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We consider a system of two-level quantum quasi-spins and gauge bosons put ona 3+1D lattice. As a model of neural network of the brain functions, thesespins describe neurons quantum-mechanically, and the gauge bosons describesweights of synaptic connections. It is a generalization of the Hopfield modelto a quantum network with dynamical synaptic weights. At the microscopic level,this system becomes a model of quantum brain dynamics proposed by Umezawa etal., where spins and gauge field describe water molecules and photons,respectively. We calculate the phase diagram of this system under quantum andthermal fluctuations, and find that there are three phases; confinement,Coulomb, and Higgs phases. Each phase is classified according to the ability tolearn patterns and recall them. By comparing the phase diagram with that ofclassical networks, we discuss the effect of quantum fluctuations and thermalfluctuations (noises in signal propagations) on the brain functions.

Neural Network for Quantum Brain Dynamics: 4D CP$^1$+U(1) Gauge Theory on Lattice and its Phase Structure