精确可解非对称神经网络模型的模式存储、分支和高阶相关结构
Pattern Storage, Bifurcations and Higher-Order Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model
摘要：
精确可解的非对称权值神经网络模型是罕见的，精确解仅在一些平均场方法中可用。在这篇文章中，我们找到了任意大小的非对称自旋玻璃模型的精确解析解，并对其动力学和统计性质进行了全面的研究。该网络具有离散时间演化方程，二进制发射率，可以由任何分布的噪声驱动。我们发现的条件和固定的联合概率分布的膜电位和放电率的解析表达式。的条件概率分布的发射率允许我们引入一个新的学习规则来存储安全，在存在噪声，点和循环吸引子，在内容可寻址存储器领域的重要应用。此外，我们研究了神经元动力学的分支结构的网络。我们推导出解析的例子余维1和余维2分叉图的网络，它描述了如何神经元动力学的变化与外部刺激。特别是，我们发现，该网络可能会经历多稳态制度，振荡行为引起的不对称突触连接，和各种形式的自发性破坏之间的过渡。另一方面，联合概率分布使我们能够分析计算网络的高阶相关结构，这揭示了神经元的制度，在统计上，膜电位和放电率是同步或异步的。我们的结果是有效的网络组成的任意数量的神经元，但完整性，我们也推导出网络方程的平均场极限，我们研究分析其局部分叉。所有的分析结果得到了广泛的验证，数值模拟。
英文摘要：
Exactly solvable neural network models with asymmetric weights are rare, and exact solutions are available only in some mean-field approaches. In this article we find exact analytical solutions of an asymmetric spin-glass-like model of arbitrary size and we perform a complete study of its dynamical and statistical properties. The network has discrete-time evolution equations, binary firing rates and can be driven by noise with any distribution. We find analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. The conditional probability distribution of the firing rates allows us to introduce a new learning rule to store safely, under the presence of noise, point and cyclic attractors, with important applications in the field of content-addressable memories. Furthermore, we study the neuronal dynamics in terms of the bifurcation structure of the network. We derive analytically examples of the codimension one and codimension two bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. In particular, we find that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry-breaking. On the other hand, the joint probability distributions allow us to calculate analytically the higher-order correlation structure of the network, which reveals neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks composed of an arbitrary number of neurons, but for completeness we also derive the network equations in the mean-field limit and we study analytically their local bifurcations. All the analytical results are extensively validated by numerical simulations.
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