Bifurcations and Chaos in Three-Coupled Ramp-Type Neurons

Abstract

The bifurcations and chaos in autonomous systems of two- and three-coupled ramp-type neurons are considered. An asymmetric piecewise linear function is employed for the output function of neurons in order to examine changes in the bifurcations from a sigmoid output function to a ramp output function. Steady solutions in the systems are obtained exactly and they undergo discontinuous bifurcations because the systems are piecewise linear. Periodic solutions and homoclinic/heteroclinic orbits in the systems are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. The bifurcations of the periodic solutions are calculated with the Poincaré maps and the Jacobian matrices, which are also derived rigorously. A stable periodic solution in a two-neuron oscillator of the Wilson–Cowan type with three couplings remains in the case of a ramp neuron. A chaotic attractor of Rössler type emerges in a network of three ramp neurons with six couplings, which is due to two saddle-focuses. The network consists of the two-neuron oscillator and one bypass neuron connected through three couplings. One-dimensional Poincaré maps show the generation of the chaotic attractor through a cascade of period-doubling bifurcations. Further, multiple homoclinic orbits based at a saddle are generated from the destabilization of two focuses when asymmetry in the output function is large. This homoclinicity causes qualitative change in the bifurcations of the periodic solutions as the output function of neurons changes from sigmoid to ramp.