Estimating Noise Correlations Across Continuous Conditions With Wishart Processes

Abstract: The signaling capacity of a neural population depends on the scale and orientation of its covariance across trials. Estimating this "noise" covariance is challenging and is thought to require a large number of stereotyped trials. New approaches are therefore needed to interrogate the structure of neural noise across rich, naturalistic behaviors and sensory experiences, with few trials per condition. Here, we exploit the fact that conditions are smoothly parameterized in many experiments and leverage Wishart process models to pool statistical power from trials in neighboring conditions. We demonstrate that these models perform favorably on experimental data from the mouse visual cortex and monkey motor cortex relative to standard covariance estimators. Moreover, they produce smooth estimates of covariance as a function of stimulus parameters, enabling estimates of noise correlations in entirely unseen conditions as well as continuous estimates of Fisher information--a commonly used measure of signal fidelity. Together, our results suggest that Wishart processes are broadly applicable tools for quantification and uncertainty estimation of noise correlations in trial-limited regimes, paving the way toward understanding the role of noise in complex neural computations and behavior.

• The paper introduces an innovative method using Wishart processes to estimate noise correlations in neural populations, particularly across continuous experimental conditions.
• This approach leverages the smooth variation across conditions and combines Gaussian processes with Wishart processes to accurately model covariance structures even with limited trial data.
• The method outperforms standard estimators on simulated and real neural data, allowing robust noise covariance estimation with fewer trials and enabling predictions for unseen conditions.

Estimating Noise Correlations Across Continuous Conditions With Wishart Processes

The paper at hand presents an innovative method for estimating noise covariance in neural populations using Wishart processes, particularly when conditions vary continuously. Traditionally, estimating noise covariance has been challenging due to the high dimensionality of neural data and the limited number of trials. Conventional estimators often struggle as the number of trials per condition is usually less than the number of recorded neurons. The authors propose leveraging the intrinsic smoothness across continuous conditions to make accurate noise covariance estimates feasible even in trial-limited scenarios.

Core Contributions

The paper's primary innovation is the application of Wishart process models to infer noise correlations across smoothly parameterized experimental conditions. Unlike conventional methods that rely heavily on simplistic assumptions or large numbers of trials, Wishart processes capitalize on the assumption that noise features change smoothly over neighboring conditions. These processes allow the estimation of covariance matrices even when whole conditions remain unsampled.

1. Model Framework: The method employs Gaussian processes to model the smooth variation in neural responses and Wishart processes to model covariance structures. This innovative approach extends Bayesian methods for covariance estimation to situations with rich and naturalistic behavior.
2. Mathematical Rigor: Wishart processes model covariance matrices by generating them as products of Gaussian process vectors. This formulation leverages smoothness assumptions inherent in many neural experiments, allowing the pooling of trial information across neighboring conditions.
3. Theoretical Implications: The authors demonstrate that the method outperforms standard estimators, especially in cases where the number of trials per condition is small relative to the number of neurons. The model's ability to interpolate and extrapolate noise covariance across unseen conditions significantly aids in understanding complex neural computations.
4. Practical Applications: The method was tested on both simulated data and real neural datasets from mouse visual cortex and monkey motor cortex. In these applications, the Wishart process models showed superior performance over traditional methods like the empirical covariance and the Ledoit-Wolf estimator, providing robust noise covariance estimates with fewer trials.

Implications for Neural Data Analysis

1. Enhanced Covariance Estimation: This method allows neuroscientists to infer noise correlations with improved precision and fewer data requirements, contributing to more accurate models of neural encoding and decoding.
2. Versatility: By enabling covariance estimation over continuous parameter spaces, these models can be applied across different sensory modalities or motor tasks, wherever smooth parameterization exists.
3. Theoretical Understanding: The continuous estimation of Fisher information, afforded by the model, facilitates a deeper understanding of the neural code fidelity across conditions—linking noise properties directly to behavioral performance predictions.
4. Predictive Capacity: The ability to estimate noise covariance for unseen conditions represents a significant advancement, offering a predictive dimension that was previously unattainable with standard methods. This aspects offers prospects for real-time applications in neuroprosthetics and adaptive neural coding models.

Future Directions

While this methodology has shown promise, further investigation is warranted to generalize it to more complex stimulus spaces, such as those containing high-dimensional or naturalistic stimuli. Additionally, future work could explore alternative noise models, such as multivariate Poisson processes, that may better capture neural spiking data inherent stochastic properties. The potential to integrate more sophisticated variational inference techniques could also enhance the estimation accuracy and computational efficiency.

In summary, the paper introduces a mathematically elegant and practically impactful methodology for estimating noise covariance in neural systems. By leveraging the natural smoothness of neural response properties, the proposed Wishart process models offer a robust framework for neural data analysis, opening new avenues in the understanding and application of complex neural data.