Hamiltonian formalism for self-organization of formal neurons
Sakari Inawashiro, Yoshihide Tamori, Shogo Miyake, Jousuke Kuroiwa · Journal of Physics A Mathematical and General · 1996
The self-organization of formal neurons due to external inputs is investigated in a two-layered model of a neural network. The synaptic connections are modified by a Hebbian rule. A steady state of the connections is attained after repeated inputs of a set of the patterns, and self-organization is accomplished. Equations describing the response of the output neurons in the steady state are transformed into the form of mean field equations for an Ising spin system. The mean field equations contain each self-field at a lattice site which is proportional to the spin average at the same site. A response property of the neural network is determined by a spin structure at a fixed low temperature. We show that a Hamiltonian of the Ising spin system and self-consistency conditions give the mean field equations. Based on the Hamiltonian, we propose a self-consistent Monte Carlo simulation as a practical method of finding a spin structure, i.e. a response property of the neural network. The self-fields are self-consistently determined in the Monte Carlo procedure. The result of the self-consistent Monte Carlo simulation qualitatively agrees with a numerical solution of the mean field equations in a simple case of self-organization.