Convergence and periodicity in a delayed network of neurons with threshold nonlinearity

Abstract

We consider an artificial neural network where the signal transmission is of a digital (McCulloch-Pitts) nature and is delayed due to the finite switching speed of neurons (amplifiers). The discontinuity of the signal transmission functions, however, makes it difficult to apply the existing dynamical systems theory which usually requires continuity and smoothness. Moreover, observe that the dynamics of the network completely depends on the connection weights, we distinguish several cases to discuss the behaviors of their solutions. We show that the dynamics of the model can be understood in terms of the iterations of a one-dimensional map. As, a result, we present a detailed analysis of the dynamics of the network starting from non-oscillatory states and show how the connection topology and synaptic weights determine the rich dynamics.

Keywords

• Neural networks
• feedback
• McCulloch-Pitts nonlinearity
• one-dimensional map
• convergence
• periodic solution.