Period-doublings to chaos in a simple neural network

The author considers a discrete-time neural network which consists of only two neurons with the sigmoidal nonlinear function as the neuron activation function and has no external inputs and no time delay. He treats the simple network as a one-parameter family of two-dimensional maps with the neuron gain as the parameter, and mathematically proves the existence of period-doublings to chaos in the network with an excitatory neuron and an inhibitory neuron. Specifically, it is proved that, for a certain class of singular connection weight matrices, the simple neural network is dynamically equivalent to a one-parameter full family of (one-dimensional) S-unimodal maps on the interval which is well-known to become chaotic through the period-doubling route as the parameter varies