Generalized relative entropy in functional magnetic resonance imaging

The generalized Kullback–Leibler distance Dq (q is the Tsallis parameter) is shown to be an useful measure for analysis of functional magnetic resonance imaging (fMRI) data series. This generalized form of entropy is used to evaluate the “distance” between the probability functions p1 and p2 of the signal levels related to periods of stimulus and non-stimulus in event-related fMRI experiments. The probability densities of the mean distance D̄q (averaged over the N epochs of the entire experiment) are obtained through numerical simulations for different values of signal-to-noise ratio (SNR) and found to be fitted very well by Gamma distributions (χ2<0.0008) for small values of N (N<30). These distributions allow us to determine the sensitivity and specificity of the method by construction of the receiver operating characteristic (ROC) curves. The performance of the method is also investigated in terms of the parameters q and L (number of signal levels) and our results indicate that the optimum choice is q=0.8 and L=3. The entropic index q is found to exert control on both sensitivity and specificity of the method. As q (q>0) is raised, sensitivity increases but specificity decreases. Finally, the method is applied in the analysis of a real event-related fMRI motor stimulus experiment and the resulting maps show activation in primary and secondary motor brain areas.