Parametrically coupled sine map networks

Abstract

Due to an inversion symmetry and a bimodal nature, the sine map, defined in R, exhibits bistability which is subject to an attractor-merging crisis. Such dynamical characteristics of the sine map suggest that it would be a promising candidate as a building block of an artificial neural network that can produce a binary associative memory. To test this idea, we designed and studied two types of Parametrically Coupled Sine Map Network. In the PCSMN-1, the amplitude of the sine function is used to control the bistability, whereas, in the PCSMN-2, an offset parameter that is added to the sine function is used to control the bistability. In both networks, the weight matrix is determined by the Hebbian rule, and coupling between the maps is accomplished via their control or bifurcation parameters; the control parameter of each sine map is continuously updated by the weighted sum of the state variables of all maps connecting to it. In the numerical study of their associative dynamics, the networks exhibited unique, desirable dynamical features that are missing in fixed-point based networks such as the Hopfield network. In addition to the potential advantage of such dynamical features in computational applications, they bring to mind some recent experimental findings concerning the neuronal activities in the brain.