Nonequilibrium statistical mechanics of neural networks

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We have studied the properties of neural networks of associative memory from the view point of the statistical behavior of equilibrium and dynamical states of attractor networks. We have concerned with the statistical mechanical aspect of the network behaviors and made full use of the concept of phase transitions. A particular emphasis has been put on the analysis of model systems having relevance to biological neural networks for which use of nonequilibrium statistical mechanics is indispensable due to the lack of energy functions of the systems. We have obtained the following results:1.Rich dynamical behaviors of stochastic Ising spin neural networks with asymmetric synaptic connections have been explored in the light of nonequilibrium phase transitions, which can be viewed as a direct generalization of the thermodynamic phase transitions.2.The relationship between Ising spin and analog neural networks has been elucidated and comparison of the network performances between the two networks has been made in terms of the storage capacity and the number density of the spurious states.3.A new method we refer to as "Self-consistent signal-to-noise analysis"(SCSNA) has been proposed, which is capable of evaluating the storage capacity of analog networks with a general type of transfer functions. The validity and powerfulness of the method has been confirmed.4.Applications of the SCSNA to the evaluation of the storage capacity of analog networks with nonmonotonic transfer functions has led to a remarkable enhancement of the storage capacity and the occurrence of a novel type of retrieval states ensuring errorless memory retrieval under the local learning rule of Hebb type. The new finding means that using analog networks with a certain type of nonmonotonic transfer functions considerably improves network performances as associative memory.