Neural representation of probabilities for Bayesian inference

Abstract

Bayesian models are often successful in describing perception and behavior, but the neural representation of probabilities remains in question. There are several distinct proposals for the neural representation of probabilities, but they have not been directly compared in an example system. Here we consider three models: a non-uniform population code where the stimulus-driven activity and distribution of preferred stimuli in the population represent a likelihood function and a prior, respectively; the sampling hypothesis which proposes that the stimulus-driven activity over time represents a posterior probability and that the spontaneous activity represents a prior; and the class of models which propose that a population of neurons represents a posterior probability in a distributed code. It has been shown that the non-uniform population code model matches the representation of auditory space generated in the owl's external nucleus of the inferior colliculus (ICx). However, the alternative models have not been tested, nor have the three models been directly compared in any system. Here we tested the three models in the owl's ICx. We found that spontaneous firing rate and the average stimulus-driven response of these neurons were not consistent with predictions of the sampling hypothesis. We also found that neural activity in ICx under varying levels of sensory noise did not reflect a posterior probability. On the other hand, the responses of ICx neurons were consistent with the non-uniform population code model. We further show that Bayesian inference can be implemented in the non-uniform population code model using one spike per neuron when the population is large and is thus able to support the rapid inference that is necessary for sound localization.

Fig. 1

Models for the neural implementation of Bayesian inference a The non-uniform population code model of Fischer and Peña (2011) proposes that the neural activity encodes the likelihood (red) and the prior is represented in the distribution of preferred directions of midbrain neurons. b The sampling hypothesis predicts that neural activity represents samples from the posterior. A histogram of the neural activity over time approximates the posterior. c A population may directly encode the posterior in the activities of neurons that are uniformly distributed

Fig. 2

Spontaneous and stimulus-driven activity in the midbrain (a) Auditory midbrain neurons in the owl have spatially restricted receptive fields (black) and low spontaneous activity. The spontaneous activity (red) is lower than the mean stimulus-driven activity (blue). (b) The mean spontaneous firing rate is significantly lower than the mean stimulus-driven firing rate, in contrast to the prediction of the sampling hypothesis. The mean stimulus-driven firing rate was computed using the Gaussian prior from the Bayesian model (gray) and the uniform distribution used in data collection (black)

Fig. 3

Variability over time in midbrain responses (a–f) The variability over time in median-filtered (black) membrane potential responses is higher for correlated sounds (a,d) than for uncorrelated sounds (b,e), in contrast to the prediction of the sampling hypothesis. This occurs for stimulus conditions where spiking occurs (a–c) and where spiking is absent (d–f). Error bars in (c) and (f) are bootstrap standard deviations

Fig. 4

Testing the representation of the posterior a The Bayesian model shows that the posterior shifts toward prior as BC decreases. b Preferred ITD of midbrain neurons doesn't shift toward zero as BC decreases. c Best ITD for perfectly correlation sounds (BC = 1) and decorrelated sounds (BC = 0.25 for intracellular, BC = 0.3 for extracellular). The dashed line is the regression line. d Predicted responses at the best ITD as BC varies needed to represent the posterior. e Measured responses at the best ITD as BC varies in the optic tectum (adapted from Saberi et al. 1998)

Fig. 5

Instantaneous inference a Responses of neurons in the model of Fischer and Peña (2011) for correlated sounds (top; BC = 1), decorrelated sounds (middle, BC = 0.5) and uncorrelated sounds (bottom; BC = 0). Jitter was added to the firing rates for display. b The population vector decoder matches the Bayesian estimate when single spikes per neuron are used. c Accurate neural inference with the population vector and single spikes per neuron requires a large population