Approximate Synchronization of Memristive Hopfield Neural Networks

Asymptotic synchronization is one of the essential differences between artificial neural networks and biologically inspired neural networks due to mismatches from the dynamical update of weight parameters and heterogeneous activations. In this paper, a new concept of approximate synchronization is proposed and investigated for Hopfield neural networks coupled with nonlinear memristors. It is proved that global solution dynamics are robustly dissipative and a sharp ultimate bound is acquired. Through a priori uniform estimates on the interneuron differencing equations, it is rigorously and analytically shown that approximate synchronization to any prescribed small gap at an exponential convergence rate of the memristive Hopfield neural networks occurs if an explicitly computable threshold condition is satisfied by the interneuron coupling strength parameter. The main result is also extended to Hopfield neural networks with Hebbian learning rules for a broad range of applications in unsupervised learning. The contribution of this approximate synchronization framework and the analytic methodology in this work advance the exploration of asymptotic dynamics for more AI mathematical models.