Quantifying causal (effective) interactions between different brain regions are very important in neuroscience research. Many conventional methods estimate effective connectivity based on linear models. However, using linear connectivity models may oversimplify the functions and dynamics of the brain. In this paper, we propose a causal relationship estimator called nonlinear Causal Relationship Estimation by Artificial Neural Network (nCREANN) that identifies both linear and nonlinear components of effective connectivity in the brain. Furthermore, it can distinguish between these two types of connectivity components by calculating the linear and nonlinear parts of the network input-output mapping. The nCREANN performance has been verified using synthesized data and then it has been applied on EEG data collected during rest in children with autism spectrum disorder (ASD) and typically developing (TD) children. The results show that overall linear connectivity in TD subjects is higher, while the nonlinear connectivity component is more dominant in ASDs. We suggest that our findings may represent different underlying neural activation dynamics in ASD and TD subjects. The results of nCREANN may provide new insight into the connectivity between the interactive brain regions.