  In this paper, we analyze the stability of feedback interconnections of a
linear time-invariant system with a neural network nonlinearity in discrete
time. Our analysis is based on abstracting neural networks using integral
quadratic constraints (IQCs), exploiting the sector-bounded and
slope-restricted structure of the underlying activation functions. In contrast
to existing approaches, we leverage the full potential of dynamic IQCs to
describe the nonlinear activation functions in a less conservative fashion. To
be precise, we consider multipliers based on the full-block Yakubovich / circle
criterion in combination with acausal Zames-Falb multipliers, leading to linear
matrix inequality based stability certificates. Our approach provides a
flexible and versatile framework for stability analysis of feedback
interconnections with neural network nonlinearities, allowing to trade off
computational efficiency and conservatism. Finally, we provide numerical
examples that demonstrate the applicability of the proposed framework and the
achievable improvements over previous approaches.
