Summary: In this paper, we are concerned with a class of recurrent neural networks with unbounded time-varying delays. Based on the geometrical configuration of activation functions, the phase space \(\mathbb R^n\) can be divided into several \(\Phi_\eta\)-type subsets. Accordingly, a new set of regions \(\Omega_\eta\) are proposed, and rigorous mathematical analysis is provided to derive the existence of equilibrium point and its local \(\mu\)-stability in each \(\Omega_\eta\). It concludes that the \(n\)-dimensional neural networks can exhibit at least \(3^n\) equilibrium points and \(2^n\) of them are \(\mu\)-stable. Furthermore, due to the compatible property, a set of new conditions are presented to address the dynamics in the remaining \(3^n-2^n\) subset regions. As direct applications of these results, we can get some criteria on the multiple exponential stability, multiple power stability, multiple log-stability, multiple log-log-stability and so on. In addition, the approach and results can also be extended to the neural networks with \(K\)-level nonlinear activation functions and unbounded time-varying delays, in which there can store \((2K+1)^n\) equilibrium points, \((K+1)^n\) of them are locally \(\mu\)-stable. Numerical examples are given to illustrate the effectiveness of our results.