In order to extract useful information from data streams, incremental learning has been introduced in more and more data mining algorithms. For instance, a self-organizing incremental neural network (SOINN) has been proposed to extract a topological structure that consists of one or more neural networks to closely reflect the data distribution of data streams. However, SOINN has the tradeoff between deleting previously learned nodes and inserting new nodes, i.e., the stability-plasticity dilemma. Therefore, it is not guaranteed that the topological structure obtained by the SOINN will closely represent data distribution. For solving the stability-plasticity dilemma, we propose a Gaussian membership-based SOINN (Gm-SOINN). Unlike other SOINN-based methods that allow only one node to be identified as a "winner" (the nearest node), the Gm-SOINN uses a Gaussian membership to indicate to which degree the node is a winner. Hence, the Gm-SOINN avoids the topological structure that cannot represent the data distribution because previously learned nodes overly deleted or noisy nodes inserted. In addition, an evolving Gaussian mixture model is integrated into the Gm-SOINN to estimate the density distribution of nodes, thereby avoiding the wrong connection between two nodes. Experiments involving both artificial and real-world data sets indicate that our proposed Gm-SOINN achieves better performance than other topology learning methods.