A numerical study of the Kullback-Leibler distance in functional magnetic resonance imaging

Abstract

The Kullback-Leibler (or relative entropy) is applied in the analysis of functional magnetic resonance (fMRI) data series. Our study is designed for event-related (ER) experiments, where a brief stimulus is presented and a long period of rest is followed. In particular, this relative entropy is used as a measure of the distance between the probability distributions p1 and p2 of the signal levels related to stimulus and non-stimulus. In order to avoid undesirable divergences of the Kullback-Leibler distance, a small positive parameter d is introduced in the definition of the probability functions in such a way that it does not bias the comparison between both distributions. Numerical simulations are performed so as to determine the probability densities of the mean Kullback-Leibler $\overline{D}$ (throughout the N epochs of the whole experiment). For small values of N (N < 30), such probability densities $f(\overline{D})$ are found to be fitted very well by Gamma distributions (c2 < 0.0009). The sensitivity and specificity of the method are evaluated by construction of the receiver operating characteristic (ROC) curves for some values of signal-to-noise ratio (SNR). The functional maps corresponding to real data series from an asymptomatic volunteer submitted to an ER motor stimulus is obtained by using the proposed technique. The maps present activation in primary and secondary motor brain areas. Both simulated and real data analyses indicate that the relative entropy can be useful for fMRI analysis in the information measure scenario.