There is an increasing interest in using physiologically plausible models in fMRI analysis. These models do raise new mathematical problems in terms of parameters estimation and interpretation of the measured data. We present some theoretical contributions in this area, using different variations of the Balloon Model (Buxton98,Friston00,Buxton04) as example models. We propose 1) a method to analyze the models dynamics and their stability around equilibrium, 2) a new way to derive least square energy gradient for parameter estimation, 3) a quantitative measurement of parameter estimation efficiency, and 4) a statistical test for detecting voxel activations. We use these methods in a visual perception checker-board experiment. It appears that the different hemodynamic models considered better capture some features of the response than linear models. In particular, they account for small nonlinearities observed for stimulation durations between 1 and 8 seconds. Nonlinearities for stimulation shorter than one second can also be explained by a neural habituation model (Buxton04), but further investigations should assess whether they are rather not due to nonlinear effects of the flow response. Moreover, the tools we have developed prove that statistical methods that work well for the GLM can be nicely adapted to nonlinear models. The activation maps obtained in both frameworks are comparable
