Fuzzy synchronization likelihood with application to attention-deficit/hyperactivity disorder

Synchronization as a measure of quantification of similarities in dynamic systems is an important concept in many scientific fields such as nonlinear science, neuroscience, cardiology, ecology, and economics. When interdependencies and connections of coupled dynamic systems are not directly accessible and measurable such as those of the neurons of the brain, quantification of similarities between their time series outputs is the best possible way to detect the existent interdependencies among them. In recent years, Synchronization Likelihood (SL) has been used as one of the most suitable algorithms in highly nonlinear and non-stationary systems. In this method, the likelihood of patterns is measured statistically, and then it is determined which patterns of the time series are similar to each other considering a threshold. But the degree of similarities is not considered in the decision. In this paper, a new measure of synchronization, fuzzy SL, is presented using the theory of fuzzy logic and Gaussian membership functions. The new fuzzy SL is compared with the conventional SL using both a standard problem from the chaos literature and a complicated real life neurological diagnostic problem, that is, the EEG-based diagnosis of Attention-Deficit/Hyperactivity Disorder (ADHD). The results of ANOVA analysis indicate the interdependencies measured by the fuzzy SL are more reliable than the conventional SL for discriminating ADHD patients from healthy individuals.