Neurons in many brain areas exhibit high trial-to-trial variability, with spike counts that are overdispersed relative to a Poisson distribution. Recent work (Goris, Movshon, & Simoncelli, 2014 ) has proposed to explain this variability in terms of a multiplicative interaction between a stochastic gain variable and a stimulus-dependent Poisson firing rate, which produces quadratic relationships between spike count mean and variance. Here we examine this quadratic assumption and propose a more flexible family of models that can account for a more diverse set of mean-variance relationships. Our model contains additive gaussian noise that is transformed nonlinearly to produce a Poisson spike rate. Different choices of the nonlinear function can give rise to qualitatively different mean-variance relationships, ranging from sublinear to linear to quadratic. Intriguingly, a rectified squaring nonlinearity produces a linear mean-variance function, corresponding to responses with a constant Fano factor. We describe a computationally efficient method for fitting this model to data and demonstrate that a majority of neurons in a V1 population are better described by a model with a nonquadratic relationship between mean and variance. Finally, we demonstrate a practical use of our model via an application to Bayesian adaptive stimulus selection in closed-loop neurophysiology experiments, which shows that accounting for overdispersion can lead to dramatic improvements in adaptive tuning curve estimation.

A firing rate map, also known as a tuning curve, describes the nonlinear relationship between a neuron's spike rate and a low-dimensional stimulus (e.g., orientation, head direction, contrast, color). Here we investigate Bayesian active learning methods for estimating firing rate maps in closed-loop neurophysiology experiments. These methods can accelerate the characterization of such maps through the intelligent, adaptive selection of stimuli. Specifically, we explore the manner in which the prior and utility function used in Bayesian active learning affect stimulus selection and performance. Our approach relies on a flexible model that involves a nonlinearly transformed gaussian process (GP) prior over maps and conditionally Poisson spiking. We show that infomax learning, which selects stimuli to maximize the information gain about the firing rate map, exhibits strong dependence on the seemingly innocuous choice of nonlinear transformation function. We derive an alternate utility function that selects stimuli to minimize the average posterior variance of the firing rate map and analyze the surprising relationship between prior parameterization, stimulus selection, and active learning performance in GP-Poisson models. We apply these methods to color tuning measurements of neurons in macaque primary visual cortex.

A sizable literature has focused on the problem of estimating a low-dimensional feature space capturing a neuron's stimulus sensitivity. However, comparatively little work has addressed the problem of estimating the nonlinear function from feature space to a neuron's output spike rate. Here, we use a Gaussian process (GP) prior over the infinite-dimensional space of nonlinear functions to obtain Bayesian estimates of the "nonlinearity" in the linear-nonlinear-Poisson (LNP) encoding model. This offers flexibility, robustness, and computational tractability compared to traditional methods (e.g., parametric forms, histograms, cubic splines). Most importantly, we develop a framework for optimal experimental design based on uncertainty sampling. This involves adaptively selecting stimuli to characterize the nonlinearity with as little experimental data as possible, and relies on a method for rapidly updating hyperparameters using the Laplace approximation. We apply these methods to data from color-tuned neurons in macaque V1. We estimate nonlinearities in the 3D space of cone contrasts, which reveal that V1 combines cone inputs in a highly nonlinear manner. With simulated experiments, we show that optimal design substantially reduces the amount of data required to estimate this nonlinear combination rule.