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This letter introduces the notion of a matrix measure flow as a tool for analyzing the stability of neural networks with time-varying weights. Given a matrix flow -- for example, one induced by gradient-based adaptation -- the matrix measure flow tracks the progression of an associated matrix measure (or logarithmic norm). We show that for certain matrix flows of interest in computational neuroscience and machine learning, the associated matrix measure flow obeys a simple inequality. In the context of neuroscience, synapses -- the connections between neurons in the brain -- are constantly being updated. This plasticity subserves many important functions, such as memory consolidation and fast parameter adaptation towards real-time control objectives. However, synaptic plasticity poses a challenge for stability analyses of recurrent neural networks, which typically assume fixed synapses. Matrix measure flows allow the stability and contraction properties of recurrent neural networks with plastic synapses to be systematically analyzed. This in turn can be used to establish the robustness properties of neural dynamics, including those associated with problems in optimization and control. We consider examples of synapses subject to Hebbian and/or Anti-Hebbian plasticity, as well as covariance-based and gradient-based rules.