Hence, is an SPD matrix.
Furthermore, the k-th channel of X can be written as:
( Σ 5xx+H(* H(.))
(33)
where (0) denotes the I-th channel of the input descriptor. Since X and (k.0) are SPD matrices, and according to the above proof, X!) is also an SPD matrix. Therefore, (() is multi-channel SPD matrix.
A.4 SPD exp(.) Operation
Proof: Since is symmetric, we know that for any integer k, the powers X are also symmetric. The matrix exponential of X is defined by the following power series:
X
exp
(34)
k!
k=0
Each term in this series involves symmetric matrix X and the sum of symmetric matrices remains symmetric. Therefore, exp(X) is symmetric.
Since X is symmetric, it can be diagonalized as: X = QAQT, where Q is an orthogonal matrix (i.e., Q Q = 7) and A is diagonal matrix containing the eigenvalues ,2, An of X. Because X positive definite, all eigenvalues A are positive, i.c.. X > 0 for all i.
The matrix exponential exp(X) is then given by:
exp(X) Q @xppAAQT
(35)
where exp(A) ) is the diagonal matrix with entries exp(A1), exp(A)).ex((22)..... nential function satisfies exp().) for E R, each eigenvalue of exp( X) is positive. Thus, exp(XX has strictly positive eigenvalues, and since it is symmetric, it is also positive definite.
exp(A,). Since the expo-
25

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A.5 SPD Padding
[:]
Operation
Given a SPD matrices X E Symi and a small positive value Ps the assemble matrix y
0
diag(p, X)= =
is a SPD matrix.
X
Proof: First, Y is a symmetric, since Y
p
0 X
Y. Then, to show that Y is
X
positive definite, we need verify that for any non-zero vector
21
R N++ the quadratic
22
form Ty2 strictly positive.
We compute the quadratic form:
Yz
[2]
22
p 0
X
z 22
zîp 22 X22.
(36)
Since > 0, the term 27p 2 0, and it is strictly positive if 21 + O.
Furthermore, since X Sym is positive definite, meaning 2/ x2 0 for any non-zero
22 R
21 22
Thus, for any non-zero vector = we have:
TY= = 2ip + 22 X22 >
(37)
which proves Y E Sym is SPD matrix.
A.6 Dataset
Table 4: The summarization the HAR and HBC datasets.
Dataset
# of sequences
# of classes
mean of lengths
# of joints/ROIs
UTKinect
199
10
29
20
Florece 3D Actions
215
19
15
HDM05
686
14
248
HCP-WM
17,296
8
39
360
ADNI
250
177
116
OASIS
1,247
390
160
PPMI
209
4
198
116
ABIDE
1,025
200
116
For HAR dataset. We evaluate the performance of the proposed GeoDynamics on three benchmark HAR datasets: the Florence 3D Actions dataset [61], the HDM05 database (53), and the UTKinect- Action3D (UTK) dataset [75). The Florence 3D Actions dataset includes 9 activities (answer phone, bow, clap, drink, read watch, sit down, stand up, tie lace, wave), performed by 10 subjects, with each activity repeated 2 to times, resulting in a total of 215 samples. These actions are represented by the motion of 15 skeletal joints. For the HDM05 dataset, we follow the protocol outlined in [7I]. selecting 14 action classes (clap above head, deposit floor, elbow to knee, grab high, hop both legs, jog, kick forward, lie down on floor, rolate both arms buckward, up, throw basketball). The sequences, captured using VICON cameras, result in 686 samples, each represented by 31 skeletal number of joints and higher intra-class variability make this dataset particularly challenging. Finally, the UTKinect-Action3D dataset consists of 10 action classes (carry, clap hands, pick up, pull, push, sit down, stand up, throw, walk, wave hands), captured using stationary Microsoft Kinect camera. Each action was performed twice by 10 subjects, yielding 199 samples total.
down chair, sneak, squat, stand
significattyy more than in the Florence dataset. The increased
For HBC dataset. We select one dataset of healthy young adults and four disease-related human brain datasets for evaluation: the Human Connectome Project-Young Adult Working Memory (HCP. WM) [79), Alzheimer's 's Discase Neuroimaging Initiative (ADNI) [52] Open Access Series of Imaging
26

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Studies (OASIS) [46], Parkinson's 's Progression Markers Initiative (PPMI) [50], and the Autism Brain Imaging Data Exchange (ABIDE). We selected a total of 1,081 subjects from the HCP-WM dataset. The working memory task included both 2-back and O-back conditions, with stimuli featuring images of bodies, places, faces, and tools, interspersed with fixation periods. The specific task events are: 2bk-body, Obk-face, 2bk-tool, Obk-body, Obk-place, 2bk-face, Obk-tool, and 2bk-place. Brain activity was parcellated into 360 regions based on the multi-modal parcellation from [24]. For the OASIS (924 subjects) and ABIDE (1,025 subjects) datasets, which are binary-class datasets, one class represents disease group and the other represent healthy controls. In the ADNI dataset, subjects are categorized based on clinical outcomes into distinct cognitive status groups: cognitively normal (CN), subjective memory concern (SMC), early-stage mild cognitive impairment (EMCI), late-stage mild cognitive impairment (LMCD, and Alzheimer's Disease (AD). For population analysis, we group CN, SMC, and EMCI into a "CN-like" group, while LMCI and AD form the "AD-like" group. This grouping enables detailed analysis of cognitive decline and disease progression. The PPMI dataset consists of four classes: normal control, scans without evidence of dopaminergic deficit (SWEDD), prodromal Parkinson's 's disease, and Parkinson's disease (PD). This classification supports the study of different stages of Parkinson's 's progression. We employ Automated Anatomical Labeling (A (AAL) atlas [67] (116 (116 brain regions) on ADNI, PPMI, ABIDE datasets, while Destrieux atlas (18] (160 brain regions) is used in OASIS to verify the scalability of the models.
A.7 Comparison Methods and Experimental Results
We roughly summarize the comparison methods for HBC into two categories: spatial models and sequential models.
Spatial models. The spatial models are essential for understanding brain dynamics. Traditional GNNs like graph convolutional network (GCN) 44 and graph isomorphism network (GIN) [77 are selected for their ability to effectively capture diffusion patterns and isomorphism encoding in structured data. Subgraph-bass based GNNs, such as Moment-GNN [4]. emphasize subgraph structures, enabling the identification of localized patterns that might be overlooked by traditional GNNs. Expressive GNNs, including graph substructure network (GSN) i and GNNAsKernel (GNN-AK) [8]] are chosen for their enhanced expressivity through subgraph isomorphism counting and local subgraph encoding, which could be crucial for distinguishing subtle differences in complex systems.
A manifold-based model like the symmetric positive definite network (SPDNet) 16 is adopted for its ability manage high-dimensional manifold data, making it suitable for more complicated datasets.
