A state-space model of the hemodynamic approach: Nonlinear filtering of BOLD signals

In this paper, a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent (BOLD) responses. The proposed method constitutes an alternative to the recently proposed Friston [Neuroimage 16 (2002) 513] method and has some advantages over it. The procedure is based on recent groundbreaking time series analysis techniques that have been, in this case, adopted to characterize hemodynamic responses in functional magnetic resonance imaging (fMRI). This work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons. First, our model includes physiological noise. Previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation. Secondly, by using the innovation method and the local linearization filter, not only the parameters, but also the underlying states of the system generating responses can be estimated. These states can include things like a flow-inducing signal triggered by neuronal activation, de-oxyhemoglobine, cerebral blood flow and volume. Finally, radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach, which could be essential to understanding those brain areas indirectly related to the stimulus. Hence, thirdly, by inferring about the radial basis parameters, we are able to perform a blind deconvolution, which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also, to implicitly reconstruct the underlying synaptic dynamics, induced experimentally, which caused these states variations. From this study, we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps (SPM), nonlinear BOLD phenomena and unspecific input temporal sequences must be included in the fMRI analysis.