Systematic fluctuation expansion for neural network activity equations

Neural Comput. 2010 Feb;22(2):377-426. doi: 10.1162/neco.2009.02-09-960.


Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.

Publication types

  • Letter
  • Research Support, N.I.H., Intramural

MeSH terms

  • Action Potentials / physiology*
  • Algorithms
  • Animals
  • Artificial Intelligence
  • Brain / physiology*
  • Humans
  • Mathematical Computing*
  • Mathematical Concepts
  • Nerve Net / physiology*
  • Neural Networks, Computer*
  • Neurons / physiology*