Two graph-based brain network analysis models for disease diagnosis. BrainGNN [49]. an in- terpretable brain graph neural network for iMRI analysis, and ContrastPool (76. contrastive
dual-attention block and a differentiable graph pooling method.
Additionally, a traditional multi-layer perceptron (MLP) serves as a model due to its efficiency and versatility across various domains.
For all spatial models, following the optimal settings described in 660 we use the vectorized static functional connectivity (FC) as graph embeddings and the static FC matrices (N N) adjacency matrices, where only the top 10% of edges are retained through thresholding to ensure sparsity. The input of SPDNet is the original N x N FC matrices.
For dynamic-FC models (STAGIN 45 and NeuroGraph [60]. the thresholded dynamic FC matrices serve as the graph, NeuroGraph serve the vectorized FC the embedding and STAGIN incorporates BOLD signals as part of the embedding, alongside its unique embedding construction method. For GeoDynamics, we use the dynamic FC matrices the input, resulting in T x N x N matrices.
Sequential models. The sequential models are selected analyze temporal dynamics in time-series BOLD signals. ID-CNN is chosen for its ability to capture temporal patterns through convolutional operations. RNN (59] and LSTM [I]] are included for their proficiency in modeling sequential data and capturing long-range dependencies. MLP-Mixer both temporal and spatial features, offering a comprehensive view by integrating information across different dimensions. Transformer (88 is chosen for its powerful attention mechanisms, which allow to capture global dependencies in sequential data. Brain network transformer (BNT) 40 is a
[5] is selected for its capability to mix
tailored approach specifically designed for brain network analysis. Lastly, the state-space model
27

---

(SSM), represented by Mamba (25 is selected for its advanced state-space modeling abilities that effectively capture system dynamics over time.
For the sequential models, the inputs are the BOLD signals (N x 7).
Note, the inputs for all comparison methods align with the recent work presented in [21], ensuring fairness in the evaluation process.
We further conducted experiments using three brain network analysis models on disease-based datasets, including ADNI, OASIS, PPMI, and ABIDE. The diagnostic accuracies of 10-fold cross- validation are presented in Table 5 compared methods.
It is clear that our GeoDynamics consistently outperforms all the
Table 5: Diagnostic accuracies on three popular brain network analysis models.
ADNI
OASIS
PPMI
ABIDE
BrainGNN BNT ContrastPool
76.57 + 10.01
86.07
67.88 + 10.32
62.24 = 4.44 69.99 5.37
79.68 6.15
86.07 # 3.19
64.55 16.80
80.08 = 5.01
89.02 # 4.22
69.78 + 7.36
70.72 + 3.45
A.8 Hyperparameters, Inference time and Number of Parameters
We list the detailed hyperparameters of each model in Table 6 We summarize the inference time and the number of parameters of each mode on HCP-WM dataset (N = 360, 7 =9)) in Table all the experiments are conducted on NVIDIA RTX 6000Ada GPUs. We can observe that our method efficiently utilized the parameters compared to most counterpart methods. Compared to Mamba (vanilla SSM), our method requires more time in the final step due to the logarithmic mapping, which involves the computationally expensive Singular Value Decomposition (SVD). However, it is more efficient than SPDNet (a manifold-bas model), as we leverage convolution operations. As a result, the overall computational cost remains manageable.
Table Hyperparameter settings for different models. N denotes the number brain regions. 'M-SGD' and 'M MM-Aam' represent Stochastic Gradient Descent (SGD) and Adam optimizers, respectively, equipped with manifold-awar aware updates that enforce geometric constraints (e.g., orthogonality).
Model
ID-CNN
RNN
LSTM
Mixer
Mamba
GCN
GIN
Optimizer
Adam
Adam
Adam
Adam
Adam
Adam
Adam
Adam
Learning rate
10-4
10-4
10-4
10-4
10-4
xxx 10-5
10-4
10-4
Weight decay
5 10-1
5xx 10-1
x
10-+
x 10
5 10-1
0
x 10-4
5 x 10-1
Batch size
64
64
64
16
64
64
Epochs
300
300
300
Hidden dim
1024
1024
1024
1024
1024
1024
1024
1024
Network layer
2
2
2
2
Model
GSN
MGNN
GNN-AK
SPDNet
MLP
STAGIN
NeuroGraph
GeoDynamics
Oplimizer
Adam
Adam
Adam
M-SGD
Adam
Adam
Adam
M-Adam
Learning rate
10
10-2
10-3
10-3
10-1
5 x 10-
10-1
x 10-5
Weight decay
5 x 10-4
0
10-5
5 10-1
10-3
5 10-4
0
Batch size
16
16
128
32
64
16
16
Epochs
300
1000
100
100
300
100
100
300
Hidden dim
1024
128
[N. 32]
128
N
Network layer
2
3
2
Table 7: Model inference time (ms/item) and the number of parameters (M) comparison across various architectures on HCP- WM dataset.
GCN
GIN
GSN
MGNN
GNN- -KK
SPDNet
MLP
ID-CNN
Time (ms)
2.29
2.28
3,40
2.23
38.18
27.05
2.67
0.93
Paru (M)
1.79
3.89
0.92
4,94
290.3
0.19
66.9
2.22
RNN
LSTM
Mixer
TF
Mamba
NeuroGraph
STAGIN
GeoDynamicx
Time (ms)
0.87
0.91
0.91
1.21
0.33
39.79 0.29
20.92
2.51
Para (M)
1.19
14.45
6.78
12.98
27.05
1.17
14.60
28

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A.9 Discussion
We expect our manifold-based deep model facilitate our understanding on brain behavior in the following ways.
(I) Enhance the prediction accuracy. A plethora of neuroscience findings indicate that fluctuation of functional connectivities exhibits self-organized spatial-temporal patterns. Following this notion, conceptualize that well-defined mathematical modeling of intrinsic data geometry of evolving functional connectivity (FC) matrices might be the gateway to enhance prediction accuracy. Our experiments have shown that respecting the intrinsic data geometry in method development leads to significantly higher prediction accuracy for cognitive states, as demonstrated in Fig. 3
(2) Enhance the model explainability. We train the deep model parameterize the transition of FC matrices on the Riemannian manifold (Eq. [1].. By doing so, we are able to analyze the temporal behaviors with respect to each cognitive state using post-hoe complex system approaches such as dynamic mode decomposition, stability analysis.
(3) Provide high-order geometric attention mechanism that is beyond node-wise or link-wise focal patterns. Conventional methods often employ attention components for each region or link in the brain network separately, thus lacking the high-order attention maps associated with neural circuits (i.e., a set of links representing a sub-network). contrast, the SPD-preserving attention mechanism (Eq. 16 in our method operates the Riemannian manifold, taking the entire brain network into account. As shown in Fig. our method has identified not only links but also sub-networks relevant to cognitive states and disease outcomes.
