In this paper a novel flexible kernel independent component analysis (FKICA) algorithm is defined and its local stability on feature space is discussed. In the FKICA algorithm, the shape of nonlinear activation function in the learning algorithm varies depending on the Gaussian exponent, which is properly selected according to the kurtosis of estimated source in feature space. In the framework of the natural gradient in Stiefel manifold, the FKICA algorithm is visited and some results about its local stability analysis are presented