NeuroKoop: Koopman-Inspired Neural Dynamics

• NeuroKoop is a family of Koopman-inspired frameworks that transform nonlinear neural dynamics into approximately linear latent representations.
• It combines advanced lifting techniques with linear evolution to enable robust long-term forecasting, control, and causal interpretation across neuroimaging modalities.
• Applications include task fMRI, epilepsy neurostimulation, dynamic causal discovery, and multimodal connectome fusion, offering both theoretical insights and practical utility.

NeuroKoop denotes a family of Koopman-inspired learning frameworks that combine nonlinear representation learning with approximately linear evolution in a lifted, embedded, or fused latent space. In the literature, the name has been attached to several related but non-identical constructions: a Diffusion-Maps/Koopman pipeline for task fMRI (Gallos et al., 2023), a deep Koopman plus model-predictive-control framework for closed-loop electrical neurostimulation in epilepsy (Liang et al., 2021), a neural Koopman prior for forecasting and variational data assimilation under irregular sampling (Frion et al., 2023), a Koopman-inspired dynamic causal discovery model for Granger-causal inference (Adesunkanmi et al., 2024), and a graph-based structural-functional connectome fusion model for prenatal drug exposure classification (Mazumder et al., 22 Aug 2025). A closely related variant, the Neural Koopman Machine, applies the same linearization principle to longitudinal multimodal Alzheimer’s disease forecasting (Hrusanov et al., 5 Dec 2025). Across these works, the common objective is to transform nonlinear dynamics into coordinates where propagation is linear or bilinear enough to support long-horizon prediction, control, causal interpretation, or multimodal integration.

1. Conceptual scope and recurring mathematical template

At the level of mathematical form, NeuroKoop methods repeatedly instantiate the same design pattern: construct observables or latent coordinates from high-dimensional inputs, evolve those coordinates with a finite-dimensional approximation of the Koopman operator, and decode, project, or interpret the result in the original domain. Depending on the application, the lifted dynamics take forms such as $z_{t+1} \approx K z_t$, $z_{t+1} = K z_t + B u_t$, $z_t = \sum_{k=1}^p A_k z_{t-k}$, or conditioned latent updates driven by attention or subject covariates (Gallos et al., 2023, Liang et al., 2021, Adesunkanmi et al., 2024, Mazumder et al., 22 Aug 2025, Hrusanov et al., 5 Dec 2025).

Variant	Domain	Core lifted dynamics
Diffusion Maps + Koopman	Task fMRI	DM coordinates with EDMD and Koopman modes
Deep Koopman + MPC	Epileptic EEG neurostimulation	$z_{t+1}=K z_t + B u_t$
Neural Koopman prior	Forecasting and data assimilation	$z_{t+1}\approx K z_t$ or $K^{\Delta t}=\exp(\Delta t L)$
NKDCD	Nonlinear Granger causality	$z_t=\sum_{k=1}^p A_k z_{t-k}$
NeuroKoop connectome fusion	SBM/FNC classification	$Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$

A common misconception is that these frameworks assume the original neural system is linear. They do not. The linearity is imposed only in the chosen observable, embedded, or latent coordinate system, while the lifting and projection stages remain nonlinear and are implemented by Diffusion Maps, MLP autoencoders, GCN encoders, or hierarchical attention modules.

2. State lifting, manifold coordinates, and latent observable design

The most explicit manifold-based NeuroKoop construction is the fMRI framework built on Diffusion Maps. Starting from high-dimensional fMRI time series $x_1,\dots,x_N \in \mathbb{R}^M$ with $M \approx 111$ regions, it defines the Gaussian kernel

$z_{t+1} = K z_t + B u_t$0

forms the affinity matrix $z_{t+1} = K z_t + B u_t$1, row-normalizes to the Markov matrix $z_{t+1} = K z_t + B u_t$2, and solves

$z_{t+1} = K z_t + B u_t$3

The embedding is then

$z_{t+1} = K z_t + B u_t$4

with truncation at $z_{t+1} = K z_t + B u_t$5 using a spectral-gap or parsimonious criterion (Gallos et al., 2023). In that formulation, the learned coordinates are not generic neural features but diffusion eigenfunctions adapted to the observed task-dependent geometry of the data cloud.

