Non-parametric temporal modeling of the hemodynamic response function via a liquid state machine

Standard methods for the analysis of functional MRI data strongly rely on prior implicit and explicit hypotheses made to simplify the analysis. In this work the attention is focused on two such commonly accepted hypotheses: (i) the hemodynamic response function (HRF) to be searched in the BOLD signal can be described by a specific parametric model e.g., double-gamma; (ii) the effect of stimuli on the signal is taken to be linearly additive. While these assumptions have been empirically proven to generate high sensitivity for statistical methods, they also limit the identification of relevant voxels to what is already postulated in the signal, thus not allowing the discovery of unknown correlates in the data due to the presence of unexpected hemodynamics. This paper tries to overcome these limitations by proposing a method wherein the HRF is learned directly from data rather than induced from its basic form assumed in advance. This approach produces a set of voxel-wise models of HRF and, as a result, relevant voxels are filterable according to the accuracy of their prediction in a machine learning framework. This approach is instantiated using a temporal architecture based on the paradigm of Reservoir Computing wherein a Liquid State Machine is combined with a decoding Feed-Forward Neural Network. This splits the modeling into two parts: first a representation of the complex temporal reactivity of the hemodynamic response is determined by a universal global "reservoir" which is essentially temporal; second an interpretation of the encoded representation is determined by a standard feed-forward neural network, which is trained by the data. Thus the reservoir models the temporal state of information during and following temporal stimuli in a feed-back system, while the neural network "translates" this data to fit the specific HRF response as given, e.g. by BOLD signal measurements in fMRI. An empirical analysis on synthetic datasets shows that the learning process can be robust both to noise and to the varying shape of the underlying HRF. A similar investigation on real fMRI datasets provides evidence that BOLD predictability allows for discrimination between relevant and irrelevant voxels for a given set of stimuli.