Statistical mechanics and dynamical analysis of the Hopfield model

The present thesis describes the Hopfield Model with the objective of analysing its phase transitions and quantifying its associative memory capacity using tools of statistical mechanics. The model was introduced in 1982 by John Joseph Hopfield and it is the representative model for defining how symmetric neural networks operate as associative memories. The work begins by connecting simplified concepts of neurobiology with mathematical modelling, presenting the main models for neural networks necessary for the development of the Hopfield network. It then focuses on a particular class of neurons: binary neurons for recurrent networks. Through examples and mathematical proofs, the elements that influence the convergence of the network, such as symmetry of the synaptic interactions matrix and Lyapunov functions, are set. The thesis then provides a statistical mechanics analysis of the Hopfield model, considering first a limited number of stored patterns and secondly generalising to an extensive number of them, highlighting differences and constraints of the model. In the end, a Python implementation of the Hopfield model is displayed where the learned version of corrupted binary images is recovered. The results confirm that the model is able to recall correctly uncompleted or noisy patterns in defined limits of capacity.