Autoencoder-based variants replace spectral manifold coordinates with learned latent observables. In the epilepsy framework, an encoder $z_{t+1} = K z_t + B u_t$6 and decoder $z_{t+1} = K z_t + B u_t$7 are each implemented as 3-layer fully connected networks, with tanh activations on the first two layers and a linear third layer, and the latent evolution is forced to obey $z_{t+1} = K z_t + B u_t$8; an optional bias term $z_{t+1} = K z_t + B u_t$9 can be included (Liang et al., 2021). In the neural Koopman prior, the encoder $z_t = \sum_{k=1}^p A_k z_{t-k}$0 and decoder $z_t = \sum_{k=1}^p A_k z_{t-k}$1 similarly impose $z_t = \sum_{k=1}^p A_k z_{t-k}$2 and $z_t = \sum_{k=1}^p A_k z_{t-k}$3, but the framework adds linearity, reconstruction, prediction, and orthogonality terms, and extends naturally to irregular sampling by introducing a generator $z_t = \sum_{k=1}^p A_k z_{t-k}$4 such that $z_t = \sum_{k=1}^p A_k z_{t-k}$5 and $z_t = \sum_{k=1}^p A_k z_{t-k}$6 (Frion et al., 2023).

Causal-discovery and multimodal-neuroimaging variants specialize the lift to the structure of the input. NKDCD uses an MLP encoder $z_t = \sum_{k=1}^p A_k z_{t-k}$7 with two hidden layers of sizes $z_t = \sum_{k=1}^p A_k z_{t-k}$8 and leaky-ReLU activations, together with a symmetric decoder $z_t = \sum_{k=1}^p A_k z_{t-k}$9, so that a sparse linear $z_{t+1}=K z_t + B u_t$0-lag dynamics can be estimated in the lifted space (Adesunkanmi et al., 2024). NeuroKoop for connectome fusion instead begins from two graphs with 53 nodes each, one structural and one functional, uses k-nearest-neighbor sparsification with $z_{t+1}=K z_t + B u_t$1, constructs node features from rows of the structural matrix $z_{t+1}=K z_t + B u_t$2 and the functional correlation matrix $z_{t+1}=K z_t + B u_t$3, and maps them through two parallel GCN encoders to latent embeddings $z_{t+1}=K z_t + B u_t$4 with $z_{t+1}=K z_t + B u_t$5 before cross-modal attention and Koopman-like refinement (Mazumder et al., 22 Aug 2025). A related multimodal variant for Alzheimer’s disease partitions 44 input features into five biologically defined groups—genetic, CSF, PET, MRI, and demographics—and encodes each group separately before fusion, so that interpretability is tied to known biological structure rather than only to an abstract latent coordinate system (Hrusanov et al., 5 Dec 2025).

3. Forecasting, reconstruction, and out-of-sample dynamics

In task fMRI, NeuroKoop is explicitly formulated as a reduced-order modeling pipeline for long-term out-of-sample prediction. After embedding the trajectory $z_{t+1}=K z_t + B u_t$6 in $z_{t+1}=K z_t + B u_t$7, the reduced dynamics are modeled either by a feedforward neural network $z_{t+1}=K z_t + B u_t$8 trained by minimizing

$z_{t+1}=K z_t + B u_t$9

or by a Koopman approximation fitted through Extended DMD,

$z_{t+1}\approx K z_t$0

The pre-image problem is then solved either through Geometric Harmonics, using Nyström extension of the DM eigenvectors, or directly through Koopman modes via

$z_{t+1}\approx K z_t$1

The embedding dimension was set to $z_{t+1}\approx K z_t$2 after examining the spectral gap; prediction error was measured by out-of-sample RMSE and $z_{t+1}\approx K z_t$3 norm over the last 80 of 360 time points; and the first five DM coordinates explained more than 90% of the variance, indicating intrinsic dimensionality $z_{t+1}\approx K z_t$4 for the visuo-motor task. Typical RMSE values for four key regions were 0.49, 0.50, 0.60, and 0.59 for FNN+GH; 0.51, 0.55, 0.58, and 0.63 for Koopman; and 0.55, 0.70, 0.65, and 0.70 for the naive random-walk baseline in the Left Calcarine Sulcus, Left Middle Occipital, Right Lingual Gyrus, and Left Superior Occipital respectively (Gallos et al., 2023). A central conclusion of that study is that Koopman on DM observables gives, for practical purposes, results equivalent to the FNN-GH route while bypassing both nonlinear decoder training and GH extrapolation.

