Biases in neural population codes with a few active neurons

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.