Attractors in Neural Networks with Innnite Gain

We study a system of equations with discontinuous right hand side, which arise as models of gene and neural networks. Associated to the system is a graph of dynamics, which can be used to deene a Morse decomposition of the invariant set of the ow on the set of rays through the origin ((5]). We study attractors in R 4 which lie in a set of orthants in the form of gure eight. Trajectories can follow one of the two loops of gure eight. We nd that if the attractor is symmetric with respect to these two loops, then the only possible attractor is a periodic orbit which traverses both loops once. We show that without the symmetry the set of admissible attractors include periodic orbits which follow one loop k times and other loop once, for any k. However, we also show that no trajectory in an attractor can traverse both loops more then once in a row.