The neural Koopman prior generalizes the forecasting problem to long-term continuous reconstruction and variational data assimilation. Its training objective combines

$z_{t+1}\approx K z_t$5

into

$z_{t+1}\approx K z_t$6

and uses either discrete powers $z_{t+1}\approx K z_t$7 or continuous propagators $z_{t+1}\approx K z_t$8 when observations are irregularly sampled. It supports a weakly constrained variational objective over the full trajectory,

$z_{t+1}\approx K z_t$9

and a strongly constrained objective over the initial latent condition,

$K^{\Delta t}=\exp(\Delta t L)$0

Empirically, the model achieved MSE of approximately $K^{\Delta t}=\exp(\Delta t L)$1 on a simulated 3-D fluid-flow task, remained stable across training frequencies, and—with orthogonality regularization—rolled dynamics backwards with MSE of approximately $K^{\Delta t}=\exp(\Delta t L)$2 versus approximately $K^{\Delta t}=\exp(\Delta t L)$3 without. On Sentinel-2 forest reflectance sequences, it improved forecasting, assimilation-forecasting, and interpolation metrics relative to LSTM or DMD in several settings, including an assimilation-forecasting MSE of approximately $K^{\Delta t}=\exp(\Delta t L)$4 in Fontainebleau and $K^{\Delta t}=\exp(\Delta t L)$5 in Orléans for the NeuroKoop+orth variant (Frion et al., 2023). Although that work is not limited to neuroscience, it sharpens a core NeuroKoop theme: linear latent propagation is used not only for forecasting but also as a differentiable prior for inverse problems.

4. Control-oriented NeuroKoop and intervention design

The epilepsy neurostimulation framework extends NeuroKoop from passive prediction to active control. It begins from a nonlinear controlled system

$K^{\Delta t}=\exp(\Delta t L)$6

learns a latent autoencoder representation $K^{\Delta t}=\exp(\Delta t L)$7, imposes

$K^{\Delta t}=\exp(\Delta t L)$8

and alternates between gradient-based updates of the encoder-decoder parameters and closed-form ridge-regularized least-squares estimation of $K^{\Delta t}=\exp(\Delta t L)$9 and $z_t=\sum_{k=1}^p A_k z_{t-k}$0:

$z_t=\sum_{k=1}^p A_k z_{t-k}$1

Training uses a reconstruction loss and a multi-step prediction loss,

$z_t=\sum_{k=1}^p A_k z_{t-k}$2

after which the learned latent linear dynamics are embedded inside a convex quadratic MPC problem with state, input, and input-increment constraints (Liang et al., 2021).

The empirical motivation for this construction is computational as much as predictive. On synthetic Jansen–Rit data, 10-step-ahead prediction yielded MSE of approximately 31.6 for VAR(5), approximately 4.21 for a GRU-Cell, and approximately 1.58 for the Koopman model. On the Epileptor system, the reported MSEs were approximately 0.053 for VAR(5), approximately 0.035 for Koopman, and approximately 0.010 for GRU, with the GRU using much larger parameter counts. On real 84-channel iEEG, a 10-order VAR achieved test MSE of approximately 0.100, the Koopman model approximately 0.072, and LSTM/GRU models with 400k parameters overfit to a test error of approximately 0.125. In closed-loop simulation, Koopman-MPC suppressed seizure dynamics with runtime of approximately 0.022 s per step on the Jansen–Rit model and approximately 0.015 s per step on the Epileptor model, compared with approximately 0.114 s and approximately 0.113 s per step for RNN-MPC; the corresponding optimization was reported to be solvable in $z_t=\sum_{k=1}^p A_k z_{t-k}$3 ms on CPU because the latent dynamics are linear in $z_t=\sum_{k=1}^p A_k z_{t-k}$4 (Liang et al., 2021).

