A spherical Hopfield model

Abstract

We introduce a spherical Hopfield-type neural network involving neurons and\npatterns that are continuous variables. We study both the thermodynamics and\ndynamics of this model. In order to have a retrieval phase a quartic term is\nadded to the Hamiltonian. The thermodynamics of the model is exactly solvable\nand the results are replica symmetric. A Langevin dynamics leads to a closed\nset of equations for the order parameters and effective correlation and\nresponse function typical for neural networks. The stationary limit corresponds\nto the thermodynamic results. Numerical calculations illustrate our findings.\n