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We study pairwise Ising models for describing the statistics of
multi-neuron spike trains, using data from a simulated cortical
network. We explore efficient ways of finding the optimal couplings
in these models and examine their statistical properties. To do
this, we extract the optimal couplings for subsets of size up to
$200$ neurons, essentially exactly, using Boltzmann learning. We
then study the quality of several approximate methods for finding
the couplings by comparing their results with those found from
Boltzmann learning. Two of these methods -- inversion of the Thouless-Anderson-Palmer
equations and an approximation proposed by Sessak and Monasson --
are remarkably accurate. Using these approximations for larger
subsets of neurons, we find that extracting couplings using data
from a subset smaller than the full network tends systematically to
overestimate their magnitude.  This effect is described
qualitatively by infinite-range spin glass theory for the normal
phase. We also show that a globally-correlated
input to the neurons in the network lead to a small increase in the
average coupling. However, the pair-to-pair variation of the
couplings is much larger than this and reflects intrinsic properties
of the network. Finally, we study the quality of these models by
comparing their entropies with that of the data.  We find that they
perform well for small subsets of the neurons in the network, but
the fit quality starts to deteriorate as the subset size grows,
signalling the need to include higher order correlations to describe
the statistics of large networks.