A related control line, not restricted to neural measurements, is Neural Koopman Lyapunov Control. There the lift is learned together with a bilinear controlled model

$z_t=\sum_{k=1}^p A_k z_{t-k}$5

an autoencoder, and a Control Lyapunov Function network

$z_t=\sum_{k=1}^p A_k z_{t-k}$6

The lifted discrete dynamics are estimated in EDMD style, Lyapunov decrease is enforced through a soft loss and an SMT-based falsifier, and the learner-falsifier loop continues until the falsifier returns “UNSAT.” The resulting controller is tied to Sontag’s formula and carries asymptotic-stability guarantees for the learned bilinear representation. Simulations on the inverted pendulum, Van der Pol oscillator, cart-pole, and spacecraft rendezvous drove 10 random initial states to the origin in every case (Zinage et al., 2022). This related framework shows that the NeuroKoop program has a strong control-theoretic branch in which stabilizability is learned jointly with the embedding.

5. Causal discovery, connectome fusion, and multimodal biomarker forecasting

NeuroKoopman Dynamic Causal Discovery (NKDCD) recasts nonlinear Granger-causal inference as sparse linear dynamics in a learned lifted space. For multivariate series $z_t=\sum_{k=1}^p A_k z_{t-k}$7, it learns $z_t=\sum_{k=1}^p A_k z_{t-k}$8, imposes

$z_t=\sum_{k=1}^p A_k z_{t-k}$9

decodes by $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$0, and trains with four mean-square terms:

$Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$1

plus a sparsity-inducing group-lasso penalty on the lag blocks $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$2. Granger causality is then declared by block nonzero structure: variable $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$3 Granger-causes variable $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$4 if any block $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$5. On finance data with linear ground truth, NKDCD recovered AUROC of approximately 99.9% and AUPR of approximately 98%, matching sparse VAR. On Lorenz-96, it reached AUROC up to 99.8% for $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$6 and 98.7% for $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$7 at long sequence length. On simulated fMRI BOLD with 5 brain regions and 2,400 time points, NKDCD obtained AUROC of approximately 95% versus 79% for sparse-VAR. On the DREAM3 gene-regulatory benchmark, it yielded AUROC of approximately 80–81% and AUPR of approximately 25–51%, outperforming sparse-VAR and cMLP/cLSTM baselines (Adesunkanmi et al., 2024).

The 2025 NeuroKoop connectome-fusion model targets classification rather than time-series forecasting, but it preserves the central Koopman idea of latent linearizable evolution. Structural input is obtained from a 53-dimensional SBM loading vector $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$8 with structural matrix $Z_{t+1}=U_\psi(Z_t)\odot M_\psi(c)$9; functional input is an FNC matrix $x_1,\dots,x_N \in \mathbb{R}^M$0 formed from pairwise Pearson correlations of rs-fMRI ICA time courses. Two GCN encoders produce $x_1,\dots,x_N \in \mathbb{R}^M$1, a bidirectional cross-modal attention layer yields $x_1,\dots,x_N \in \mathbb{R}^M$2, and the latent fusion is refined by

$x_1,\dots,x_N \in \mathbb{R}^M$3

where $x_1,\dots,x_N \in \mathbb{R}^M$4 is a scalar working-memory score. After $x_1,\dots,x_N \in \mathbb{R}^M$5 steps, global average pooling and an MLP head produce the exposure probability. Training minimizes a binary cross-entropy term plus an adversarial alignment loss with $x_1,\dots,x_N \in \mathbb{R}^M$6, using stratified 5-fold cross-validation on 860 subjects, evenly split between 430 prenatal-drug-exposed cases and 430 controls. Reported performance was 82.33 ± 1.97% accuracy, 82.49 ± 1.78% precision, 82.09 ± 3.42% recall, and 82.26 ± 2.15% $x_1,\dots,x_N \in \mathbb{R}^M$7, compared with 77.45 ± 2.12% accuracy for BrainNN, 75.67 ± 1.30% for Joint DCCA, 74.53 ± 1.86% for Joint GCN, and 68.94 ± 1.23% for random forest. Removing working-memory scores reduced accuracy from 82.33% to 80.16%, while removing the Koopman layer reduced performance to approximately 70.1%. The strongest structural-to-functional attention edges differed between controls and exposed subjects, with controls showing broadly distributed SMN–VSN–CON–DMN couplings and exposed subjects showing more focused DMN–CON–CBN coupling (Mazumder et al., 22 Aug 2025).

