fMRI hemodynamic response function estimation in autoregressive noise by avoiding the drift

The measured functional magnetic resonance imaging (fMRI) time series is typically corrupted by instrumental drift and physiological noise due to respiration and heartbeat giving rise to temporal correlation in the signals. Most methods proposed so far for nonparametric hemodynamic response function (HRF) estimation in fMRI data do not account for these confounding effects, and thus produce biased and inefficient estimates. The aim of this paper is to address this issue by modeling the noise in fMRI time series using an autoregressive model of order p (AR(p)). Making use of a semiparametric model to characterize the fMRI time series and the AR(p) to model the temporally correlated noise, a generalized least squares (GLS) estimator for voxelwise consistent nonparametric HRF estimation is derived. The proposed estimation method is a three-stage procedure that relies on first-order differencing to remove drift and a novel structured covariance estimator for the AR noise based on Cholesky decomposition to derive the best linear unbiased estimator (BLUE) of the HRF. We also establish the asymptotic consistency of the proposed estimator. Simulation results show that the proposed method generates more accurate HRF estimates compared to existing methods. When applied to real fMRI data, it demonstrates the effectiveness in uncovering the brain response temporal dynamics for both event-related and block-design paradigms. Our approach removes the two types of noise in fMRI data simultaneously, thus providing efficient estimation of brain hemodynamic responses, while allowing for flexible characterization of the shape and timing of the voxelwise HRF.