Towards establishing a meaningful and practical dynamics results for the unified RNN model

Since the unified model of continuous-time recurrent neural networks, i.e., continuous-time uniformly pseudo-projection-anti-monotone net, can jointly cover almost all of the known continuous-time recurrent neural network individuals, and the critical condition can characterize the intrinsic bounded line of stability and un-stability of a recurrent neural network, thus, it is quite meaningful to study thoroughly the critical dynamics behaviors of the unified model. In the present paper, the critical global convergence and stability are obtained for the continuous-time uniformly pseudo-projection-anti-monotone net, and the critical analysis here does not need any extra requirements, namely, only require the unified model to satisfy the critical condition that one given discriminant matrix is nonnegative definite. The results can jointly cover, simplify and extend most of those known diverse dynamics results validated for each recurrent neural network individual, including those critical analysis and let alone those non-critical analysis, thus can eliminate the redundant among those known results directly. The approach adopted here launches a visible step towards establishing a meaningful and practical unified theory for the continuous-time RNNs.