A.10 Limitations and Future Work
First, the use of matrix exponentials and logarithmic operations (core to our Riemannian framework), requires eigenvalue Cholesky decompositions, which are computationally intensive. This leads to higher complexity compared to standard Euclidean approaches, especially as the number of nodes (i.e., brain regions) increases. To address this challenge, we can leverage parallel computing to mitigate the computational burden.
Second, the effectiveness of manifold based modeling depends on the assumption that input SPD matrices lie within well-behaved region of the manifold (e.g., within a geodesic ball), which ensures the uniqueness and stability of the Fréchet mean. When this assumption is violated, such as in highly noisy or poorly conditioned data (rank-deficient), the performance of the method may degrade.
Third, interpretability remains a challenge. While the geometric framework captures intrinsic data structure more faithfully, understanding how specific patterns relate clinical cognitive outcomes is still an open area of research.
In future work, plan to: (1) Optimize runtime further through low-rank approximations manifold- aware pruming. (2) Extend the framework to handle multi-modal neuroimaging data (c.g., EEG, MEG). (3) Explore other interpretable models to bridge the gap between mathematical complexity and clinical insight.
A.11 Impact Statement
This work bridges the fields of machine learning and neuroscience by introducing geometric deep learning-based state-space model (GeoDynamics) for understanding brain dynamics and their relationship behavior and cognition. By leveraging the unique properties of Riemannian geometry, our model offers a holistic view brain state evolution as a self-organized dynamical system, addressing critical challenges in functional neuroimaging studies and enabling applications in early diagnosis of neurodegenerative diseases such as Alzheimer's is and Parkinson's ss
Beyond neuroscience, GeoDynamics demonstrates broad applicability in capturing spatio-temporal dynamies across diverse domains, as evidenced by its promising performance in human action recognition benchmarks. This highlights the potential of our approach to impact fields ranging from healthcare to computer vision, offering tools for scalable, robust, and interpretable analysis of complex systems.
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Reviews:
[{"review_id": "DyWWzWwC3y", "reviewer": "Reviewer_tBEG", "summary": "The authors designed a geometric neural state-space neural networks (GeoDynamics) to model the functional connectivity (FC) among different brain areas. They benchmarked their model against fifteen other models (sequential and spatial types of model) and two datasets (HBC and HAR).", "questions": "L44 and Figure 1b legend: Does the \"black solid box\" indicate Euclidean space or Manifold space?\n\nL251: So it is a 1 out 8 classification task?\n \nWhere is the Discussion of this paper? Section 4.3 is not the real Discussion.", "limitations": "Yes (in supplementary)", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The benchmark against other models and coverage of datasets are comprehensive.\n\nThe presentation of Figures/Tables are clear.\n\nMy main concern is the improvement of current model over previous ones. In L325, the author said that \"Our proposed GeoDynamics demonstrates a much better performance in these tasks\". I believe they mean ADNI, OASIS and PPMI datasets. Are those improvement really big enoguh? For example, the Accuracy increase from 80.4 to 81.2 in ADNI (+1%), from 89.26 to 89.6 in OASIS (0.38%), and from 73.31 to 71.35 in PPMI (worse than previous models). What are the hyper-parameters of GeoDynamics?\n\nFurthermore, are those values (Acc, Pre, and F1) shown in previous models copied from original papers or the authors reproduced previous models by themself?", "was_revised": true, "final_justification": "The authors have answered my questions.", "rebuttal": "**We thank the reviewers for their constructive feedback and positive assessment of our manuscript. Below is a brief summary of our work, followed by our responses to the key concerns raised.**\n\n**Q1: Are those improvement really big enough? (the Accuracy increase from 80.4 to 81.2 in ADNI (+1%), from 89.26 to 89.6 in OASIS)**\n\n**A1:** Thank you for your thoughtful comment. While the observed accuracy gain may appear modest (e.g., 80.4% → 81.2% on ADNI), even small improvements can be meaningful in clinical or population-scale applications, especially when models are already operating near performance ceilings. To assess the robustness of this improvement, we conducted a paired t-test over the 10-fold cross-validation results. For ADNI, the improvement was statistically significant (p<0.05), suggesting the gain is unlikely due to random variation. For OASIS, however, the improvement is not signifiant at the level p<0.05. We also note important differences between the datasets. ADNI is more heterogeneous, spanning broader disease stages, acquisition protocols, and demographics—conditions under which modest gains are more likely to emerge and generalize. In contrast, OASIS is more homogeneous with higher baseline performance, making further improvement inherently more challenging.\n\nWe have added this statistical analysis (Table 1 '*' denotes the significant improvement ($p<0.05$)) to the revised version and will include it in the final manuscript.\n\n**Q2: What are the hyper-parameters of GeoDynamics?**\n\n**A2:** The hyper-parameters are as follows: Learning rate: $5\\times10^{-5}$, Weight decay=0, Batch size=16, Epochs=300, Hidden dim=number of brain region, network layer=2, window size=15. The full configuration is summarized in Table 5 of the original manuscript for reproducibility. We have conducted an ablation study in terms of difference window size (Appendix A.9 in the original manuscript). We will ensure this information remains clearly accessible in the final version.\n\n**Q3: Are those values (Acc, Pre, and F1) shown in previous models copied from original papers or the authors reproduced previous models by themself?**\n\n**A3:** Thank you for your question. All results reported for the baseline models—including accuracy, precision, and F1-score—were reproduced by us using publicly available implementations or reimplementations based on the descriptions in the original papers. To ensure a fair comparison, we applied the same 10-fold cross-validation splits, consistent preprocessing, and identical evaluation metrics across all models. This uniform setup ensures that performance differences reflect the intrinsic modeling capabilities rather than external confounding factors. \n\nWe would like to clarify that the complete benchmarking code was already uploaded at the time of the initial submission to the anonymous GitHub repository (name \"Comparison methods\" folder) linked in the original manuscript. We respectfully encourage the reviewers to revisit the provided link for verification purposes. This inclusion complies with the rebuttal policy, as no new materials have been introduced. To further enhance transparency and reproducibility, we will make this information more prominent in the final version. Thank you so much.\n\n**Q4: L44 and Figure 1b legend: Does the \"black solid box\" indicate Euclidean space or Manifold space?