A related multimodal forecasting variant, the Neural Koopman Machine, addresses longitudinal Alzheimer’s disease progression. It groups 44 features into five modalities, learns modality-specific encoders, forms refined latent states over a sliding window of $x_1,\dots,x_N \in \mathbb{R}^M$8 visits, computes temporal attention $x_1,\dots,x_N \in \mathbb{R}^M$9, feature-group attention $M \approx 111$0, and a gated fusion control $M \approx 111$1, and evolves the latent state by

$M \approx 111$2

with spectral regularization

$M \approx 111$3

On ADNI with subject-stratified 5-fold cross-validation, it reported Pearson $M \approx 111$4, Spearman $M \approx 111$5, MAE $M \approx 111$6, and RMSE $M \approx 111$7, outperforming LSTM, AttnRNN, TimesNetTiny, XGBoost, Neural ODE, and EDMD with RBF basis on most reported metrics. Attention and saliency analyses attributed predictive importance to MRI networks in the Default Mode, Limbic, and Temporal-Parietal systems, hippocampal and ventricular volumes, CSF p-tau, FDG-PET, APOE4, and ICV, with DefaultA regions BA39 and BA10 highlighted for ADAS13 (Hrusanov et al., 5 Dec 2025). This related line shows how Koopman-style latent linearization can be integrated with biologically structured attention for personalized biomarker forecasting.

6. Interpretability, misconceptions, and future directions

Interpretability in NeuroKoop is not a single mechanism but a family of operator-theoretic readouts. In the Diffusion-Maps fMRI model, Koopman modes $M \approx 111$8 are described as physically interpretable spatio-temporal patterns, each corresponding to a global brain-wide pattern evolving at rate $M \approx 111$9 (Gallos et al., 2023). In the neural Koopman prior, the eigenstructure of $z_{t+1} = K z_t + B u_t$00 is used to expose growth and decay rates together with frequencies, and orthogonality regularization is introduced to stabilize long-term and even backward reconstruction (Frion et al., 2023). In NKDCD, interpretability is combinatorial: zero versus nonzero lag blocks provide an explicit Granger-causal graph (Adesunkanmi et al., 2024). In multimodal clinical forecasting, interpretability is driven by average attention weights, gradient-based saliency, and modality-aware grouping rather than only by eigenmodes (Hrusanov et al., 5 Dec 2025).

A second recurring point is that “linear” does not mean “simplistic.” The Koopman operator approach in the fMRI ROM was reported to give, for practical purposes, equivalent results to the FNN-GH scheme while removing the need to train a nonlinear decoder and to perform Geometric Harmonics lifting (Gallos et al., 2023). In closed-loop epilepsy control, the benefit of latent linearity is operational: it converts the downstream MPC problem into a convex quadratic program with a unique global optimum and markedly lower runtime than RNN-MPC (Liang et al., 2021). This suggests that, in NeuroKoop systems, the practical value of linearization is often less about exact spectral theory in the abstract and more about identifiability, optimization geometry, and long-horizon numerical stability.

The literature also indicates several extension paths. The task-fMRI framework explicitly proposes application to EEG/MEG, high-resolution fMRI, multi-subject generalization, and closed-loop control by selecting modes of interest (Gallos et al., 2023). The epilepsy study points to TMS, tDCS, VNS, and network-level control, and frames online Koopman re-estimation as a route to controllability and observability analyses in neural systems (Liang et al., 2021). The connectome-fusion model proposes more expressive operators, self-supervised or contrastive pretraining, time-resolved fMRI, analytic Koopman modes, and integration of additional modalities such as diffusion MRI and EEG (Mazumder et al., 22 Aug 2025). Outside neuroscience proper, the same design principle has already been transferred to mesh-free operator learning for nonlinear PDEs in the Koopman Neural Operator, where linear latent propagation, FFT truncation, and reconstruction losses are used to obtain mesh-independent and long-term predictions with slower error growth than several neural-operator baselines (Xiong et al., 2023). Taken together, these works indicate that NeuroKoop is best understood not as one fixed architecture but as a general strategy: choose observables well, linearize where possible, and exploit that linearization for prediction, inference, or control without discarding domain structure.