TY - JOUR

T1 - The role of fluctuations in coarse-grained descriptions of neuronal networks

AU - Cai, David

AU - Tao, Louis

AU - Shkarayev, Maxim S.

AU - Rangan, Aaditya V.

AU - Mclaughlin, David W.

AU - Kovačič, Gregor

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

KW - Fokker-planck equation

KW - Integrate-and-fire neuronal network

KW - Kinetic theory

KW - Meandriven limit

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U2 - 10.4310/CMS.2012.v10.n1.a14

DO - 10.4310/CMS.2012.v10.n1.a14

M3 - Review article

AN - SCOPUS:84855468370

VL - 10

SP - 307

EP - 354

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -