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, point-neuron 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 kinetic-theoretic 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.
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This work was partly supported by the National Science Foundation through grants DMS-0506287 and DMS-0636358.
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