Effects of Low and High Neuron Activation Gradients on the Dynamics of a Simple 3D Hopfield Neural Network

Abstract

In this paper, the effects of low and fast response speeds of neuron activation gradient of a simple 3D Hopfield neural network are explored. It consists of analyzing the effects of low and high neuron activation gradients on the dynamics. By considering an imbalance of the neuron activation gradients, different electrical activities are induced in the network, which enable the occurrence of several nonlinear behaviors. The significant sensitivity of nontrivial equilibrium points to the activation gradients of the first and second neurons relative to that of the third neuron is reported. The dynamical analysis of the model is done in a wide range of the activation gradient of the second neuron. In this range, the model presents areas of periodic behavior, chaotic behavior and periodic window behavior through complex bifurcations. Interesting behaviors such as the coexistences of two, four, six and eight disconnected attractors, as well as the phenomenon of coexisting antimonotonicity, are reported. These singular results are obtained by using nonlinear dynamics analysis tools such as bifurcation diagrams and largest Lyapunov exponents, phase portraits, power spectra and basins of attraction. Finally, some analog results obtained from PSpice-based simulations further verify the numerical results.