连续随机网络的动态平均场理论
Dynamic Mean-Field Theory for Continuous Random Networks
AI总结：
本文研究了连续随机网络的平均场近似动力学，发展了严格的连续版本平均场理论和双副本系统，通过最大李雅普诺夫指数推导出网络临界性条件，并在实数线和p-adic线两种架构下验证了与临界大脑假设相符的连续相变行为。
摘要：
本文研究了连续随机网络的平均场近似的动力学。这些网络是由高斯噪声驱动的随机积分微分方程。积分算子中的核是广义高斯随机变量的实现。该方程控制着被解释为神经活动的宏观状态的时间演化，该状态依赖于位置和时间。位置是可测空间的一个元素。这样的网络对应于由动量生成泛函给出的统计场论（STF）。上述网络的离散版本曾出现在自旋玻璃中，并作为人工神经网络（NNs）的模型。每个离散网络对应于一个格点SFT，其中作用量包含有限数量的神经元，且每个神经元有两个标量场。在本文中，我们发展了数学上严格的、连续版本的平均场理论近似和双副本系统，使我们能够通过最大李雅普诺夫指数推导出连续随机网络的临界性条件。我们使用了两种基本架构；在第一种架构中，神经元空间是实数线，神经元被组织在一层中；在第二种架构中，神经元空间是p-adic线，神经元被组织成无限的、分形的、树状结构。我们还研究了一个具有连续相变的连续高斯网络玩具模型。这种行为与临界大脑假设相吻合，该假设认为某些生物神经元网络在相变附近工作。
英文摘要：
This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. The position is an element of a measurable space. Such a network corresponds to a statistical field theory (STF) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks (NNs). Each of these discrete networks corresponds to a lattice SFT, where the action contains a finite number of neurons and two scalar fields for each neuron. In this article, we develop mathematically rigorous, continuous versions of the mean-field theory approximation and the double-copy system that allow us to derive a condition for the criticality of continuous stochastic networks via the largest Lyapunov exponent. We use two basic architectures; in the first one, the space of neurons is the real line, and then the neurons are organized in one layer; in the second one, the space of neurons is the p-adic line, and then the neurons are organized in an infinite, fractal, tree-like structure. We also studied a toy model of a continuous Gaussian network with a continuous phase transition. This behavior matches the critical brain hypothesis, which states that certain biological neuronal networks work near phase transitions.
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