## Nonlinear estimation of BOLD responses in fMRI time-series

The relationship between stimulus, neural activation, and blood oxygenation level dependent (BOLD) response has been studied in many research groups all over the world, but it is not yet thoroughly understood. It has been found that the shape of the BOLD response varies across subjects and also within subject depending on the stimulus and active cortical area [1]. Studies have shown that the amplitude of the BOLD response is nonlinear with respect to stimulus duration [2], i.e., response to a stimulus of long duration cannot be predicted from the response to a short stimulus. If the stimuli are separated by at least 4 s, linear deconvolution has, however, found to be effective [4,3].

The aim of this research is to study the possibilities to model the BOLD response as an output of a nonlinear dynamic system. Solution of the analysis of fMRI then reduces to the parameters of such a system. Estimation of the parameters necessitate extensive calculations and may need implementation of parallel computing systems. The temporal nonlinearity of the BOLD response is studied from the estimation theory point of view using Gauss-Newton -method and regularization theory.

Another interesting issue to study is the effect of adjacent active voxels to hemodynamic response in certain area. The aim of this is to study whether the hemodynamic response can be assumed to be independent of these adjacent responses or not, and if not, how this dependence should be taken into account in calculations and analysis. This spatial nonlinearity is studied from the estimation theory point of view as well as the temporal nonlinearity.

Finally the methods for studying spatial and temporal nonlinearities are combined to one comprehensive model of the BOLD response.

The linear least squares (LS) estimation method is illustrated in the figure below. The quality of the LS-fit depends on the basis functions that are used and on the accuracy of the timing of the stimuli and scanning. This linear method, however, cannot distinguish single events very well when they have occurred in short time interval.

[2] R. M. Birn, Z. S. Saad, and P. A. Bandettini. Spatial heterogeneity of the nonlinear dynamics in the fMRI BOLD response. NeuroImage, 14(4):817-826, 2001

[3] G. H. Glover. Deconvolution of impulse response in event-related BOLD fMRI. NeuroImage, 9:416-429, 1999

[4] A. L. Vazquez, and D. C. Noll. Nonlinear aspects of the BOLD response in functional MRI. NeuroImage, 7(2):108:118, 1998