  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.
