Similarities and Differences in the Behavior of the Excitatory and Inhibitory Analog Hopfield Neural Networks with Three Time Delays

Hopfield neural networks were widely studied due to their capability of simulating properties of the brain and its sub-networks, as well as their extensive applications for signal and image processing, pattern recognition, and optimization. While a significant number of investigations were devoted to the stability analysis of the Hopfield neural networks, less attention was paid to the analysis of the network behavior in the instability regions. The purpose of this study is to investigate the activity patterns that are generated by the Hopfield neural network composed of three sub-networks with three time delays and coupled by excitatory or inhibitory connections. It is found that both types of neural networks (excitatory and inhibitory) demonstrate synchronous behavior at the steady-state. An increase of the time delay in one or two inhibitory sub-networks results in multiresonances, discrete oscillation periods, and a period-tripling bifurcation. The novel findings are: (1) while the stability threshold of the purely excitatory network does not depend on the time delays, the threshold for purely inhibitory network nonmonotonically depends on the time delays and sub-network configuration; (2) numerical and analytical studies demonstrate that in the neural network that contains a sub-network with much smaller time delay than those in other sub-networks, the interval until the output saturation dramatically decreases compared to the neural network with larger and identical time delays; (3) numerical and analytical studies reveal the existence of quasi-steady-state sub-levels during neural network output development until saturation, which are connected to the neural network structure; and (4) maximum oscillation amplitudes and oscillation periods in the purely inhibitory neural network with three time delays as 2D functions of the time delays demonstrate qualitatively more complex behavior than the neural network with two time delays.