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Multiple sclerosis is an inflammatory autoimmune demyelinating disease that is characterized by lesions in the central nervous system. Typically, magnetic resonance imaging (MRI) is used for tracking disease progression. Automatic image processing methods can be used to segment lesions and derive quantitative lesion parameters. So far, methods have focused on lesion segmentation for individual MRI scans. However, for monitoring disease progression, \textit{lesion activity} in terms of new and enlarging lesions between two time points is a crucial biomarker. For this problem, several classic methods have been proposed, e.g., using difference volumes. Despite their success for single-volume lesion segmentation, deep learning approaches are still rare for lesion activity segmentation. In this work, convolutional neural networks (CNNs) are studied for lesion activity segmentation from two time points. For this task, CNNs are designed and evaluated that combine the information from two points in different ways. In particular, two-path architectures with attention-guided interactions are proposed that enable effective information exchange between the two time point's processing paths. It is demonstrated that deep learning-based methods outperform classic approaches and it is shown that attention-guided interactions significantly improve performance. Furthermore, the attention modules produce plausible attention maps that have a masking effect that suppresses old, irrelevant lesions. A lesion-wise false positive rate of 26.4% is achieved at a true positive rate of 74.2%, which is not significantly different from the interrater performance.