  Functional MRI (fMRI) has become the most common method for investigating the
human brain. However, fMRI data present some complications for statistical
analysis and modeling. One recently developed approach to these data focuses on
estimation of computational encoding models that describe how stimuli are
transformed into brain activity measured in individual voxels. Here we aim at
building encoding models for fMRI signals recorded in the primary visual cortex
of the human brain. We use residual analyses to reveal systematic nonlinearity
across voxels not taken into account by previous models. We then show how a
sparse nonparametric method [J. Roy. Statist. Soc. Ser. B 71 (2009b) 1009-1030]
can be used together with correlation screening to estimate nonlinear encoding
models effectively. Our approach produces encoding models that predict about
25% more accurately than models estimated using other methods [Nature 452
(2008a) 352--355]. The estimated nonlinearity impacts the inferred properties
of individual voxels, and it has a plausible biological interpretation. One
benefit of quantitative encoding models is that estimated models can be used to
decode brain activity, in order to identify which specific image was seen by an
observer. Encoding models estimated by our approach also improve such image
identification by about 12% when the correct image is one of 11,500 possible
images.
