Symmetric discrete universal neural networks

Given the class of symmetric discrete weight neural networks with finite state set {0, 1}, we prove that there exist iteration modes under these networks which allow to simulate in linear space arbitrary neural networks (non-necessarily symmetric). As a particular result we prove that an arbitrary symmetric neural network can be simulated by a symmetric one iterated sequentially, with some negative diagonal weights. Further, considering only the synchronous update we prove that symmetric neural networks with one refractory state are able to simulate arbitrary neural networks.