**\n\n**A4**: In Fig.1b, the \"black solid box\" denotes the vanilla SSMs, which operate in Euclidean space. We agree that this distinction could be made more explicit, and we will revise the figure legend in the final version to clearly indicate that the “black solid box” corresponds to Euclidean space.\n\n**Q5: L251: So it is a 1 out 8 classification task?**\n\n**A5:** Yes, this is an 8-class classification task. We selected 1,081 subjects from the HCP-WM dataset, where the working memory paradigm included both 2-back and 0-back conditions across four stimulus categories: bodies, places, faces, and tools, along with fixation periods. The eight specific task events are: 2bk-body, 0bk-face, 2bk-tool, 0bk-body, 0bk-place, 2bk-face, 0bk-tool, and 2bk-place. A detailed description of the dataset and task design is provided in Appendix A.6 (Table 3, Page 18) of the original manuscript.\n\n**Q6: Where is the Discussion of this paper? Section 4.3 is not the real Discussion.**\n\n**A6:** Thank you for pointing this out. We have revised Section 4.3 to more accurately reflect its focus on **Model Validation**. The main discussion, including limitations, future directions, and broader implications, is provided in **Appendix A.10-A12 (Page 22 -23)** of the original manuscript. We will clarify this organization in the final version to improve readability and structure.\n\n**Q7: Format**\n\n**A7:**  Sorry about that. The overlapping of line numbers (e.g., 112+113, 177+178, 197+198, 233+234) results from our use of *\\vspace* commands to condense vertical spacing. We will correct this and restore proper spacing in the final version of the manuscript.\n\n**Thank you for your time and consideration. We are committed to incorporating all suggestions and improvements into the final version of the manuscript.**"}, {"review_id": "hGmyKjaMoj", "reviewer": "Reviewer_nqTq", "summary": "This paper introduces GeoDynamics, a geometric state-space neural network designed to model time-varying functional connectivity (FC) in the brain. Unlike standard SSMs which operate in Euclidean space, GeoDynamics constrains both system inputs, hidden states, and outputs to lie on the Symmetric Positive Definite (SPD) manifold, motivated by the observation that brain FC matrices are naturally SPD. The transition dynamics are defined using group actions (particularly of the orthogonal group) and Riemannian constructs like the Fréchet mean and the Stein metric. The model is evaluated on both human brain datasets and a human action recognition benchmark.", "questions": "1. Model components: Could you explicitly define what constitutes the input $X(t)$, the hidden state $S(t)$, and the observed output $Y(t)$ in the context of your neuroscience application? It is currently unclear how the functional connectivity matrices relate to these quantities.\n2. Equation 3 confusion: What is the set over which you are computing the Fréchet mean in Equation 3? $S^{(k-1)}$ appears to be a single SPD matrix, not a set. What is $\\tilde{A}$? Ostensibly it is a set of weights (scalars).\n3. Group action inconsistency: You define the translation $\\mathcal{T}$ using the action of the orthogonal group $\\mathbb{O}$ on the SPD manifold, but then apply it to the Fréchet mean of SPD matrices. If the argument to $\\mathcal{T}$ must be orthogonal, this would be undefined unless the mean is trivially the identity. Can you clarify?\n4. On Fréchet mean and uniqueness: What are the guarantees on the existence and uniqueness of the Fréchet mean under your assumptions? Do you enforce closeness on the manifold during training?\n5. Experimental focus: Why include human action recognition? How does this task align with your main objective of modeling brain dynamics? Could the model be evaluated more thoroughly on neuroscience-relevant tasks?\n\nMy score could increase if the authors:\n\n- Sharply revise the manuscript to clarify definitions and model structure,\n- Cleanly separate results from interpretation,\n- Remove redundant text and clarify inconsistent terminology,\n- Provide more focused experimental evaluation and justification for each task.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n- Principled use of Riemannian geometry: The authors leverage interesting manifold-aware constructs (e.g., the Fréchet mean, group actions, Stein metric) to build the model in a way that respects the geometry of SPD matrices.\n- Theoretical grounding: There are rigorous definitions and group-theoretic justifications, and the paper includes formal results and claims with proofs in the appendix.\n- Potentially generalizable: While aimed at brain data, the method could be extended to any domain where SPD-valued time series arise.\n\n### Weaknesses\n\n- Major clarity issues throughout:\n    - Core concepts such as system input, hidden state, and output are not clearly mapped to the neuroscience application (e.g., is the FC the input or the hidden state?).\n    - Equations (e.g., Equation 3) are difficult to parse and are not explained with sufficient care.\n    - Basic elements of the SSM formalism (e.g., matrices A, B, C, D) are not introduced for readers unfamiliar with control theory.\n    - Redundant or imprecise definitions (e.g., repeated definitions of group actions) obscure the main argument.\n- Lack of focus in narrative: The inclusion of the human action recognition dataset seems like an afterthought and dilutes the focus of the paper, which is ostensibly about modeling brain dynamics.\n- Experimental presentation is weak: The structure of the experimental section is muddled. Results are intermixed with speculative interpretation (”Remarks”), and important clarifications (e.g., what is being predicted, what the task is) are missing.\n- Terminological inconsistency: Terms like \"system state\" vs. \"hidden state\" are used interchangeably, it would help to stick to one of them.\n- Foundational confusion: Some parts of the model formulation appear self-inconsistent (e.g., the domain of the group action $\\mathcal{T}$ vs. its usage in Equation 3), and the writing does not help to disambiguate.", "was_revised": true, "final_justification": "I had major concerns about the clarity and readability of the manuscript, especially unclear mathematical notation. The authors have provided significant clarification in their rebuttal, and made a commitment to increase the clarity and focus of their manuscript.\n\nThey have also conducted additional ablation experiments and benchmarks. \n\nTherefore, I have increased my initial score (2) to a (4).", "rebuttal": "**We appreciate your recognition of our method's theoretical grounding and geometric formulation. Below we respond to each of your concerns and will incorporate all suggestions into the final version.**\n\n\n**Q1: Core concepts such as system input, hidden state, and output are not clearly mapped to the neuroscience application (e.g., is the FC the input or the hidden state?)**\n\n**A1:** Sorry for the unclear description. The FC matrices $X(t)$ are treated as the system input in our model. The hidden state $S(t)$ represents the internal geometric representation on the manifold (learned from $X(t)$ that summarizes historical FC dynamics, and the system output $Y(t)$ corresponds to the predicted FC state used for downstream inference tasks (e.g., brain state identification). We have updated **Sec. 3.1-Overview** to clearly define the mapping between the neuroscience data and the model variables. These changes will be reflected in the revised manuscript to improve clarity and align the mathematical formulation with the neuroscience context.