Characterizing Neural Manifolds' Properties and Curvatures using Normalizing Flows

Abstract: Neuronal activity is found to lie on low-dimensional manifolds embedded within the high-dimensional neuron space. Variants of principal component analysis are frequently employed to assess these manifolds. These methods are, however, limited by assuming a Gaussian data distribution and a flat manifold. In this study, we introduce a method designed to satisfy three core objectives: (1) extract coordinated activity across neurons, described either statistically as correlations or geometrically as manifolds; (2) identify a small number of latent variables capturing these structures; and (3) offer an analytical and interpretable framework characterizing statistical properties by a characteristic function and describing manifold geometry through a collection of charts. To this end, we employ Normalizing Flows (NFs), which learn an underlying probability distribution of data by an invertible mapping between data and latent space. Their simplicity and ability to compute exact likelihoods distinguish them from other generative networks. We adjust the NF's training objective to distinguish between relevant (in manifold) and noise dimensions (out of manifold). Additionally, we find that different behavioral states align with the components of the latent Gaussian mixture model, enabling their treatment as distinct curved manifolds. Subsequently, we approximate the network for each mixture component with a quadratic mapping, allowing us to characterize both neural manifold curvature and non-Gaussian correlations among recording channels. Applying the method to recordings in macaque visual cortex, we demonstrate that state-dependent manifolds are curved and exhibit complex statistical dependencies. Our approach thus enables an expressive description of neural population activity, uncovering non-linear interactions among groups of neurons.

• The paper introduces Normalizing Flows to capture non-linear neural manifold geometry and curvature.
• It leverages volume-preserving bijective mappings to extract latent neural structures, surpassing traditional PCA.
• The approach revealed behavior-dependent curved manifolds in macaque visual cortex data through advanced statistical analysis.

Characterizing Neural Manifolds' Properties and Curvatures using Normalizing Flows

Introduction

The paper "Characterizing Neural Manifolds' Properties and Curvatures using Normalizing Flows" (2506.12187) introduces a novel methodology to analyze neuronal activity, which is represented as low-dimensional manifolds in high-dimensional neuron space. Traditional methods often employ linear techniques such as PCA, which impose limitations due to assumptions of Gaussian data distributions and flat manifolds. The paper employs Normalizing Flows (NFs), an advanced neural network paradigm, to address these limitations by learning the data's distribution through invertible mappings. This approach facilitates the analysis of complex, state-dependent neural manifolds, capturing their curvature and statistical dependencies.

Method and Implementation

The methodology integrates NFs to achieve three core objectives:

1. Extract coordinated neural activity, geometrically and statistically.
2. Identify latent variables representing the underlying neural structure.
3. Offer an interpretative framework for characterizing statistical properties via characteristic functions and manifold geometry through local charts.

Architectural Approach

Utilizing volume-preserving constraints, the flow architecture ensures that the Jacobian's determinant remains constant, thus simplifying likelihood calculations in a multi-dimensional space. The architecture adopts the following key features:

• Normalizing Flows: Enable bijective mapping between data and latent spaces, preserving volume and allowing exact likelihood computation.
• Curvature and Statistical Analysis: Bivariate Gaussian mixtures provide flexibility in modeling multimodal neural activity, reflecting behavior states as different curved manifolds.

This methodology moves beyond simplistic Gaussian and linear models by incorporating non-linear interactions, crucial for understanding neurophysiological signals.

Practical Applications

Application to Visual Cortex Data

The authors applied their method to macaque visual cortex data, demonstrating that combinations of latent variables align with behaviorally distinct states. The NFs identified curved manifolds and revealed rich, non-linear statistical interactions among neuron groups. This showcase highlights the utility of NFs in uncovering state-dependent manifold properties, supporting substantial deviations from Gaussian assumptions.

Analytical Insights

By approximating the NF mappings with quadratic functions, the methodology provides analytical tractability. This simplification is pivotal for interpreting manifold geometry and evaluating statistical dependencies. The quadratic approximation allowed the calculation of cumulants, revealing dimensional dependencies and highlighting manifold curvature, captured by Riemannian metrics.

Geometrical and Statistical Interpretations

The paper's approach interlinks statistical correlations and geometry, revealing neural representations with characteristic curvature signatures. The computed sectional and scalar curvatures, derived from a quadratic mapping approximation, offered insights into how neural interactions project onto behavioral states.

Conclusion

This research advances the interpretative capabilities of large-scale neural recordings, empowering AI applications to discern non-linearities in neural manifolds. The proposed methodology's strength lies in its ability to marry sophisticated generative models with differential geometry principles, enhancing the understanding of high-dimensional neural data through meaningful dimension reduction and manifold characterization.

Future Directions

Potential further developments include extending the framework to accommodate dynamic manifold changes over time, adapting to larger datasets, and exploring more diverse neural activity. This foundation promises significant implications for neural decoding, brain-computer interfaces, and the broader field of computational neuroscience.