Contraction Analysis of Hopfield Neural Networks with Hebbian Learning

Abstract

This paper investigates stability conditions of continuous-time Hopfield and firing-rate neural networks by leveraging contraction theory. First, we present a number of useful general algebraic results on matrix polytopes and products of symmetric matrices. Then, we give sufficient conditions for strong and weak Euclidean contractivity, i.e., contractivity with respect to the ℓ2 norm, of both models with symmetric weights and (possibly) non-smooth activation functions. Our contraction analysis leads to contraction rates which are logoptimal in almost all symmetric synaptic matrices. Finally, we use our results to propose a firing-rate neural network model to solve a quadratic optimization problem with box constraints.

Cluster Synchronization as a Mechanism of Free Recall in Working Memory Networks

“…− a i1,i2 (a i1,i2 − g a o i1,i2 ) (16), which is forward invariant [33]. We now discuss the existence and stability of equilibria within this set.…”

“…Attractor neural networks, dynamical networks that evolve towards a stable pattern, have been employed for understanding the mechanisms of memory [8], [9], [10], [11], [12], [13], including WM [7], and the display of bursty limit cycle replay of encoded memories [14]. The most studied attractor neural network model is the Hopfield model, capable of storing different patterns as stable equilibria [15], [16]. The abstract Hopfield model represents a memory network with a fully connected graph, symmetric weights, and capable of storing only binary values.…”