\n\n**Q2: Equations (e.g., Equation 3) are difficult to parse and are not explained with sufficient care**\n\n**A2:** Eq. 3 is derived as the manifold-based extension of Eq. 1. The key distinction lies in the underlying geometry of the system: \n- Eq. 1 corresponds to a vanilla SSM defined in Euclidean space, where system evolution and observations are governed by standard linear operations.\n- Eq. 3 reformulates the SSM on the Riemannian manifold of symmetric positive definite (SPD) matrices, where FC matrices are naturally embedded.\n\nIn this manifold setting, standard operations (e.g., averaging, updating states) must be replaced with geometrically consistent operations, including:\n\n- the Fréchet mean $\\mathcal{F}$,  which generalizes the concept of averaging to Riemannian spaces, and\n- the translation operator $\\mathcal{T}$, which allows us to evolve system states and outputs across the manifold.\n\nWe will revise Sec. 3.1 to walk readers through these concepts more carefully and to explain each term in Eq. 3 with explicit reference to their role in modeling the time-varying FC matrices on the SPD manifold.\n\n**Q3: Basic elements of the SSM formalism (e.g., matrices A, B, C, D) are not introduced for readers unfamiliar with control theory.**\n\n**A3:** We agree that clarity around the role of A,B,C,D is important, especially for readers unfamiliar with control theory. Fortunately, it should be an easy fix in the final version by highlighting the existing write-ups. As noted in Sec. 2.1 (line 110-117), we have already introduced the standard linear state-space model and defined each matrix. In our proposed model, these parameters retain similar functional roles but operate on SPD manifolds through geometry-aware operations (e.g., Fréchet mean and manifold translation), as described in Sec. 3.1 and Eq. (3). We will make this connection between the classical and manifold-based formulations more explicit in the revised version.\n\n**Q4: About group action definitions** \n\n**A4:** Thank you for these insightful comments. We have unified and clarified our definitions in Section 2.2 as follows:\n- 1. **Group action on SPD matrices:**\n$ g \\cdot X := gXg^\\top,\n   \\quad g \\in \\mathbb{G},; X \\in \\mathcal M.$\n   Here, $\\mathbb{G}$ denotes the **general linear group** of full‑rank $N\\times N$ matrices.\n- 2. **Restriction to orthogonal subgroup:**\n   We then restrict to the **orthogonal subgroup**\n   $\\mathbb{O}\\subset\\mathbb{G}$, consisting of all $N\\times N$ orthogonal matrices that preserve the Stein metric.\n- 3. **Translation operator:**\n $\n   \\mathcal T: \\mathcal M \\times \\mathbb{O} \\to \\mathcal M,\n   \\quad\n   \\mathcal T(X,g) = gXg^\\top.\n$\n   In this definition, the first argument $X$ may be *any* SPD matrix (e.g., the output of the Fréchet mean), and the second argument $g$ is an orthogonal matrix.\n\n- 4. **Application in Eq. (3):**\n$S^{(k)} = \\mathcal T\\bigl(\\mathcal F(\\cdots),,\\widetilde A\\bigr),$\n   which denotes the action of the orthogonal matrix $\\widetilde A \\in \\mathbb{O}$ on the SPD output of $\\mathcal F$.  Since $\\mathcal F(\\cdot)$ always returns an SPD matrix, $\\mathcal T(\\mathcal F(\\cdot),\\widetilde A)$ is well‑defined.\n\nWe have updated Lines 126–133 accordingly to remove any redundant or unnecessary repetitions. Thanks.\n\n**Q5: Equation 3 confusion: What is the set over which you are computing the Fréchet mean in Equation 3? S^(k-1) appears to be a single SPD matrix, not a set. What is  Ostensibly it is a set of weights (scalars).**\n\n**A5:** Thank you for this insightful question. We appreciate the opportunity to clarify Eq. 3, particularly the structure and interpretation of the weighted Fréchet mean.  In Eq. 3 the Fréchet mean is not taken over a singleton {$S^{(k-1)}$}, but over a sliding window of past states (and inputs) with their associated weights. Concretely, letting the window length be $\\tau$, we compute \n$$\nS^{(k)} = \\mathcal{T} \\left(\n\\mathcal{F}_1(\\cdot), \\, \n\\mathcal{F}_2(\\cdot)\n\\right)\n$$\n\n$\\mathcal{F}_1(\\cdot) = \\mathcal{F}\\left(\n\\left\\{ S_j \\right\\}_{j=k-\\tau}^{k-1},\n\\left\\{ A_j \\right\\}_{j=k-\\tau}^{k-1}\n\\right)$\n\n$\\mathcal{F}_2(\\cdot) = \\mathcal{F}\\left(\n\\left\\{ X_j \\right\\}_{j=k-\\tau}^{k},\n\\left\\{ B_j \\right\\}_{j=k-\\tau}^{k}\n\\right)$\n\nThe full expression is too long to display clearly in a single line, so we break it down into individual components for clarity:\n\n| Fréchet mean  | First Argument                         | Second Argument                        |\n|-----------------|----------------------------------------|----------------------------------------|\n| $\\mathcal{F}_1(\\cdot)$ |  {$S_j$} $_{j={k-\\tau}}^{k-1}$            |   {$A_j$} $_{j={k-\\tau}}^{k-1}$            |\n| $\\mathcal{F}_2(\\cdot)$ |  {$X_j$} $_{j={k-\\tau}}^{k}$              |   {$B_j$} $_{j={k-\\tau}}^{k}$              |\n\n\n so that the set  {$S_j$} $_{j={k-\\tau}}^{k-1}$ contains the $\\tau$ most‑recent system–state SPD matrices (window size), with transition matrix $\\{A_j\\}$, \n\nthe set  {$X_j$} $_{j={k-\\tau}}^{k}$ contains the same window of inputs plus the current $X^k$. We have updated Eq. 3 and the related text to make this explicit. We have also conducted an ablation study in terms of difference window size (Appendix A.9 in the original manuscript). We will make it more clear in the final version, thanks. \n\n\n**Q6: What are the guarantees on the existence and uniqueness of the Fréchet mean under your assumptions? Do you enforce closeness on the manifold during training?**\n\n**A6:** Under the affine‑invariant metric, SPD(n) is a Hadamard manifold (complete, simply‑connected, non‑positive curvature) [Afsari, B. (2011). Riemannian $𝐿^𝑝$ center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society]. Hence, there exists a unique Fréchet mean given a set of FC matrices. We initialize each translation matrix $g$ close to the identity (so $g \\thickapprox I$) and train with small learning rates using our manifold‑aware optimizers (M‑Adam) (Table 5 in the original manuscript). This guarantees that each update exp(Δ) moves only a short geodesic distance on $\\mathbb O(n)$, keeping all points for the Fréchet mean within a convex neighborhood.\n\n**Q7: Could the model be evaluated more thoroughly on neuroscience-relevant tasks?**\n\n**A7:**  We appreciate this constructive suggestion and agree that the core manuscript should remain squarely focused on modeling brain dynamics. To address this, we will: (1) Move the HAR experiment to a concise “**Method Validation**” section in the Supplement, so that the main text is entirely devoted to neuroscience applications. (2) Enrich our neuroscience evaluation by adding three EEG‑based benchmarks, such as Emotion recognition on the SEED [1], DEAP [2], and HCI [3] datasets, where our geometry‑aware SSM outperforms baseline methods (as below). We are committed to include the following EEG results in the final version. \n\n| **Mamba**     | **SEED**      | **DEAP**        | **HCI**       | \n|------------------|-----------------|-----------------|-----------------|\n| **Acc**         | 81.62 ± 0.48    | 88.98 ± 1.79    | 86.89 ± 4.56    | \n| **Pre**         | 79.76 ± 0.95    | 88.10 ± 1.83    | 75.51 ± 17.15    | \n| **F1**          | 76.31 ± 0.45    | 87.79 ± 2.07    | 69.78 ± 14.23    | \n| **Mamba**     | **SEED**      | **DEAP**        | **HCI**       | \n|------------------|-----------------|-----------------|-----------------|\n| **SPDNet**     | **SEED**      | **DEAP**        | **HCI**       | \n| **Acc**         | 85.32 ± 1.41    | 90.23 ± 2.40    | 93.16 ± 2.73    | \n| **Pre**         | 82.68 ± 1.88    | 88.69 ± 3.64    | 84.62 ± 6.21    | \n| **F1**          | 80.16 ± 1.92    | 88.48 ± 3.28    | 76.78 ± 2.54    | \n|------------------|-----------------|-----------------|-----------------|\n| **GeoDynamics**     | **SEED**      | **DEAP**        | **HCI**       | \n| **Acc**         | 92.51 ± 2.39    | 93.92 ± 1.46    | 95.23 ± 3.59    | \n| **Pre**         | 92.62 ± 2.36    | 93.94 ± 1.46    | 95.05 ± 3.74    | \n| **F1**          | 92.48 ± 2.39    | 93.91 ± 1.46    | 94.92 ± 4.05    | \n\n[1] Zheng, W. L., et al  (2015). Investigating critical frequency bands and channels for EEG-based emotion recognition with deep neural networks. IEEE TAMD.