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# 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

#### 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