Synchronization, Multiresonance Phenomena, and Discrete Oscillation Periods in a Hopfield Neural Network with Two Time Delays

Abstract

Hopfield neural network attracts particular attention as it serves as a relatively simple mathematical model that describes some properties of the brain function. We investigate analog Hopfield neural networks with two time delays. It is shown that the neural network with all inhibitory connections demonstrates growing oscillations after exceeding the threshold, and oscillations become synchronous after a relatively short period of time of the order of the larger time delay. The oscillation amplitude of the neural network as function of the time delay in one subnetwork demonstrates resonance-like phenomena with multiple peaks. The oscillation period of the neural network with two time delays shows discrete structure and changes within relatively narrow intervals in contrast to the oscillation period of the neural network with one time delay, which shows continuous changes of oscillation amplitude and period with time delay. Multiresonance behavior is sustained for both inhibitory and excitatory connections of the second subnetwork, while the first subnetwork possesses only inhibitory connections. The oscillation period of the subnetwork with smaller time delay and equal coupling strengths dominates the whole neural network activity. Turning on and off smaller time delay in one subnetwork allows for the control of the oscillation period and demonstrates memory-like behavior. Mechanisms of the simulated phenomena are disclosed and their similarity to brain function is discussed.