\n\n[2] Koelstra, S., et al. (2011). Deap: A database for emotion analysis; using physiological signals. IEEE TAC.\n\n[3] Soleymani, et al (2011). A multimodal database for affect recognition and implicit tagging. IEEE TAC. \n\n**Q8: Terminological inconsistency: Terms like \"system state\" vs. \"hidden state\" are used interchangeably, it would help to stick to one of them.**\n\n**A8:** Thank you for the suggestion. We have standardized the terminology throughout the manuscript by replacing all occurrences of both “system state” and “hidden state” with the single term “system state”. This change has been applied in Sections 2, 3, and all figure captions to ensure consistency.\n\n**Thanks again for your feedback.**"}, {"review_id": "2tZvknbzut", "reviewer": "Reviewer_bcvk", "summary": "The goal of this paper is to analyze functional MRI (fMRI) data to study human brain with respect to clinical conditions or some cognitive tasks being performed by the patient. There are two claimed innovations in this paper. Firstly, it uses a state-space model on the SPDM manifold to model the times series data. Specifically, the approach is to transform the data into a time series of brain connectivity matrices, and to model it as a time series on SPDM manifold. To analyze this time series, the paper formulates a Markov decision process that models a hidden state, a transition matrix, and an output that classifies the state. The paper user geometry of SPDM to formulate these elements intrinsically on the manifold. Secondly, the inference is performed using a neural-network architecture. It defines a convolution operation on SPDMs for use in forming convolution layers in the network. \n\nThere is a very extensive experimental support to the approach in the paper. It compares several methods on datasets for classifying cognitive tasks and neurodegenerative diseases from fMRI data. The results are generally superior to the existing methods.", "questions": "Please clarify the novelty of your approach to the part papers on dynamic brain functional connectivity.", "limitations": "I don't see any note on the limitations of the model/method proposed here.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1.\tIt utilizes the geometry of SPDMs to respect the structure of brain connectivity. Several papers have utilized this representation to constrain analysis to interpretable solutions. \n2.\tIt implements a deep-learning version of the state space models on nonlinear manifolds. \n3.\tThe experimental results are both extensive and impressive. \n\nWeaknesses: \n\n1.\tThe modeling of brain functional connectivity using a time-series of SPD matrices is not new, a number of papers have used this formulation. See for example, Dai et al, Analyzing Dynamical Brain Connectivity as Trajectories on Space of Covariance Matrices, 2020. Implementing this SSM using neural networks has some novelty to it. \n2.\tThe computational cost of performing analysis on SPDM manifold is expensive. The matrix computations grow much faster than the size of matrices. The method will not scale as well if the number of nodes in the brain increases. \n3.\tEven though the empirical results are impressive, this formulation is hard to analyze theoretically due to nonlinearity of the underlying manifold.", "was_revised": true, "final_justification": "In view of the discussion with the authors and their response to my comments, I retain my original (very high) rating of this paper.", "rebuttal": "**We sincerely appreciate the reviewers' constructive feedback and thoughtful critiques. Below, we provide point-by-point responses to address each concern and outline the improvements made in the revised manuscript.**\n\n**Q1: Please clarify the novelty of your approach to the part papers on dynamic brain functional connectivity.**\n\n**A1** Thank you for pointing this out. We fully agree that modeling brain functional connectivity as a trajectory of SPD matrices has been explored in prior work, such as Dai et al. (2020). Our contribution builds upon this foundation but introduces a novel integration of *geometric deep learning and state space modeling (SSM)* within the Riemannian manifold framework.\n\nSpecifically, our approach differs in three important ways (Line 71-94 in the original manuscript):\n\n- **Structural novelty**: We formulate dynamic FC modeling as a learning task on the Riemannian manifold of SPD matrices, treating each FC matrix as a point on the manifold and encoding whole-brain interactions more holistically than Euclidean approximations.\n\n\n- **Behavioral modeling via SSM**: Unlike prior approaches that either lack temporal modeling or rely on RNNs, we adopt a continuous-time state space model (SSM) to capture latent neural dynamics. This dual formulation of state and observation equations enables a principled mapping between brain states and observed neural signals over time.\n\n\n- **Geometric–neural integration**: We replace traditional Euclidean algebra in SSMs with Riemannian geometry, enhancing the model's ability to handle the curvature and structure of SPD space. This is paired with a geometric-adaptive attention mechanism that improves efficiency and generalization (Sec. 3).\n\nWhile Dai et al. focused on geometry-aware trajectory analysis (a valuable contribution), our work adds a temporal modeling layer via SSM and neural parametrization to learn both latent dynamics and manifold structure in an end-to-end fashion. To our knowledge, our work is the first framework to unify these components within a deep geometric setting for functional brain analysis.\n\nWe have validated our method not only on multiple brain datasets (e.g., ADNI, OASIS, HCP, PPMI, ABIDE), but also on human action recognition tasks, demonstrating generalizability across domains. We will make more clearer in the final version.\n\n**Q2: The computational cost of performing analysis on SPDM manifold is expensive. The matrix computations grow much faster than the size of the matrices. The method will not scale as well if the number of nodes in the brain increases.**\n\n**A2:** Thank you for highlighting this important point. We acknowledge that operations on the SPD manifold—especially matrix logarithms and exponentials—can become computationally expensive as the number of brain regions increases (we have discussed this limitation (Appendix A.11-Page 22) in the original manuscript). This is a general challenge for manifold-based methods. To address this, we currently leverage **parallel computing and batched matrix operations** to significantly reduce the computational burden during training and inference. Additionally, since many SPD operations are independent across time steps or subjects, they are naturally suited to GPU acceleration and distributed computing.