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The accurate segmentation and tracking of cells in microscopy image sequences is an important task in biomedical research, e.g., for studying the development of tissues, organs or entire organisms. However, the segmentation of touching cells in images with a low signal-to-noise-ratio is still a challenging problem. In this paper, we present a method for the segmentation of touching cells in microscopy images. By using a novel representation of cell borders, inspired by distance maps, our method is capable to utilize not only touching cells but also close cells in the training process. Furthermore, this representation is notably robust to annotation errors and shows promising results for the segmentation of microscopy images containing in the training data underrepresented or not included cell types. For the prediction of the proposed neighbor distances, an adapted U-Net convolutional neural network (CNN) with two decoder paths is used. In addition, we adapt a graph-based cell tracking algorithm to evaluate our proposed method on the task of cell tracking. The adapted tracking algorithm includes a movement estimation in the cost function to re-link tracks with missing segmentation masks over a short sequence of frames. Our combined tracking by detection method has proven its potential in the IEEE ISBI 2020 Cell Tracking Challenge (http://celltrackingchallenge.net/) where we achieved as team KIT-Sch-GE multiple top three rankings including two top performances using a single segmentation model for the diverse data sets.