Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

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

Pages: 307 – 354

DOI: http://dx.doi.org/10.4310/CMS.2012.v10.n1.a14

Authors

David Cai (Department of Mathematics and Institute for Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Gregor Kovačič (Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, New York)

David W. McLaughlin (Courant Institute of Mathematical Sciences, New York University, New York)

Aaditya V. Rangan (Courant Institute of Mathematical Sciences, New York University, New York)

Maxim S. Shkarayev (Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, New York)

Louis Tao (College of Life Sciences, Peking University, Beijing, China)

Abstract

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.

Keywords

integrate-and-fire neuronal network, kinetic theory, Fokker-Planck equation, mean driven limit

2010 Mathematics Subject Classification

82C31, 82C32, 92C20, 94C15

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