A spherical Hopfield model

A spherical Hopfield-type neural network is introduced, involving neurons and patterns that are continuous variables. Both the thermodynamics and dynamics of this model are studied. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate these findings.