The role of fluctuations in coarse-grained descriptions of neuronal networks

David Cai, Louis Tao, Maxim S. Shkarayev, Aaditya V. Rangan, David W. Mclaughlin, Gregor Kovačič

Research output: Contribution to journalReview articlepeer-review


This paper reviews our recent work addressing the role of both synaptic-input and connectivity-architecture fluctuations in coarse-grained descriptions of integrate-and-fire (I&F) pointneuron network models. Beginning with the most basic coarse-grained description, the all-to-all coupled, mean-field model, which ignores all fluctuations, we add the effects of the two types of fluctuations one at a time. To study the effects of synaptic-input fluctuations, we derive a kinetictheoretic description, first in the form of a Boltzmann equation in (2+1) dimensions, simplifying that to an advection-diffusion equation, and finally reducing the dimension to a system of two (1+1)- dimensional kinetic equations via the maximum entropy principle. In the limit of an infinitely-fast conductance relaxation time, we derive a Fokker-Planck equation which captures the bifurcation between a bistable, hysteretic operating regime of the network when the amount of synaptic-input fluctuations is small, and a stable regime when the amount of fluctuations increases. To study the effects of complex neuronal-network architecture, we incorporate the network connectivity statistics in the mean-field description, and investigate the dependence of these statistics on the statistical properties of the neuronal firing rates for three network examples with increasingly complex connectivity architecture.

Original languageEnglish (US)
Pages (from-to)307-354
Number of pages48
JournalCommunications in Mathematical Sciences
Issue number1
StatePublished - Mar 2012


  • Fokker-planck equation
  • Integrate-and-fire neuronal network
  • Kinetic theory
  • Meandriven limit

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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