Bifurcations and Chaos in Periodically Forced Coupled Ramp Neurons

Abstract

Bifurcations and chaos in periodically forced coupled ramp neurons (rectified linear units) are examined as one of minimal chaotic neural networks. The first ramp neuron connects to the second ramp neuron excitatorily while the second ramp neuron connects to the first neuron inhibitorily. When the first ramp neuron is forced by a rectangular wave, a stable periodic solution is generated. The system is piecewise linear and its periodic solutions are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. Their Poincaré maps and Jacobian matrices are also derived rigorously, by which the bifurcations of the periodic solutions are calculated. The periodic solution undergoes period-doubling bifurcations, which leads to the generation of a chaotic attractor. A period-three solution is also generated through saddle-node bifurcations, which is characteristic of a chaotic system. A one-dimensional phase return map is derived and it is shown that the bifurcations of its fixed points agree with the bifurcations of the multiple periodic solutions of the original forced system. Further, qualitatively the same bifurcations and chaos occur when the first ramp neuron is replaced by a step neuron, in which a chaotic attractor is also generated through a border collision bifurcation. The linear response and the dead zone of the ramp neurons are essential for the emergence of chaos.