NeuralFieldManifold#
Recover low-dimensional manifold geometry from noisy, autocorrelated time series
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.
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.
Installation#
pip install -e .
# or with dev/docs extras
pip install -e ".[dev,docs]"
Requires: Python ≥ 3.10, PyTorch, JAX, NumPy, SciPy, scikit-learn.
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
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},
}