Methods for computing the maximum performance of computational models of fMRI responses

Computational neuroimaging methods aim to predict brain responses (measured e.g. with functional magnetic resonance imaging [fMRI]) on the basis of stimulus features obtained through computational models. The accuracy of such prediction is used as an indicator of how well the model describes the computations underlying the brain function that is being considered. However, the prediction accuracy is bounded by the proportion of the variance of the brain response which is related to the measurement noise and not to the stimuli (or cognitive functions). This bound to the performance of a computational model has been referred to as the noise ceiling. In previous fMRI applications two methods have been proposed to estimate the noise ceiling based on either a split-half procedure or Monte Carlo simulations. These methods make different assumptions over the nature of the effects underlying the data, and, importantly, their relation has not been clarified yet. Here, we derive an analytical form for the noise ceiling that does not require computationally expensive simulations or a splitting procedure that reduce the amount of data. The validity of this analytical definition is proved in simulations, we show that the analytical solution results in the same estimate of the noise ceiling as the Monte Carlo method. Considering different simulated noise structure, we evaluate different estimators of the variance of the responses and their impact on the estimation of the noise ceiling. We furthermore evaluate the interplay between regularization (often used to estimate model fits to the data when the number of computational features in the model is large) and model complexity on the performance with respect to the noise ceiling. Our results indicate that when considering the variance of the responses across runs, computing the noise ceiling analytically results in similar estimates as the split half estimator and approaches the true noise ceiling under a variety of simulated noise scenarios. Finally, the methods are tested on real fMRI data acquired at 7 Tesla.Author summary: Encoding computational models in brain responses measured with fMRI allows testing the algorithmic representations carried out by the neural population within voxels. The accuracy of a model in predicting new responses is used as a measure of the brain validity of the computational model being tested, but the result of this analysis is determined not only by how precisely the model describes the responses but also by the quality of the data. In this article, we evaluate existing approaches to estimate the best possible accuracy that any computational model can achieve conditioned to the amount of measurement noise that is present in the experimental data (i.e. the noise ceiling). Additionally we introduce a close form estimation of the noise ceiling that does not require computationally or data expensive procedures. All the methods are compared using simulated and real fMRI data. We draw conclusions over the impact of regularization procedures and make practical recommendations on how to report the results of computational models in neuroimaging.