Analysis of multiple quasi-periodic orbits in recurrent neural networks

In this paper we consider a recurrent neural network model consisting of two neurons and analyze its stability using the associated characteristic model. In order to analyze the multiple quasi-periodic orbits, the strong resonance of this system, in particular that known as the R2 bifurcation, is also studied. In the case of two neurons, one necessary condition that yields the bifurcation is found. In addition, the direction of the R2 bifurcation is determined by applying normal form theory and the center manifold theorem. The simple conditions for ensuring the existence of multiple quasi-periodic orbits are given. The strong resonance phenomenon is analyzed using numerical simulations and is related with the codimension-two bifurcation of the high-iteration map.