\n\nLooking ahead, we plan to explore more **scalable approximations**, such as low-rank factorizations, tangent-space projections, or spectral truncation of covariance matrices, which can maintain performance while further improving efficiency.\nWe appreciate the reviewer’s comment and will clarify these scalability considerations in the revised manuscript.\n\n**Q3: Even though the empirical results are impressive, this formulation is hard to analyze theoretically due to nonlinearity of the underlying manifold.**\n\n**A3:** Thank you for raising this insightful point. We agree that theoretical analysis becomes more complex when working with nonlinear Riemannian manifolds such as the space of SPD matrices. This is a known limitation of many geometry-based deep models.\n\nTo mitigate this challenge, our framework is carefully designed to preserve theoretical tractability where possible. Specifically:\n\n- We **replace Euclidean algebra in state space models with Riemannian counterparts** in a principled way, allowing certain analogs of stability and controllability theory to still be studied in the manifold setting (see Section 3).\n\n- The **structure of the SSM** provides an interpretable framework, with latent dynamics governed by well-defined ODEs and output mappings, which serve as a bridge between complex geometry and system theory.\n\n- For practical robustness, we **regularize manifold operations** using exponential map truncation and intrinsic norms, which helps constrain model behavior and improve empirical generalization.\n\nWe also note that the field is actively developing new theoretical tools for analyzing learning dynamics on manifolds (e.g., Riemannian optimization and tangent space approximations). We believe our work contributes to this ongoing effort by demonstrating a successful application in the neuroscience domain, and future work will aim to formalize the theoretical underpinnings further.\n\nWe will clarify this discussion in the revised manuscript.\n\n**Q4: I don't see any note on the limitations of the model/method proposed here.**\n\n**A4:**  Thank you for your important observation. Due to space constraints in the main text, we have included a detailed discussion of the model’s limitations in **Appendix A.11 (Page 22)** of the original manuscript. This section outlines key limitations, such as the computational overhead introduced by Riemannian operations and challenges related to noisy or rank-deficient input data.  We will make this cross-reference more prominent in the revised version to ensure readers are aware of these important considerations. Thank you so much."}, {"review_id": "fUnBaBE03O", "reviewer": "Reviewer_X13L", "summary": "This paper introduces a neural network based on a state-space model that operates on the Riemannian manifold of SPD matrices. The proposed network is tailored for processing sequences of functional connectivity matrices, which are modeled as SPD matrices that capture the dynamics of brain activity. The proposed network is evaluated on various benchmarks involving brain dynamics and action recognition.", "questions": "Specific comments and suggestions:\n- Line 74: specify which “Riemannian manifold”\n- Line 80: “manifold instance” => sample/point on the manifold\n- Sec. 2.2 - “Riemannian geometry algebra” => “Riemannian geometry”\n- Sec. 2.2, line 120: the notation in parenthesis is not clear. My suggestion is to replace \\mathcal{M} => sym^+_N.\n- Sec. 2.2 - only the Stein metric (aka Jensen-Bregman logdet divergence) [10] is mentioned and used but there are many other Riemanian metrics on the SPD manifold, which have been proposed and studied extensively. It is important to mention the alternatives, e.g., the affine-invariant metric [*] and log Euclidean [**], and discuss why the particular logdet divergence was chosen.\n\n[*] Pennec, X., Fillard, P., & Ayache, N. (2006). A Riemannian framework for tensor computing. International Journal of computer vision, 66, 41-66.\n\n[**] Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006). Log‐Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 56(2), 411-421.", "limitations": "Some limitations are mentioned in the appendix (A.11) but I find it quite hidden, and some of the discussion could be moved to the main text", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The addressed problems in neuroscience are important and central\n- The proposed network is new and nicely combines concepts and tools from different fields, where the internal state and output updates are performed “on the manifold” using translation and the weighted Riemannian mean (Eq. 3)\n- The experiments demonstrate the benefits of the proposed network compared to baselines in a very broad range of applications\n\nWeaknesses:\n- The presentation and writing need some improvement, specifically, the treatment of the Riemannian geometry of SPD matrices (see details below)\n- Implementations details are missing - for example, details on how the Frechet mean is computed\n- The computational complexity is mentioned only in Appendix A.11 in the context of limitations without details. From practical standpoint, this is an important issue that should be better discussed. Specifically, what is the computational complexity of the algorithm and how does it scale with the size of the FC matrices?", "was_revised": true, "final_justification": "The authors’ response adequately addresses my concerns, particularly regarding the lack of detail, the treatment of Riemannian geometry, the choice of metric, and the discussion of limitations. Provided that all these changes are incorporated into the revised paper, and after considering the other reviews and responses, I have decided to raise my score to 5: accept.", "rebuttal": "**We sincerely thank the reviewer for their positive and constructive feedback. We are glad that the reviewer recognizes the importance of the problems we address in neuroscience, as well as the novelty of our neural network model based on state-space dynamics on the SPD manifold. We appreciate the encouraging comments on how our method combines tools from different fields and performs intrinsic updates on the manifold. We also thank the reviewer for pointing out areas for improvement. In the following, we respond to each concern and suggestion in detail.**\n\n\n**Q1: The presentation and writing need some improvement, specifically, the treatment of the Riemannian geometry of SPD matrices. (see details above)**\n\n**A1:** Thank you for your detailed and insightful comments regarding the use and presentation of Riemannian geometry on the SPD manifold. We have carefully revised the manuscript to address the clarity and terminology issues you highlighted:\n\n- **Line 74**: We now explicitly specify the manifold as the space of $N \\times N$ symmetric positive definite (SPD) matrices, denoted $\\text{Sym}^+_N$, a well-studied Riemannian manifold in information geometry and medical imaging.\n\n- **Line 80:** We have replaced “manifold instance” with “a point on the manifold” to align with standard terminology in differential geometry.\n\n- **Section 2.2:** We have corrected “Riemannian geometry algebra” to “Riemannian geometry” throughout. In line 120, we now clarify the notation by replacing $\\mathcal{M}$ with $\\text{Sym}^+_N$ to improve precision.