Brain mapping with transcranial magnetic stimulation using a refined correlation ratio and Kendall's tau

Transcranial magnetic stimulation provides a mean to stimulate the brain non-invasively and painlessly. The effect of the stimulation hereby depends on the stimulation coil used and on its placement. This paper presents a mapping algorithm based on the assumption of a monotonous functional relationship between the applied electric field strength at the representation point of a muscle and the evoked motor potential. We combine data from coil characteristics, coil placement, and stimulation outcome to calculate a likelihood map for the representation of stimulated muscles in the brain. Hereby, correlation ratio (CR) and Kendall's rank coefficient tau are used to find areas in the brain where there is most likely a functional or monotonous relationship between electric field strength applied to this area and the muscle response. First results show a good accordance of our method with mapping from functional magnetic resonance imaging. In our case, classical evaluation of CR with binning is impossible, because sample data sets are too small and data are continuous. We therefore introduce a refined CR formula based on a Parzen windowing of the X-data to solve the problem. In contrast to usual windowing approaches, which require numeric integration, it can be evaluated directly in O(n2) time. Hence, its advantage lies in fast evaluation while maintaining robust applicability to small sample sets. We suggest that the presented formula can generally be used in CR-related problems where sample size is small and data range is continuous.