Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay

Abstract

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge–Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.