\n\n- **Metric choice**: Thank you for your insightful comment. We agree that many Riemannian metrics have been proposed for the SPD manifold, including the affine-invariant Riemannian metric (AIRM) [Pennec et al., 2006] and the log-Euclidean metric [Arsigny et al., 2006]. In the revised manuscript (Sec. 2.2), we have included a discussion of these alternatives and clarified our rationale for selecting the Stein metric (also known as Jensen-Bregman log-det divergence).\n\nCompared to AIRM and log-Euclidean, the Stein metric offers several benefits for our application:\n\n○ Like AIRM and log-Euclidean, it respects the intrinsic Riemannian geometry of SPD matrices.\n\n○ It is invariant under congruent transformations, which is important for cross-subject comparisons in functional connectivity.\n\n○ It avoids the computational cost of matrix logarithms and square roots.\n\nTo empirically support this choice, we conducted an ablation study on the ADNI dataset, comparing classification accuracy and runtime across three manifold-based metrics:\n\n\n| Metric        | Formulation                                                                                                         | Accuracy (%) | Runtime (ms) |\n| ------------- | ------------------------------------------------------------------------------------------------------------------- | ------------ | ------------ |\n| AIRM          | $\\|\\log(\\mathbf{X}^{-1/2} \\mathbf{Y} \\mathbf{X}^{-1/2})\\|\\_F\\$                                                      | 81.51        | 1.525        |\n| Log-Euclidean | $\\|\\log(\\mathbf{X}) - \\log(\\mathbf{Y})\\|\\_F\\$                                                                       | 83.10        | 1.488        |\n| Stein (ours)  | \\$\\sqrt{ \\log\\det\\left( \\frac{\\mathbf{X} + \\mathbf{Y}}{2} \\right) - \\frac{1}{2} \\log\\det(\\mathbf{X} \\mathbf{Y}) }\\$ | **85.49**    | **1.375**    |\n\nAs shown above, the Stein metric achieves the best trade-off between classification performance and computational efficiency. These findings, combined with its theoretical strengths, support our decision to adopt the Stein metric as the default geometry-aware distance measure in our model. The manuscript has been updated to reflect this comparison and justification.\n\nThese clarifications and revisions have been incorporated into Section 2.2 and we believe they improve both the rigor and readability of the presentation. Thank you again for your constructive feedback.\n\n**Q2: Implementations details are missing - for example, details on how the Frechet mean is computed**\n\n**A2:** Thank you for your insightful comment. We now provide a detailed explanation of how the weighted Fréchet mean is computed in our implementation:\n1. $L$ is initialized as the identity matrix so that $F=L L^{\\top}+\\varepsilon I$ remains SPD. (the complexity is $O(N^3)$)\n2. The algorithm minimizes the weighted sum of squared LogDet distances over the SPD matrices in $X$​ by updating $L$ using an optimizer (Adam). This differentiates the log-determinant functions concerning $F$ while respecting the SPD manifold step typically involves structure. (the complexity is $O(N^3)$)\n3. The process stops when the change in loss falls below a given tolerance, ensuring convergence. (the complexity is $O(N^3)$)\n\nIn addition, the translation is defined as: $\\mathcal{T}(X,g):=g X g^{\\top}$. The implementation details are provided below.\n\n1. Reshapes vector $g$ as a row vector;\n\n2. From the row vector, a lower triangular matrix $L$ is constructed by iterating over each row, extracting the required slice of $g$, and padding with zeros; (need interaing N rows, the complexity is $O(N^2)$);\n\n3. A skew-symmetric matrix is formed by subtracting the transpose of the lower triangular matrix: $G_{skew}​= L-L^⊤$;\n\n4. The skew-symmetric matrix is then mapped into a rotation matrix in $\\mathbb O(n)$ using the matrix exponential: $G=exp(G_{skew})$; (involves inverting and multiplying operation, so the complexity is $O(N^3)$).\n\n5. The translated SPD matrix is computed by conjugating $X$ with $G$: $\\mathcal{T}(X,G):=G X G^{\\top}$. (including two matrix multiplications, the complexity is $O(N^3)$).\n\nWe will incorporate the implementation details in the final version, thank you so much.\n\n**Q3: The computational complexity is mentioned only in Appendix A.11 in the context of limitations without details. From a practical standpoint, this is an important issue that should be better discussed. Specifically, what is the computational complexity of the algorithm and how does it scale with the size of the FC matrices?**\n\n**A3:** Thank you for this constructive comment. If the FC matrices are $N\\times N$, and both the input $X$ and the convolution kernel $K$ are $N\\times N$, then each output element requires an inner product over $N^2$ elements. Since the convolution is computed at each spatial location—amounting to $N^2$ output elements—the overall time complexity for one time step is $O(N^4)$. When considering  $T$ time points, the total complexity becomes: $O(T \\cdot N^4)$. nn.DataParallel in PyTorch can achieve parallel computing, which distributes the computation across multiple GPUs. We have included the inference time in Table 5 and will add this analysis in the final version.\n\n**Q4: Some limitations are mentioned in the appendix (A.11) but I find it quite hidden, and some of the discussion could be moved to the main text**\n\n**A4:** Thank you for your helpful suggestion. It should be an easy fix by highlighting the existing write-ups. In the revised version, we will move a concise summary of the key limitations to the main text (end of Section 4.3), including aspects such as computational cost, assumptions about manifold structure, and interpretability challenges. We will retain the detailed discussion in the appendix for completeness but make cross-references more prominent to ensure these important considerations are accessible to readers. \n\n**Thanks again for your feedback, if the reviewer has additional questions, we would be glad to provide further clarifications.**"}]

Author rebuttal:

Meta-review: The authors propose a neural network model that integrates the temporal modeling of state-space models with geometric modeling, i.e. Riemannian manifold of SPD matrices, to learn a geometric representation of sequential data with inherent network structure. Their model is primarily targeted at modeling functional connectivity matrices in neural recordings (fMRI, EEG). GeoDynamics is demonstrated to have superior performance in multiple neuroscience datasets compared to a variety of baselines.

Four expert reviewers reviewed the paper. 
The reviewers generally agree on the novelty of the proposed network model, the principled integration of Riemannian geometry in the approach, the significance of the neuroscience application studied, and the experimental results.
There were multiple concerns regarding clarity of the presentation, however, the reviewers agreed these were addressed by the detailed explanations during the rebuttal. 
Additional experiments conducted during the rebuttal on EEG datasets were also received with positively.
After the rebuttal, the reviewers recommend acceptance. 

The AC strongly encourages the authors to take into account all feedback from the reviewers regarding clarity and definitions, as discussed during the rebuttal period, to improve the final version of the manuscript.
Decision: Accept (poster)