An approximation to cluster size distribution of two Gaussian random fields conjunction with application to FMRI data

In brain mapping, the regions of the brain that are 'activated' by a task or external stimulus are detected by thresholding an image of test statistics. Often the experiment is repeated on several different subjects or for several different stimuli on the same subject, and the researcher is interested in the common points in the brain where 'activation' occurs in all test statistic images. The conjunction is thus defined as those points in the brain that show 'activation' in all images. We are interested in which parts of the conjunction are noise, and which show true activation in all test statistic images. We would expect truly activated regions to be larger than usual, so our test statistic is based on the volume of clusters (connected components) of the conjunction. Our main result is an approximate P-value for this in the case of the conjunction of two Gaussian test statistic images. The results are applied to a functional magnetic resonance experiment in pain perception.