Schwab David J, Nemenman Ilya, Mehta Pankaj
Department of Physics and Lewis-Sigler Institute, Princeton University, Princeton, New Jersey 08540, USA.
Departments of Physics and Biology, Emory University, Atlanta, Georgia 30322, USA.
Phys Rev Lett. 2014 Aug 8;113(6):068102. doi: 10.1103/PhysRevLett.113.068102. Epub 2014 Aug 7.
The joint probability distribution of states of many degrees of freedom in biological systems, such as firing patterns in neural networks or antibody sequence compositions, often follows Zipf's law, where a power law is observed on a rank-frequency plot. This behavior has been shown to imply that these systems reside near a unique critical point where the extensive parts of the entropy and energy are exactly equal. Here, we show analytically, and via numerical simulations, that Zipf-like probability distributions arise naturally if there is a fluctuating unobserved variable (or variables) that affects the system, such as a common input stimulus that causes individual neurons to fire at time-varying rates. In statistics and machine learning, these are called latent-variable or mixture models. We show that Zipf's law arises generically for large systems, without fine-tuning parameters to a point. Our work gives insight into the ubiquity of Zipf's law in a wide range of systems.
生物系统中多自由度状态的联合概率分布，例如神经网络中的放电模式或抗体序列组成，通常遵循齐普夫定律，即在秩-频率图上观察到幂律。这种行为已被证明意味着这些系统处于一个独特的临界点附近，在该点熵和能量的广延部分恰好相等。在这里，我们通过解析和数值模拟表明，如果存在影响系统的波动的未观察到的变量（一个或多个），例如导致单个神经元以随时间变化的速率放电的共同输入刺激，那么类似齐普夫的概率分布就会自然出现。在统计学和机器学习中，这些被称为潜变量或混合模型。我们表明，对于大型系统，齐普夫定律通常会出现，无需将参数微调至某一点。我们的工作深入了解了齐普夫定律在广泛系统中的普遍性。
