Differential interference contrast (DIC) microscopy is widely used for observing unstained biological samples that are otherwise optically transparent. Combining this optical technique with machine vision could enable the automation of many life science experiments; however, identifying relevant features under DIC is challenging. In particular, precise tracking of cell boundaries in a thick ( ) slice of tissue has not previously been accomplished. We present a novel deconvolution algorithm that achieves the state-of-the-art performance at identifying and tracking these membrane locations. Our proposed algorithm is formulated as a regularized least squares optimization that incorporates a filtering mechanism to handle organic tissue interference and a robust edge-sparsity regularizer that integrates dynamic edge tracking capabilities. As a secondary contribution, this paper also describes new community infrastructure in the form of a MATLAB toolbox for accurately simulating DIC microscopy images of in vitro brain slices. Building on existing DIC optics modeling, our simulation framework additionally contributes an accurate representation of interference from organic tissue, neuronal cell-shapes, and tissue motion due to the action of the pipette. This simulator allows us to better understand the image statistics (to improve algorithms), as well as quantitatively test cell segmentation and tracking algorithms in scenarios, where ground truth data is fully known.