Modeling correlated noise is necessary to decode uncertainty

Highlights

• Information in a cortical response is best characterized by a probability distribution.

• Accurate decoding of this distribution requires accounting for correlated noise.

• Correlated noise is most relevant when it mimics a stimulus-driven response.

• Including this noise in fMRI forward models enables the estimation of cortical uncertainty.

Abstract

Brain decoding algorithms form an important part of the arsenal of analysis tools available to neuroscientists, allowing for a more detailed study of the kind of information represented in patterns of cortical activity. While most current decoding algorithms focus on estimating a single, most likely stimulus from the pattern of noisy fMRI responses, the presence of noise causes this estimate to be uncertain. This uncertainty in stimulus estimates is a potentially highly relevant aspect of cortical stimulus processing, and features prominently in Bayesian or probabilistic models of neural coding. Here, we focus on sensory uncertainty and how best to extract this information with fMRI. We first demonstrate in simulations that decoding algorithms that take into account correlated noise between fMRI voxels better recover the amount of uncertainty (quantified as the width of a probability distribution over possible stimuli) associated with the decoded estimate. Furthermore, we show that not all correlated variability should be treated equally, as modeling tuning-dependent correlations has the greatest impact on decoding performance. Next, we examine actual noise correlations in human visual cortex, and find that shared variability in areas V1-V3 depends on the tuning properties of fMRI voxels. In line with our simulations, accounting for this shared noise between similarly tuned voxels produces important benefits in decoding. Our findings underscore the importance of accurate noise models in fMRI decoding approaches, and suggest a statistically feasible method to incorporate the most relevant forms of shared noise.

Introduction

What sensory stimulus evoked this particular pattern of cortical activity? This question lies at the heart of brain decoding algorithms. Most fMRI decoders will estimate or ‘decode’ a single stimulus value that is, according to some underlying model, most consistent with the observed data. In truth, however, there is rarely just one stimulus that provides a plausible explanation. Rather, different stimuli may all be somewhat consistent with the measured response. The main reason for this imprecision is variability (or noise), which causes even the same stimulus to elicit different activity patterns each time the stimulus is presented. By the same token, variability allows the same response pattern to be evoked by a range of different stimuli. From a noisy response pattern, therefore, the stimulus that elicited the pattern cannot be inferred with perfect precision. Rather, there is some degree of uncertainty in the predictions, and this uncertainty may vary from one decoded activity pattern to the next. Importantly, while uncertainty may stem from imprecise measurements, it can also be of neural origin. Since neural responses themselves are inherently noisy (Dean, 1981, Schiller et al., 1976), cortical activity cannot encode stimulus information with perfect precision, and this imprecision can, moreover, fluctuate over time. Uncertainty, thus, is an important feature of cortical stimulus representations, providing a window on the fidelity of neural stimulus processing from one moment to the next.

How can uncertainty be measured from cortical activity patterns? Mathematically speaking, the presence of uncertainty means that cortical information is most accurately characterized by a probability distribution over all possible stimuli. The wider this distribution, the larger is the range of stimuli that could have evoked the observed pattern of cortical activity, and hence, the higher is the uncertainty. In order to measure uncertainty, therefore, we should estimate probability distributions. But how can this be achieved, and why is this not possible with conventional decoding algorithms? Recall that uncertainty largely stems from noise in the data. This noise turns the causal relationship between stimuli and responses from a deterministic to a stochastic one, described by probabilities rather than fixed outcomes. To estimate probabilities, therefore, a decoding algorithm should capture this noisy, stochastic relationship. Mathematical models that describe the causal link between stimuli and cortical activity are typically known as forward or generative models, and have become a popular tool to describe (and extract information from) fMRI activity. Importantly, however, most models to date assume that noise is simply independent between fMRI voxels (e.g. Kay et al., 2008, Brouwer and Heeger, 2009, Serences et al., 2009, Jehee et al., 2012, Ester et al., 2013). Contrary to this assumption, mounting evidence suggests that variability is instead correlated in cortex (Arcaro et al., 2015, Bair et al., 2001, de Zwart et al., 2008, Henriksson et al., 2015, Parkes et al., 2005, Rosenbaum et al., 2016, Smith and Kohn, 2008, Zohary et al., 1994). It is well known that given incorrect assumptions of independence, decoding algorithms may fail to fully characterize the probability distribution encoded in cortical activity, despite producing reasonable estimates of the most likely presented stimulus (Domingos and Pazzani, 1997, Niculescu-Mizil and Caruana, 2005, Zhang, 2004).

Thus, the ability to measure stimulus distributions from cortical activity patterns hinges on having an appropriate model of the noise correlations in the data. But since the number of these correlations increases quadratically with the number of voxels in an fMRI data set, estimating them individually and without any guiding principles is often statistically impossible. Here, we propose a simpler approach, based on the notion that not all shared noise is equally important. Specifically, as others have argued before (Abbott and Dayan, 1999, Averbeck and Lee, 2006, Moreno-Bote et al., 2014, Smith and Kohn, 2008), correlated noise is most detrimental when it is indistinguishable from the stimulus-driven response. That is, when noise is correlated between similarly-tuned voxels, their joint activation can either indicate the presence of a mutually preferred stimulus, or that of shared noise. A decoder ignorant of the possibility of such correlated noise would tend to conclude that the voxels were activated by their preferred stimulus. Accordingly, we reasoned, this “naïve” decoder would incorrectly assign high probabilities to the stimuli preferred by these voxels, without considering interpretations consistent with shared noise.

To quantify these intuitions, we will first demonstrate in simulations that an accurate characterization of probability distributions is possible if, specifically, those correlations are accounted for that align with similarities in voxel tuning preferences. Correlations that do not have such tuning-dependent structure, on the other hand, may be safely ignored. We then examine noise correlations in fMRI measurements from human visual cortex, and find that these correlations contain the relevant tuning-related structure. Finally, we show that a decoding model that takes these tuning-dependent noise correlations into account provides an accurate window onto trial-by-trial fluctuations in the uncertainty in cortical stimulus representations. These findings exemplify the importance of incorporating noise correlations in forward models of neuroimaging data, and suggest a simple, statistically feasible approach to do so.