Optimal Neuronal Tuning for a Finite Stimulus Space

Abstract

A central question in sensory neuroscience is how neurons should be tuned to optimally encode a set of stimuli. Classical theory assumes an infinite or continuous stimulus space and derives broad or narrow tuning curves depending on the decoding scheme. Here, we consider the realistic case of a finite and discrete stimulus space. We derive analytically the optimal tuning curve shape and width that maximizes the mutual information between stimuli and neural responses for a finite number of stimuli. Our results show that the optimal tuning width depends strongly on the number of distinct stimuli in the environment and the noise level, and that classical results emerge as limiting cases. These findings have implications for understanding the diversity of tuning curve widths observed across sensory cortices and for theories of population coding.

Introduction

The question of optimal neural coding has fascinated neuroscientists and theorists for decades. How should a neuron's tuning curve — its response as a function of stimulus value — be shaped to maximize information transmission? Classic analyses by Paradiso, Seung, and Sompolinsky, among others, have addressed this for continuous stimulus spaces. However, real-world sensory environments contain finite sets of biologically relevant stimuli.

We extend the theory of optimal tuning to finite stimulus spaces and derive closed-form solutions for optimal tuning width as a function of the number of stimuli and noise characteristics, providing a unifying framework for understanding tuning diversity in the brain.

Key Findings

• Optimal tuning width is a non-monotonic function of stimulus set size.
• For small stimulus sets, broad tuning maximizes information; for large sets, narrow tuning is preferred.
• Classical continuous-space results emerge as limiting cases of the finite-space theory.
• Predictions align with empirical observations of tuning diversity across sensory areas.

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