# Biases in neural population codes with a few active neurons

## TL;DR

This paper shows that biases in decoding neural population codes naturally emerge when only a few neurons are active, even with optimal decoding methods.

## Contribution

The study reveals that biases in population coding arise naturally with few active neurons and introduces a technique to estimate these biases.

## Key findings

- Biases in decoding emerge naturally when only a few neurons are active.
- Biases can be attractive or repulsive depending on the stimulus value.
- An approximation technique for Bayesian estimator decoders is introduced.

## Abstract

Throughout the brain information is coded in the activity of multiple neurons at once, so called population codes. Population codes are a robust and accurate way of coding information. One can evaluate the quality of population coding by trying to read out the code with a decoder, and estimate the encoded stimulus. In particular when neurons are noisy, coding accuracy has extensively been evaluated in terms of the trial-to-trial variation in the estimate. While most decoders yield unbiased estimators if many neurons are actived, when only a few neurons are active, biases readily emerge. That is, even after averaging, a systematic difference between the true stimulus and its estimate remains. We characterize the shape of this bias for different encoding models (rectified cosine tuning and von Mises functions) and show that it can be both attractive or repulsive for different stimulus values. Biases appear for maximum likelihood and Bayesian decoders. The biases have a non-trivial dependence on noise. We also introduce a technique to estimate the bias and variance of Bayesian least square decoders. The work is of interest to those studying neural populations with a few active neurons.

The way information is represented in neurons is a fundamental property of computation in neural systems. In most brain regions, a single stimulus leads to the activity of multiple neurons. Such a so called population code combines high representational capacity with robustness against neural death and noise. Numerous studies have studied decoders of noisy population activity that minimize the trial-to-trial variance in the estimate, thereby revealing fundamental limits to the code. However, decoding can also be biased, that is, even in the limit of an infinite number of observations, a difference between the estimate and the actual value of a stimulus remains. Biases have been occasionally studied and appeared to emerge only in special situations (sub-optimal decoders, the presence of multiple stimuli, or non-uniform stimulus priors). Here we show that biases already emerge naturally when only a small population of neurons is active. The bias emerges for all common decoding methods. The biases have a complex dependence on neural tuning curves and noise, but we develop an effective approximation technique for the Bayesian estimator decoder. The work is of importance for studying how neuron populations with few active neurons encode information.