## Mean-field approximations for coupled
populations of generalized linear model spiking neurons with Markov
refractoriness

Neural Computation 21,
1203-1243.

There has recently been a great deal of interest in inferring network
connectivity from the spike trains in populations of neurons. One
class of useful models which can be fit easily to spiking data is
based on generalized linear point process models from statistics. Once
the parameters for these models are fit, the analyst is left with a
nonlinear spiking network model with delays, which in general may be
very difficult to understand analytically. Here we develop mean-field
methods for approximating the stimulus-driven firing rates (both in
the time-varying and steady-state case), auto- and cross-correlations,
and stimulus-dependent filtering properties of these networks. These
approximations are valid when the contributions of individual network
coupling terms are small and, hence, the total input to a neuron is
approximately Gaussian. These approximations lead to deterministic
ordinary differential equations that are much easier to solve and
analyze than direct Monte Carlo simulation of the network activity.
These approximations also provide analytical way to evaluate the
linear input-output filter of neurons and how the filters are
modulated by network interactions and some stimulus feature. Finally,
in the case of strong refractory effects, the mean-field
approximations in the generalized linear model become inaccurate;
therefore we introduce a model that captures strong refractoriness,
retains all of the easy fitting properties of the standard generalized
linear model, and leads to much more accurate approximations of mean
firing rates and cross-correlations that retain fine temporal
behaviors.

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