There are two aspects in functional magnetic resonance imaging (fMRI) data that make them awkward to analyse with traditional multivariate methods - high order and high dimension. The first of these refers to the tensorial nature of observations as array-valued elements instead of vectors. Although this can be circumvented by vectorizing the array, doing so simultaneously loses all the structural information in the original observations. The second aspect refers to the high dimensionality along each dimension making the concept of dimension reduction a valuable tool in the processing of fMRI data. Different methods of tensor dimension reduction are currently gaining popularity in literature, and in this paper we apply two recently proposed methods of tensorial independent component analysis to simulated task-based fMRI data. Additionally, as a preprocessing step we introduce a novel extension of PCA for tensors. The simulations show that when extracting a sufficiently large number of principal components, the tensor methods find the task signals very reliably, something the standard temporal independent component analysis (tICA) fails in.