Gauged neural network: Phase structure, learning, and associative memory

A gauge model of neural network is introduced, which resembles the Z(2) Higgs lattice gauge theory of high-energy physics. It contains a neuron variable Sx=±1 on each site x of a 3D lattice and a synaptic-connection variable Jxμ=±1 on each link (x,x+μ^)(μ=1,2,3). The model is regarded as a generalization of the Hopfield model of associative memory to a model of learning by converting the synaptic weight between x and x+μ^ to a dynamical Z(2) gauge variable Jxμ. The local Z(2) gauge symmetry is inherited from the Hopfield model and assures us the locality of time evolutions of Sx and Jxμ and a generalized Hebbian learning rule. At finite “temperatures”, numerical simulations show that the model exhibits the Higgs, confinement, and Coulomb phases. We simulate dynamical processes of learning a pattern of Sx and recalling it, and classify the parameter space according to the performance. At some parameter regions, stable column-layer structures in signal propagations are spontaneously generated. Mutual interactions between Sx and Jxμ induce partial memory loss as expected.