Information-Theoretic Bounds and Approximations in Neural Population Coding

TL;DR

This paper develops and validates asymptotic bounds and approximation formulas for mutual information in neural population coding, enabling efficient and accurate analysis of high-dimensional neural data.

Contribution

It introduces novel asymptotic bounds and convex optimization methods for approximating mutual information in large neural populations.

Findings

Asymptotic formulas are highly accurate for large neural populations.

Optimization of population density is a convex problem, enabling efficient solutions.

In some cases, the approximation formulas exactly match the true mutual information.

Abstract

While Shannon's mutual information has widespread applications in many disciplines, for practical applications it is often difficult to calculate its value accurately for high-dimensional variables because of the curse of dimensionality. This paper is focused on effective approximation methods for evaluating mutual information in the context of neural population coding. For large but finite neural populations, we derive several information-theoretic asymptotic bounds and approximation formulas that remain valid in high-dimensional spaces. We prove that optimizing the population density distribution based on these approximation formulas is a convex optimization problem which allows efficient numerical solutions. Numerical simulation results confirmed that our asymptotic formulas were highly accurate for approximating mutual information for large neural populations. In special cases, the approximation formulas are exactly equal to the true mutual information. We also discuss techniques of variable transformation and dimensionality reduction to facilitate computation of the approximations.