Attractive periodic sets in discrete-time recurrent networks (with emphasis on fixed-point stability and bifurcations in two-neuron networks)

We perform a detailed fixed-point analysis of two-unit recurrent neural net works with sigmoid-shaped transfer functions. Using geometrical arguments i n the space of transfer function derivatives, we partition the network stat e-space into distinct regions corresponding to stability types of the fixed points. Unlike in the previous studies, we do not assume any special form of connectivity pattern between the neurons, and all free parameters are al lowed to vary. We also prove that when both neurons have excitatory self-co nnections and the mutual interaction pattern is the same (i.e., the neurons mutually inhibit or excite themselves), new attractive fixed points are cr eated through the saddle-node bifurcation. Finally, for an N-neuron recurre nt network, we give lower bounds on the rate of convergence of attractive p eriodic points toward the saturation values of neuron activations, as the a bsolute values of connection weights grow.