# Periodic Neural Codes and Sound Localization in Barn Owls

## TL;DR

This paper introduces periodic neural codes inspired by barn owl sound localization, analyzes their properties, and explores how they can be completed to convex codes, providing insights into neural coding and localization errors.

## Contribution

The study defines and characterizes periodic codes, introduces methods for convex completion, and applies algebraic tools to analyze neural codes related to sound localization.

## Key findings

- Periodic codes are generally not convex but can be completed to convex codes with noise.
- The convex closure of a periodic code can be explicitly described.
- The probability of convex closure formation is higher for sparser codes.

## Abstract

Inspired by the sound localization system of the barn owl, we define a new class of neural codes, called periodic codes, and study their basic properties. Periodic codes are binary codes with a special patterned form that reflects the periodicity of the stimulus. Because these codes can be used by the owl to localize sounds within a convex set of angles, we investigate whether they are examples of convex codes, which have previously been studied for hippocampal place cells. We find that periodic codes are typically not convex, but can be completed to convex codes in the presence of noise. We introduce the convex closure and Hamming distance completion as ways of adding codewords to make a code convex, and describe the convex closure of a periodic code. We also find that the probability of the convex closure arising stochastically is greater for sparser codes. Finally, we provide an algebraic method using the neural ideal to detect if a code is periodic. We find that properties of periodic codes help to explain several aspects of the behavior observed in the sound localization system of the barn owl, including common errors in localizing pure tones.