### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 307-354 |

Number of pages | 48 |

Journal | Communications in Mathematical Sciences |

Volume | 10 |

Issue number | 1 |

State | Published - Mar 2012 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*10*(1), 307-354.

**The role of fluctuations in coarse-grained descriptions of neuronal networks.** / Cai, David; Tao, Louis; Shkarayev, Maxim S.; Rangan, Aaditya V.; Mclaughlin, David W.; Kovačič, Gregor.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 10, no. 1, pp. 307-354.

}

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

UR - http://www.scopus.com/inward/record.url?scp=84855468370&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855468370&partnerID=8YFLogxK

M3 - Article

VL - 10

SP - 307

EP - 354

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -