复杂神经系统和高维数据的统计力学
Statistical mechanics of complex neural systems and high dimensional data
摘要：
神经科学的最新实验进展为神经网络的巨大复杂性开辟了新的前景。数据的激增在两个平行的方面对我们提出了挑战。首先，我们如何形成足够的理论框架来理解动态网络过程如何在广泛不同的时空尺度上合作，以解决重要的计算问题？第二，我们如何从高维数据集中提取有意义的神经元系统模型？为了帮助在这些挑战中，我们给出了一个收集的想法和理论方法在统计物理学，计算机科学和神经生物学的交叉点产生的教学审查。我们介绍了相关的副本和腔的方法，这起源于统计物理的强大的方法来定量分析许多相互作用的自由度的大型高度异质系统。我们还介绍了密切相关的概念，图形模型中的消息传递，它起源于计算机科学作为一个分布式算法能够解决大型推理和优化问题，涉及许多耦合变量。然后，我们将展示如何统计物理学和计算机科学的观点可以应用在各种各样的背景下，在理论神经科学和数据分析中出现的问题。一路上，我们讨论自旋眼镜，学习理论，噪声中的结构错觉，随机矩阵，降维和压缩感知，所有这些都在副本方法的统一形式主义中。此外，我们回顾最近的概念之间的联系，在图形模型的消息传递，神经计算和学习。总的来说，这些想法说明了统计物理学和计算机科学如何提供一个镜头，通过它我们可以发现深埋在神经元网络动力学复杂性中的新兴计算功能。
英文摘要：
Recent experimental advances in neuroscience have opened new vistas into the immense complexity of neuronal networks. This proliferation of data challenges us on two parallel fronts. First, how can we form adequate theoretical frameworks for understanding how dynamical network processes cooperate across widely disparate spatiotemporal scales to solve important computational problems? And second, how can we extract meaningful models of neuronal systems from high dimensional datasets? To aid in these challenges, we give a pedagogical review of a collection of ideas and theoretical methods arising at the intersection of statistical physics, computer science and neurobiology. We introduce the interrelated replica and cavity methods, which originated in statistical physics as powerful ways to quantitatively analyze large highly heterogeneous systems of many interacting degrees of freedom. We also introduce the closely related notion of message passing in graphical models, which originated in computer science as a distributed algorithm capable of solving large inference and optimization problems involving many coupled variables. We then show how both the statistical physics and computer science perspectives can be applied in a wide diversity of contexts to problems arising in theoretical neuroscience and data analysis. Along the way we discuss spin glasses, learning theory, illusions of structure in noise, random matrices, dimensionality reduction, and compressed sensing, all within the unified formalism of the replica method. Moreover, we review recent conceptual connections between message passing in graphical models, and neural computation and learning. Overall, these ideas illustrate how statistical physics and computer science might provide a lens through which we can uncover emergent computational functions buried deep within the dynamical complexities of neuronal networks.
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