NeuralFieldManifold

Recover low-dimensional manifold geometry from noisy, autocorrelated time series

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Overview

Standard manifold learning methods break down on signals with strong temporal autocorrelation — oscillations, 1/f noise, and nonstationarity corrupt the geometry that dimensionality reduction is meant to reveal.

NeuralFieldManifold provides a principled solution grounded in dynamical systems theory: model the signal as a (time-varying) autoregressive process, then exploit the analytical link between oscillatory spectral structure and the topology of delay embeddings. The core theoretical result is that K sustained oscillatory modes produce a K-torus in the lag-embedded state space — and aperiodic background only adds thickness, not topology.

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Physics-Informed Reconstruction of the Manifold

To recover the predicted toroidal geometry from real, nonstationary neural recordings, we introduce DeepLagField — a deep learning model that estimates time-varying autoregressive (TVAR) coefficients with physics-informed constraints. It features time-varying AR estimation via a neural network backbone with adaptive order selection and physics-informed losses — bounded energy, temporal smoothness of coefficients, and autoregressive reconstruction error.

_{The input local field potential (LFP) signal is processed by two coupled modules. The order block predicts a soft selection over candidate autoregressive orders, producing a sparse mask that determines the effective lag set. Conditioned on this mask, the dynamic block outputs time-varying autoregressive coefficients $\{\phi_k(t)\}$, enabling a locally stationary TVAR representation.}

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Installation

pip install -e .

# or with dev/docs extras
pip install -e ".[dev,docs]"

Requires: Python ≥ 3.10, PyTorch, JAX, NumPy, SciPy, scikit-learn.

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Quick Start

from NeuralFieldManifold.generators import sinusoid, tvar
from NeuralFieldManifold.embedders import embed
from NeuralFieldManifold.models import DeepLagEmbed

# generate a synthetic time-varying AR signal
coeffs = sinusoid(T=10000, order=4)
x = tvar(coeffs, noise_std=0.1)

# delay embedding
z = embed(x, m=6, tau=15)

# learn time-varying AR coefficients + automatic order selection
import torch
model = DeepLagEmbed(seq_len=600, max_ar_order=6)
coeffs_hat, p_logits, p_hard, x_hat = model(
    torch.tensor(x[:600], dtype=torch.float32).unsqueeze(0)
)

Gallery

_{Monkey LFP manifold after lag embedding}

_{Mouse LFP manifold reconstruction}

_{Mouse EEG manifold reconstruction}

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Citation

If you use this package in your research, please cite:

@inproceedings{fallah2026neuralfieldmanifold,
  title     = {{NeuralFieldManifold}: Reconstruction of {LFP} Manifold with Lag Embedding},
  author    = {Fallah, Kasra and Chen, Haoyu Novak and Singha, Rudramani and Kong, Eunji and Turi, Georgo and Losonczy, Attila and Zabeh, Erfan},
  booktitle = {Preprint},
  year      = {2026},
  url       = {https://arxiv.org/abs/XXXX.XXXXX},
}

Contributors

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