Multistability and convergence in delayed neural networks

We present the existence of 2n2n stable stationary solutions for a general nn-dimensional delayed neural networks with several classes of activation functions. The theory is obtained through formulating parameter conditions motivated by a geometrical observation. Positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established. The theory is also extended to the existence of 2n2n limit cycles for the nn-dimensional delayed neural networks with time-periodic inputs. It is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self-feedback time lags are small for the neurons with negative self-connection weights. Our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network. Four numerical simulations are presented to illustrate our theory.