Approximating network dynamics: Some open problems

Discrete-time finite-state dynamical systems on networks are often conceived as tractable approximations to more detailed ODE-based models of natural systems. Here we review research on a class of such discrete models $N$ that approximate certain ODE models $M$ of mathematical neuroscience. In particular, we outline several open problems on the dynamics of the models $N$ themselves, as well as on structural features of ODE models $M$ that allow for the construction of discrete approximations $N$ whose predictions will be consistent with those of $M$.