Nonequilibrium statistical mechanics of neural networks

本文从吸引子网络的平衡态和动力学态的统计行为出发，研究了联想记忆神经网络的性质。我们关注网络行为的统计力学方面，并充分利用相变的概念。一个特别强调的是已经放在分析的模型系统具有相关的生物神经网络的非平衡统计力学的使用是必不可少的，由于缺乏能源功能的系统。我们得到了以下结果：1.从非平衡相变的角度研究了具有非对称突触连接的随机Ising自旋神经网络的丰富动力学行为，这可以看作是热力学相变的直接推广。2.阐明了Ising自旋与模拟神经网络之间的关系，并从存储容量和伪态数密度两个方面比较了两种网络的性能。3.提出了一种新的“自洽信噪比分析”（SCSNA）方法，其能够用一般类型的传递函数来评估模拟网络的存储容量。4.将SCSNA应用于具有非单调传递函数的模拟网络的存储容量评估，在Hebb型局部学习规则下，存储容量得到了显著提高，并出现了一种新的保证无错记忆检索的检索状态。这一新发现意味着，使用具有某种非单调传递函数的模拟网络，可以大大提高网络的联想记忆性能。

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.